the formation of the oort cloud and the primitive galactic environment

8/13/2019 The Formation of the Oort Cloud and the Primitive Galactic Environment

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ICARUS 129, 106119 (1997)ARTICLE NO . IS975754

The Formation of the Oort Cloud and the PrimitiveGalactic Environment

Julio A. Ferna ndezDepartamento de Astronom a, Facultad de Ciencias, Tristan Nar v aja 1674, 11200 Monte v ideo, Uruguay

E-mail: [email protected]

Received October 11, 1996; revised April 7, 1997

could have been the source from which the external Oort cloudhas been replenished through the age of the Solar System.We analyze the conditions of formation of the Oort cloud

An interesting by-product of our scenario of a much denserfrom icy planetesimals scattered by the accreting outer planets.galactic environment is that not only bodies from the accretionThe combined effect of planetary and external perturbationszones of Uranus and Neptune could nd their way to the Oortis considered to be the mechanism of transfer from the comets

cloud, but also a signicant number of residual planetesimalsbirthplace in the planetary region to the Oort cloud reservoir. from the Jupiter and Saturn accretion zones could have beenIf the main external perturbers from the primitive galacticincorporated into the Oort reservoir. The physicalchemicalenvironment were similar to the current ones (namely, passingnature of new comets may present different signatures ac-stars and the tidal force of the galactic disk), the resulting Oortcording to their different birthplaces, thus constituting relevantcloud would have probably been too loosely bound to havepieces of information to learn about the galactic environmentwithstood the disrupting effect of penetrating encounters within which the Solar System formed. 1997 Academic Pressgiant molecular clouds. An additional problem is that most of

the objects formed in the outer planetary region are found tobe nally ejected by Saturn or Jupiter, and not by Neptune or

1. INTRODUCTIONUranus, thus making the whole process of transfer of bodiesfrom the planetary region to the Oort cloud very inefcient.

Comets are thought to be very pristine bodies, leftovJupiter and Saturn perturbations are so strong that most bodiesof the formation of the jovian planets. Oort (1950) arguscattered by these planets are very likely to overshoot the

narrow energy range of the present Oort cloud to interstellar that bodies scattered by planetary perturbations to almospace. interstellar distances would be decoupled from the plan

It is shown that the combined action of planetary and external tary region by perturbations from passing stars. This conforces would have produced a more tightly bound comet reser- tion is essential forobtaining long dynamical lifetimes, sovoir if the Solar System formed within a much denser galactic to keep a large fraction of primordial bodies still revolvenvironment, perhaps a molecular cloud and/or an open cluster. around the Sun, since planetary perturbations would leThis seems to be the way in which most stars form. Moreover, to ultimate ejection within time scales considerably shorthe time scales of formation of Uranus and Neptune could well

than the age of the Solar System. Stellar perturbatiohave been very short (a few times 10 7 years or even less), aswould also be responsible for randomizing the orbtheir non-negligible contents of hydrogen and helium suggest,planes of the scattered planetesimals. The main featurewhich would give stronger support to the idea that the massiveOorts theory is thus the existence of a huge reservoirscattering of planetesimals in the outer planetary region was

produced while the Solar System was still within its natal envi- icy bodies swarming around the Sun in a spherical structuronment. It is found that a much stronger external eld, caused at distances of several times 10 4 AU. Oort also argued either by other members of an open cluster or by the tidal force stellar perturbations were also responsible for bringof the molecular cloud itself, could have produced a much more some members of the comet cloud back to the planetastrongly bound Oort cloud at distances of a few thousand AU. region where they may become detectable as new coFurthermore, a widened energy range for the Oort cloud reser- ets. The Oort cloud theory has been able to explain sevevoir would have increased the probability of trapping bodies dynamical features of comets, in particular the randoscattered by Jupiter and Saturn there, thus making the transfer

orientations of the orbital planes of long-period comprocess much more efcient. The strong external perturbationsand the concentration of original orbital energiesthat drove comets to a much more tightly bound Oort cloud(which are usually expressed in terms of the reciproceased to act shortly afterward, as the molecular cloud (or thesemimajor axis 1/ a) in the narrow energy range of nopen cluster) dissipated, thus preventing the formed comet

cloud from being disrupted. Such a tightly bound comet cloud parabolic orbits: 10 4 AU 1 Eorig

0 (negative v

106


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OORT CLOUD FORMATION AND PRIMITIVE GALACTIC ENVIRONMENT

of E orig , or positive values of aorig , are for elliptical orbits). plemented by numerical simulations that illustrate hocomets scattered by the jovian planets become trapped The word original refers to the orbit the comet has

before entering the planetary region, i.e., before being the Oort reservoir by stellar perturbations (e.g., Weissman1979, Ferna ndez 1980). More complete numerical simperturbed by the planets.

Planetary perturbations act on near-parabolic comets tions including the tidal force of the galactic disk werelater carried out by Duncan et al. (1987).when they are close to perihelion, causing mainly a change

in the comets orbital energy. This is a stochastic process So far, most studies on Oort cloud formation have implicitly assumed that the eld of external perturbers hin which the comet receives a kick at every perihelion

passage at a different energy level (it can either gain or experienced little change from the early Solar System upto now. But calculations show that the Sun may have exlose energy). The result is a random walk in energy phase

space. If no other forces act on the comet, the ultimate fate rienced radial excursions of more than 10 3 parsecs (pit moved around the galactic center to a zone where twill invariably be ejection to interstellar space (neglecting

collisions with planets or the Sun or sublimation by the surface density of molecular gas falls off very steeply. Thhas probably modulated the strength of the tidal forceSuns radiation). It is the classical diffusion problem with

a cliff at one of the extremes that has been widely studied the galactic disk (e.g., Hut and Tremaine 1985). Mateseal. (1995) have further estimated how radial variationsby several authors (e.g., van Woerkom 1948, Lyttleton and

Hammersley 1963). Yet, if other forces act on the comet the galactocentric distance (and, thus, in the local densityof the galactic disk), as the Solar System revolves arouwhen it is far away from the Sun (for instance, stellar

perturbations), then its perihelion distance may increase the galactic center, modulate the rate of comets injected

into the planetary region. The fundamental question, hobeyond the reach of planetary perturbations. The cometwill remain in such a loosely bound orbit until the external ever, is: What was the galactic environment in which th

Solar System formed, since the buildup of the Oort cloperturbers send it back to the planetary region or eject itto interstellar space. probably occurred soon after the accreting jovian plane

acquired substantial masses?Oort suggested that comets andasteroidsmighthave hada common origin in the asteroid belt, though the different It is well known that most stars tend to form in clusters

within molecular clouds (e.g., Lada et al. 1993), sphysical nature of the rocky asteroids and the icy cometswas pointed out by Kuiper (1951) shortly afterward. Later then probable that this was the way in which the Sola

System formed. However interesting this possibility mstudies by Safronov (1969) and Ferna ndez (1978) showedthat the UranusNeptune region was the most likely source be, very little attention has been paid to it until now. Mott-

mann (1977) argued that the late heavy bombardment of comets, since the more modest perturbations of theseplanets would ensure that a large percentage of the scat- the surfaces of the terrestrial planets, about 4 109

ago, was triggered by a very close stellar passage atered comets would fall in the Oort region, namely, theregion in energy space where stellar perturbations are early epoch when the Sun formed part of an open cluster.

