Mon. Not. R. Astron. Soc. 000, 1–5 (2009) Printed 17 December 2009 (MN LATEX style file v2.2)

Planetesimal collisions in binary systems

S.-J. Paardekooper? and Z.M. LeinhardtDAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Draft version 17 December 2009

ABSTRACTWe study the collisional evolution of km-sized planetesimals in tight binary star sys-tems to investigate whether accretion towards protoplanets can proceed despite thestrong gravitational perturbations from the secondary star. The orbits of planetesi-mals are numerically integrated under the influence of the two stars and gas drag.The masses and orbits of the planetesimals are allowed to evolve due to collisions withother planetesimals and accretion of collisional debris. In addition, the mass in debriscan evolve due to plantesimal-planetesimal collisions and the creation of new plantes-imals. We show that it is possible for km-sized planetesimals to grow by two ordersof magnitude in size if the efficiency of planetesimal formation from small debris isrelatively low.

Key words: planets and satellites: formation –planetary systems: protoplanetarydiscs –stars: individual: γ Cephei –stars: individual: α Centauri

1 INTRODUCTION

We consider the problem of planetesimal evolution in tightbinary star systems such as γ Cephei and α Centauri. Todate several hundred extrasolar planets have been detected– 20% of which orbit the primary of a binary or multiplesystem (Desidera & Barbieri 2007). In most of these casesthe stars are widely separated and do not significantly af-fect the evolution of planets, but γ Cephei, Gl86, HD41004A,and possibly HD196885 are examples of tight binary systemswith a semi-major axis, ab, around 20 AU, that have Jupitermass planets. γ Cephei has the most extreme binary param-eters of the known tight binaries with detected planets. Theprimary star is a K giant (MA? ∼ 1.4 M) the secondary isa M dwarf (MB? ∼ 0.4 M), the binary has a high eccen-tricity (eb = 0.4), and a small ab of 20 AU. The planet is1.6 MJ with an a ∼ 2 AU (Neuhauser et al. 2007). Due toits extreme characteristics γ Cephei is often considered oneof the most stringent tests for planet formation models.

Previous work by Quintana et al. (2007) has shown thatplanets can form in γ Cephei if planetary embryos can formwithin the stable region < 3 AU (Wiegert & Holman 1997).However, work on the earlier stage of planet formation, theevolution of planetesimals, has shown that it is very difficultto grow planetesimals that are initially in the km size regimebecause of the secular perturbations from the secondary staron the planetesimals and the gas disc. The coupled effect ofperturbations from the binary and the gas drag from thedisc causes differential orbital phasing of the planetesimalsand high speed destructive impacts between planetesimals

? E-mail: S.Paardekooper@damtp.cam.ac.uk

of different sizes. Thebault et al. (2006) suggest that thedestructive collisions prohibit the growth of planetesimalsin tight binary systems like γ Cephei.

Although planets have yet to be detected in theα Centauri system, the proximity of the system and theroughly solar mass of both stellar components make thesystem a prime target for finding an Earth mass planetin the habitable zone (Guedes et al. 2008). However, theα Centauri system is more extreme than γ Cephei with amass ratio close to one (MA? = 1.1 M and MB? = 0.93M), eb = 0.52, and ab = 23.4 AU. Thebault et al. (2008,2009) found that impact speeds between different sized plan-etesimals would be above the disruption threshold due to dif-ferential orbital phasing, and planetesimals could not growin the habitable zone in either α Centauri A or B.

In this letter we show that planetesimal growth from1 to 100 km is possible in these tight binaries when thecollisional evolution of the planetesimal disc is taken intoaccount. Disruptive collisions will prevent differential orbitalphasing from being set up and, at the same time, create areservoir of small debris that can be accreted onto the leftover planetesimals.

