connecting the formation of stars and planets. ii
TRANSCRIPT
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
Revista Mexicana de Astronomıa y Astrofısica 57 217ndash231 (2021)
copy 2021 Instituto de Astronomıa Universidad Nacional Autonoma de Mexico
httpsdoiorg1022201ia01851101p2021570116
CONNECTING THE FORMATION OF STARS AND PLANETS IICOUPLING THE ANGULAR MOMENTUM OF STARS WITH THE
ANGULAR MOMENTUM OF PLANETS
L M Flor-Torres1 R Coziol1 K-P Schroder1 D Jack1 and J H M M Schmitt2
Received February 6 2020 accepted January 7 2021
ABSTRACT
A sample of 46 stars host of exoplanets is used to search for a connectionbetween their formation process and the formation of the planets rotating aroundthem Separating our sample into two stars hosting high-mass exoplanets (HMEs)and low-mass exoplanets (LMEs) we found the former to be more massive and torotate faster than the latter We also found the HMEs to have higher orbital angularmomentum than the LMEs and to have lost more angular momentum throughmigration These results are consistent with the view that the more massive thestar and the higher its rotation the more massive was its protoplanetarys disk androtation and the more efficient was the extraction of angular momentum from theplanets
RESUMEN
Una muestra de 46 estrellas que albergan exoplanetas se usa para la busquedade una conexion entre su proceso de formacion y el proceso de formacion de losplanetas que las orbitan Separamos nuestra muestra en dos estrellas que alberganexoplanetas de baja masa (LME) y de alta masa (HME) encontramos que lasestrellas con HMEs rotan preferentemente alrededor de estrellas con alta masa ycon mayor rotacion que las estrellas con LMEs Tambien encontramos que lasHMEs tienen un momento angular orbital mas alto que las LMEs y que perdieronmas momento angular durante su migracion Nuestros resultados son congruentescon un modelo en el cual las estrellas mas masivas con alta rotacion forman discosprotoplanetarios mas masivos que tambien rotan mas rapido y que ademas son maseficientes en disipar el momento angular de sus planetas
Key Words planetary systems mdash stars formation mdash stars fundamental parame-ters mdash stars rotation
1 INTRODUCTION
The discovery of gas giant planets rotating very closeto their stars (hot Jupiter or HJs) has forced us toreconsider our model for the formation of planetsaround low mass stars by including in an ad hocway large scale migration Since this did not happenin the Solar System it raises the natural question ofunderstanding under what conditions large scale mi-gration could be triggered in a proto-planetary disk(PPD) By stating such a question we adopt thesimplest view that there is only one universal pro-cess for the formation of planets following the col-
1Departamento de Astronomıa Universidad de Guanajua-to Guanajuato Gto Mexico
2Hamburger Sternwarte Universitat Hamburg HamburgGermany
lapse of dense regions in a molecular cloud (McKee ampOstriker 2007 Draine 2011 Champion 2019) Thisreduces the problem to a more specific one whichis how do we include migration in a well developedmodel like the core collapse scenario (the standardmodel) which explains in details how the Solar Sys-tem formed (Safronov 1969 Wetherill amp W 1989Wurm amp Blumm 1996 Poppe amp Blum 1997 Klahramp Brandner 2006 Hilke amp Sari 2011 de Pater ampLissauer 2015 Raymond amp Morbidelli 2020)
In the literature two migration mechanisms arefavored for HJs (Dawson amp Johnson 2018 Raymondamp Morbidelli 2020) The first is disk migration (egBaruteau et al 2014 Armitage 2020 and referencestherein) which proposes that a planet loses its or-
217
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
218 FLOR-TORRES ET AL
bit angular momentum by tidal interactions with thePPD while the second high-eccentricity migration(eg Rasio amp Ford 1996 Weidenschilling amp Marzari1996 Marzari amp Weidenschilling 2002 Chatterjee etal 2008 Nagasawa et al 2008 Beauge amp Nesvorny2012) suggests that a planet interacting with otherplanets gains a higher eccentricity which brings itclose to its star where it reaches equilibrium by tidalinteractions (a process know as circularization) Interms of migration these two mechanisms might sug-gest that massive disks somehow amplified the levelof migration compared to what happened in the So-lar System because more massive PPDs either in-crease the intensity of interactions of the planetswith their disks or favor the formation of a highernumber of planets Within the standard model thiswould suggest that what counts is whether the PPDfollows the minimum mass model with a mass be-tween 001 to 002 M or the maximum mass modelwith a mass above 05 M (Armitage 2010) Thereare few clues which could help us to determine whichpath the PPD of the Solar System followed (andstrong difficulties compared to direct observationsof PPD see Figure 2 and discussion in Raymondamp Morbidelli 2020) One is the total mass of theplanets which represents only 01 the mass of theSun This implies the Solar System PPD has lostan important amount of its mass after the formationof the planets Another clue is that 99 of the an-gular momentum of the Solar System is located inthe planets suggesting that the initial angular mo-mentum of the PPD might have been conserved inthe planets However this is obviously not the casewhen large scale migration occurs so what was thedifference
If the initial angular momentum of the PPDpasses to the planets then one could use the orbitalrotation momentum in exoplanetary systems to testdifferent scenarios connecting the formation of theplanets to the formation of their stars For examplehow is the angular momentum of the PPD coupledto the angular momentum of the stars Since largescale migration represents a loss of angular momen-tum of the planets (at least by a factor 10) whatwas the initial angular momentum of the PPD whenit formed and how does this compared to the initialmass of the PPD Does this influence the masses ofthe planets and their migration The answers arenot trivial considering that the physics involved isstill not fully understood
In particular we know that the angular momen-tum is not conserved during the formation of starsThis is obvious when one compares how fast the Sun
rotates with how fast its rotation should have beenassuming the angular momentum of the collapsingmolecular cloud where it formed was conserved Ac-tually working the math (a basic problem but quiteinstructive see course notes by Alexander 2017) theSun effective angular momentum j = JM isasymp 106 times lower than expected Intriguingly jis also 103 lower than the angular momentum of itsbreaking point jb the point where the centripetalforce becomes stronger than gravity (McKee amp Os-triker 2007) If that were not true then no starswhatever their mass would be able to form In factobservations revealed that in general the angularmomentum of stars with spectral type O5 to A5traces a power law J prop Mα with α asymp 2 with typ-ical Jlowast values that are exactly ten times lower thantheir breaking point However how universal thisldquolawrdquois and how stars with different masses get toit is unexplained (Wesson 1979 Brosche 1980 Car-rasco et al 1982 God lowski et al 2003) To compli-cate the matter it is clear now that lower mass starslater than A5 do not follow this law their spin go-ing down exponentially (cf Figure 6 in Paper I)For low-mass stars McNally (1965) Kraft (1967)and Kawaler (1987) suggested a steeper power lawJ prop M57 which implies that they lose an extraamount of angular momentum as their mass goesdown What is interesting is that low-mass stars arealso those that form PPDs and planets which ledsome researchers to speculate that there could be alink between the two
To explain how low-mass stars lose their angu-lar momentum different mechanisms are consideredThe most probable one is stellar wind (Schatzman1962 Mestel 1968) which is related to the convectiveenvelopes of these stars This is how low-mass starswould differ from massive ones However whetherthis mechanism is sufficient to explain the break inthe J minusM relation is not obvious because it ignoresthe possible influence of the PPD (the formation ofa PPD seems crucial see de la Reza amp Pinzon 2004)This is what the magnetic braking model takes intoaccount (Wolff amp Simon 1997 Tassoul 2000 Uzden-sky et al 2002) Being bombarded by cosmic raysand UV radiation from ambient stars the matter ina molecular cloud is not neutral and thus permeableto magnetic fields This allows ambipolar diffusion(the separation of negative and positive charges) toreduce the magnetic flux allowing the cloud to col-lapse Consequently a diluted field follows the mat-ter through the accretion disk to the star forming itsmagnetic field (McKee amp Ostriker 2007) This alsoimplies that the accretion disk (or PPD) stays con-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 219
nected to the star through its magnetic field as longas it exists that is over a period that although briefincludes the complete phase of planet formation andmigration According to the model of disk-lockinga gap opens between the star and the disk at a dis-tance Rt from the star and matter falling betweenRt and the radius of corotation Rco (where the Ke-plerian angular rotation rate of the PPD equals thatof the star) follows the magnetic field to the polesof the star creating a jet that transports the angu-lar momentum out In particular this mechanismwas shown to explain why the classic T-Tauris rotatemore slowly than the weak T-Tauris (Ray 2012)How this magnetic coupling could influence the plan-ets and their migrations on the other hand is stillan open question (Matt amp Pudritz 2004 Fleck 2008Champion 2019)
To investigate further these problems we starteda new project to observe host stars of exoplanetsusing the 12 m robotic telescope TIGRE which isinstalled near our department at the La Luz Ob-servatory (in central Mexico) In paper I we ex-plained how we succeeded in determining in an effec-tive and homogeneous manner the physical charac-teristics ( Teff log g [MH] [FeH] and V sin i) of ainitial sample of 46 bright stars using iSpec (Blanco-Cuaresma 2014 2019) In this accompanying articlewe now explore the possible links between the phys-ical characteristics of these 46 stars and the physicalcharacteristics of their planets in order to gain newinsight about a connection between the formation ofstars and their planets
2 SAMPLE OF EXOPLANETS
Our observed sample consists of 46 stars hosts of59 exoplanets which were selected from the revisedcompendium of confirmed exoplanets in the Exo-planet Orbit Database3 In Table 1 the dominantplanet (Column 2) in each stellar system is iden-tified by the same number (Column 1) which wasused in Paper I to identify their stars In Column 3and 4 we repeat the magnitude and distance of thehost stars as they appeared in Table 1 of Paper IThis is followed by the main properties of the planetsas reported in the exoplanet Orbit Database mass(Column 5) radius (Column 6) period (Column 7)major semi-axis (Column 8) and eccentricity (Col-umn 9) Note that an eccentricity of zero could meanthe actual eccentricity is not known The last twocolumns identify the detection method and the dis-tinction between high mass and low mass exoplanet(as explained below)
3httpexoplanetsorg
Among our sample one exoplanet 39 (Hn-Peg b) was found to have a mass above the browndwarf (BD) lower limit of 13 MJup (Spiegel et al2011 Burgasser 2008) Note that Hn-Peg b was de-tected by imaging (identified as Im in Column 11)and its huge distance from its host star (Column 8)is more typical of BDs than of exoplanets (the otherexoplanet detected by imaging is 34 HD 115383 bat 435 AU from it stars but with a mass of only4 MJup) Another exoplanet whose mass is close tothe lower mass limit for BDs is 31 (XO-3 b) butsince it is located very near its star at 005 AUthere is no difficulty in accepting its classification asan exoplanet Due to the small size of our samplewe cannot assess how the formation of exoplanetswith masses above the BD limit can differ from theformation of the other exoplanets Consequently al-though we kept Hn-Peg in our sample of host starswe did not include its ldquoexoplanetrdquo in our statisticalanalysis
Another point of importance that can be notedin Table 1 is the fact that we have only 14 exoplan-ets detected by RV This is most probably due tothe fact that we required the radius of the planetsto be known which is easier to determine by the Trmethod Curiously only three of the 14 exoplanetsdetected by RV show the trend to be located fartherfrom their stars than the Tr exoplanets as observedin the literature (and as it is obvious in the Exo-planet Orbit Database) Two are close to the iceline and thus are rather warm than hot and onewith 386 MJup is at the same distance as Jupiterfrom the Sun This implies that any bias introducedby the different detection methods RV vs Tr can-not be explored thoroughly in our present analysis
Although the variety of the characteristics of ex-oplanets cannot be addressed with our present sam-ple we can however separate our sample of exoplan-ets into two groups based on their masses For ouranalysis this distinction is important in order to testhow the mass of the exoplanet is related to the massof its host star To use a mass limit that has a phys-ical meaning we choose 12 MJup which is the massabove which self-gravity in a planet becomes strongerthan the electromagnetic interactions (Padmanab-han 1993 Flor-Torres et al 2016 see demonstrationin Appendix A) Using this limit we separated oursample into 22 high mass exoplanets (HMEs) and 23low mass exoplanets (LMEs) This classification isincluded in Table 1 in the last column
