brownian motion in planetary migration

8/14/2019 Brownian Motion in Planetary Migration

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arXiv:ast

ro-ph/0607203v1

10Jul2006

Accepted to ApJ: July 10, 2006Preprint typeset using LATEX style emulateapj v. 6/22/04

BROWNIAN MOTION IN PLANETARY MIGRATION

Ruth A. Murray-Clay1 & Eugene I. Chiang1,2

Accepted to ApJ: July 10, 2006

ABSTRACTA residual planetesimal disk of mass 10100M remained in the outer solar system following the

birth of the giant planets, as implied by the existence of the Oort cloud, coagulation requirementsfor Pluto, and inefficiencies in planet formation. Upon gravitationally scattering planetesimal debris,planets migrate. Orbital migration can lead to resonance capture, as evidenced here in the Kuiperand asteroid belts, and abroad in extra-solar systems. Finite sizes of planetesimals render migrationstochastic (noisy). At fixed disk mass, larger (fewer) planetesimals generate more noise. Extremenoise defeats resonance capture. We employ order-of-magnitude physics to construct an analytictheory for how a planets orbital semi-major axis fluctuates in response to random planetesimal scat-terings. The degree of stochasticity depends not only on the sizes of planetesimals, but also on theirorbital elements. We identify the conditions under which the planets migration is maximally noisy.To retain a body in resonance, the planets semi-major axis must not random walk a distance greaterthan the resonant libration width. We translate this criterion into an analytic formula for the reten-tion efficiency of the resonance as a function of system parameters, including planetesimal size. We

verify our results with tailored numerical simulations. Application of our theory reveals that captureof Resonant Kuiper belt objects by a migrating Neptune remains effective if the bulk of the primordialdisk was locked in bodies having sizes < O(100) km and if the fraction of disk mass in objects withsizes 1000 km was less than a few percent. Coagulation simulations produce a size distribution ofprimordial planetesimals that easily satisfies these constraints. We conclude that stochasticity did notinterfere with nor modify in any substantive way Neptunes ability to capture and retain ResonantKuiper belt objects during its migration.

Subject headings: celestial mechanicsKuiper beltdiffusionplanets and satellites: formationsolar system: formation

1. INTRODUCTION

Planet formation by coagulation of planetesimals isnot perfectly efficientit leaves behind a residual disk of

solids. Upon their coalescence, the outer planets of oursolar system were likely embedded in a 10100M disk ofrock and ice containing the precursors of the Oort cloud(Dones et al. 2004) and the Kuiper belt (see the reviewsby Chiang et al. 2006; Cruikshank et al. 2006; Levison etal. 2006). The gravitational back-reaction felt by planetsas they scatter and scour planetesimals causes the plan-ets to migrate (Fernandez & Ip 1984; Murray et al. 1998;Hahn & Malhotra 1999; Gomes, Morbidelli, & Levison2004). Neptune is thought to have migrated outwardand thereby trapped Kuiper belt objects (KBOs) intoits exterior mean-motion resonances, both of low-ordersuch as the 3:2 (Malhotra 1995) and of high-order suchas the 5:2 (Chiang et al. 2003; Hahn & Malhotra 2005).

Likewise, Jupiters inward migration may explain the ex-istence of Hilda asteroids in 2:3 resonance with the gas gi-ant (Franklin et al. 2004). A few pairs of extra-solar plan-ets, locked today in 2:1 resonance (Vogt et al. 2005; Lee etal. 2006), may have migrated to their current locationswithin parent disks composed of gas and/or planetesi-mals. Orbital migration and resonant trapping of dustgrains may also be required to explain non-axisymmetric

1 Center for Integrative Planetary Sciences, Astronomy Depart-ment, University of California at Berkeley, Berkeley, CA 94720,USA2 Alfred P. Sloan Research Fellow

Electronic address: [email protected], echiang@astron

structures observed in debris disks surrounding stars 10100 Myr old (e.g., Wyatt 2003; Meyer et al. 2006).

Only when orbital migration is sufficiently smooth and

slow can resonances trap bodies. The slowness crite-rion requires migration to be adiabatic: Over the timethe planet takes to migrate across the width of the res-onance, its resonant partner must complete at least afew librations. Otherwise the bodies speed past reso-nance (e.g., Dermott, Malhotra, & Murray 1988; Chiang2003; Quillen 2006). Smoothness requires that changesin the planets orbit which are incoherent over timescalesshorter than the libration time do not accumulate un-duly. Orbital migration driven by gravitational scatter-ing of discrete planetesimals is intrinsically not smooth.A longstanding concern has been whether Neptunes mi-gration was too noisy to permit resonance capture andretention (see, e.g., Morbidelli, Brown, & Levison 2003).