Healso argued that such an encounter also tilted the orbitastrong enough to decouple bodies from the planetary re-gion before ejection to interstellar space by planetary per- planes of the jovian planets by 8 with respect to

solar spin axis. Hills (1982) later assumed that the Soturbations occurs.But not only passing stars can perturb comets moving on System and an inner comet cloud formed during the early

collapsing stages of the nebula within a very dense near-parabolic orbits. Galactic tidal forces and penetratingencounters with giant molecular clouds (GMCs) can exert cluster. Tremaine (1991) attributed the twist of the orbital

angular momentum vector of the planets to torques dan even larger dynamical effect. Biermann (1978) sug-gested that molecular clouds can play a fundamental role to nearby mass concentrations within the solar nebula or

asymmetric infall of material. Gaidos (1995) has furtin the dynamical evolution of Oort cloud comets, and Na-

pier and Clube (1979) argued that GMCs can disrupt the analyzed the dynamical consequences of Solar System for-mation within a dense galactic environment. He sets couter portions of theOort cloud. Tidal forcesof thegalactic

disk are more intense than those of the galactic nucleus, straints on the local density of external perturbers fromthe current orbital inclinations of Uranus and Neptunas was shown by Byl (1983). The most important dynamical

effect is to change the comets perihelion distance, so it Gaidos also refers to the formation of a transient comecloud at 3000 AU from residual planetesimals scattecan be removed from or injected back into the planetary

region. It can be shown that the tidal force of the galactic by Saturn, but he argues that it would have promptly beeneroded by the strong tidal eld of the dense environmdisk is more intense at mid-galactic latitudes [cf. eq.(8)],

which is reected in a greater concentration of the aph- and frequent stellar encounters.We think that all the dynamical consequences of telion points of the observed long-period comets there

(Delsemme 1987). formation of the Solar System within a dense galactic enronment, in particular concerning the buildup of the comThe dynamical studies of the Oort cloud have been com-


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108 JULIO A. FERNA NDEZ

cloud, have not been thoroughly explored. Our increasing Uranus and Neptune was dominated by small, kilometer-sized planetesimals.body of observational data showing that stars form within

molecular clouds, and usually in clusters of different sizes, Ferna ndez and Ip (1984, 1996) studied numericallyaccretion and scattering of bodies in the UranusNeptugives relevance to this subject. Furthermore, some recent

studies (e.g., Lissauer et al. 1995,Pollack et al. 1996) suggest zone. One interesting and unexpected result was that torbit of Neptune, and to a lesser extent those of Uranthat the outer planets formed on time scales much shorter

than thought before (e.g., Safronov 1969), so the buildup and Saturn, experienced an outward drift due to exchangeof angular momentum with the interacting planetesimaof the Oort cloud might have been a very early episode

in the Solar Systems lifetime, probably when the Solar The angular momentum gained by the orbital expansionof these planets was compensated by a small drift inwSystem was still within its natal environment. The aim of

this paper is to further discuss the dynamical consequences of the massive Jupiter. These numerical models are sugges-tive in that initial masses two to three times the combinof a dense galactic environment on the formation of the

Oort cloud. masses of Uranus and Neptune (i.e., 60100M ) required to form these planets; the unaccreted solid matrial was lost to the inner planetary region or to interstel2. ACCRETION OF THE JOVIAN PLANETSspace. Therefore, the accretion of Uranus and Neptuseems to have been very inefcient in their late stagIt is widely agreed that the mostly gaseous Jupiter and

Saturn had to form before the dispersal of the hydrogen which can be explained as due to the increasing probabilityof ejection of interacting planetesimals by rapidly growand helium of the primitive nebula on a very short time

scale of a few million years (e.g., Lissauer 1987). There protoplanets (say, masses a few M ) as comparedcollisional accretion.is strong observational support for a rapid dissipation of

circumstellar gas around pre-main-sequence, low-mass The much larger population in the outer planetary zonepresumably had to include many massive bodies, in adstars, as the detection of naked T Tauri stars with ages

approximatelya million yearsold suggests (see,e.g., Walter tion to proto-Uranus and proto-Neptune. Stern (1991) hasargued that Triton, the PlutoCharon binary system, aet al. 1988). Recent radio CO observations by Zuckerman

et al. (1995) conrm that the molecular gas surrounding the tilt of Uranusand Neptunes spin axes are fossil recordsof a substantial population of 1000-km-sized objects. Tyoung solar-type stars tends to dissipate very quickly, per-

haps in only a few million years. Uranus and Neptune late heavy bombardment of the terrestrial planets thatlasted until 3800 myr ago might be explained by a soumay have essentially formed by collisional accumulation

of planetesimals over longer time scales. We do not yet of long-lived projectiles in the outer planet zone (Wetherill1975, Ferna ndez and Ip 1983). The above discussion have good theoretical or numerical models to assess how

much longer these time scales were in comparison with suggests that there was a massive scattering by the joviaplanets of the residual population left after the formatithose of Jupiter and Saturn. Ferna ndez and Ip (1996)

showed that embryo planets of Marss size, initially spread of Uranus and Neptune and that it likely occurred earlyin the history of the Solar System.in the outer planetary region, can grow to Neptune-size

planets over time scales of 12 108 years. The fact that3. EXTERNAL PERTURBERS IN THE SUNSUranus and Neptune contain nonnegligible fractions of

NEIGHBORHOODhydrogen and helium, perhaps amounting to somethingbetween 1 M and 2M (Hubbard 1989, Hubbard et al.