According to Fortney et al (2007) the mass-radius relation of exoplanets shows a trend for ex-oplanets above 10 MJup to have a constant radius
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
220 FLOR-TORRES ET AL
TABLE 1
PHYSICAL PARAMETERS THE PLANETS IN OUR SAMPLE
Id Star and Magnitude Distance Mp Rp Period ap ep Detection Planetary
Planet (V) (pc) (Mjup) (Rjup) (days) (AU ) Method type
1 KELT-6 c 103 2424 371 268 12760 239 021 RV HME
2 HD 219134 h 56 65 028 080 21980 306 037 RV LME
3 KEPLER-37 b 98 640 001 003 1337 010 0 Tr LME
4 HD 46375 b 78 296 023 102 302 004 005 RV LME
5 HD 75289 b 64 291 047 103 351 005 002 RV LME
6 HD 88133 b 80 738 030 100 342 005 008 RV LME
7 HD 149143 b 79 734 133 105 407 005 001 RV HME
8 HAT-P-30 b 104 2153 071 134 281 004 004 Tr LME
9 KELT-3 b 98 2113 142 133 270 004 0 Tr HME
10 KEPLER-21 b 83 1089 002 015 279 004 002 Tr LME
11 KELT-2A b 87 1346 149 131 411 005 019 Tr HME
12 HD86081 b 87 1042 150 108 200 004 006 RV HME
13 WASP-74 b 98 1498 097 156 214 004 0 Tr LME
14 HD 149026 b 81 760 036 072 288 004 0 Tr LME
15 HD 209458 b 76 484 069 138 352 005 001 Tr LME
16 BD-10 3166 b 100 846 046 103 349 005 001 RV LME
17 HD 189733 b 76 198 114 114 222 003 0 Tr LME
18 HD 97658 b 77 216 002 020 949 008 008 Tr LME
19 HAT-P-7 b 105 3445 174 143 220 004 0 Tr HME
20 KELT-7 b 85 1372 129 153 273 004 0 Tr HME
21 HAT-P-14 b 100 2241 220 120 463 006 010 Tr HME
22 WASP-14 b 97 1628 734 128 224 004 009 Tr HME
23 HAT-P-2 b 87 1282 874 095 563 007 052 Tr HME
24 WASP-38 b 94 1368 271 108 687 008 003 Tr HME
25 HD 118203 b 81 925 214 105 613 007 029 RV HME
26 HD 2638 b 94 550 048 104 344 004 004 RV LME
27 WASP-13 b 104 2290 049 137 435 005 0 Tr LME
28 WASP-34 b 103 1326 059 122 432 005 004 Tr LME
29 WASP-82 b 101 2778 124 167 271 004 0 Tr HME
30 HD 17156 b 82 783 320 110 2122 016 068 Tr HME
31 XO-3 b 99 2143 1179 122 319 005 026 Tr HME
32 HD 33283 b 80 901 033 099 1818 017 046 RV LME
33 HD 217014 b 55 155 047 190 423 005 001 RV LME
34 HD 115383 b 52 175 400 096 minus 435 0 Im HME
35 HAT-P-6 b 105 2775 106 133 385 005 0 Tr LME
36 HD 75732 d 60 126 386 274 48670 545 003 RV HME
37 HD 120136 b 45 157 584 106 331 005 008 RV HME
38 WASP-76 b 95 1953 092 183 181 003 0 Tr LME
39 Hn-Peg b 60 181 1600 110 minus 795 0 Im BD
40 WASP-8 b 99 902 224 104 816 008 031 Tr HME
41 WASP-69 b 99 500 026 106 387 005 000 Tr LME
42 HAT-P-34 b 104 2511 333 111 545 007 044 Tr HME
43 HAT-P-1 b 99 1597 053 132 447 006 000 Tr LME
44 WASP-94 A b 101 2125 045 172 395 006 000 Tr LME
45 WASP-111 b 103 3005 185 144 231 004 000 Tr HME
46 HAT-P-8 b 104 2128 134 150 308 004 000 Tr HME
An in front of the name of the planet identifies multiple planetary systems
(see also Fortney et al 2010) In fact models ofexoplanet structures (eg Baraffe et al 1998 20032008) predict an inflection point in the mass-radiusrelation where the radius starts to decrease instead
of increasing as the mass increases Actually thisis what we observe in brown dwarfs (BDs) Physi-cally therefore this inflection should be located near12 MJup where self-gravity becomes stronger than
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 221
M(MJup)
R(R
Ju
p)
1eminus03 1eminus02 1eminus01 1e+00 1e+01 1e+02
00
50
10
02
00
50
10
02
00
12
MJup
13
MJup
BDHMEsLMEs
Fig 1 The M -R diagram for exoplanets in the Exo-planet Orbit Database The vertical line at 12 MJup
corresponds to the mass criterion we used to separatethe exoplanets in LMEs and HMEs The other verticallime is the lower mass limit for the BDs 13 MJup
the electromagnetic interaction and where the objectstarts to collapse (which is the case of BDs) How-ever if massive HJs have more massive envelopesof liquid metallic hydrogen (LMH) than observedin Jupiter (as suggested by JUNO Guillot et al2018 Kaspi et al 2018 Iess et al 2018 Adriani etal 2018) their structures might resist gravity (atleast for a while) due to the liquid state being in-compressible pushing the collapse of the radius toslightly higher masses (models and observations ver-ifying this prediction can be found in Hubbard et al1997 Dalladay-Simpson et al 2016 Flor-Torres etal 2016) This possibility however is still contro-versial and we only use the mass limit in our anal-ysis to separate our sample of exoplanets accordingto their masses On the other hand the possibil-ity of massive LMH envelopes in the HMEs mighthave some importance since these exoplanets wouldbe expected to have higher magnetic fields than theLMEs which consequently could affect their inter-actions with the PPD and nearby host stars (this fitsthe case of HD 80606 b a HME studied by de Witet al 2016) leading possibly to different migrationbehaviors Likewise we might also consider the pos-sibility of inflated radii in the LMEs since they areso close to their stars although less obvious is theeffect of inflated radius on migration (one possibilityis that they circularize more rapidly)
To test further our distinction in mass we tracein Figure 1 the mass-radius relation of 346 exoplan-ets from the Exoplanet Orbit Database Based onvisual inspection it is not clear how to separate the
HJs due to inflated radii One quantitative criterionis the mass-radius relation In Figure 1 we tracedthree M -R relations adopting 12 MJup to distin-guish LMEs form HMEs Below this limit we findthe relation
lnRLME = (0496plusmn0018) lnMLME+(0407plusmn0034)(1)
The slope is positive and the correlation coefficient ishigh r2 = 075 which implies that the radius withinthis range of masses continually increases with themass This relation is also fully consistent with whatwas previously reported by Valencia et al (2006) andChen amp Kipping (2017) Above the lower mass limitfor BDs we find the relation
lnRBD = (minus0117plusmn0111) lnMBD +(0455plusmn0404)(2)
with a negative slope and a weaker correlation coeffi-cient of r2 = 018 but sufficient to indicate that thetrend is for the radius to decrease with the massThis is as expected for objects where self-gravityis stronger than the electromagnetic repulsion (BDsnot having enough mass to ignite fusion in their corecannot avoid the effect of gravitational collapse) Fi-nally in between these two mass limits defining theHMEs we obtain the relation
lnRHME=(minus0044plusmn0030)lnMHME+(0229plusmn0031)(3)
which has an almost nil slope and a very weak coef-ficient of correlation r2 = 002 consistent with nocorrelation This range of mass therefore is fullyconsistent with HJs near the inflection point extend-ing this region over a decade in mass (as previouslynoted by Hatzes amp Rauer 2015) For our analysisthese different M -R relations are sufficient to justifyour separation between LMEs and HMEs (note thatthis physical criterion has never been used in the lit-erature the only study which uses a limit close toours is Sousa et al 2011)
3 CONNECTING THE STARS TO THEIRPLANETS
In Paper I we verified that the rotational velocityof a star V sin i decreases with the temperatureTeff In Figure 2 we reproduce this graphic but thistime distinguishing between stars hosting LMEs andHMEs We observe a clear trend for the HMEs tobe found around hotter and faster rotator stars thanthe LMEs Considering the small number of stars inour sample we need to check whether this result isphysical or due to an observational bias For exam-ple one could suggest that HMEs are easier to detect
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
222 FLOR-TORRES ET AL
6500 6000 5500 5000
010
20
30
40
50
Teff (K)
V s
in i (
km
s)
HMEsLMEsBDSun
Fig 2 Star rotational velocity vs temperature sepa-rating stars hosting HMEs and LMEs The position ofthe Sun is included as well as the star with a BD ascompanion
than LMEs through the RV than Tr method aroundhotter (more massive) and faster rotator stars Tocheck for observational biases we trace first in Fig-ure 3a the distributions of the absolute V magnitudefor the stars hosting LMEs and HMEs We do finda trend for the stars hosting HMEs to be more lumi-nous than the stars hosting LMEs This is confirmedby a non-parametric Mann-Whitney test with a p-value of 00007 (Dalgaard 2008) However in Fig-ure 3b we also show that the reason why the starswith HMEs are more luminous is because they arelocated farther out and this is independent of thedetection method Consequently the trend for theHMEs to be found around hotter and faster rotatorstars than the LMEs does not depend on the methodof detection but is a real physical difference Con-sidering the mass-temperature relation on the mainsequence this suggests that the more massive exo-planets in our sample rotate around more massivestars
In Figure 4 we compare the physical character-istics of the stars that host HMEs with those thathost LMEs Figure 4a shows a clear trend for therotational velocity to be higher in the stars hostingHMEs than LMEs In Table 2 we give the resultsof non-parametric Mann-Whitney (MW) tests Theuse of non-parametric tests is justified by the factthat we did not find normal distributions for our data(established by running 3 different normality tests)In Column 6 we give the p-values of the tests at alevel of confidence of 95 in Column 7 the signifi-cance levels (low medium and high ) and
$
amp
(
)+
-
)- )$- 01- 01$-2
322
22
422
amp22
56789lt9-
+-
Fig 3 a) Absolute magnitude in V separately for starshosting HMEs and LMEs b) Distance of the stars alsoseparated by the detection method radial velocity RVor transit Tr The bars correspond to the medians andinterquartile ranges
in Column 8 the acceptance or not of the differencesobserved Note that a non-parametric test comparesthe ranks of the data around the medians not themeans In the case of V sin i the MW test confirmsthe difference of medians at a relatively high level ofconfidence (stars 22 33 and 36 were not considereddue to their large uncertainties) In Figure 4b wesee the same trend for Teff on average also with asignificant difference in median in Table 2
In Figure 4c the difference in mass is obviousand confirmed by the MW test at the highest levelof confidence Stars with HMEs are more massivethan stars with LMEs On the other hand we findno difference in the distribution of the metallicityAlthough the medians and means reported in Ta-ble 2 look different the p-value of the MW test isunequivocal being much higher than 005 The me-dian [FeH] for our sample is 007 dex and the meanis 012 dex with a standard deviation of 024 dexThese values are comparable to those reported in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
218 FLOR-TORRES ET AL
bit angular momentum by tidal interactions with thePPD while the second high-eccentricity migration(eg Rasio amp Ford 1996 Weidenschilling amp Marzari1996 Marzari amp Weidenschilling 2002 Chatterjee etal 2008 Nagasawa et al 2008 Beauge amp Nesvorny2012) suggests that a planet interacting with otherplanets gains a higher eccentricity which brings itclose to its star where it reaches equilibrium by tidalinteractions (a process know as circularization) Interms of migration these two mechanisms might sug-gest that massive disks somehow amplified the levelof migration compared to what happened in the So-lar System because more massive PPDs either in-crease the intensity of interactions of the planetswith their disks or favor the formation of a highernumber of planets Within the standard model thiswould suggest that what counts is whether the PPDfollows the minimum mass model with a mass be-tween 001 to 002 M or the maximum mass modelwith a mass above 05 M (Armitage 2010) Thereare few clues which could help us to determine whichpath the PPD of the Solar System followed (andstrong difficulties compared to direct observationsof PPD see Figure 2 and discussion in Raymondamp Morbidelli 2020) One is the total mass of theplanets which represents only 01 the mass of theSun This implies the Solar System PPD has lostan important amount of its mass after the formationof the planets Another clue is that 99 of the an-gular momentum of the Solar System is located inthe planets suggesting that the initial angular mo-mentum of the PPD might have been conserved inthe planets However this is obviously not the casewhen large scale migration occurs so what was thedifference
If the initial angular momentum of the PPDpasses to the planets then one could use the orbitalrotation momentum in exoplanetary systems to testdifferent scenarios connecting the formation of theplanets to the formation of their stars For examplehow is the angular momentum of the PPD coupledto the angular momentum of the stars Since largescale migration represents a loss of angular momen-tum of the planets (at least by a factor 10) whatwas the initial angular momentum of the PPD whenit formed and how does this compared to the initialmass of the PPD Does this influence the masses ofthe planets and their migration The answers arenot trivial considering that the physics involved isstill not fully understood
In particular we know that the angular momen-tum is not conserved during the formation of starsThis is obvious when one compares how fast the Sun