In N-body simulations of migration within planetesimaldisks (Hahn & Malhotra 1999; Gomes et al. 2004; Tsiga-nis et al. 2005), N O(104) is still too small to producethe large, order-unity capture efficiencies seemingly de-manded by the current census of Resonant KBOs.

At the same time, the impediment against resonancecapture introduced by inherent stochasticity has been ex-ploited to explain certain puzzling features of the Kuiperbelt, most notably the Classical (non-Resonant) beltsouter truncation radius, assumed to lie at a heliocentricdistance of 48 AU (Trujillo & Brown 2001; Levison &Morbidelli 2003). If Neptunes 2:1 resonance capturedKBOs and released them en route, Classical KBOs couldhave been transported (combed) outwards to popu-
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2 Murray-Clay & Chiang

late the space interior to the final position of the 2:1resonance, at a semi-major axis of 47.8 AU (Levison &Morbidelli 2003). As originally envisioned, this scenariorequires that 3M be trapped inside the 2:1 resonanceso that an attendant secular resonance suppresses growthof eccentricity during transport. It further requires thatthe degree of stochasticity be such that the migrationis neither too smooth nor too noisy. Whether these re-

quirements were actually met remain open questions.3Stochastic migration has also been studied in gas disks,

in which noise is driven by density fluctuations in tur-bulent gas. Laughlin, Steinacker, & Adams (2004) andNelson (2005) propose that stochasticity arising fromgas that is unstable to the magneto-rotational instabil-ity (MRI) can significantly prolong a planets survivaltime against accretion onto the parent star. The spec-trum of density fluctuations is computed by numericalsimulations of assumed turbulent gas.

In this work, we study stochastic changes to a planetsorbit due to planetesimal scatterings. The planetsBrownian motion arises from both Poisson variationsin the rate at which a planet encounters planetesimals,

and from random fluctuations in the mix of planet-planetesimal encounter geometries. How does the vigorof a planets random walk depend on the masses and or-bital properties of surrounding planetesimals? We an-swer this question in 2 by constructing an analytictheory for how a migrating planets semi-major axisfluctuates about its mean value. We employ order-of-magnitude physics, verifying our assertions whenever fea-sible by tailored numerical integrations. Because theproperties of planetesimal disks during the era of plane-tary migration are so uncertain, we consider a wide vari-ety of possibilities for how planetesimal semi-major axesand eccentricities are distributed. One of the fruits ofour labors will be identification of the conditions under

which a planets migration is maximally stochastic.Apportioning a fixed disk mass to fewer, larger plan-etesimals renders migration more noisy. How noisy is toonoisy for resonance capture? What limits can we placeon the sizes of planetesimals that would keep captureof Resonant KBOs by a migrating Neptune a viable hy-pothesis? These questions are answered in 3, where wewrite down a simple analytic formula for the retentionefficiency of a resonance as a function of disk proper-ties, including planetesimal size. Quantifying the sizespectrum of planetesimals is crucial for deciphering thehistory of planetary systems. Many scenarios for the evo-lution of the Kuiper belt implicitly assume that most ofthe mass of the primordial outer solar system was lockedin planetesimals having sizes of

O(100) km, like those ob-

served today (see, e.g., Chiang et al. 2006 for a critiqueof these scenarios). By contrast, coagulation simulationsplace the bulk of the mass in bodies having sizes of O(1)km (Kenyon & Luu 1999). For ice giant formation toproceed in situin a timely manner in the outer solar sys-

3 While Classical KBOs do have semi-major axes that extendup to 48 AU, the distribution of their perihelion distances cuts offsharply at distances closer to 45 AU (see, e.g, Figure 2 of Chiang etal. 2006). Interpreted naively (i.e., without statistics), the absenceof bodies having perihelion distances of 4548 AU and eccentricitiesless than 0.1 smacks of observational bias and motivates us to re-visit the problem of whether an edge actually exists, or at leastwhether the edge bears any relation to the 2:1 resonance.

tem, most of the primordial disk may have to reside insmall, sub-km bodies (Goldreich, Lithwick, & Sari 2004).