For a body to be stored in the Oort cloud, its periheli1995), suggests that they grew fast enough to be able todistance has to be raised above Neptunes orbital radcapture gas from the nebula before its dispersal by theby at least 1015 AU. For comets in the outer plastrong T Tauri wind.Earth-size embryo planets in theouter

zone, this condition should be fullled when q planet region might have already been able to maintain mentioned, a comet of initial orbital energy E 0 extended dense atmospheres of hydrogen and helium (Lis-(given in AU 1) will random walk in energy space dusauer et al. 1995) favoring their later growth. Pollack et al.planetary perturbations, experiencing an energy change(1996) have recently developed a sophisticated numericalduring each perihelion passage. Since the energy chanmodel for the accretion of the jovian planets, taking intoare stochastic, the number of revolutions required for taccount both the gas and planetesimal accretion rates.comet to reach a parabolic orbit (1/ a 0) is of the They nd that it might have taken about 1.6 107 years

for Uranus to reach its present size (not much longer thanthe growth times of Jupiter and Saturn), while for Neptune N

(1/a0)22t

, the corresponding formation time might have been about3.7 107 years. The authors nd that these formation timescales could have been even shorter if the accretion of where

t is the typical energy change per passage, c


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FIG. 1. Distribution of original reciprocal semimajor axes of the observed new and young long-period comets from Marsden and Williams(1996) catalog with (1/ a)orig 10 3 AU 1 (i.e., comets with aorig 103 AU or original hyperbolic orbits with aorig 0). The thick histogramthe subsample of comets of quality class 1 (as dened in the Marsden and Williams catalog) and perihelion distances q 2 AU which are presuless affected by non-gravitational forces.

puted as the root mean square (rms) of a large sample of 6 10 5 AU 1 denes the lower limit of the Oort regat aorig 1.7 104 AU.individual energy changes (see Ferna ndez 1981, Duncan

et al. 1987). t is a function of the planets mass and semima- A passing star of mass M and relative velocity Vimpart an impulsive change in the comets velocity relatjor axis and of the encounter velocity of the comet with

the planet. Therefore, for near-parabolic comets there will to the Sun given bybe a strong dependence of t on the comets periheliondistance and inclination. Equation (1) is valid provided1/a0 t . v vc v

2GM V D cD 2c DD 2 , If after N revolutions the comet has not been decoupled

from the planetary region, it will be ejected to interstellarspace. Therefore, to be stored in the Oort cloud a comet where G is the gravitational constant; vc and v aof energy (1/ a0) will have to experience a change impulses received by the comet and the Sun from

q q within N revolutions. Strictly speaking, a will passing star, and D c and D are the distances of clochange due to planetary perturbations on each perihelion approach of the star to the comet and to the Sun, resppassage, so we should expect that the change in q required tively (see, e.g., Ferna ndez and Ip 1991). For distantto remove the comet from the planetary region should counters, vc and v become nearly parallel, so the mooccur before N revolutions. If N is smaller, a 0 is larger, so lus of v in Eq. (2) can be approximately expressed bstrictly speaking Eq. (1) will give us a lower limit for thesemimajor axis a0 of comets likely to be removed from the

v2GMr cos

VD 2, planetary region before ejection occurs.

The distribution of the original orbital energies E orig(1/aorig ) of new and young comets clearly shows a spike

in the energy range 6 10 5 AU 1 E orig 0 (Fig. 1). where r is the heliocentric distance of the comet andthe angle between D and r. We adopt in the followiThe spike is equally outstanding when we limit the sample

to the best-determined orbits as described in the gure time-average heliocentric distance r a (1 e2 /2)(valid for a near-parabolic orbit of eccentricity e caption. The observed sharp boundary at the energy level


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The cumulative change in the orbital velocity of the equation that splits v into two contributions, from cand distant encounters (cf. Ferna ndez and Ip 1991)comet during time span T will be given bylead to a decrease in the computed value of a by no than 2040% . This variation does not qualitatively chav 2* T D maxD min v 2 s(D ) dD , (4) our discussion.

At present, the tidal force of the galactic disk is mairesponsible for changes in the comets angular momentuwhere D max and D min are the maximum and minimum(and, thus, in q) (Byl 1983, Morris and Muller 1986, Hedistances of closest approach of passing stars to the Sunand Tremaine 1986). The change in q in one orbital revexpected during T . D max can be taken as innity withouttion due to this force is given by (e.g., Byl 1983, Fertoo much error. D min (2n* T )

1/2 , where n* is the stellar dez 1992)ux in the Suns neighborhood of about 7 stars myr 1 pass-ing through a circle of 1-pc radius, assuming a relativevelocity V 30 km sec 1 (Ferna ndez and Ip 1991). s(D )dD 2n*D dD is the rate of stellar passages with

qq 1 6 G aP cos sin 2 (2GM q) 1/2 , impact parameters in the range ( D , D dD ). Let us

adopt for the stars mass M M . The average change inthe transverse velocity of the comet is where is the density of the galactic disk in the S

neighborhood, P is thecomets orbital period, is the( v *)

2T v

2* cos

2 v 2*. (5) between the orbital plane and the plane perpendicular the galactic disk that contains the radius vector Suncom

The corresponding change in the perihelion distance is r , and is the galactic latitude of r (which, for a given by parabolic orbit, is very close to the direction of the aphelio

point). As seen, the greatest dynamical effect is attainfor 45 , in agreement with the observed concenq

q 2

( v *)Tv T

(6)tion of aphelion points at mid-galactic latitudes (cf. Stion 1).

There is some question about the best value of . where v T (2GM q)1/2 /r is the transverse velocity of thethe comparison of different gravitational potential modecomet (assumed to be in a near-parabolic orbit).of the Galaxy with velocity dispersions of tracer stLet us set T N P in the integral of Eq. (4), whereBahcall (1984) derived 0.185M pc 3, while KuN is given by Eq. (1) and P 2 (GM ) 1/2 a3/2 is theand Gilmore (1989) obtained a lower value of 0.comets orbital period, and adopt for v the approximatepc 3. Matese et al. (1995) used a density 0.25M expression of Eq. (3), which is a reasonable assumption,at the most recent plane crossing if dark matter is preseexcept for the rare, very close encounters. In the followingbut it could be as low as 0.13 M pc 3 in a no-dark-mwe consider the case of a body in Neptunes zone for whichmodel. Moreover, they found a quasiperiodic variationq 30 AU and t 2.25 10 5 AU 1 (cf. Table II). If thegalactic disk density between 0.05M and0.15M we substitute the derived value of v * in Eqs. (5) and (6), as the Solar System circles the galactic center at varywe obtain after introducing the corresponding numericalgalactocentricdistances. Weadopt in thefollowing an avevaluesage value of 0.15M pc 3.

If we assume that the comet remains more or less wqq N ,stellar 0.0226 t2.25 10 5 AU 1 1 q30 AU 1/2 the same a during N revolutions, the total change will linearly (assuming also that the orientation of the come

apsidal line and the vertical distance to the galactic m a104 AU

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.(7)

plane do not change signicantly during N revolutiso we have

Since the body is at the edge of the planetary region, weassume that a change q 0.5q is enough to place the q

q N qq 1 N . body beyond the perturbing inuence of the planets (Ever-hart 1968).