rotates with how fast its rotation should have beenassuming the angular momentum of the collapsingmolecular cloud where it formed was conserved Ac-tually working the math (a basic problem but quiteinstructive see course notes by Alexander 2017) theSun effective angular momentum j = JM isasymp 106 times lower than expected Intriguingly jis also 103 lower than the angular momentum of itsbreaking point jb the point where the centripetalforce becomes stronger than gravity (McKee amp Os-triker 2007) If that were not true then no starswhatever their mass would be able to form In factobservations revealed that in general the angularmomentum of stars with spectral type O5 to A5traces a power law J prop Mα with α asymp 2 with typ-ical Jlowast values that are exactly ten times lower thantheir breaking point However how universal thisldquolawrdquois and how stars with different masses get toit is unexplained (Wesson 1979 Brosche 1980 Car-rasco et al 1982 God lowski et al 2003) To compli-cate the matter it is clear now that lower mass starslater than A5 do not follow this law their spin go-ing down exponentially (cf Figure 6 in Paper I)For low-mass stars McNally (1965) Kraft (1967)and Kawaler (1987) suggested a steeper power lawJ prop M57 which implies that they lose an extraamount of angular momentum as their mass goesdown What is interesting is that low-mass stars arealso those that form PPDs and planets which ledsome researchers to speculate that there could be alink between the two
To explain how low-mass stars lose their angu-lar momentum different mechanisms are consideredThe most probable one is stellar wind (Schatzman1962 Mestel 1968) which is related to the convectiveenvelopes of these stars This is how low-mass starswould differ from massive ones However whetherthis mechanism is sufficient to explain the break inthe J minusM relation is not obvious because it ignoresthe possible influence of the PPD (the formation ofa PPD seems crucial see de la Reza amp Pinzon 2004)This is what the magnetic braking model takes intoaccount (Wolff amp Simon 1997 Tassoul 2000 Uzden-sky et al 2002) Being bombarded by cosmic raysand UV radiation from ambient stars the matter ina molecular cloud is not neutral and thus permeableto magnetic fields This allows ambipolar diffusion(the separation of negative and positive charges) toreduce the magnetic flux allowing the cloud to col-lapse Consequently a diluted field follows the mat-ter through the accretion disk to the star forming itsmagnetic field (McKee amp Ostriker 2007) This alsoimplies that the accretion disk (or PPD) stays con-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 219
nected to the star through its magnetic field as longas it exists that is over a period that although briefincludes the complete phase of planet formation andmigration According to the model of disk-lockinga gap opens between the star and the disk at a dis-tance Rt from the star and matter falling betweenRt and the radius of corotation Rco (where the Ke-plerian angular rotation rate of the PPD equals thatof the star) follows the magnetic field to the polesof the star creating a jet that transports the angu-lar momentum out In particular this mechanismwas shown to explain why the classic T-Tauris rotatemore slowly than the weak T-Tauris (Ray 2012)How this magnetic coupling could influence the plan-ets and their migrations on the other hand is stillan open question (Matt amp Pudritz 2004 Fleck 2008Champion 2019)
To investigate further these problems we starteda new project to observe host stars of exoplanetsusing the 12 m robotic telescope TIGRE which isinstalled near our department at the La Luz Ob-servatory (in central Mexico) In paper I we ex-plained how we succeeded in determining in an effec-tive and homogeneous manner the physical charac-teristics ( Teff log g [MH] [FeH] and V sin i) of ainitial sample of 46 bright stars using iSpec (Blanco-Cuaresma 2014 2019) In this accompanying articlewe now explore the possible links between the phys-ical characteristics of these 46 stars and the physicalcharacteristics of their planets in order to gain newinsight about a connection between the formation ofstars and their planets
2 SAMPLE OF EXOPLANETS
Our observed sample consists of 46 stars hosts of59 exoplanets which were selected from the revisedcompendium of confirmed exoplanets in the Exo-planet Orbit Database3 In Table 1 the dominantplanet (Column 2) in each stellar system is iden-tified by the same number (Column 1) which wasused in Paper I to identify their stars In Column 3and 4 we repeat the magnitude and distance of thehost stars as they appeared in Table 1 of Paper IThis is followed by the main properties of the planetsas reported in the exoplanet Orbit Database mass(Column 5) radius (Column 6) period (Column 7)major semi-axis (Column 8) and eccentricity (Col-umn 9) Note that an eccentricity of zero could meanthe actual eccentricity is not known The last twocolumns identify the detection method and the dis-tinction between high mass and low mass exoplanet(as explained below)
3httpexoplanetsorg
Among our sample one exoplanet 39 (Hn-Peg b) was found to have a mass above the browndwarf (BD) lower limit of 13 MJup (Spiegel et al2011 Burgasser 2008) Note that Hn-Peg b was de-tected by imaging (identified as Im in Column 11)and its huge distance from its host star (Column 8)is more typical of BDs than of exoplanets (the otherexoplanet detected by imaging is 34 HD 115383 bat 435 AU from it stars but with a mass of only4 MJup) Another exoplanet whose mass is close tothe lower mass limit for BDs is 31 (XO-3 b) butsince it is located very near its star at 005 AUthere is no difficulty in accepting its classification asan exoplanet Due to the small size of our samplewe cannot assess how the formation of exoplanetswith masses above the BD limit can differ from theformation of the other exoplanets Consequently al-though we kept Hn-Peg in our sample of host starswe did not include its ldquoexoplanetrdquo in our statisticalanalysis
Another point of importance that can be notedin Table 1 is the fact that we have only 14 exoplan-ets detected by RV This is most probably due tothe fact that we required the radius of the planetsto be known which is easier to determine by the Trmethod Curiously only three of the 14 exoplanetsdetected by RV show the trend to be located fartherfrom their stars than the Tr exoplanets as observedin the literature (and as it is obvious in the Exo-planet Orbit Database) Two are close to the iceline and thus are rather warm than hot and onewith 386 MJup is at the same distance as Jupiterfrom the Sun This implies that any bias introducedby the different detection methods RV vs Tr can-not be explored thoroughly in our present analysis
Although the variety of the characteristics of ex-oplanets cannot be addressed with our present sam-ple we can however separate our sample of exoplan-ets into two groups based on their masses For ouranalysis this distinction is important in order to testhow the mass of the exoplanet is related to the massof its host star To use a mass limit that has a phys-ical meaning we choose 12 MJup which is the massabove which self-gravity in a planet becomes strongerthan the electromagnetic interactions (Padmanab-han 1993 Flor-Torres et al 2016 see demonstrationin Appendix A) Using this limit we separated oursample into 22 high mass exoplanets (HMEs) and 23low mass exoplanets (LMEs) This classification isincluded in Table 1 in the last column
According to Fortney et al (2007) the mass-radius relation of exoplanets shows a trend for ex-oplanets above 10 MJup to have a constant radius
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
220 FLOR-TORRES ET AL
TABLE 1
PHYSICAL PARAMETERS THE PLANETS IN OUR SAMPLE
Id Star and Magnitude Distance Mp Rp Period ap ep Detection Planetary
Planet (V) (pc) (Mjup) (Rjup) (days) (AU ) Method type
1 KELT-6 c 103 2424 371 268 12760 239 021 RV HME
2 HD 219134 h 56 65 028 080 21980 306 037 RV LME
3 KEPLER-37 b 98 640 001 003 1337 010 0 Tr LME
4 HD 46375 b 78 296 023 102 302 004 005 RV LME
5 HD 75289 b 64 291 047 103 351 005 002 RV LME
6 HD 88133 b 80 738 030 100 342 005 008 RV LME
7 HD 149143 b 79 734 133 105 407 005 001 RV HME
8 HAT-P-30 b 104 2153 071 134 281 004 004 Tr LME
9 KELT-3 b 98 2113 142 133 270 004 0 Tr HME
10 KEPLER-21 b 83 1089 002 015 279 004 002 Tr LME
11 KELT-2A b 87 1346 149 131 411 005 019 Tr HME
12 HD86081 b 87 1042 150 108 200 004 006 RV HME
13 WASP-74 b 98 1498 097 156 214 004 0 Tr LME
14 HD 149026 b 81 760 036 072 288 004 0 Tr LME
15 HD 209458 b 76 484 069 138 352 005 001 Tr LME
16 BD-10 3166 b 100 846 046 103 349 005 001 RV LME
17 HD 189733 b 76 198 114 114 222 003 0 Tr LME
18 HD 97658 b 77 216 002 020 949 008 008 Tr LME
19 HAT-P-7 b 105 3445 174 143 220 004 0 Tr HME
20 KELT-7 b 85 1372 129 153 273 004 0 Tr HME
21 HAT-P-14 b 100 2241 220 120 463 006 010 Tr HME
22 WASP-14 b 97 1628 734 128 224 004 009 Tr HME
23 HAT-P-2 b 87 1282 874 095 563 007 052 Tr HME
24 WASP-38 b 94 1368 271 108 687 008 003 Tr HME
25 HD 118203 b 81 925 214 105 613 007 029 RV HME
26 HD 2638 b 94 550 048 104 344 004 004 RV LME
27 WASP-13 b 104 2290 049 137 435 005 0 Tr LME
28 WASP-34 b 103 1326 059 122 432 005 004 Tr LME
29 WASP-82 b 101 2778 124 167 271 004 0 Tr HME
30 HD 17156 b 82 783 320 110 2122 016 068 Tr HME
31 XO-3 b 99 2143 1179 122 319 005 026 Tr HME
32 HD 33283 b 80 901 033 099 1818 017 046 RV LME
33 HD 217014 b 55 155 047 190 423 005 001 RV LME
34 HD 115383 b 52 175 400 096 minus 435 0 Im HME
35 HAT-P-6 b 105 2775 106 133 385 005 0 Tr LME
36 HD 75732 d 60 126 386 274 48670 545 003 RV HME
37 HD 120136 b 45 157 584 106 331 005 008 RV HME
38 WASP-76 b 95 1953 092 183 181 003 0 Tr LME
39 Hn-Peg b 60 181 1600 110 minus 795 0 Im BD
40 WASP-8 b 99 902 224 104 816 008 031 Tr HME
41 WASP-69 b 99 500 026 106 387 005 000 Tr LME
42 HAT-P-34 b 104 2511 333 111 545 007 044 Tr HME
43 HAT-P-1 b 99 1597 053 132 447 006 000 Tr LME
44 WASP-94 A b 101 2125 045 172 395 006 000 Tr LME
45 WASP-111 b 103 3005 185 144 231 004 000 Tr HME
46 HAT-P-8 b 104 2128 134 150 308 004 000 Tr HME
An in front of the name of the planet identifies multiple planetary systems
(see also Fortney et al 2010) In fact models ofexoplanet structures (eg Baraffe et al 1998 20032008) predict an inflection point in the mass-radiusrelation where the radius starts to decrease instead
of increasing as the mass increases Actually thisis what we observe in brown dwarfs (BDs) Physi-cally therefore this inflection should be located near12 MJup where self-gravity becomes stronger than
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 221
M(MJup)
R(R
Ju
p)
1eminus03 1eminus02 1eminus01 1e+00 1e+01 1e+02
00
50
10
02
00
50
10
02
00
12
MJup
13
MJup
BDHMEsLMEs
Fig 1 The M -R diagram for exoplanets in the Exo-planet Orbit Database The vertical line at 12 MJup
corresponds to the mass criterion we used to separatethe exoplanets in LMEs and HMEs The other verticallime is the lower mass limit for the BDs 13 MJup
the electromagnetic interaction and where the objectstarts to collapse (which is the case of BDs) How-ever if massive HJs have more massive envelopesof liquid metallic hydrogen (LMH) than observedin Jupiter (as suggested by JUNO Guillot et al2018 Kaspi et al 2018 Iess et al 2018 Adriani etal 2018) their structures might resist gravity (atleast for a while) due to the liquid state being in-compressible pushing the collapse of the radius toslightly higher masses (models and observations ver-ifying this prediction can be found in Hubbard et al1997 Dalladay-Simpson et al 2016 Flor-Torres etal 2016) This possibility however is still contro-versial and we only use the mass limit in our anal-ysis to separate our sample of exoplanets accordingto their masses On the other hand the possibil-ity of massive LMH envelopes in the HMEs mighthave some importance since these exoplanets wouldbe expected to have higher magnetic fields than theLMEs which consequently could affect their inter-actions with the PPD and nearby host stars (this fitsthe case of HD 80606 b a HME studied by de Witet al 2016) leading possibly to different migrationbehaviors Likewise we might also consider the pos-sibility of inflated radii in the LMEs since they areso close to their stars although less obvious is theeffect of inflated radius on migration (one possibilityis that they circularize more rapidly)
To test further our distinction in mass we tracein Figure 1 the mass-radius relation of 346 exoplan-ets from the Exoplanet Orbit Database Based onvisual inspection it is not clear how to separate the
HJs due to inflated radii One quantitative criterionis the mass-radius relation In Figure 1 we tracedthree M -R relations adopting 12 MJup to distin-guish LMEs form HMEs Below this limit we findthe relation
lnRLME = (0496plusmn0018) lnMLME+(0407plusmn0034)(1)
The slope is positive and the correlation coefficient ishigh r2 = 075 which implies that the radius withinthis range of masses continually increases with themass This relation is also fully consistent with whatwas previously reported by Valencia et al (2006) andChen amp Kipping (2017) Above the lower mass limitfor BDs we find the relation
lnRBD = (minus0117plusmn0111) lnMBD +(0455plusmn0404)(2)
with a negative slope and a weaker correlation coeffi-cient of r2 = 018 but sufficient to indicate that thetrend is for the radius to decrease with the massThis is as expected for objects where self-gravityis stronger than the electromagnetic repulsion (BDsnot having enough mass to ignite fusion in their corecannot avoid the effect of gravitational collapse) Fi-nally in between these two mass limits defining theHMEs we obtain the relation