In 4, in addition to summarizing our findings, we ex-tend them in a few directions. The main thrust of thispaper is to analyze how numerous, small perturbationsto a planets orbit accumulate. We extend our analysisin 4 to quantify the circumstances under which a sin-gle kick to the planet from an extremely large planetesi-

mal can disrupt the resonance. We also examine pertur-bations exerted directly on Resonant KBOs by ambientplanetesimals.

2. STOCHASTIC MIGRATION: AN

ORDER-OF-MAGNITUDE THEORY

We assume the planets eccentricity is negligibly small.We decompose the rate of change of the planets semi-major axis, ap, into average and random components,

ap = ap,avg + ap,rnd . (1)

The average component (signal) arises from any globalasymmetry in the way a planet scatters planetesimals,e.g., an asymmetry due to systematic differences betweenplanetesimals inside and outside a planets orbit. Therandom component (noise) results from chance varia-tions in the numbers and orbital elements of planetesi-mals interacting with the planet. By definition, ap,rndtime-averages to zero. We assume that ap,avg(t) i s aknown function of time t, and devote all of 2 to thederivation of ap,rnd.

Each close encounter between the planet and a singleplanetesimal lasting time te causes the planets semi-major axis to change by ap. Expressions for ap andte depend on the planetesimals orbital elements. Wedefine x a ap as the difference between the semi-major axes of the planetesimal and of the planet, b > 0as the impact parameter of the encounter, and u

ea

as the planetesimals random (epicyclic) velocity, wherea, e, and are the semi-major axis, eccentricity, andmean angular velocity of the planetesimal, respectively.We assume that |x| ap. Encounters unfold differentlyaccording to how |x| and b compare with the planets Hillradius,

RH = ap

Mp

3M

1/3, (2)

and according to how u compares with the Hill velocity,

vH pRH . (3)Here Mp and M are the masses of the planet and ofthe star, respectively, and p is the angular velocity of

the planet. See Table A1 for a listing of frequently usedsymbols.

In 2.1, we calculate ap for a single encounter witha planetesimal having |x| RH. In 2.2, we repeat thecalculation for |x| RH. In 2.3, we provide formulaefor the root-mean-squared (RMS) random velocity dueto cumulative encounters, a2p,rnd1/2, and identify whichcases of those treated in 2.12.2 potentially yield thestrongest degree of stochasticity in the planets migra-tion.


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Stochastic Migration 3

2.1. Single Encounters with |x| RH: Non-HorseshoesWe calculate the change in the planets semi-major

axis, ap, resulting from an encounter with a single plan-etesimal having |x| RH. We treat planetesimals onorbits that do not cross that of the planet in 2.1.1 andthose that do cross in 2.1.2. Throughout, refers to thechange in a quantity over a single encounter, evaluated

between times well before and well after the encounter.2.1.1. Non-Crossing Orbits

Planetesimals on orbits that do not cross that of theplanet have

|x| > ae , (4)which corresponds to

|x|/RH > u/vH . (5)Our plan is to relate ap to x by conservation of en-ergy, calculate e using the impulse approximation, andfinally generate x from e by conservation of the Ja-cobi integral.

By conservation of energy,

GMMp

2ap GMm

2a

= 0 , (6)

where m Mp is the mass of the planetesimal. We havedropped terms that account for the potential energies ofthe planet and of the planetesimal in the gravitationalfield of the ambient disk. These are small because thedisk mass is of order Mp M and because the diskdoes not act as a point mass but is spatially distributed.Equation (6) implies

ap mMp

apa

2a . (7)

Since |ap| |a| and ap a, we have x a andap m

Mpx . (8)

The impulse imparted by the planet changes the ec-centricity of the planetesimal by e. An encounter forwhich |x| is more than a few times RH imparts an impulseper mass4

u GMpb2

te . (9)

The impact parameter b is limited by

|x| ae b |x| + ae . (10)Since ae |(x)|.

order-of-magnitude results. The change in the eccentric-ity of the planetesimal is hence

e upap

MpM

apx

2. (13)

When |e| < (the pre-encounter) e, the change e canbe either positive or negative, depending on the true

anomaly of the planetesimal at the time of encounter.If |e| > e, then e > 0. When |x| RH, |e| attainsits maximum value of (Mp/M)1/3; i.e., u vH.5

To calculate the corresponding change in the planetes-imals semi-major axis, x, we exploit conservation ofthe Jacobi integral, CJ. That a conserved integral ex-ists relies on the assumption that in the frame rotatingwith the planet, the potential (having a centrifugal termplus gravitational contributions due to the star, planet,and disk) is time-stationary; the Jacobi integral is simplythe energy of the planetesimal (test particle) evaluatedin that frame. To the same approximation embodied inEquation (6),