If close encounters occur during the diffusion of thecomet, say stars approaching the Sun to distances 2 r Substituting N by Eq. (1) and ( q /q)1 by Eq. (8)

introducing numerical values for the averages cos3a, the semimajor axis computed from Eq. (7) will besomewhat overestimated. The use of a more complete 2/ and sin 2 2/3, we nally obtain


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FIG. 2. Relative change in the perihelion distance of a comet as a function of its semimajor axis for different external perturbers. Dashedcurves are for perturbers from the present galactic environment. Solid curves are for assumed perturbers in an early galactic environment.

ment similar to the current one, icy planetesimals scatterqq N ,tide 0.329 t2.25 10 5 AU 1 2 q30 AU 1/2 by the jovian planets would have given rise to a loos

bound Oort cloud with an inner radius of a 1.3 AU. We note that the denition of an inner radius do a

104 AU

3/2

.

(10)

not imply that no comets with smaller semimajor axes cbe trapped in the Oort cloud. Since we introduced averavalues in Eq. (10), there will always be favorable circThe error committed by averaging sine and cosine factorsstances to produce captures of comets with a 1.3in the above equation fortunately has only little inuenceAU. Nevertheless, we should expect that the number on the computed values of a. For instance, changes bycomets trapped in the Oort cloud falls off below that limabout 50 % in the product cos sin 2 would lead toFrom numerical experiments, Duncan et al. (1987changes of only 30% in the computed a.tained a substantial fraction of Oort cloud comets wThe dashed curves of Fig. 2 show the change q /q as aa 300013,000 AU, which amounts to about two-thfunction of the semimajor axis a for stellar perturbationsof the total Oort cloud population in the range 300050,0[Eq. (7)] and for the tidal force of the galactic disk in theAU. Yet, this discrepancy with our computed inner radipresent solar neighborhood [Eq. (10)]. From the plots we

cannot be considered to be very signicant given the diffsee that changes q 0.5q due to the tidal force of the ent procedures employed. Part of the discrepancy mgalactic disk are reached for a semimajor axis a 1.3 arise from small differences in the adopted numerical v104 AU. The effect of stellar perturbations is much smaller;ues (for instance, the density of the galactic disk, the ranthe condition q 0.5q is reached only for a 3.4 104of initial q of the scattered comets). If we consider soAU. Therefore, the tidal force of the galactic disk plays,extra effects, such as an enhanced role of Jupiter aat present, themajor role in injectingcomets into theplane-Saturn in the scattering of bodies (cf. Section 5), the ctary region, which conrms previous results (e.g., Heislerture efciency in the inner core may have decreased beland Tremaine 1986, Morris and Mueller 1986) and is inthe fraction estimated by Duncan et al. Therefore, wegood agreement with the observed maximum 1/ a0 valueconclude that under the current galactic conditions tof the spike of original reciprocal semimajor axes shownfraction of comets trapped in the range 300013,000 in Fig. 1.

If the early Solar System was within a galactic environ- would lie somewhere between a few percent an


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65% . The rest of the comet population would have and Uranus and Neptune were probably well on the wayto reaching their present sizes and locations. That wbeen trapped in more loosely bound orbits with a

13,000 AU. probably also the time when most of the residual somatter in the accretion zones of Jupiter and Saturn andThe picture of a loosely bound Oort cloud, however,

presents some difculties with which to deal. The rst signicant fraction of the residual solid mass in the Uranuand Neptune zones were ejected (cf. Section 2).difculty has to do with the disrupting effect of penetrating

encounters with GMCs over the age of the Solar System Let us now analyze what would have been the consequences on the formation of the Oort cloud if the ea(Napier and Staniucha 1982, Bailey 1983, Hut and Tre-

maine 1985). But there is a second important issue that Sun would have been within such a dense galactic environment. Let us assume rst that the Sun was within a molewe analyze below: No matter in which part of the planetary

region the residual planetesimals were originally located, lar cloud.Theaverage density of a molecular cloud is about50 H2 molecules cm 3 (Blitz 1993), which correspondJupiter and Saturn were, at the end, the planets that took

control of the dynamical evolution of most bodies. And, a mass density of mc 2.5M pc 3. If we next assumethe Solar System formed at a distance s( Rmc ) fromas we discussed earlier, the probability that Jupiter and

Saturn placed a comet in a weakly bound, Oort-type orbit center of the cloud (assumed to be spherical of radiusand of uniform density), the tidal force acting on a bois very low.at a radial distance s from the Sun is given by

4. THE EARLY GALACTIC ENVIRONMENT OF THESOLAR SYSTEM F

dF

ds s

2GM s s s3

8

3G

mc s,

Near-infrared imaging surveys of nearby GMCs haveshed new light on the way stars form. Young stars appear where M s is the mass enclosed within the sphere of radiuembedded in dense cores of gas (mainly molecular hydro- The rate of change of angular momentum of a comegen) and dust, which in itself is an indication that stars a distance r from the Sun is given byform within molecular clouds (Lada et al. 1993, Lada 1995).Furthermore, they do not seem to form in isolation, but dH

dt F t r

12

dF ds

r 2 sin 2 cos , in groups of different sizes and compactness, ranging fromvery poor clusters of a few members to very rich clustersof hundreds of stars. Kroupa (1995) raises the interesting

where F t is the transverse component of the tidal forceissue of observations showing that the proportion of wideis the angle between r and the direction from the Subinaries (separations from a few AU to 1800 AU) amongthe center of the natal molecular cloud, s r cpre-main-sequence stars is about 1.5 times larger than onand is the angle between the plane containing the radthe main sequence. From this nding Kroupa concludesvector r and the center of the molecular cloud and that most galactic eld stars may have formed as binarycomets orbital plane.systems in clusters of best-t parameters: 200 binary sys-

We note that the drag force due to the comets motitems and half-mass radius of 0.8 pc. Perturbations amongthrough the gas of the molecular cloud is negligible. cluster binaries would lead to thedissolution of many pairs,a comet nucleus of radius Rc and density c , and assuleaving the fraction of binaries of about 60 % observed inthat Epsteins drag regime applies (i.e., that the mean frgalactic eld stars.path of the gas molecules is large as compared with It is accordingly reasonable to propose that the Sun alsodimensions of the body), we can dene the stoppformedwithina molecular cloud and, perhaps, a star clustertime, t s , i.e., the time required to reduce the com(we will leave aside here the intriguing and very exciting

velocity by a factor 1/ e, as (see, e.g., Weidenschilling 1possibility that the Sun had a primitive companion thatescaped before the cluster dissolved). This primitive galac-tic environment could have lasted at most a few 10 7 years, t s

cR c mc v

, the time required for a GMC to dissipate in the galacticmedium (Blitz 1993). If the Sun happened to form withina rich cluster, then the residence time could have extended where v is the mean thermal velocity of the gas. I

introduce the numerical values Rc 105 cm, c to approximately a hundred million years, the average life-time of open clusters (Lynga 1982). Although a time span g cm 3 for a standard comet nucleus, mc 2.5M