2 DIFFERENTIAL ORBITAL PHASING

In the absence of any gas, a tight eccentric binary companionwill excite eccentricities of particles orbiting the primary.For particles starting on circular orbits, their eccentricitywill oscillate around the forced eccentricity ef :

ef(a) =5

4

a

ab

eb

1− e2b

, (1)

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2 S.-J. Paardekooper and Z.M. Leinhardt

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0.00

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e

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0.00

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e

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Figure 1. Eccentricity vs. semi-major axis of collisionless, mass-

less test particles in the γ Cephei system after 100 binary orbits

in a gas disc with a constant gas density of ρg = 1.4 · 10−9 gcm−3. Colour indicates size in km; the black curve with the large

oscillations denotes a gas-free case.

with amplitude 2ef , where a is the semi-major axis of theparticle (Heppenheimer 1978). Since there is no dissipativeforce in the system, the oscillations do not damp and theirspatial frequency increases with time, which eventually leadsto orbital crossing of neighbouring particles (Thebault et al.2006). This is illustrated by the black curve in Fig. 1. As soonas particles at an eccentricity maximum can collide withparticles at an eccentricity minimum the resulting encountervelocities are too high for accretion to occur.

The presence of a gas disc and the associated aerody-namic gas drag on the particles can effectively damp theoscillations leading to a well-defined equilibrium eccentric-ity distribution e0 = e0(a) (see Paardekooper et al. 2008).The time scale for this equilibrium to be reached is the secu-lar time scale. Therefore, it takes many binary orbits beforeplanetesimals are on their equilibrium orbits. Once the par-ticles have reached their equilibrium orbits particles of thesame size have orbits that are in phase and, therefore, theywill not suffer from high-speed collisions with each other(Marzari & Scholl 2000). However, since the magnitude ofthe gas drag force depends on the particle size, each par-ticle size will have a different e0(a), a phenomenon that iscalled differential orbital phasing (Thebault et al. 2006). InFig. 1 particles of a single size (and colour) follow phasedorbits with no oscillations. The residual oscillations that canbe seen for the largest particles will eventually be dampedcompletely by gas drag. However, for particles of differingsize there is a considerable spread in eccentricity. The dif-ference in eccentricity magnitude can easily exceed 0.01 forparticles of 1 and 5 km. This implies that particles of differ-ent size will undergo high-speed collisions, which are usuallydestructive. As a result, it is difficult to form planetary coresin tight binary systems if the only growth mechanism is ac-cretionary planetesimal-planetesimal collisions.

3 COLLISION TIME SCALE

3.1 Single star case

Consider a population of equal-sized bodies of mass Maround a single star at 1 AU. The collision time scale isgiven by

τc =1

πR2n∆v, (2)

where R is the radius of the bodies, n the number den-sity and ∆v the velocity dispersion. The latter will be ofthe order of the escape velocity, vesc =

p2GM/R. Fo-

cusing on a = 1 AU around a Solar type star we havevesc = 4.5 · 10−5(R/km)aΩ, where Ω is the angular velocity.The number density n is given by n = Σs/(∆zM), where Σs

is the surface density of solids and ∆z is the vertical extentof the particle disc. We can write ∆z ≈ ai, with i the maxi-mum inclination, which we can assume to be small. In fact,we expect 2i ≈ e ≈ vesc/aΩ. Plugging this all in we find forthe collision time scale:

τc ≈ 11750R

km

17 g cm−2

Σs

ρp

3 g cm−3Ω−1, (3)

where ρp is the bulk density of the particles. This means thatfor typical parameters at 1 AU, a km-sized planetesimal willundergo a collision once every few thousand years.

3.2 Binary star case

Perturbations due to a coplanar, eccentric binary will in-crease the eccentricity of the planetesimals. Differential or-bital phasing can easily give rise to the velocity dispersionentering the collision time scale will go up by two ordersof magnitude compared to the single star case. Keeping allother parameters the same, a km-sized planetesimal will nowundergo a collision every 10-20 orbits at 1 AU. If we take thebinary companion to have a semi-major axis of 20 AU, thiscollision time scale amounts to 25 % of a binary orbit. Thisshould be compared to the time scale for a particle to reachits equilibrium orbit in the binary system, which happens ona secular time scale (many binary orbits). Therefore, phys-ical collisions are of crucial importance for the evolution ofthe planetesimal population. It is important to stress thatnot only, as has been realised before, are collisions destruc-tive when differential orbital phasing happens, they can ac-tually prevent this size-dependent orbital structure from be-ing set-up in the first place. It is therefore necessary to takecollisions into account when studying the effect of differen-tial orbital phasing on planetesimal accretion.

Another consequence of having many possibly catas-trophic collisions in the system is a large amount of smalldebris. This small debris can act as a source of new planetes-imals or be accreted onto remaining larger bodies. All theseingredients need to be taken into account in a consistentmodel of planetesimal accretion in binary systems.