lnRHME=(minus0044plusmn0030)lnMHME+(0229plusmn0031)(3)
which has an almost nil slope and a very weak coef-ficient of correlation r2 = 002 consistent with nocorrelation This range of mass therefore is fullyconsistent with HJs near the inflection point extend-ing this region over a decade in mass (as previouslynoted by Hatzes amp Rauer 2015) For our analysisthese different M -R relations are sufficient to justifyour separation between LMEs and HMEs (note thatthis physical criterion has never been used in the lit-erature the only study which uses a limit close toours is Sousa et al 2011)
3 CONNECTING THE STARS TO THEIRPLANETS
In Paper I we verified that the rotational velocityof a star V sin i decreases with the temperatureTeff In Figure 2 we reproduce this graphic but thistime distinguishing between stars hosting LMEs andHMEs We observe a clear trend for the HMEs tobe found around hotter and faster rotator stars thanthe LMEs Considering the small number of stars inour sample we need to check whether this result isphysical or due to an observational bias For exam-ple one could suggest that HMEs are easier to detect
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
222 FLOR-TORRES ET AL
6500 6000 5500 5000
010
20
30
40
50
Teff (K)
V s
in i (
km
s)
HMEsLMEsBDSun
Fig 2 Star rotational velocity vs temperature sepa-rating stars hosting HMEs and LMEs The position ofthe Sun is included as well as the star with a BD ascompanion
than LMEs through the RV than Tr method aroundhotter (more massive) and faster rotator stars Tocheck for observational biases we trace first in Fig-ure 3a the distributions of the absolute V magnitudefor the stars hosting LMEs and HMEs We do finda trend for the stars hosting HMEs to be more lumi-nous than the stars hosting LMEs This is confirmedby a non-parametric Mann-Whitney test with a p-value of 00007 (Dalgaard 2008) However in Fig-ure 3b we also show that the reason why the starswith HMEs are more luminous is because they arelocated farther out and this is independent of thedetection method Consequently the trend for theHMEs to be found around hotter and faster rotatorstars than the LMEs does not depend on the methodof detection but is a real physical difference Con-sidering the mass-temperature relation on the mainsequence this suggests that the more massive exo-planets in our sample rotate around more massivestars
In Figure 4 we compare the physical character-istics of the stars that host HMEs with those thathost LMEs Figure 4a shows a clear trend for therotational velocity to be higher in the stars hostingHMEs than LMEs In Table 2 we give the resultsof non-parametric Mann-Whitney (MW) tests Theuse of non-parametric tests is justified by the factthat we did not find normal distributions for our data(established by running 3 different normality tests)In Column 6 we give the p-values of the tests at alevel of confidence of 95 in Column 7 the signifi-cance levels (low medium and high ) and
$
amp
(
)+
-
)- )$- 01- 01$-2
322
22
422
amp22
56789lt9-
+-
Fig 3 a) Absolute magnitude in V separately for starshosting HMEs and LMEs b) Distance of the stars alsoseparated by the detection method radial velocity RVor transit Tr The bars correspond to the medians andinterquartile ranges
in Column 8 the acceptance or not of the differencesobserved Note that a non-parametric test comparesthe ranks of the data around the medians not themeans In the case of V sin i the MW test confirmsthe difference of medians at a relatively high level ofconfidence (stars 22 33 and 36 were not considereddue to their large uncertainties) In Figure 4b wesee the same trend for Teff on average also with asignificant difference in median in Table 2
In Figure 4c the difference in mass is obviousand confirmed by the MW test at the highest levelof confidence Stars with HMEs are more massivethan stars with LMEs On the other hand we findno difference in the distribution of the metallicityAlthough the medians and means reported in Ta-ble 2 look different the p-value of the MW test isunequivocal being much higher than 005 The me-dian [FeH] for our sample is 007 dex and the meanis 012 dex with a standard deviation of 024 dexThese values are comparable to those reported in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 219
nected to the star through its magnetic field as longas it exists that is over a period that although briefincludes the complete phase of planet formation andmigration According to the model of disk-lockinga gap opens between the star and the disk at a dis-tance Rt from the star and matter falling betweenRt and the radius of corotation Rco (where the Ke-plerian angular rotation rate of the PPD equals thatof the star) follows the magnetic field to the polesof the star creating a jet that transports the angu-lar momentum out In particular this mechanismwas shown to explain why the classic T-Tauris rotatemore slowly than the weak T-Tauris (Ray 2012)How this magnetic coupling could influence the plan-ets and their migrations on the other hand is stillan open question (Matt amp Pudritz 2004 Fleck 2008Champion 2019)
To investigate further these problems we starteda new project to observe host stars of exoplanetsusing the 12 m robotic telescope TIGRE which isinstalled near our department at the La Luz Ob-servatory (in central Mexico) In paper I we ex-plained how we succeeded in determining in an effec-tive and homogeneous manner the physical charac-teristics ( Teff log g [MH] [FeH] and V sin i) of ainitial sample of 46 bright stars using iSpec (Blanco-Cuaresma 2014 2019) In this accompanying articlewe now explore the possible links between the phys-ical characteristics of these 46 stars and the physicalcharacteristics of their planets in order to gain newinsight about a connection between the formation ofstars and their planets
2 SAMPLE OF EXOPLANETS
Our observed sample consists of 46 stars hosts of59 exoplanets which were selected from the revisedcompendium of confirmed exoplanets in the Exo-planet Orbit Database3 In Table 1 the dominantplanet (Column 2) in each stellar system is iden-tified by the same number (Column 1) which wasused in Paper I to identify their stars In Column 3and 4 we repeat the magnitude and distance of thehost stars as they appeared in Table 1 of Paper IThis is followed by the main properties of the planetsas reported in the exoplanet Orbit Database mass(Column 5) radius (Column 6) period (Column 7)major semi-axis (Column 8) and eccentricity (Col-umn 9) Note that an eccentricity of zero could meanthe actual eccentricity is not known The last twocolumns identify the detection method and the dis-tinction between high mass and low mass exoplanet(as explained below)
3httpexoplanetsorg
Among our sample one exoplanet 39 (Hn-Peg b) was found to have a mass above the browndwarf (BD) lower limit of 13 MJup (Spiegel et al2011 Burgasser 2008) Note that Hn-Peg b was de-tected by imaging (identified as Im in Column 11)and its huge distance from its host star (Column 8)is more typical of BDs than of exoplanets (the otherexoplanet detected by imaging is 34 HD 115383 bat 435 AU from it stars but with a mass of only4 MJup) Another exoplanet whose mass is close tothe lower mass limit for BDs is 31 (XO-3 b) butsince it is located very near its star at 005 AUthere is no difficulty in accepting its classification asan exoplanet Due to the small size of our samplewe cannot assess how the formation of exoplanetswith masses above the BD limit can differ from theformation of the other exoplanets Consequently al-though we kept Hn-Peg in our sample of host starswe did not include its ldquoexoplanetrdquo in our statisticalanalysis
Another point of importance that can be notedin Table 1 is the fact that we have only 14 exoplan-ets detected by RV This is most probably due tothe fact that we required the radius of the planetsto be known which is easier to determine by the Trmethod Curiously only three of the 14 exoplanetsdetected by RV show the trend to be located fartherfrom their stars than the Tr exoplanets as observedin the literature (and as it is obvious in the Exo-planet Orbit Database) Two are close to the iceline and thus are rather warm than hot and onewith 386 MJup is at the same distance as Jupiterfrom the Sun This implies that any bias introducedby the different detection methods RV vs Tr can-not be explored thoroughly in our present analysis
Although the variety of the characteristics of ex-oplanets cannot be addressed with our present sam-ple we can however separate our sample of exoplan-ets into two groups based on their masses For ouranalysis this distinction is important in order to testhow the mass of the exoplanet is related to the massof its host star To use a mass limit that has a phys-ical meaning we choose 12 MJup which is the massabove which self-gravity in a planet becomes strongerthan the electromagnetic interactions (Padmanab-han 1993 Flor-Torres et al 2016 see demonstrationin Appendix A) Using this limit we separated oursample into 22 high mass exoplanets (HMEs) and 23low mass exoplanets (LMEs) This classification isincluded in Table 1 in the last column
According to Fortney et al (2007) the mass-radius relation of exoplanets shows a trend for ex-oplanets above 10 MJup to have a constant radius
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
220 FLOR-TORRES ET AL
TABLE 1
PHYSICAL PARAMETERS THE PLANETS IN OUR SAMPLE
Id Star and Magnitude Distance Mp Rp Period ap ep Detection Planetary
Planet (V) (pc) (Mjup) (Rjup) (days) (AU ) Method type
1 KELT-6 c 103 2424 371 268 12760 239 021 RV HME
2 HD 219134 h 56 65 028 080 21980 306 037 RV LME
3 KEPLER-37 b 98 640 001 003 1337 010 0 Tr LME
4 HD 46375 b 78 296 023 102 302 004 005 RV LME
5 HD 75289 b 64 291 047 103 351 005 002 RV LME
6 HD 88133 b 80 738 030 100 342 005 008 RV LME
7 HD 149143 b 79 734 133 105 407 005 001 RV HME
8 HAT-P-30 b 104 2153 071 134 281 004 004 Tr LME
9 KELT-3 b 98 2113 142 133 270 004 0 Tr HME
10 KEPLER-21 b 83 1089 002 015 279 004 002 Tr LME
11 KELT-2A b 87 1346 149 131 411 005 019 Tr HME
12 HD86081 b 87 1042 150 108 200 004 006 RV HME
13 WASP-74 b 98 1498 097 156 214 004 0 Tr LME
14 HD 149026 b 81 760 036 072 288 004 0 Tr LME
15 HD 209458 b 76 484 069 138 352 005 001 Tr LME
16 BD-10 3166 b 100 846 046 103 349 005 001 RV LME
17 HD 189733 b 76 198 114 114 222 003 0 Tr LME
18 HD 97658 b 77 216 002 020 949 008 008 Tr LME
19 HAT-P-7 b 105 3445 174 143 220 004 0 Tr HME
20 KELT-7 b 85 1372 129 153 273 004 0 Tr HME
21 HAT-P-14 b 100 2241 220 120 463 006 010 Tr HME
22 WASP-14 b 97 1628 734 128 224 004 009 Tr HME
23 HAT-P-2 b 87 1282 874 095 563 007 052 Tr HME
24 WASP-38 b 94 1368 271 108 687 008 003 Tr HME
25 HD 118203 b 81 925 214 105 613 007 029 RV HME
26 HD 2638 b 94 550 048 104 344 004 004 RV LME
27 WASP-13 b 104 2290 049 137 435 005 0 Tr LME
28 WASP-34 b 103 1326 059 122 432 005 004 Tr LME
29 WASP-82 b 101 2778 124 167 271 004 0 Tr HME
30 HD 17156 b 82 783 320 110 2122 016 068 Tr HME
31 XO-3 b 99 2143 1179 122 319 005 026 Tr HME
32 HD 33283 b 80 901 033 099 1818 017 046 RV LME
33 HD 217014 b 55 155 047 190 423 005 001 RV LME
34 HD 115383 b 52 175 400 096 minus 435 0 Im HME
35 HAT-P-6 b 105 2775 106 133 385 005 0 Tr LME
36 HD 75732 d 60 126 386 274 48670 545 003 RV HME
37 HD 120136 b 45 157 584 106 331 005 008 RV HME
38 WASP-76 b 95 1953 092 183 181 003 0 Tr LME
39 Hn-Peg b 60 181 1600 110 minus 795 0 Im BD
40 WASP-8 b 99 902 224 104 816 008 031 Tr HME
41 WASP-69 b 99 500 026 106 387 005 000 Tr LME
42 HAT-P-34 b 104 2511 333 111 545 007 044 Tr HME
43 HAT-P-1 b 99 1597 053 132 447 006 000 Tr LME
44 WASP-94 A b 101 2125 045 172 395 006 000 Tr LME
45 WASP-111 b 103 3005 185 144 231 004 000 Tr HME
46 HAT-P-8 b 104 2128 134 150 308 004 000 Tr HME
An in front of the name of the planet identifies multiple planetary systems
(see also Fortney et al 2010) In fact models ofexoplanet structures (eg Baraffe et al 1998 20032008) predict an inflection point in the mass-radiusrelation where the radius starts to decrease instead
of increasing as the mass increases Actually thisis what we observe in brown dwarfs (BDs) Physi-cally therefore this inflection should be located near12 MJup where self-gravity becomes stronger than
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 221
M(MJup)
R(R
Ju
p)
1eminus03 1eminus02 1eminus01 1e+00 1e+01 1e+02
00
50
10
02
00
50
10
02
00
12
MJup
13
MJup
BDHMEsLMEs
Fig 1 The M -R diagram for exoplanets in the Exo-planet Orbit Database The vertical line at 12 MJup
corresponds to the mass criterion we used to separatethe exoplanets in LMEs and HMEs The other verticallime is the lower mass limit for the BDs 13 MJup
the electromagnetic interaction and where the objectstarts to collapse (which is the case of BDs) How-ever if massive HJs have more massive envelopesof liquid metallic hydrogen (LMH) than observedin Jupiter (as suggested by JUNO Guillot et al2018 Kaspi et al 2018 Iess et al 2018 Adriani etal 2018) their structures might resist gravity (atleast for a while) due to the liquid state being in-compressible pushing the collapse of the radius toslightly higher masses (models and observations ver-ifying this prediction can be found in Hubbard et al1997 Dalladay-Simpson et al 2016 Flor-Torres etal 2016) This possibility however is still contro-versial and we only use the mass limit in our anal-ysis to separate our sample of exoplanets accordingto their masses On the other hand the possibil-ity of massive LMH envelopes in the HMEs mighthave some importance since these exoplanets wouldbe expected to have higher magnetic fields than theLMEs which consequently could affect their inter-actions with the PPD and nearby host stars (this fitsthe case of HD 80606 b a HME studied by de Witet al 2016) leading possibly to different migrationbehaviors Likewise we might also consider the pos-sibility of inflated radii in the LMEs since they areso close to their stars although less obvious is theeffect of inflated radius on migration (one possibilityis that they circularize more rapidly)