( 1.7 10 22 g cm 3), and v 3.5 104 cm sec 1of a few tens of millions to one hundred of millions of years represents only a small fraction of the Solar System mean temperature of the molecular cloud of 10 K), w

obtain t s 8.3 1021 sec ( 2.6 1014 years); i.e.age, it may nevertheless cover key episodes of its historywhen Jupiter and Saturn probably formed (cf. Section 1), longer than the age of the known universe. Even if we


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consider the density of a core within the molecular cloud tive Suns neighborhood would have been ncl V cl 15 stars pc 2 myr 1.of 104 H 2 cm 3 we obtain a corresponding time t s

1.3 1012 years, still longer than the age of the universe. We can make use of Eqs. (3) and (4) for stellar encounters. Even in an open cluster like the one assumed heWe can therefore neglect gas drag effects on the comets

motion with total condence. the interstar distances will generally be much greater thathe comets semimajor axis at which perturbations by clTaking into account thatter stars can decouple the orbit from the planetary regioprovided that the considered time scale is not longer th

3 107 years, the average lifetime of a comet widH GM q2

1/2dqq

, (14)of a few thousand AU under the gravitational control Neptune. The mean separation between cluster starsd (1/ cl)1/3 8.4 104 AU, whereas the closest approand considering again that the tidal force of the natal mo-to the Sun expected during 3 107 years is Dlecular cloud acts during a period N P , where P is the(2 cl ) 1/2 6.9 103 AU. Therefore, very few star comets orbital period and N is given by Eq. (1), then aftersages will be expected at distances smaller than the comsubstituting these expressions into Eq. (12), we obtain fordistance, so we can still use Eq. (3) for distant encountthe change in the perihelion distance over N revolutions(actually, if we consider the very close stellar encountethe expressionthe computedchange v c willbe larger, so our result shobe considered as a lower limit). It may be argued t

the impulse approximation described by Eq. (3), whq

q N ,mc43

2 G mc r 2NP cos sin 2 (GM q) 1/2 . (15) assumes the comet to be at rest during the stars passa

breaks down for the low encounter velocities of clusstars; however, Brunini and Ferna ndez (1996) have fSubstitutingby theappropiate numericalvaluesand con-that the impulse formula is a good approximation, evensidering averages cos 2/ and sin 2 2/3 wethe cases in which the encounter time is on the ordernally obtainthe comets orbital period. We accordingly use Eq. (3) the time being to obtain order-of-magnitude estimatthough admitting that more accurate numerical integrq

q N ,mc 3.65 t2.25 10 5 AU 1 2 q30 AU 1/2(16) tions will be a more appropriate way to address this pr

lem in a follow-up study.By introducing the numerical values discussed before a

104 AU

3/2

.Eqs. (3) and (4), we obtain

From Eq. (16) we nd that the tidal force of the natal qq N ,cluster 4.528 t2.25 10 5 AU 1 1 q30 AUmolecular cloud can remove comets from the planetary

region for semimajor axes as small as a few 10 3 AU. Theresults are plotted in Fig. 2. a

104 AU

5/2

.Therefore, a galactic environment much more crowdedthan the present one might have had dramatic conse-quences in the buildup of the Oort cloud. A tightly bound Since the relative velocities within open clusters are abOort cloud with a radius of a few thousand AU might be 30 times smaller than in the Suns neighborhood at presea consequence of such an early environment where the the dynamical effect will be much stronger, which is cSun possibly formed. Since most stars tend to form in rmed by Eq. (17). According to these results, if the clusters within molecular clouds, we consider this not to would have been a member of an open cluster like be an ad hocassumption, but based on strongobservational one described here, the strong perturbations of other clugrounds. We can further speculate that had the Solar Sys- ter stars would have decoupled comets from the planetatem formed in a galactic environment like the current one, region for a of a few thousand AU (see Fig. 2).far fewer comets in the Oort cloud would have surviveduntil the present epoch. 5. THE CONTRIBUTION OF THE JOVIAN PLANETS

Let us now assume that the Solar System formed within TO THE OORT CLOUD: A REASSESSMENT OF THEa cluster of stellar density cl 15 pc 3, which is within ROLE PLAYED BY JUPITER AND SATURNthe range observed in open clusters (Lynga 1982). For acluster in virial equilibrium we nd a rms relative velocity It has long been argued that the UranusNeptune region

was the source of Oort cloud comets and that Neptunof V cl

1 km sec 1. Therefore the stellar ux in the primi-


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TABLE Iperturbations, and to a lesser degree Uranuss perturba-Probability That a Given Jovian Planet Willtions, were the main driving force in transferring comets

Control the Dynamical Evolution of a Bodyfrom near-circular orbits within the planetary region toStarting in Neptunes Accretion Zonenear-parabolic orbits (Safronov 1969, Ferna ndez 1978).

The reason for this is that the typical energy change per Planet p1perihelion passage that a low-inclination comet experi-

Jupiter 0.60ences under the gravitational control of Neptune ( 2 Saturn 0.2410 5 AU 1) is somewhat smaller than the binding energiesUranus 0.06of Oort cloud comets ( 36 10 5 AU 1). Therefore,Neptune 0.10a comet random walking in the energy space under the

gravitational control of Neptune (or Uranus) is very likelyto fall in the energy range of the Oort cloud before beingejected. Conversely, Jupiters perturbations are so strong fore, p will rapidly increase for smaller heliocentric

tances and larger planet masses, the largest p being fo( 1.5 10 3 AU 1) that comets under its gravitationalinuence will very likely overshoot the Oort energy range piter.