3.3 Two-dimensional disc

In order to reduce the computational cost of the models, wewill work in a two-dimensional (2D) geometry. The collisiontime scale is then given by

τc,2D =1

2Rn∆v=

2.4 · 10−4

e

„R

km

«217 g cm−2

Σs

ρp

3 g cm−3Ω−1,

(4)

where n is the surface number density and e is a measureof the velocity dispersion (e ≈ 4.5 · 10−5 if ∆v = vesc for 1km sized bodies). The collision time scale for a 2D disc isartificially short; we have to correct for this by choosing alow surface density in the simulations.

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Planetesimal collisions in binary systems 3

4 MODEL DESIGN

From the above discussion it is clear that the effect of colli-sions, on the size distribution as well as on the orbital ele-ments of the planetesimals, can not be ignored. As a result,we use a simple model that incorporates the three necessarycomponents: gas, planetesimals and small dust.

4.1 Gas disc

For simplicity, we take the gas disc to be static and circular.Although the gas disc is expected to become eccentric underthe influence of the binary companion (Paardekooper et al.2008; Kley & Nelson 2008) the qualitative outcome of themodel presented here is not affected by the assumption ofzero eccentricity. Paardekooper et al. (2008) showed that ifthe gas disc does not follow the forced eccentricity, differ-ential orbital phasing will occur, which always occured inthe full hydrodynamical simulations. A circular gas disc is,therefore, a reasonable starting point.

The gas density used in this work is assumed to be con-stant with a. The value is chosen to be close to the MinimumMass Solar Nebula at 1 AU, ρ1 = 1.4 · 10−9 g cm−3. Theresulting gas drag force is

~Fdrag = −3ρgCd

8ρpR|~v − ~vg|(~v − ~vg), (5)

with Cd the drag coefficient. We take Cd = 0.4, appropriatefor spherical bodies.

4.2 Small dust

We embed the gas disc with a population of tightly coupledsmall particles, which are assumed to move on circular or-bits together with the gas. They can be accreted onto exist-ing planetesimals, but also form new planetesimals, and arecreated in destructive collisions between planetesimals. Thesmall dust particles are distributed over typically 32 radialbins. At the start of the simulation, the mass in dust is equalto Md = MminN , where Mmin is the mass of a planetesimalof the minimum size we consider (usually 1 km), and N is aninput parameter specifying the total mass. Note that N isalso the maximum number of particles we could have at anyone time in the simulation. The dust mass inside a radialbin is smeared out to give a smooth surface density Σd.

4.3 Planetesimals

We model planetesimals as test particles, moving under theinfluence of gravity from both stars as well as gas drag.Planetesimals have a minimum size of 1 km (everythingsmaller is taken to be small dust), and can grow by low-velocity collisions with each other and by accreting smalldust. Collisions are detected by assigning an inflated radiusRinfl ∝ R to the particles and testing whether two particlesoverlap. The planetesimal disc is characterised by the prod-uct RinflN , which, together with ∆v, sets the collision timescale. Since ∆v is set by the gravitational perturbations dueto the binary companion and is independent of N and Rinfl,we expect simulations with the same value for RinflN to givesimilar results, a well-known result of collision models.

4.4 Collision outcomes

We use the velocity-dependent catastrophic disruption cri-terium of Stewart & Leinhardt (2009) (strong particle ver-sion) to determine the size of the largest remnant when acollision is detected. We then use the technique described inWyatt & Dent (2002) to predict the second-largest remnant.The total mass minus the largest remnant is assumed to fol-low a power law size distribution with index −1.93, fromwhich the number of bodies larger than R, N(> R), can bederived. The size for which N(> R) = 2 is the size we takefor the second largest remnant (Wyatt & Dent 2002). Anymass that is below the minimum particle size is added tothe small dust.

4.5 Planetesimal formation

Contrary to all previous studies, we do not let all planetesi-mals appear at t = 0, but instead have them form from thesmall dust present in the system. The efficiency of planetes-imal formation is a major unknown in this model. We takea very simple approach and say that:

dMA

dt= 2εpRinfl,min

Md

MminΣdaΩ, (6)

where εp is an efficiency factor, Rinfl,min is the inflated ra-dius of the minimum planetesimal size, and MA is the massavailable (in a certain radial dust bin) to form planetesimals.As soon as MA > Mmin, we create a new planetesimal. Notethat for fixed RinflN , MA ∝ N , so that the number of plan-etesimals at any time is proportional to N . Increasing Nwhile keeping RinflN constant effectively increases the ’res-olution’ of the simulation.