1

2 CJ = E pJ= GM

ap

1

2(a/ap)+

(a/ap)(1 e2)

(14)

far from encounter, where E and J are the energy andangular momentum per mass of the planetesimal, respec-tively. Taking the differential of (14) yields, to leadingorder,

3

4

x

ap

2+

3

2

x

ap 1

2e2

x

ap (e2) = 0 . (15)

Since |x|/ap > e > e2 (non-crossing condition) and(e2) < (x/ap)2 (by Equation [13] and the condition

|x| > RH), Equation (15) reduces to

x 2a2p

3x(e2) . (16)

We combine Equations (8) and (16) to find

ap mMp

a2px

(e2) . (17)

Equation (17) takes two forms depending on how |e|compares with (the pre-encounter) e. If

|x| > RH vHu

1/2

, (18)

then |e| < e, (e2) 2ee, and Equation (17) be-comes

ap mM

a4px3

e . (19)

5 When |x| 2RH, the encounter pulls the planetesimal into theplanets Hill sphere. The planetesimal accelerates in a complicatedway and exits the Hill sphere in a random direction with u oforder the planets escape velocity at the Hill radius, vH (Petit &Henon 1986). The planetesimals eccentricity is boosted by e (Mp/M)1/3. The encounter time is typically the time required tocomplete a few orbits around the planet, te 2/p. Since teand e match, to order of magnitude, Equations (12) and (13)for |x| RH, we do not treat RH |x| 2RH as an explicitlydifferent case.


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The right-hand side is extremized for |x| RH(vH/u)1/2:

max |ap| mMp

RH

ape

RH

5/2 e and (e2) (e)2, and Equation (17)becomes

ap mMpM2

a6px5

. (22)

Equation (22) agrees with the more careful solution ofHills problem by Henon & Petit (1986). The right-handside is extremized for |x| RH:

max |ap| mMp

RH . (23)

In summary, if x/RH > u/vH, then (a) the planetesi-mals orbit does not cross (nc) that of the planet, (b)

te 1p

, (24)

and (c)

ap =

ap,nc1 mMpM2

a6px5

,

if RH x (vH/u)1/2RH;

ap,nc2 mM

a4px3

e,

if x (vH/u)1/2RH.

(25)

2.1.2. Crossing Orbits

Encounters with planetesimals on orbits that cross thatof the planet, i.e., those with

|x|/RH < u/vH , (26)differ from encounters with non-crossing planetesimalsin two key respects. First, the relative velocity of thetwo bodies is dominated by the planetesimals random(epicyclic) velocity rather than the Keplerian shear. Sec-ond, the planetesimals impact parameter, b, may differsignificantly from |x|. The impact parameter may takeany value bmin < b ae , (27)

where bmin is the impact parameter below which the plan-etesimal collides with the planet. Because crossing orbitsallow for encounters with many different geometries, out-comes of these encounters can vary dramatically. Here werestrict ourselves to estimating the maximum |ap| thatcan result from an orbit-crossing encounter. In 2.3.2,we argue this restriction is sufficient for our purposes.

When u > vH, the eccentricity of the planetesimalcan change by at most |e| e. Such a change corre-sponds to an order-unity rotation of the direction of theplanetesimals random velocity vector, and requires that

b GMp/u2. The change in the planetesimals specific

energy over the encounter is approximately

GM

2a

1

2v2

+

GM

r

, (28)

where v is the velocity of the planetesimal relative tothe star (in an inertial frame of reference) and r is thedistance between the planetesimal and the star. Now rfor an encounter with b GMp/u2 is at most GMp/u2 2a

GMpu2

, (29)

the second term on the right-hand side of Equation (28)is negligible compared to the first, and the maximum|a| over an encounter is

max |a| a2

GMau ae . (30)

By Equation (7) and a ap,

max |ap| m

Mp ape . (31)

We have verified Equation (31) by numerical orbit in-tegrations. We could also have arrived at Equation (31)through Equation (15), which yields |x| ape for cross-ing orbits when |e| e.