Numerical simulations show that bodies starting at Neto a hyperbolic orbit. We now deem it necessary to recon-sider some aspects of this scenario. tunes zone indeed evolve in such a way that most of the

end up ejected by Jupiter. For instance, numerical simuAs residual planetesimals of Neptunes zone are scat-tered, there is a statistical increase in theencounter velocity tions by Duncan et al. (1995) show that about one-t

of objects starting in Neptunes zone end up as visiu due to Fermis acceleration mechanism (Arnold 1965),and also due to secular perturbations by the other planets. Jupiter family comets. Ferna ndez and Gallardo (1

have repeated these calculations using O piks two-A body can be ejected in a parabolic orbit if the encountervelocity reaches the value u ( 2 1)v cir , where v cir is algorithmandfound very similar results. In particular, the

results show that almost 50 % of the sample falls undethe (circular) orbital velocity at Neptunes distance. Now,before that happens the bodys perihelion can go down to gravitational control of Jupiter and is eventually ejected

by this planet, unless a collision with a planet or the SUranuss orbit. The minimum perihelion distance qmin abody can reach is occurs rst. No more than 1520 % of the bodies sta

in Neptunes inuence zone continue under its control untthey are ejected. And these are results for the current Sol

qmin (1 U )2

1 2U U 2, (18) System: If we assume early conditions where Jupiter a

Saturn were already formed while Uranus and Neptuwere still accreting material, the contribution of the twhere U u/ v cir . Equation (18) shows us that for velocities outermostplanets turns out to be somewhat lower (FernaU 0.3, i.e., signicantly smaller than that required for dez and Ip 1981). Furthermore, if the birthplaces of Uranescape, the body can reach Uranuss zone. and Neptune were closer to the Sun, and therefore Once the body reaches Uranus zone, it can be subject Jupiter and Saturn, as the numerical experiments of Feto strong perturbations by both Neptune andUranus. Even na ndez and Ip (1984, 1996) show, a larger fraction of rethoughclose encounters with Neptune could be more prob- ual bodies of their accretion zones would fall under able at the beginning, because one of the nodes of the gravitational control of their closer giant neighbors Jupibodys orbit should be close to Neptunes orbit, secular and Saturn. The probability p1 that a body startingperturbations by the planets will change the orientation of low-inclination, low-eccentricity orbit in the accretion zothe nodes and apsidal line of the bodys orbit, so close of Neptune falls under the dynamical control of a givinteractions with either Uranus or Neptune can occur. The jovian planet is given in Table I. The computed probabprobabilities of close interactions with one of the planets ties are average values for the different scenarios describcan be expressed by O piks (1951) equation above. They have been derived from different results otained analytically and numerically by Ferna ndez (1Ferna ndez and Ip (1981, 1984, 1996), and some new num p

2U sin i U x

, (19)ical experiments by means of O piks two-body code.scatter in the average probabilities is about 20 % .

For bodies starting in Uranus zone we obtained resuwhere is the radius of the cross section for strong interac-tions and U x is the component of the encounter velocity similar to the previous ones for Neptunes zone.

As mentioned, comets scattered outward can be trappin the radial direction. is proportional to the gravitationalradius for collision expressed in units of the radius of the in the Oort cloud. Let p2 be the probability that a co

random walking in the energy space falls in the narrplanets heliocentric orbit (assumed to be circular). There-


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TABLE II E (1/a)par (1/a) l , where (1/a) par 0 is the eTypical Energy Changes of Near-Parabolic of a parabolic comet and E l (1/ a)l is the largest bin

Comets in Low-Inclination Orbits energy (minimum semimajor axis) of a scattered comthat has its perihelion decoupled from the planetary regiPlanet t(AU 1) before ejection by planetary perturbations. Therefoa l 1 / E is the minimum semimajor axis of Oort clJupiter 1.50 10 3

Saturn 3.49 10 4 comets, so comets diffusing to semimajor axes a aUranus 3.67 10 5 be incorporated into the Oort cloud (since we work wNeptune 2.25 10 5 average values, al cannot be taken as a sharp bound

in actuality we should expect to have a transition regbetween no captures and full captures).

Results of poort as a function of the minimum semimenergy range E of Oort cloud comets. If we assume that axis a l are shown in Fig. 3. We can see that for the classthe distribution of energy changes per perihelion passage Oort cloud, whose observed width is E 6 10 5of near-parabolic orbits follows a Gaussian distribution (a l 17,000 AU) (cf. Section 3), Neptune is clearly(Kerr 1961), the probability p2 is approximately given by main driving force in placing bodies there, in agreem

with previous results (Safronov 1969, Ferna ndez 1This isbecause even though about 60 % of the planetesi p2

12

E

0e z

2 /2 dz , (20)of Neptunes accretion zone fall under the gravitation

control of Jupiter (cf. Table 1), the latter planet has a vewhere z 3/2 / t (Ferna ndez 1981), is the energy low probability of placing bodies in the Oort cloud (change per perihelion passage, and t is the standard devia- than 2% ). The situation changes when we move to mtion of the Gaussian energy distribution (that we adopted tightly bound models of the Oort cloud, which meanas the typical energy change). The adopted values of t for widening of E . For instance, for a 4000 AU Sanear-parabolic comets in low-inclination orbits (random and to a lesser degree Jupiter, places in the Oort cloinclinations in the range 0 i 30 ) and perihelia close a signicant fraction of the total mass (about 30 % )to the jovianplanetcontrolling thedynamicalevolutionare extremely compact models of the Oort cloud ( a listed in Table II. They are taken from Ferna ndez (1981). AU), Saturn and in second place Jupiter become the ma

The values quoted in Table II do not take into account contributors of bodies to the Oort cloud. This was poinclose encounters; however, Ferna ndez (1981) showed that out earlier by Weissman (1994), who suggested that a widclose encounters played a very important role in the ejec- energy range for the Oort cloud would result in a greationprocess,mainly whenlow-inclination orbits are consid- efciency of trapping comets there as compared with ejered. The distribution of energy changes takes the form tion, in particular for the case of Saturn. Accordingof a Gaussian distribution with long tails corresponding to Weissman, if this greater trapping efciency were also strong perturbations (Everhart 1968). Ferna ndez showed plied to other solar systems, it would help to explain that in this case typical energy changes are about three to seeming scarcity of interstellar comets.four times larger than the values quoted in Table II. Tomake allowance forcloseencounters, we computed p2 from 6. DISCUSSIONEq. (20) taken as the standard deviation *t 3.5 t .We should bear in mind that this is a rough approximation, As mentioned, one of the problems with the existesince the distribution departs now from a Gaussian one of a loosely bound Oort cloud is its survival during billbecause of the long tails. Nevertheless, our results show a of years (e.g., Napier and Staniucha 1982, Bailey 19

fairly good agreement with those found numerically by Hut and Tremaine (1985) have found a half-life of 3Ferna ndez and Ip (1981), so we do not expect variations years for a comet with a 25,000 AU, which is by more than a factor 2 to 3 in more detailed studies stringent constraint. Yet this still leaves little energy ran(though we deem it extremely interesting to do this in the for building up a comet cloud, since comets must hnear future). a 17,000 AU or E 6 10 5 AU 1 to be effect

The probability that a given jovian planet will place a decoupled from the planetary region before ejection body coming from Neptunes accretion zone in the Oort planetary perturbations (cf. Fig. 1). But, on the other hancloud is expressed as for a 25,000 AU or E 4 10 5, only 20% o

comets will survive throughout the Solar System lifetim poort p1 p2 . (21) Therefore, the storage of comets in the Oort cloud will

much more efcient for the energy range 6 1E 4 10 5 AU 1, i.e., for a narrow energy wThe energy width can be approximately expressed as


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FIG. 3. Probability of placing a body coming from Neptunes accretion zone in the Oort cloud by a given jovian planet, as a function of thminimum semimajor axis of the Oort cloud reservoir (the smaller the minimum semimajor axis, the wider the energy range of the Oort cloud reservoir).