For equal masses in km-sized planetesimals and smalldust, the collision time scale and the planetesimal formationtime scale are related through τp = τce/εp. If we considerthe single star case, with e ≈ 10−5 and τc ≈ 103 yr, and ifwe expect planetesimal formation to proceed on a time scaleof τp > 104 yr, we must have εp ∼ 10−6. This is the valueused for the standard model discussed below.

4.6 Dust accretion

Planetesimals can accrete dust at a rate

dM

dt= 2εdRinflΣd|~v − ~vg|, (7)

where Σd is the surface density of small dust. An efficiencyfactor εd accounts for the possibility that not all dust createdin collisions is available for accretion, and that the efficiencyof accreting small bodies may be smaller than 1. Note thatΣd ∝ N , so that the amount of dust accreted per planetes-imal is the same if N is changed but the product RinflN iskept constant.

5 RESULTS

We start by describing a simulation that has N = 106,Rinfl/ab = 10−5R/km, εp = 10−6, εd = 1 and constantρg = 1.4 ·10−9 g cm−3. Note that this is a very unfavourablecase for accretion dominated planetesimal collisions sincegas drag is very strong throughout the disc. The minimum

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4 S.-J. Paardekooper and Z.M. Leinhardt

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Figure 2. Eccentricity vs. semi-major axis of test particles in the

γ Cephei system after 1000 binary orbits. Color indicates size inkm.

0 200 400 600 800 1000t (Pb)

0.0

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0.4

0.6

0.8

M/M

tot

0

20

40

60

80

Rm

ax, R

mea

n (k

m)

Figure 3. Evolution of maximum particle size (solid curve, toppanel), mean particle size (dashed curve, top panel), planetesimalmass fraction (solid curve, bottom panel) and dust mass fraction

(dashed curve, bottom panel) for the γ Cephei system.

particle size is 1 km and there are no planetesimals at t = 0.Binary parameters are those of the γ Cephei system. Figure2 shows the resulting (a, e) distribution after 1000 binaryorbits (∼ 90000 yr). It is clear that significant accretion hastaken place in the inner parts of the disc, where particleshave grown from 1 km to 50 km. Outside a/ab = 0.1, whichcorresponds to 2 AU, perturbations due to the binary aretoo strong for accretion to occur.

Inside 2 AU, planetesimals grow to sizes up to 70 km.The top panel of Fig. 3 shows the time evolution of the max-imum and mean planetesimal size over the whole disc. Themean size is dominated by the large number of small plan-etesimals in the outer disc, and stays between 10 and 20 km.The maximum particle size goes up rapidly by dust accre-tion (smooth parts of the curve) and by accreting collisions(jumps). The largest planetesimal is destroyed a few timesas well, but in general the trend is to grow to larger sizes.After ∼ 600 binary orbits, there is no source of small dustremaining from which to create new planetesimals, and thecollision time scale goes up.

0.02 0.04 0.06 0.08 0.10 0.12a/ab

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Figure 4. Eccentricity vs. semi-major axis of test particles in theα Centauri system after 1000 binary orbits. Color indicates size

in km. Top panel: α Centauri A. Bottom panel: α Centauri B.

The maximum size that can be reached in this scenariois limited by the total amount of solid material present in thedisc. In the 2D approximation, this mass has to be artificiallylow in order to end up with a realistic collision time scale.For a disc that has twice the solid material, keeping all otherparameters the same, growth up to 100 km was observed.Increasing N by a factor of 2 while decreasing Rinfl by thesame factor, which amounts to increasing the resolution ofthe simulation, showed growth up to 150 km. In these higherresolution runs more accreting collisions are observed thandepicted in Fig. 3. This is because the system goes throughphases of low particle number density, for which the collisionstatistics in the lower resolution runs are not optimal. In thissense, Fig. 3 represents a worst-case scenario, where manylow-velocity collisions are missed, and adding particles, whilekeeping the collision time scale the same, will only favourplanetesimal growth more.