To test further our distinction in mass we tracein Figure 1 the mass-radius relation of 346 exoplan-ets from the Exoplanet Orbit Database Based onvisual inspection it is not clear how to separate the
HJs due to inflated radii One quantitative criterionis the mass-radius relation In Figure 1 we tracedthree M -R relations adopting 12 MJup to distin-guish LMEs form HMEs Below this limit we findthe relation
lnRLME = (0496plusmn0018) lnMLME+(0407plusmn0034)(1)
The slope is positive and the correlation coefficient ishigh r2 = 075 which implies that the radius withinthis range of masses continually increases with themass This relation is also fully consistent with whatwas previously reported by Valencia et al (2006) andChen amp Kipping (2017) Above the lower mass limitfor BDs we find the relation
lnRBD = (minus0117plusmn0111) lnMBD +(0455plusmn0404)(2)
with a negative slope and a weaker correlation coeffi-cient of r2 = 018 but sufficient to indicate that thetrend is for the radius to decrease with the massThis is as expected for objects where self-gravityis stronger than the electromagnetic repulsion (BDsnot having enough mass to ignite fusion in their corecannot avoid the effect of gravitational collapse) Fi-nally in between these two mass limits defining theHMEs we obtain the relation
lnRHME=(minus0044plusmn0030)lnMHME+(0229plusmn0031)(3)
which has an almost nil slope and a very weak coef-ficient of correlation r2 = 002 consistent with nocorrelation This range of mass therefore is fullyconsistent with HJs near the inflection point extend-ing this region over a decade in mass (as previouslynoted by Hatzes amp Rauer 2015) For our analysisthese different M -R relations are sufficient to justifyour separation between LMEs and HMEs (note thatthis physical criterion has never been used in the lit-erature the only study which uses a limit close toours is Sousa et al 2011)
3 CONNECTING THE STARS TO THEIRPLANETS
In Paper I we verified that the rotational velocityof a star V sin i decreases with the temperatureTeff In Figure 2 we reproduce this graphic but thistime distinguishing between stars hosting LMEs andHMEs We observe a clear trend for the HMEs tobe found around hotter and faster rotator stars thanthe LMEs Considering the small number of stars inour sample we need to check whether this result isphysical or due to an observational bias For exam-ple one could suggest that HMEs are easier to detect
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
222 FLOR-TORRES ET AL
6500 6000 5500 5000
010
20
30
40
50
Teff (K)
V s
in i (
km
s)
HMEsLMEsBDSun
Fig 2 Star rotational velocity vs temperature sepa-rating stars hosting HMEs and LMEs The position ofthe Sun is included as well as the star with a BD ascompanion
than LMEs through the RV than Tr method aroundhotter (more massive) and faster rotator stars Tocheck for observational biases we trace first in Fig-ure 3a the distributions of the absolute V magnitudefor the stars hosting LMEs and HMEs We do finda trend for the stars hosting HMEs to be more lumi-nous than the stars hosting LMEs This is confirmedby a non-parametric Mann-Whitney test with a p-value of 00007 (Dalgaard 2008) However in Fig-ure 3b we also show that the reason why the starswith HMEs are more luminous is because they arelocated farther out and this is independent of thedetection method Consequently the trend for theHMEs to be found around hotter and faster rotatorstars than the LMEs does not depend on the methodof detection but is a real physical difference Con-sidering the mass-temperature relation on the mainsequence this suggests that the more massive exo-planets in our sample rotate around more massivestars
In Figure 4 we compare the physical character-istics of the stars that host HMEs with those thathost LMEs Figure 4a shows a clear trend for therotational velocity to be higher in the stars hostingHMEs than LMEs In Table 2 we give the resultsof non-parametric Mann-Whitney (MW) tests Theuse of non-parametric tests is justified by the factthat we did not find normal distributions for our data(established by running 3 different normality tests)In Column 6 we give the p-values of the tests at alevel of confidence of 95 in Column 7 the signifi-cance levels (low medium and high ) and
$
amp
(
)+
-
)- )$- 01- 01$-2
322
22
422
amp22
56789lt9-
+-
Fig 3 a) Absolute magnitude in V separately for starshosting HMEs and LMEs b) Distance of the stars alsoseparated by the detection method radial velocity RVor transit Tr The bars correspond to the medians andinterquartile ranges
in Column 8 the acceptance or not of the differencesobserved Note that a non-parametric test comparesthe ranks of the data around the medians not themeans In the case of V sin i the MW test confirmsthe difference of medians at a relatively high level ofconfidence (stars 22 33 and 36 were not considereddue to their large uncertainties) In Figure 4b wesee the same trend for Teff on average also with asignificant difference in median in Table 2
In Figure 4c the difference in mass is obviousand confirmed by the MW test at the highest levelof confidence Stars with HMEs are more massivethan stars with LMEs On the other hand we findno difference in the distribution of the metallicityAlthough the medians and means reported in Ta-ble 2 look different the p-value of the MW test isunequivocal being much higher than 005 The me-dian [FeH] for our sample is 007 dex and the meanis 012 dex with a standard deviation of 024 dexThese values are comparable to those reported in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
220 FLOR-TORRES ET AL
TABLE 1
PHYSICAL PARAMETERS THE PLANETS IN OUR SAMPLE
Id Star and Magnitude Distance Mp Rp Period ap ep Detection Planetary
Planet (V) (pc) (Mjup) (Rjup) (days) (AU ) Method type
1 KELT-6 c 103 2424 371 268 12760 239 021 RV HME
2 HD 219134 h 56 65 028 080 21980 306 037 RV LME
3 KEPLER-37 b 98 640 001 003 1337 010 0 Tr LME
4 HD 46375 b 78 296 023 102 302 004 005 RV LME
5 HD 75289 b 64 291 047 103 351 005 002 RV LME
6 HD 88133 b 80 738 030 100 342 005 008 RV LME
7 HD 149143 b 79 734 133 105 407 005 001 RV HME
8 HAT-P-30 b 104 2153 071 134 281 004 004 Tr LME
9 KELT-3 b 98 2113 142 133 270 004 0 Tr HME
10 KEPLER-21 b 83 1089 002 015 279 004 002 Tr LME
11 KELT-2A b 87 1346 149 131 411 005 019 Tr HME
12 HD86081 b 87 1042 150 108 200 004 006 RV HME
13 WASP-74 b 98 1498 097 156 214 004 0 Tr LME
14 HD 149026 b 81 760 036 072 288 004 0 Tr LME
15 HD 209458 b 76 484 069 138 352 005 001 Tr LME
16 BD-10 3166 b 100 846 046 103 349 005 001 RV LME
17 HD 189733 b 76 198 114 114 222 003 0 Tr LME
18 HD 97658 b 77 216 002 020 949 008 008 Tr LME
19 HAT-P-7 b 105 3445 174 143 220 004 0 Tr HME
20 KELT-7 b 85 1372 129 153 273 004 0 Tr HME
21 HAT-P-14 b 100 2241 220 120 463 006 010 Tr HME
22 WASP-14 b 97 1628 734 128 224 004 009 Tr HME
23 HAT-P-2 b 87 1282 874 095 563 007 052 Tr HME
24 WASP-38 b 94 1368 271 108 687 008 003 Tr HME
25 HD 118203 b 81 925 214 105 613 007 029 RV HME
26 HD 2638 b 94 550 048 104 344 004 004 RV LME
27 WASP-13 b 104 2290 049 137 435 005 0 Tr LME
28 WASP-34 b 103 1326 059 122 432 005 004 Tr LME
29 WASP-82 b 101 2778 124 167 271 004 0 Tr HME
30 HD 17156 b 82 783 320 110 2122 016 068 Tr HME
31 XO-3 b 99 2143 1179 122 319 005 026 Tr HME
32 HD 33283 b 80 901 033 099 1818 017 046 RV LME
33 HD 217014 b 55 155 047 190 423 005 001 RV LME
34 HD 115383 b 52 175 400 096 minus 435 0 Im HME
35 HAT-P-6 b 105 2775 106 133 385 005 0 Tr LME
36 HD 75732 d 60 126 386 274 48670 545 003 RV HME
37 HD 120136 b 45 157 584 106 331 005 008 RV HME
38 WASP-76 b 95 1953 092 183 181 003 0 Tr LME
39 Hn-Peg b 60 181 1600 110 minus 795 0 Im BD
40 WASP-8 b 99 902 224 104 816 008 031 Tr HME
41 WASP-69 b 99 500 026 106 387 005 000 Tr LME
42 HAT-P-34 b 104 2511 333 111 545 007 044 Tr HME
43 HAT-P-1 b 99 1597 053 132 447 006 000 Tr LME
44 WASP-94 A b 101 2125 045 172 395 006 000 Tr LME
45 WASP-111 b 103 3005 185 144 231 004 000 Tr HME
46 HAT-P-8 b 104 2128 134 150 308 004 000 Tr HME
An in front of the name of the planet identifies multiple planetary systems
(see also Fortney et al 2010) In fact models ofexoplanet structures (eg Baraffe et al 1998 20032008) predict an inflection point in the mass-radiusrelation where the radius starts to decrease instead
of increasing as the mass increases Actually thisis what we observe in brown dwarfs (BDs) Physi-cally therefore this inflection should be located near12 MJup where self-gravity becomes stronger than
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 221
M(MJup)
R(R
Ju
p)
1eminus03 1eminus02 1eminus01 1e+00 1e+01 1e+02
00
50
10
02
00
50
10
02
00
12
MJup
13
MJup
BDHMEsLMEs
Fig 1 The M -R diagram for exoplanets in the Exo-planet Orbit Database The vertical line at 12 MJup
corresponds to the mass criterion we used to separatethe exoplanets in LMEs and HMEs The other verticallime is the lower mass limit for the BDs 13 MJup
the electromagnetic interaction and where the objectstarts to collapse (which is the case of BDs) How-ever if massive HJs have more massive envelopesof liquid metallic hydrogen (LMH) than observedin Jupiter (as suggested by JUNO Guillot et al2018 Kaspi et al 2018 Iess et al 2018 Adriani etal 2018) their structures might resist gravity (atleast for a while) due to the liquid state being in-compressible pushing the collapse of the radius toslightly higher masses (models and observations ver-ifying this prediction can be found in Hubbard et al1997 Dalladay-Simpson et al 2016 Flor-Torres etal 2016) This possibility however is still contro-versial and we only use the mass limit in our anal-ysis to separate our sample of exoplanets accordingto their masses On the other hand the possibil-ity of massive LMH envelopes in the HMEs mighthave some importance since these exoplanets wouldbe expected to have higher magnetic fields than theLMEs which consequently could affect their inter-actions with the PPD and nearby host stars (this fitsthe case of HD 80606 b a HME studied by de Witet al 2016) leading possibly to different migrationbehaviors Likewise we might also consider the pos-sibility of inflated radii in the LMEs since they areso close to their stars although less obvious is theeffect of inflated radius on migration (one possibilityis that they circularize more rapidly)
To test further our distinction in mass we tracein Figure 1 the mass-radius relation of 346 exoplan-ets from the Exoplanet Orbit Database Based onvisual inspection it is not clear how to separate the
HJs due to inflated radii One quantitative criterionis the mass-radius relation In Figure 1 we tracedthree M -R relations adopting 12 MJup to distin-guish LMEs form HMEs Below this limit we findthe relation
lnRLME = (0496plusmn0018) lnMLME+(0407plusmn0034)(1)
The slope is positive and the correlation coefficient ishigh r2 = 075 which implies that the radius withinthis range of masses continually increases with themass This relation is also fully consistent with whatwas previously reported by Valencia et al (2006) andChen amp Kipping (2017) Above the lower mass limitfor BDs we find the relation
lnRBD = (minus0117plusmn0111) lnMBD +(0455plusmn0404)(2)
with a negative slope and a weaker correlation coeffi-cient of r2 = 018 but sufficient to indicate that thetrend is for the radius to decrease with the massThis is as expected for objects where self-gravityis stronger than the electromagnetic repulsion (BDsnot having enough mass to ignite fusion in their corecannot avoid the effect of gravitational collapse) Fi-nally in between these two mass limits defining theHMEs we obtain the relation
lnRHME=(minus0044plusmn0030)lnMHME+(0229plusmn0031)(3)
which has an almost nil slope and a very weak coef-ficient of correlation r2 = 002 consistent with nocorrelation This range of mass therefore is fullyconsistent with HJs near the inflection point extend-ing this region over a decade in mass (as previouslynoted by Hatzes amp Rauer 2015) For our analysisthese different M -R relations are sufficient to justifyour separation between LMEs and HMEs (note thatthis physical criterion has never been used in the lit-erature the only study which uses a limit close toours is Sousa et al 2011)
3 CONNECTING THE STARS TO THEIRPLANETS
In Paper I we verified that the rotational velocityof a star V sin i decreases with the temperatureTeff In Figure 2 we reproduce this graphic but thistime distinguishing between stars hosting LMEs andHMEs We observe a clear trend for the HMEs tobe found around hotter and faster rotator stars thanthe LMEs Considering the small number of stars inour sample we need to check whether this result isphysical or due to an observational bias For exam-ple one could suggest that HMEs are easier to detect
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
222 FLOR-TORRES ET AL
6500 6000 5500 5000
010
20
30
40
50
Teff (K)
V s
in i (
km
s)
HMEsLMEsBDSun