E 2 10 5 AU 1, of the order of Neptunes typical powerful stellar wind coming from the forming star meethe surrounding envelope. Yet the formed comets wouenergy change. To overcome this difculty, the existence

of an inner core of the Oort cloud has been postulated as have escape velocities, so this procedure would produceinterstellar comets rather than a bound core or shell an additional reservoir (Hills 1981, Bailey 1983, Duncan

et al. 1987). As comets with smaller binding energies are comets. Duncan et al. (1987) did obtain a substantial inOort cloud population with semimajor axes in the ranejected when the Solar System meets a strong perturber

(for instance, a GMC), other comets from the inner core 300020,000 AU (about 80 % of the total populatithough their results might somewhat depend on the initgain energy (i.e., they are pumped up to occupy more

loosely bound orbits), so there is continuous replenishment conditions of the comets scattered to the Oort cloud (cf.Section 3). For instance, if the initial perihelion distanof the outer (or classical) Oort cloud. This is not quite a

steady-state process since the inner core will also be de- were concentrated around 510 AU, as discussed in Section 5, then the fraction of comets placed in theOort regiopleted with time, though its dynamical lifetime may sub-

stantially exceed the age of the Solar System. with a 300020,000 AU would drop drastically; howeour proposal for the formation of an inner core appears We next analyze how a massive inner core could form.

In situ formation(e.g., Biermann and Michel 1978)encoun- a natural by-product of the formation of the Solar Systemwithin a dense galactic environment.ters the difculty of the extremely low density of the nebula

medium at such large heliocentric distances. Grains are not Gaidos (1995) also considered the formation of a tightlybound Oort cloud at a 3000 AU in a dense galexpected to agglomerate in comet-sized bodies at distances

greater than a few hundred AU (Ferna ndez and Gallardo environment. But he argued that the same strong forcethat formed the comet cloud (either the tidal force of t1997). Cameron (1973) tried to explain the formation of

comets in satellite nebulae of the solar nebula moving in natal molecular cloud or cluster stars) disrupted it veryquickly; however, because of the short lifetime of a dehighly elliptic orbits, while Hills (1982) considered that

grains might have coagulated into comets at distances galactic environment, it is very likely that the core of tightlybound comets will survive throughout this early stage. E15 103 AU under the combined action of radiation

pressure from the proto-Sun and neighboring protostars. ternal perturbers will tend to thermalize the comet cloudpopulation, so the probability of lowering the cometaBailey (1987) tried to address the problem of the low den-

sity of the nebular material at such distances, arguing that perihelia back to planetary distances will decrease to avery small value; for instance, for a thermalized comcomets could form at the shock front produced when the


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population the probability that the perihelion distance de- have lifetimes signicantly longer, perhaps on the orderof a few hundreds of millions of years. Therefore, to answcreases to q qL , where qL is the radius of the planetary

region, is 2qL /a (Hills 1981). Comets in the core can also this question we have to know the time scales of formatioof the jovian planets, since the massive scattering of resgain energy to positive values (hyperbolic orbits). If we

assume that theSolar Systemremained within a star cluster ual planetesimals accompanied the latest stages of theirformation. The gaseous composition of Jupiter and Satuduring T 108 years, the condition for escape is that the

rms change in the comets velocity, v * [cf. eq. (4)] reaches strongly suggests a short formation time scale, probasignicantly shorter than the dissolution time of the nathe valuemolecular cloud. With respect to Uranus and Neptune, tanswer is more uncertain. Their non-negligible content

v * v esc2GM

r 1/2 , (22) hydrogen and helium suggests that they were able to grinto massive objects on a short time scale, so they costart to scatter bodies while the Sun was still within

where v esc is the escape velocity at distance r . Again, if we natal molecular cloud.take an average r 1.5a, we nd that the above condition 3. Since stronger external perturbers widen the eneris fullled for a 5800 AU. Therefore, comets in an inner range of the Oort region, did Jupiter and Saturn playcore with a 5000 AU would probably have survived more signicant role in the buildup of the Oort cloud thduring the residence time in an open cluster. The condition thought before? As shown, the answer may be positivefor survival in the natal molecular cloud is less stringent. particular, Saturn might have been a greater contributFor instance, if the Solar System remained there for 30

than Jupiter.myr, the total energy change would be (1/a) 5 10 6 4. Did Saturn and Jupiter place in the Oort cloudAU 1 (see Appendix), which is considerably smaller than signicant fraction of residual planetesimals from their owthe binding energy of the comets ( 10 4 AU 1). accretion zones? The answer to this question also depenWe should bear in mind that the scenario described in on how efcient the process of accretion of solid matterthis paper is only one among a wide range of possible these mostly gaseous planets was. Models of their interscenarios, of which the formation of the comet cloud structure show that Jupiter and Saturn may possess innaround an isolated Sun constitutes only an extreme case. cores composed of silicates and ices of about 15 M The Solar System could well have formed in a molecular (Hubbard 1989), so most of the accreted material was gcloud more or less dense than the one adopted here, as eous hydrogen and helium. The probable presence of ewell as in a more compact star cluster or in near isolation. tended gaseous envelopes might have increased the eWe have tried to describe average conditions derived from ciency of capture of planetesimals by these two planets,observations of star-forming regions. This discussion opens gas drag, fragmentation, and dissolution of bodies crossup new possibilities not foreseen until now. After nishing through their envelopes (Pollack et al. 1986). Stilthis manuscript, I received a preprint from Eggers et al. probable that a signicant fraction of the interacting bodi(1997) discussing the capture of intracluster comets by the were nally ejected by the powerful gravitational interearly Sun during a stage in which the Sun was assumed to tions of Jupiter and Saturn. If Jupiter and Saturn webe within an open cluster. able to place a large number of bodies from their o