Gas drag appears to play only a minor role in deter-mining the qualitative outcome of the model. This is mainlydue to the fact that the planetesimals are weak enough sothat any small eccentricity difference of ∼ 0.01 will lead todestructive collisions. Whether this difference is due to dif-ferential orbital phasing (under the influence of gas drag) orsimply orbital crossing (in the absence of gas) does not mat-ter. A simulation without any gas showed the same trend asin Fig. 3, with growth up to 80 km.

Crucial parameters are the efficiency of planetesimalformation and dust accretion. Accretion as shown in Fig.3 can only occur if the small debris created in catastrophiccollisions is swept up by larger bodies rather than formingnew planetesimals. Increasing εp by a factor of 10 still re-sults in accretion up to 80 km, but for a factor 100 growthstalls at 10 km. Similarly, the result depicted in Fig. 3 isrobust to changes in εd up to a factor of 10.

We consider the α Centauri system in Figs. 4 and 5.Results are shown for both binary components A and B,but since only the binary mass ratio changes between the

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Planetesimal collisions in binary systems 5

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0.0

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0.4

0.6

0.8

M/M

tot

0

20

40

60

80

Rm

ax, R

mea

n (k

m)

Figure 5. Evolution of maximum particle size (solid curve, top

panel), mean particle size (dashed curve, top panel), planetesimalmass fraction (solid curve, bottom panel) and dust mass fraction

(dashed curve, bottom panel) for the α Centauri system. Black

curves are for α Centauri A, grey curves are for α Centauri B.

two, the results are very similar. Again, we see growth upto 70 km in the inner region of the disc. Due to the factthat the system is more strongly perturbed than γ Cephei,the accretion-friendly region has shifted inward comparedto Fig. 2. The rough periodicity seen at early times in thebottom panel of Fig. 5 is due to different generations ofplanetesimals. The largest objects are formed inside 1.4 AUwith significant growth inside habitable zone < 1 AU.

6 DISCUSSION

We present the first study of planet formation in binariestaking into account the physical effects of collisions. Weshow that these effects tend to favour growth towards largeplanetesimals in the perturbed system. Two main mecha-nisms have been identified. First, frequent collisions tendto prevent planetesimals from reaching their equilibriumorbits. If collision rates are high enough, this means thatthe accretion-hostile environment due to differential orbitalphasing is never reached. Second, fragments produced byhigh-velocity collisions make up a large reservoir of materialthat is very easily reaccreted by the remaining planetesimalsas they sweep through the disc on highly eccentric orbits.

We have chosen parameters that in some respectsshould be very unfavourable for planetesimal growth. Gasdrag is strong throughout the disc in the models presentedhere. However, since gas drag plays only a minor role, chang-ing the gas density to a more realistic power law in radiusdoes not change the results. New planetesimals are formedon circular orbits and are, therefore, immediately capable ofdestroying larger bodies that are on eccentric orbits. Form-ing new planetesimals on eccentric equilibrium orbits wouldfavour more accreting collisions. On the other hand, we haveassumed dust accretion to be efficient. This efficiency de-

pends amongst others on the size distribution that is pro-duced in collisions and the vertical extent over which thefragments are distributed. Some of the fragments may belost due to radial drift, which is ignored in the current model,but radial drift may also help bringing more mass into theaccretion friendly inner regions.

We have worked in a 2D geometry for computationalreasons. Although we have tried to scale the surface densityin such a way to get realistic collision time scales, it is im-portant to realise that τc,2D evolves in a different way withparticle size than τc. It is, therefore, not possible to haverealistic collision time scales at all times. Furthermore, in-clinations of the planetesimals may be excited by collisions,resulting in i = e/2, which decreases the collision time scale.However, since most collisions are destructive, it is likelythat the remaining planetesimals will essentially orbit in theplane of the disc. Another important three-dimensional ef-fect was pointed out by Xie & Zhou (2009), who showedthat if the disc is slightly inclined with respect to the bi-nary plane, this may favour planetesimal accretion. Clearly,three-dimensional simulations are the way forward, and theresults in this letter should be interpreted as a proof of prin-ciple, that planetesimal accretion is possible in tight binarysystems under the right circumstances.

ACKNOWLEDGEMENTS

We thank Philippe Thebault for useful comments thatgreatly improved the manuscript and Debra Fischer for get-ting us interested (again). SJP and ZML are supported bySTFC Postdoctoral Fellowships.

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