Fig 2 Star rotational velocity vs temperature sepa-rating stars hosting HMEs and LMEs The position ofthe Sun is included as well as the star with a BD ascompanion
than LMEs through the RV than Tr method aroundhotter (more massive) and faster rotator stars Tocheck for observational biases we trace first in Fig-ure 3a the distributions of the absolute V magnitudefor the stars hosting LMEs and HMEs We do finda trend for the stars hosting HMEs to be more lumi-nous than the stars hosting LMEs This is confirmedby a non-parametric Mann-Whitney test with a p-value of 00007 (Dalgaard 2008) However in Fig-ure 3b we also show that the reason why the starswith HMEs are more luminous is because they arelocated farther out and this is independent of thedetection method Consequently the trend for theHMEs to be found around hotter and faster rotatorstars than the LMEs does not depend on the methodof detection but is a real physical difference Con-sidering the mass-temperature relation on the mainsequence this suggests that the more massive exo-planets in our sample rotate around more massivestars
In Figure 4 we compare the physical character-istics of the stars that host HMEs with those thathost LMEs Figure 4a shows a clear trend for therotational velocity to be higher in the stars hostingHMEs than LMEs In Table 2 we give the resultsof non-parametric Mann-Whitney (MW) tests Theuse of non-parametric tests is justified by the factthat we did not find normal distributions for our data(established by running 3 different normality tests)In Column 6 we give the p-values of the tests at alevel of confidence of 95 in Column 7 the signifi-cance levels (low medium and high ) and
$
amp
(
)+
-
)- )$- 01- 01$-2
322
22
422
amp22
56789lt9-
+-
Fig 3 a) Absolute magnitude in V separately for starshosting HMEs and LMEs b) Distance of the stars alsoseparated by the detection method radial velocity RVor transit Tr The bars correspond to the medians andinterquartile ranges
in Column 8 the acceptance or not of the differencesobserved Note that a non-parametric test comparesthe ranks of the data around the medians not themeans In the case of V sin i the MW test confirmsthe difference of medians at a relatively high level ofconfidence (stars 22 33 and 36 were not considereddue to their large uncertainties) In Figure 4b wesee the same trend for Teff on average also with asignificant difference in median in Table 2
In Figure 4c the difference in mass is obviousand confirmed by the MW test at the highest levelof confidence Stars with HMEs are more massivethan stars with LMEs On the other hand we findno difference in the distribution of the metallicityAlthough the medians and means reported in Ta-ble 2 look different the p-value of the MW test isunequivocal being much higher than 005 The me-dian [FeH] for our sample is 007 dex and the meanis 012 dex with a standard deviation of 024 dexThese values are comparable to those reported in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 221
M(MJup)
R(R
Ju
p)
1eminus03 1eminus02 1eminus01 1e+00 1e+01 1e+02
00
50
10
02
00
50
10
02
00
12
MJup
13
MJup
BDHMEsLMEs
Fig 1 The M -R diagram for exoplanets in the Exo-planet Orbit Database The vertical line at 12 MJup
corresponds to the mass criterion we used to separatethe exoplanets in LMEs and HMEs The other verticallime is the lower mass limit for the BDs 13 MJup
the electromagnetic interaction and where the objectstarts to collapse (which is the case of BDs) How-ever if massive HJs have more massive envelopesof liquid metallic hydrogen (LMH) than observedin Jupiter (as suggested by JUNO Guillot et al2018 Kaspi et al 2018 Iess et al 2018 Adriani etal 2018) their structures might resist gravity (atleast for a while) due to the liquid state being in-compressible pushing the collapse of the radius toslightly higher masses (models and observations ver-ifying this prediction can be found in Hubbard et al1997 Dalladay-Simpson et al 2016 Flor-Torres etal 2016) This possibility however is still contro-versial and we only use the mass limit in our anal-ysis to separate our sample of exoplanets accordingto their masses On the other hand the possibil-ity of massive LMH envelopes in the HMEs mighthave some importance since these exoplanets wouldbe expected to have higher magnetic fields than theLMEs which consequently could affect their inter-actions with the PPD and nearby host stars (this fitsthe case of HD 80606 b a HME studied by de Witet al 2016) leading possibly to different migrationbehaviors Likewise we might also consider the pos-sibility of inflated radii in the LMEs since they areso close to their stars although less obvious is theeffect of inflated radius on migration (one possibilityis that they circularize more rapidly)
To test further our distinction in mass we tracein Figure 1 the mass-radius relation of 346 exoplan-ets from the Exoplanet Orbit Database Based onvisual inspection it is not clear how to separate the
HJs due to inflated radii One quantitative criterionis the mass-radius relation In Figure 1 we tracedthree M -R relations adopting 12 MJup to distin-guish LMEs form HMEs Below this limit we findthe relation
lnRLME = (0496plusmn0018) lnMLME+(0407plusmn0034)(1)
The slope is positive and the correlation coefficient ishigh r2 = 075 which implies that the radius withinthis range of masses continually increases with themass This relation is also fully consistent with whatwas previously reported by Valencia et al (2006) andChen amp Kipping (2017) Above the lower mass limitfor BDs we find the relation
lnRBD = (minus0117plusmn0111) lnMBD +(0455plusmn0404)(2)
with a negative slope and a weaker correlation coeffi-cient of r2 = 018 but sufficient to indicate that thetrend is for the radius to decrease with the massThis is as expected for objects where self-gravityis stronger than the electromagnetic repulsion (BDsnot having enough mass to ignite fusion in their corecannot avoid the effect of gravitational collapse) Fi-nally in between these two mass limits defining theHMEs we obtain the relation
lnRHME=(minus0044plusmn0030)lnMHME+(0229plusmn0031)(3)
which has an almost nil slope and a very weak coef-ficient of correlation r2 = 002 consistent with nocorrelation This range of mass therefore is fullyconsistent with HJs near the inflection point extend-ing this region over a decade in mass (as previouslynoted by Hatzes amp Rauer 2015) For our analysisthese different M -R relations are sufficient to justifyour separation between LMEs and HMEs (note thatthis physical criterion has never been used in the lit-erature the only study which uses a limit close toours is Sousa et al 2011)
3 CONNECTING THE STARS TO THEIRPLANETS
In Paper I we verified that the rotational velocityof a star V sin i decreases with the temperatureTeff In Figure 2 we reproduce this graphic but thistime distinguishing between stars hosting LMEs andHMEs We observe a clear trend for the HMEs tobe found around hotter and faster rotator stars thanthe LMEs Considering the small number of stars inour sample we need to check whether this result isphysical or due to an observational bias For exam-ple one could suggest that HMEs are easier to detect
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
222 FLOR-TORRES ET AL
6500 6000 5500 5000
010
20
30
40
50
Teff (K)
V s
in i (
km
s)
HMEsLMEsBDSun
Fig 2 Star rotational velocity vs temperature sepa-rating stars hosting HMEs and LMEs The position ofthe Sun is included as well as the star with a BD ascompanion
than LMEs through the RV than Tr method aroundhotter (more massive) and faster rotator stars Tocheck for observational biases we trace first in Fig-ure 3a the distributions of the absolute V magnitudefor the stars hosting LMEs and HMEs We do finda trend for the stars hosting HMEs to be more lumi-nous than the stars hosting LMEs This is confirmedby a non-parametric Mann-Whitney test with a p-value of 00007 (Dalgaard 2008) However in Fig-ure 3b we also show that the reason why the starswith HMEs are more luminous is because they arelocated farther out and this is independent of thedetection method Consequently the trend for theHMEs to be found around hotter and faster rotatorstars than the LMEs does not depend on the methodof detection but is a real physical difference Con-sidering the mass-temperature relation on the mainsequence this suggests that the more massive exo-planets in our sample rotate around more massivestars
In Figure 4 we compare the physical character-istics of the stars that host HMEs with those thathost LMEs Figure 4a shows a clear trend for therotational velocity to be higher in the stars hostingHMEs than LMEs In Table 2 we give the resultsof non-parametric Mann-Whitney (MW) tests Theuse of non-parametric tests is justified by the factthat we did not find normal distributions for our data(established by running 3 different normality tests)In Column 6 we give the p-values of the tests at alevel of confidence of 95 in Column 7 the signifi-cance levels (low medium and high ) and
$
amp
(
)+
-
)- )$- 01- 01$-2
322
22
422
amp22
56789lt9-
+-
Fig 3 a) Absolute magnitude in V separately for starshosting HMEs and LMEs b) Distance of the stars alsoseparated by the detection method radial velocity RVor transit Tr The bars correspond to the medians andinterquartile ranges
in Column 8 the acceptance or not of the differencesobserved Note that a non-parametric test comparesthe ranks of the data around the medians not themeans In the case of V sin i the MW test confirmsthe difference of medians at a relatively high level ofconfidence (stars 22 33 and 36 were not considereddue to their large uncertainties) In Figure 4b wesee the same trend for Teff on average also with asignificant difference in median in Table 2
In Figure 4c the difference in mass is obviousand confirmed by the MW test at the highest levelof confidence Stars with HMEs are more massivethan stars with LMEs On the other hand we findno difference in the distribution of the metallicityAlthough the medians and means reported in Ta-ble 2 look different the p-value of the MW test isunequivocal being much higher than 005 The me-dian [FeH] for our sample is 007 dex and the meanis 012 dex with a standard deviation of 024 dexThese values are comparable to those reported in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
222 FLOR-TORRES ET AL
6500 6000 5500 5000
010
20
30
40
50
Teff (K)
V s
in i (
km
s)
HMEsLMEsBDSun
Fig 2 Star rotational velocity vs temperature sepa-rating stars hosting HMEs and LMEs The position ofthe Sun is included as well as the star with a BD ascompanion
than LMEs through the RV than Tr method aroundhotter (more massive) and faster rotator stars Tocheck for observational biases we trace first in Fig-ure 3a the distributions of the absolute V magnitudefor the stars hosting LMEs and HMEs We do finda trend for the stars hosting HMEs to be more lumi-nous than the stars hosting LMEs This is confirmedby a non-parametric Mann-Whitney test with a p-value of 00007 (Dalgaard 2008) However in Fig-ure 3b we also show that the reason why the starswith HMEs are more luminous is because they arelocated farther out and this is independent of thedetection method Consequently the trend for theHMEs to be found around hotter and faster rotatorstars than the LMEs does not depend on the methodof detection but is a real physical difference Con-sidering the mass-temperature relation on the mainsequence this suggests that the more massive exo-planets in our sample rotate around more massivestars
In Figure 4 we compare the physical character-istics of the stars that host HMEs with those thathost LMEs Figure 4a shows a clear trend for therotational velocity to be higher in the stars hostingHMEs than LMEs In Table 2 we give the resultsof non-parametric Mann-Whitney (MW) tests Theuse of non-parametric tests is justified by the factthat we did not find normal distributions for our data(established by running 3 different normality tests)In Column 6 we give the p-values of the tests at alevel of confidence of 95 in Column 7 the signifi-cance levels (low medium and high ) and
$
amp
(
)+
-
)- )$- 01- 01$-2
322
22
422
amp22
56789lt9-
+-
Fig 3 a) Absolute magnitude in V separately for starshosting HMEs and LMEs b) Distance of the stars alsoseparated by the detection method radial velocity RVor transit Tr The bars correspond to the medians andinterquartile ranges
in Column 8 the acceptance or not of the differencesobserved Note that a non-parametric test comparesthe ranks of the data around the medians not themeans In the case of V sin i the MW test confirmsthe difference of medians at a relatively high level ofconfidence (stars 22 33 and 36 were not considereddue to their large uncertainties) In Figure 4b wesee the same trend for Teff on average also with asignificant difference in median in Table 2
In Figure 4c the difference in mass is obviousand confirmed by the MW test at the highest levelof confidence Stars with HMEs are more massivethan stars with LMEs On the other hand we findno difference in the distribution of the metallicityAlthough the medians and means reported in Ta-ble 2 look different the p-value of the MW test isunequivocal being much higher than 005 The me-dian [FeH] for our sample is 007 dex and the meanis 012 dex with a standard deviation of 024 dexThese values are comparable to those reported in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 223
TABLE 2
STATISTICAL TESTS
HME LME MW Sign Diff
Parameter Median Mean Median Mean p-value