accretion zones in a more tightly bound Oort cloud, t7. CONCLUDING REMARKS would imply a signicant mixing of bodies from diffe

parts of the planetary region, thus providing a physicaIn building a new scenario for the formation of a more heterogeneous population of Oort cloud comets, as boditightly bound comet cloud, several key questions arise re- formed closer to the Sun would tend to have differlated to the conditions of formation of the Oort cloud andproportionsof volatilesandrock (in a sense,Oorts originthe primitive galactic environment of the Solar System: idea that the asteroid belt was the source of comets minow be partially vindicated if bodies from the accret1. Did the Solar System form in a molecular cloud and/

or an open cluster? As seen before, observations tend to zone of Jupiter were stored in a very compact Oort cloud).Futureobservationsofthechemicalnatureofnewcometfavor this formation scenario as the most common one,

since molecular clouds are observed to be star factories, will be very relevant to determining their birthplaces in theplanetary region. If we could nd new comets that we and protostars and young stars usually appear in clumps

that lead to open clusters and associations. ableto show, fromtheir chemicalcomposition,wereformedin the JupiterSaturn region, then important aspects of t2. Did the buildup of the Oort cloud take place while

the Sun was still within the natal molecular cloud? As galactic environment that surrounded the early Solar System would be highlighted. Comets become, once more, imentioned, molecular clouds have an average lifetime of

a few tens of millions of years, while open clusters can portant probes in learning how the Solar System formed.


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APPENDIX: ENERGY CHANGE OF A COMET Biermann, L., and K. W. Michel 1978. On the origin of cometary nin the presolar nebula. Moon Planets 18, 447464.ASSUMING THE SUN TO BE WITHIN A

Blitz, L. 1993. Giant molecular clouds. In Protostars and PlanetMOLECULAR CLOUD(E. H. Levy and J. I. Lunine, Eds.), pp. 125162, Univ. of AriPress, Tucson.Let us assume the Sun to be within a molecular cloud of radius Rmc

and uniform density mc . For simplicity let us also assumethat themolecu- Brunini, A., and J. A. Ferna ndez 1996. Perturbations on an extelar cloud has a spherical shape and the Sun is at a distance s( Rmc ) from Kuiper disk caused by passing stars and giant molecular clouds. Athe center. A comet of semimajor axis a and eccentricity e will experience Astrophys. 308, 988994.a change d (1/ a)/ dt given by the Gauss equation

Byl, J. 1983. Galactic perturbations of nearly-parabolic cometary orbMoon Planets 29, 121137.Cameron, A. G. W. 1973. Accumulation processes in the primitive sd (1/ a)

dt 2a 3/ 2 1/ 2 R ae sin f

1 e2B

a2 1 e2r , (23) nebula. Icarus 18, 407450.

Delsemme, A. H. 1987. Galactic tides affect the Oort cloud: An obserwhere GM , R and B are the radial and tranverse components of tional conrmation. Astron. Astrophys. 187, 913918.the perturbing force, and f and r are the true anomaly and the heliocentric Duncan, M. J., H. F. Levison, and S. M. Budd 1995. The dynamdistance of the comet, respectively. The tidal force acting on the comet structure of the Kuiper belt. Astron. J. 110, 30733081.will be F 8 /3 G mc r cos (s / s), so we have R F cos and B F

Duncan, M., T. Quinn, and S. Tremaine 1987. The formation and exsin cos , where, as before (cf. Section 4), is the angle between Fof the Solar System comet cloud. Astron. J. 94, 13301338.and r , and is the angle between the plane formed by r and F and the

Eggers, S., H. U. Keller, P. Kroupa, and W. J. Markiewicz 1997. Ocomets orbital plane.and dynamics of comets and star formation. Planet. Space SFor a quasiparabolic orbit we have 1 e2 2q /a . If we averagepress.d (1/ a)/ dt over the comets orbital period P , the rst term on the right

side of Eq. (23) vanishes (the energychanges in theoutgoingandincoming Everhart, E. 1968. Change in total energy of comets passing throughleg of the orbit cancel out), so Eq. (23) reduces to Solar System. Astron. J. 73, 10391052.

Fernandez, J. A. 1978. Mass removed by the outer planets in the eSolar System. Icarus 34, 173181.d (1/ a)

dt P 2 2 1/ 2 q 1P P 0 Br dt . (24) Ferna ndez, J. A. 1980. Evolution of comet orbits under the perturbinuence of the Giant Planets and nearby stars. Icarus 42, 406

Making the appropriate substitutions in Eq. (24) we obtain Ferna ndez, J. A. 1981. New and evolved comets in the Solar Sy Astron. Astrophys. 96, 2635.

Fernandez, J. A. 1992. Comet showers. In Chaos, Resonance and Cod (1/ a)dt P 83 2 G 1/ 2q1/2 mc sin 2 cos . (25) tiv e Dynamical Phenomena in the Solar System (S. Ferraz-Mello

pp. 239254. Kluwer, Dordrecht.Ferna ndez, J. A., and T. Gallardo 1997. The origin ofcomets. In AstTaking averages sin 2 2/3 and cos 2 / , a density mc

Comets, Meteors 96. COSPAR, in press.2.5M pc

3

, and q 30 AU, we nally obtain for the energy change Fernandez, J. A., and W.-H. Ip 1981. Dynamical evolution of a come(1/ a) during T ( P)swarm in the outer planetary region. Icarus 47, 470479.

Ferna ndez, J. A., and W.-H. Ip 1983. On the time evolution of the c(1/ a) 5.0 10 6

T 30 myr

AU 1 . (26) etary inux in the region of the terrestrial planets. Icarus 54, 37Ferna ndez, J. A., and W.-H. Ip 1984. Some dynamical aspects o

accretion of Uranus and Neptune: The exchange of orbital anguACKNOWLEDGMENTSmomentum with planetesimals. Icarus 58, 109120.

Ferna ndez, J. A., and W.-H. Ip 1991. Statistical and evolutionary aspI thank Paul Weissman who, as a referee, made very useful commentsof cometary orbits. In Comets in the Post-Halley Era (R. L. Newand remarks that helped to improve the presentation of the results,Jr., et al., Eds.), pp. 487535. Kluwer, Dordrecht.and Soenke Eggers for pointing out an error in an earlier version of

the manuscript. Ferna ndez, J. A., and W.-H. Ip 1996. Orbital expansion and resontrapping during the late accretion stages of the outer planets.

Space Sci. 44, 431439.REFERENCESGaidos, E. J. 1995. Paleodynamics: Solar System formation and the e

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the formation of the oort cloud and the primitive galactic environment

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