V sin i 747 1094 411 462 00048 yes
Teff 6186 6158 5771 5800 lt 00001 yes
Mlowast 122 121 111 105 lt 00001 yes
[FeH] 021 019 015 016 06810 ns no
Jlowast 883 143 417 733 00096 yes
Jp 417 493 104 109 lt 00001 yes
The units and scale values for the medians and means are those of Figure 4
literature for systems harboring massive HJ exoplan-ets (Sousa et al 2011 Buchhave et al 2012)
In Figures 4d and 4f once again we distinguishobvious differences in the distributions the starswith HMEs having higher angular momentum thanthe stars with LMEs and the HMEs having alsohigher angular momentum than the LMEs The MWtests in Table 2 confirmed these differences at a rela-tively high level of confidence (like for V sin i) Con-sidering Eq 2 in Paper I for the angular momen-tum of the star the statistical test confirms that theHMEs rotate around more massive and faster rotatorstars than the LMEs Considering the difference inangular momentum of the exoplanets Jp this resultseems to support the hypothesis that more massiveplanets form in more massive PPDs with higher an-gular momentum The question is then how doesthis difference affect the migration process of the dif-ferent exoplanets
In Figure 5a following Berget amp Durrance(2010) we trace the specific angular momentumof the stars jlowast = JlowastMlowast as a function of theirmasses distinguishing between stars hosting HMEsand LMEs The angular momenta of all the hoststars in our sample fall well below the theoreticalrelation proposed by McNally (1965) There is aclear distinction between the LMEs and HMEs ex-cept for the differences encountered in Figure 4 andconfirmed in Table 2 and there is no evidence thatthe stars follow a j minusM relation
On the other hand when we compare in Fig-ure 5b the angular momentum for the systemsjsys = jlowast+jp we do see a difference between systemswith LMEs and HMEs But this is expected since bydefinition the HMEs having higher masses naturallyhave a higher contribution to jsys However and de-spite being more massive than Jupiter very few of
the HME systems have a value of jsys comparable tothe Solar System Obviously this is because of thelarge scale migration they suffered To illustrate thispoint we traced in Figure 5c the possible ldquoinitialrdquoangular momentum the system could have had as-suming the exoplanets formed at the same distanceas Jupiter (5 AU) Comparing with the positions inFigure 5b the HMEs would have lost on average 89of their initial momentum compared to 86 for theLMEs Those losses are enormous Considering theloss of angular momentum of the stars and planetsit might be difficult to expect a coupling between Jpand Jlowast or even a j minusM relation
As a final test we have calculated the Pear-son correlation matrices (also explained in Dalgaard2008) for the systems with HMEs and LMEs Theresults can be found in Table 3 and Table 4 Sincethe correlation matrices are symmetrical we includeonly the lower diagonal of each showing first the ma-trix with the p-values (with alpha asymp 005) followedby the matrix for the Pearson correlation coefficient(keeping only the significant correlations marked inbold in the matrix of the p-values)
Comparing the HMEs with LMEs there are ob-vious correlations of the temperature with the massand radius in both systems which suggests that ingeneral as the temperature increases the mass andradius increase This explains therefore the strongcorrelations in both systems of the temperature withthe velocity of rotation and thus the angular momen-tum Note that these results are consistent with thegeneral relation we determined in Paper I betweenV sin i Teff and log g This suggests that the moremassive the star the faster its rotation
One difference between the two systems is thatalthough there are no correlations of the temperaturewith the surface gravity in the HMEs there is an an-
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
224 FLOR-TORRES ET AL
$
amp
(
)
+-0123-4
$56
57
amp5
amp5
amp5)
amp56
80
4
$
)
6
98amp)amp 012-ltamp4
$
6
6
=gt04
$lt5)
lt5
5
5
5)
56
Agt3
$
)
6
7
amp
9Bamp) 012-ltamp4
C4 D4
E4 F4
gt4 4
Fig 4 Comparing the physical characteristics of the stars (and planet) hosting HMEs and LMEs In each graph themedians and interquartile ranges are drawn over the data a) rotational velocity b) effective temperature c) mass ofthe star d) metallicity e) the angular momentum of the star f) the angular momentum of the orbit of the dominantplanet
ticorrelation in the LMEs Due to the smallness ofour samples it is difficult to make sense of this dif-ference physically However we note that log g inthe LMEs is also well correlated with the radius themass and the angular momentum (but not V sin i it-self) something not seen in the HMEs In Figure 2wee see that the dispersion of V sin i decreases at lowtemperatures which implies that the bi-exponentialrelation of V sin i as a function of Teff and log g be-comes tighter suggesting that the behavior of log gbecomes more ordered which could explain the anticorrelations with Teff and the correlations with Mlowastand Rlowast (and thus also with Jlowast)
It is remarkable that [FeH] in both systems isnot correlated with any of the other parameters ei-ther related to the stars or the planets For the starsthe most obvious correlations in both systems arebetween Mlowast and Rlowast or V sin i and Jlowast Note alsothat although V sin i shows no correlation with Mlowastand Rlowast in the HMEs it is correlated with Mlowast in theLMEs This might also explain why Rlowast is not corre-lated with Jlowast in the HMEs while it is in the LMEs
In the case of the planets the most important re-sult of this analysis is the (almost complete) absenceof correlations between the parameters of the planetsand the parameters of the stars (most obvious in the
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 225
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
lo
g10 (
J M
) [m
sup2s]
G0
F5
F0
A5
A0
(a)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
mM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(b)
minus04 00 02 04
10
11
12
13
14
log10 (M ) [MO]
log
10 (
Jsyste
m_5A
UM
syste
m)
[msup2
s]
G0
F5
F0
A5
A0
(c)
Fig 5 In (a) Specific angular momentum of the host stars as function of their masses The symbols are as in Figure 2The solid black line is the relation proposed by McNally (1965) for low-mass stars (spectral types A5 to G0) andthe dotted line is the extension of the relation suggested for massive stars The black triangle represents the Sun (b)specific angular momentum for the planetary systems with the inverted black triangle representing the Solar system(c) original angular momentum assuming the planets formed at 5 AU
LMEs) The only parameter that shows some corre-lation with the star parameters is the semi-axis apof the orbit of the planets in the HMEs This mightsuggests a difference in terms of circularization Tworesults also seem important The first is that Rp isonly correlated to Mp in in the HMEs which is con-sistent with the fact that the radius of the HMEs isconstant The second is that there is no correlationbetween Jp and Jlowast which is consistent with the be-havior observed in Figure 5 and which could suggestthat there is no dynamical coupling between the twoThis probably is due to important losses of angularmomentum during the formation of the stars and tomigrations of the planets
4 DISCUSSION
Although our sample is small we do find a connec-tion between the exoplanets and their stars massiveexoplanets tend to form around more massive starsthese stars being hotter (thus brighter) and rotatingfaster than less massive stars When we comparethe spin of the stars with the angular momentum of
the orbits of the exoplanets we find that in the HMEsystems both the stars and planets rotate faster thanin the LME systems This is consistent with theidea that massive stars formed more massive PPDswhich rotate faster explaining why the planets form-ing in these PPDs are found also to rotate faster
When we compare the effective angular momentaof the stars (Figure 5a) we find no evidence that theyfollow a j minusM relation in particular like the onesuggested by McNally (1965) or that the angularmomentum of the systems (Figure 5b) follows suchrelation Furthermore there is a correlation betweenJp and Jlowast which suggests that there is no dynami-cal coupling between the two This is probably dueto the important losses of angular momentum of thestars during their formation (by a factor of 106) andof the planets during their migrations (higher than80 of their possible initial values) For the planetstheir final angular momenta depend on their masseswhen their migration ends ap Assuming conse-quently that they all form more or less at the samedistance (farther than the ice line in their systems)and end their migration at the same distance from
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
226 FLOR-TORRES ET AL
TABLE 3
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 02918
[FeH] 06794 01519
V sin i 00048 06052 04121
Rlowast 00199 00785 09143 08734
Mlowast lt00001 00884 04889 01242 lt00001
Jlowast 00006 07412 01961 lt00001 02345 00162
Mp 08087 07356 03401 03785 00577 01892 07779
Rp 03861 04608 02527 09065 02122 01460 08397 01323
ap 00339 00173 06018 09447 00118 00074 03824 00074 01050
ep 06455 04758 08288 09864 04266 01871 08931 00073 00418 00122
Jp 04358 04193 04496 05536 00455 05697 07913 lt00001 02679 lt00001 00234
log g
[FeH]
V sin i 05909
Rlowast 04925
Mlowast 08402 08662
Jlowast 06736 09857 05064
Mp
Rp
ap -04539 05021 -05267 -05546 05544
ep 05550 -04373 05243
Jp -04305 09393 07272 04811
TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME Teff log g [FeH] V sin i Rlowast Mlowast Jlowast Mp Rp ap ep
log g 00002
[FeH] 03461 02153
V sin i 00097 00389 00761
Rlowast lt00001 lt00001 02374 01499
Mlowast lt00001 lt00001 00618 00187 lt00001
Jlowast 00034 00012 01880 lt00001 00077 00012
Mp 00658 01672 05022 03259 03008 01802 03683
Rp 01825 00681 05140 07583 01878 01902 09893 lt00001
ap 04458 05652 05036 09258 04294 04570 08521 00303 02162
ep 02682 03735 03691 02266 05929 03681 03979 00645 00348 02426
Jp 02301 01979 07299 02168 04060 03170 01022 lt00001 00029 07167 04335
log g -06992
[FeH]
V sin i 05506 -04537
Rlowast 07509 -09084
Mlowast 09036 -08577 05081 09407
Jlowast 05840 -06319 09312 05408 06327
Mp
Rp 07468
ap -04521
ep -04418
Jp 067921 05927
their stars (which seems to be the case as we showin Figure 6) the HMEs would have lost a slightlylarger amount of angular momentum than the LMEs
Within the scenario suggested above this might sug-gest that more massive PPDs are more efficient indissipating the angular momentum of their planets
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 227
001 002 005 010 020 050 100 200 500
1e+
40
1e+
41
1e+
42
1e+
43
1e+
44
Only Dominant planet
a (AU)
Jp (
kg
m2s
)
HME minus exoplanetorg
LME minus exoplanetorg
HME minus This work
LME minus This work
Fig 6 Angular momentum of the exoplanets in our sample with respect to their distances from their stars Forcomparison the exoplanets in our initial sample are also shown as light gray signs
One thing seems difficult to understand how-ever Considering that HMEs are more massive andlost more angular momentum during their migrationwhy did they end their migration at almost exactlythe same distance from their stars as the less mas-sive LMEs In Figure 6 the accumulation we per-ceive between 004 and 005 AU is consistent withthe well known phenomenon called the three-dayspile-up (3 days is equivalent to slightly more than004 AU assuming Kepler orbits) which is supposedto be an artifact due to selection effects of ground-based transit surveys (Gaudi 2005) However Daw-son amp Johnson (2018) in their review about migra-tion suggested that this could be physical and someauthors did propose different physical explanations(see Fleck 2008 and references therein) The modelof Fleck (2008) is particularly interesting because ittries to solve the problem using the same structureof the PPD that many authors believe explains howthe stars lose their angular momentum during theT-Tauri phase that is by magnetic braking In Ap-pendix B we do some calculations which show thatthe radius where the planets in our sample end theirmigration could be close to the co-rotation radiusthe region of the PPD where the disk rotates atthe same velocity as the star But does this im-ply that disk migration is more probable than high-eccentricity tidal migration
According to the theory of high-eccentricity tidalmigration one expects a strong dependence of thetidal evolution timescale on the final location of theorbit of the planets afinal (eg Eggleton 1998)
a prop a8final (4)
This implies that since the HMEs and LMEs havethe same afinal asymp 004 AU they should also havethe same tidal evolution timescale and thus no dif-ference would be expected comparing their eccen-tricities What stops the planet migration in thehigh-eccentricity tidal migration model is the circu-larization of the orbit through tidal interactions withthe central star Therefore assuming the same tidalevolution timescale we would expect the eccentrici-ties for the HMEs and LMEs to be all close to zero(Bolmont et al 2011 Remus et al 2014) Note thatusing our data to test whether the HMEs and LMEshave similar distributions in eccentricity is possiblydifficult because we are not certain if an eccentric-ity of zero is physical or not (meaning an absence ofdata) For the HME sample on 22 exoplanets wecount 7 (32) with zero eccentricity while in theLMEs out of 23 10 (43) have zero eccentricityThe difference seems marginal For the remainingplanets with non zero eccentricities (15 HMEs and13 LMEs) we compare in Figure 7 their distributionsThere is a weak difference with a median (mean) of
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
228 FLOR-TORRES ET AL
$
amp
amp
amp(
amp)
amp+
Fig 7 Eccentricities of the exoplanets in our sampledistinguishing between HMEs and LMEs
ep = 019 (ep = 022) for the HMEs compared toep = 004 (ep = 010) for the HMEs and LMEs AMW test yields a p-value = 00257 which suggests adifference at the lowest significance level Thereforethere seems to be a trend for the HMEs to have onaverage a higher eccentricity than the LMEs TheHMEs possibly reacted more slowly to circulariza-tion than the LMEs (de Wit et al 2016) suggest-ing possibly different structures due to their largermasses (Flor-Torres et al 2016)
5 CONCLUSIONS
Based on an homogeneous sample of 46 stars ob-served with TIGRE and analysed using iSpec westarted a project to better understand the connec-tion between the formation of the stars and theirplanets Our main goal is to check whether therecould be a coupling between the angular momentumof the planets and their host stars Here are ourconclusions
There is a connection between the stars and theirexoplanets which passes by their PPDs Massivestars rotating faster than low-mass stars had moremassive PPDs with higher angular momentum ex-plaining why they formed more massive planets ro-tating faster around their stars However in termsof stellar spins and planet orbit angular momentumwe find that both the stars and their planets havelost a huge amount of angular momentum (by morethan 80 in the case of the planets) a phenomenonwhich could have possibly erased any correlations ex-pected between the two The fact that all the planetsin our sample stop their migration at the same dis-tance from their stars irrespective of their massesmight favor the views that the process of migrationis due to the interactions of the planets with their
PPDs and that massive PPDs dissipate more angu-lar momentum than lower mass PPDs Consistentwith this last conclusion HMEs might have differ-ent structures than LMEs which made them moreresilient to circularization
We would like to thank an anonymous refereefor a careful revision of our results and for com-ments and suggestions that helped us to improveour work L M F T thanks S Blanco-Cuaresmafor discussions and support with iSpec She alsothanks CONACyT for a scholar grant (CVU 555458)and travel support (bilateral Conacyt-DFG projects192334 207772 and 278156) as well as for supportgiven by the University of Guanajuato for confer-ence participation and international collaborations(DAIP and Campus Guanajuato) This researchhas made use of the Exoplanet Orbit Database theExoplanet Data Explorer (exoplanetsorg Han et al2014) exoplanetseu (Schneider et al 2011) and theNASArsquos Astro-physics Data System
APPENDICES
A CALCULATING THE SELF-GRAVITATINGMASS LIMIT
According to Padmanabhan (1993) the formation ofstructures with different masses and sizes involvesa balance between two forces gravity Fg and elec-tromagnetic Fe For the interaction between twoprotons we get
FeFg
=κee
2r2
Gm2pr
2=κee
2
Gm2p
=
(κee
2
hc
)(hc
Gm2p
)
(A5)where κe and G are the electromagnetic and gravita-tional constants that fix the intensity of the forcese and mp the charges and masses interacting (mp isthe mass of a proton) and r the distance betweenthe sources (the laws have exactly the same mathe-matical form) Introducing the reduced Plank con-stant h = h2π and the velocity of light c (two im-portant constant in physics) the first term on theright is the fine structure constant α asymp 729times 10minus3and the second term is the equivalent for gravityαG asymp 588times 10minus39 This implies that
α
αGasymp 124times 1036 (A6)
This results leads to the well-known hierarchi-cal problem there is no consensus in physics whythe electromagnetic force should be stronger thanthe gravitational force in such extreme The reasonmight have to do with the sources of the forces in
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 229
particular the fact that in electromagnetic there aretwo types of charge interacting while there is onlyone type of mass The important consequence forthe formation of large-scale structures is that whilein the electromagnetic interaction the trend to min-imize the potential energy reduces the total chargemassive object being neutral the same trend in grav-ity is for the gravitational field to increase with themass Therefore small structures tend to be domi-nated by the electromagnetic force while large struc-tures are dominated by gravity Based on this phys-ical fact Padmanabhan calculated that there is acritical mass Mc above which the force of gravitybecomes more important than the electromagneticforce The point of importance for planet formationis that this critical mass turned out to be comparableto the mass of a gas giant planet
To realize that it suffices to compare the energyof ligation of a structure with its gravitational poten-tial energy Since the escape energy for an electronis
E0 asymp α2mec2 435times 10minus18J (A7)
a spherical body formed of N atoms would have anenergy of ligation E = N times E0 On the other handits mass would be M = Nmp and its volume wouldbe 43πR3 = Ntimes43πa30 where a0 530times10minus11 mis the Bohr radius From this we deduce that itsradius woud be R asymp N13a0 and its gravitationalpotential
EG GM2
R N53
Gm2p
a0= N53αGmeαc
2 (A8)
Now the condition for a stable object to form undergravity is E ge EG while for E lt EG the objectwould collapse under its own weight In equilibriumwe would thus have
N53αGmeαc2 = Nα2mec
2 (A9)
This yields to a maximum number of atoms Nmax asymp(ααG)32 asymp 138times 1054 and a critical mass
Mc = Nmaxtimesmp asymp 231times 1027kg asymp 12MJ (A10)
This suggests that the inflection point we observein the mass-radius relation of exoplanets could bethe critical point where an object becomes unstableunder self-gravity
B AT WHAT DISTANCE IS THECOROTATION RADIUS RCO
In their review about the migration of planets Daw-son amp Johnson (2018) claimed that the distribution
of exoplanets around their host stars is consistentwith the co-rotation radius Rco which is part of themagnetic structure connecting the star to its PPDIn the model presented by Matt amp Pudritz (2004)the authors discussed some specific aspects of thismagnetic structure that could contribute in stop-ping planet migration (see their Figure 3 and ex-planations therein) Following this model the starand disk rotate at different angular speeds exceptat Rco where the magnetic field becomes twistedazimuthally by differential rotation triggering mag-netic forces These forces would act to restore thedipole configuration conveying torques between thestar and the disk As a planet approaches Rco there-fore these torques would transfer angular momen-tum from the star to the planet stopping its mi-gration (Fleck 2008) One important aspect of thismodel is that the dumping would only work up to adistance Rout asymp 16Rco and thus one would expectthe planets to pile-up over this region that is be-tween Rout and Rco The question then is what isthe value of Rco
One way to determine this value is to assume thatwhen the wind of a newly formed star starts evap-orating the PPD Rout minusrarr Rco and the magneticpressure at Rco balances the gas pressure due to thewind PB = Pg Assuming that the intensity of themagnetic field decreases as the cube of the distancewe get B asymp BsR
3lowastr
3 where Bs is the intensity ofthe magnetic field at the surface of the star Rlowast isthe radius of the star and micro0 is the permeability ofthe vacuum This yields that
PB =B2
2micro0=B2sR
6lowast
2micro0r6 (B11)
For the gas pressure we used the expression for ldquoRampressurerdquo
Pg = nmv2 (B12)
where v is the velocity of the wind and nm is itsload that is the amount of mass transported by thewind This load then can be expressed in terms ofthe flux of matter M as
nm =M
4πr2v (B13)
Assuming equality at r = Rco we get
R4co =
2π
micro0
B2sR
6lowast
Mv (B14)
which for a star like the Sun (Bs asymp 10minus4 TM asymp 200times 10minus14Myr and v = 215times 10minus3 ms)
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
230 FLOR-TORRES ET AL
yields Rco asymp 677 times 109m or 0045 AU Accord-ing to this model therefore we could expect to seea real pile up of exoplanets independent of theirmasses (and losses of angular momentum) between004minus 007 AU which neatly fit the observations
REFERENCES
Adriani A Mura A Orton G et al 2018 Natur 555216
Alexander R 2017 Lecture 2 Protoplanetary discshttpswwwastroleacukrda5planets 2017html
Armitage P J 2010 Astrophysics of Planet Formation(Cambridge UK CUP)
2020 Astrophysics of Planet Formation (2nded Cambridge UK CUP)
Asplund M Grevesse N Sauval AJ amp Scott P 2009ARAampA 47 481
Baraffe I Chabrier G Allard F amp Hauschildt P H1998 AampA 337 403
Baraffe I Chabrier G Barman T S Allard F ampHauschildt P H 2003 AampA 402 701
Baraffe I Chabrier G amp Barman T 2008 AampA 482315
Baruteau C Crida A Paardekooper S-J et al 2014Protostars and Planets VI ed H Beuther R SKlessen C P Dullemond amp T Henning (TuczonAZ UAP) 667
Beauge C amp Nesvorny D 2012 ApJ 751 119Berget D J amp Durrance S T 2010 Journal of the
Southeastern Association for Research in Astronomy3 32
Blanco-Cuaresma S Soubiran C Heiter U amp JofreP 2014 AampA 569 111
Blanco-Cuaresma S 2019 MNRAS 486 2075Bolmont E Raymond S N amp Leconte J 2011 AampA
535 1Brosche P 1980 in Cosmology and Gravitation ed
P G Bergmann amp V De Sabbata (Boston MASpringer) 375 doi101007978-1-4613-3123-0 17
Buchhave L A Latham D W Johansen A et al2012 Natur 486 375
Burgasser A J 2008 PhT 61 70Carrasco L Roth M amp Serrano A 1982 AampA 106
89Champion J 2019 PhD Thesis Univertite de ToulouseChatterjee S Ford E B Matsumura S amp Rasio F
A 2008 ApJ 686 580Chen J amp Kipping D 2017 ApJ 834 17Dalgaard P 2008 Introductory statistics with R (New
York NY Springer) doi101007978-0-387-79054-1Dalladay-Simpson P Howie R T amp Gregoryanz E
2016 Natur 529 63Dawson R I amp Johnson J A 2018 ARAampA 56 175de la Reza R amp Pinzon G 2004 AJ 128 1812de Pater I amp Lissauer J J 2015 Planetary Sciences
(2nd ed CUP)de Wit J Lewis N K Langton J et al 2016 ApJ
820 33
Draine B T 2011 Physics of the Interstellar and Inter-galactic Medium (Princeton MA PUP)
Eggleton P P Kiseleva L G amp Hut P 1998 ApJ499 853
Fleck R C 2008 ApampSS 313 351Flor Torres L Coziol R Schroeder K-P
Caretta C A amp Jack D 2016 pp 1-5(arXiv160707922) doi105281zenodo58717httparxivorgabs160707922
Flor-Torres L Coziol R Schroeder K-P amp Jack D2020 in press
Fortney J J Marley M S amp Barnes J W 2007 ApJ659 1661
Fortney J J Baraffe I amp Militzer B 2010 in Exo-planets ed S Seager (Tuczon AZ UAP) 397
Gaudi B S 2005 ApJ 628 73God lowski W Szyd lowski M Flin P amp Biernacka
M 2003 GReGr 35 907Guillot T Miguel Y Militzer B et al 2018 Natur
555 227Han E Wang S X Wright J T et al 2014 PASP
126 827Hatzes A P amp Rauer H 2015 ApJL 810 25Hilke E S amp Sari R 2011 ApJ 728 68Hubbard W B Guillot T Lunine J I et al 1997
PhPl 4 2011Iess L Folkner W M Durante D et al 2018 Natur
555 220Kaspi Y Galanti E Hubbard W B et al 2018
Natur 555 223Kawaler S D 1987 PASP 99 1322Klahr H amp Brandner W 2006 Planet Formation The-
ory Observations and Experiments (Cambridge UKCUP)
Kraft R P 1967 ApJ 150 551Marzari F amp Weidenschilling S J 2002 Icar 156 570Matt S amp Pudritz R E 2004 ApJ 607 43McNally D 1965 Obs 85 166McKee C F amp Ostriker E C 2007 ARAampA 45 565Mestel L 1968 MNRAS 138 359Nagasawa M Ida S amp Bessho T 2008 ApJ 678 498Padmanabhan T 1993 Structure Formation in the Uni-
verse (New York NY CUP)Poppe T amp Blum J 1997 AdSpR 20 1595Rasio F A amp Ford E B 1996 Sci 274 954Ray T 2012 AampG 53 19Raymond S N amp Morbidelli A 2020 arXiv e-prints
arXiv200205756Remus F Mathis S Zahn J-P amp Lainey V 2014
IAUS 293 Formation Detection and Characteriza-tion of Extrasolar Habitable Planets ed N Haghigh-ipour (New York NY CUP) 362
Safronov V S 1969 Ann Astrophys 23 979Schatzman E 1962 AnAp 25 18Schneider J Dedieu C Le Sidaner P Savalle R amp
Zolotukhin I 2011 AampA 532 79Sousa S G Santos N C Israelian G Mayor M amp
Udry S 2011 AampA 533 141
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102
copy C
op
yri
gh
t 2
02
1 In
stitu
to d
e A
stro
no
miacutea
U
niv
ers
ida
d N
ac
ion
al A
utoacute
no
ma
de
Meacute
xic
oD
OI h
ttp
s
do
iorg
10
22
20
1i
a0
18
51
10
1p
20
21
57
01
16
COUPLING THE ANGULAR MOMENTUM OF STARS AND PLANETS 231
Spiegel D S Burrows A amp Milsom J A 2011 ApJ727 57
Tassoul J-L 2000 Stellar Rotation (New York NYCUP) doi101017CBO9780511546044
Uzdensky D A Konigl A amp Litwin C 2002 ApJ565 1191
Valencia D OrsquoConnell R J amp Sasselov D 2006 Icar181 545
R Coziol L M Flor-Torres D Jack and K-P Schroder Departamento de Astronomıa Universidad deGuanajuato Guanajuato Gto Mexico
J H M M Schmitt Hamburger Sternwarte Universitat Hamburg Hamburg Germany
Weidenschilling S J amp Marzari F 1996 Natur 384 619Wesson P S 1979 AampA 80 296Wetherill G W 1989 in The formation and evolution
of planetary systems ed H A Weaver amp L Danly(Cambridge UK CUP) p 1-24 Discussion p 24
Wolff S C amp Simon T 1997 PASP 109 759Wurm G amp Blum J 1996 AASDivision for Planetary
Meeting Abstracts 28 p 1102