2.2. Single Encounters with |x| RH: HorseshoesWhen |x| < RH, planetesimals can occupy horseshoe

orbits. A planetesimal on a horseshoe orbit for which|x| RH encounters the planet on a timescale somewhatshorter than the orbital period; by the impulse approxi-mation, such a planetesimal kicks the planet such that

ap mMa3

pb2 , (32)

where we have momentarily restricted consideration toplanetesimals having sub-Hill eccentricities (e RH/ap).From Henon and Petit (1986),

b =8

3

R3Hx2

, (33)

valid for x not too far below RH. Then

|ap| mMp

x4

R3H. (34)

The kick is maximal for maximum |x| = RH:max |ap| m

MpRH . (35)

This is the same maximum as was derived for the |x| RH, non-crossing case; see Equation (23). Thus, a co-orbital ring of planetesimals on horseshoe orbits withsub-Hill eccentricities increases the stochasticity gener-ated by planetesimals on non-horseshoe, non-crossing or-bits by a factor of at most order unity (under the assump-tion that disk properties are roughly constant within sev-eral Hill radii of the planet). For this reason, and also be-cause the horseshoe region may well have been depletedof planetesimals compared to the rest of the disk, weomit consideration of co-orbital, sub-Hill planetesimals


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for the remainder of the paper, confident that the errorso incurred will be at most order unity.

What about planetesimals on horseshoe orbits withsuper-Hill eccentricities (e RH/ap)? Upon encounter-ing the planet, such objects can have their semi-majoraxes changed by |a| > RHwhereupon they are ex-pelled from the 1:1 horseshoe resonance. Because highlyeccentric, horseshoe resonators are unstable, we neglect

consideration of them for the rest of our study.

2.3. Multiple Encounters: Cumulative Stochasticity

We now extend our analysis from individual encoun-ters to the cumulative stochasticity generated by a diskwith surface density m in planetesimals of a single massm. Note that m need not equal the total surface den-sity (integrated over all possible masses m). We willconsider size distributions in 3.6. We consider plan-etesimals with sub-Hill (u < vH) velocities (2.3.1) sep-arately from those with super-Hill (u > vH) velocities(2.3.2). Sub-Hill (non-horseshoe) planetesimals alwaysoccupy non-crossing orbits. Super-Hill planetesimals canbe crossing or non-crossing.

2.3.1. Sub-Hill Velocities (u < vH)

Consider planetesimals with sub-Hill velocities locateda radial distance x away from the planet (|x| > RH).Since u < vH, the speeds of planetesimals relative tothe planet are determined principally by Keplerian shear(Equation [12]), and the scale height of the planetesi-mals is less than RH. The planet encounters (undergoesconjunctions with) such planetesimals at a mean rate

N mm

x2 , (36)

as is appropriate for encounters in a two-dimensional ge-ometry. Over a time interval t, the planet encounters

N = Nt such planetesimals on average. Systematictrends in N with xsay, systematically more objectsencountered interior to the planets orbit than exteriorto itcause the planet to migrate along an average tra-

jectory with velocity ap,avg.Random fluctuations in (a) the number of planetesi-

mals encountered per fixed time interval and (b) the mixof planetesimals pre-encounter orbital elements causethe planet to random walk about this average trajectory.Contribution (a) is straightforward to model. The prob-ability that the planet encounters N objects located adistance x away in time t is given by Poisson statistics:

P(N) =

NN

N! eN

. (37)The variance in N is

2N

(N N)2 = N . (38)Fluctuations in N drive the planet either towards or awayfrom the star with equal probability and with typicalspeed

a2p,rnd1/2 |ap|

tN

1/2, (39)

hereafter the root-mean-squared (RMS) speed. While

a2p,rnd1/2 1/

t, the distance random walked

a2p,rnd1/2t

t.

Our assumption of Poisson statistics is reasonable. Inthe sub-Hill case, a planet-planetesimal encounter re-quires a time te 1/p (Equation [12]) to complete.Encounters separated by more than te are uncorrelatedwith one another, at least until the planet completes onerevolution with respect to the surrounding disk, i.e., atleast until a synodic time tsyn 4ap/(3p|x|) elapses.After a synodic period, it is possible, in principle, for the

planet to essentially repeat the same sequence of encoun-ters that it underwent during the last synodic period. Weassume in this paper that this does not happenthatthe orbits of planetesimals interacting with the planetare randomized on a timescale trdz < tsyn. We ex-pect this inequality to be enforced by a combination of(i) randomization of planetesimal orbits due to encoun-ters with the planet (e.g., encounters within the chaoticzone of the planet [Wisdom 1980]), (ii) phase mixing ofplanetesimals due to Keplerian shear (which occurs ontimescale tsyn for planetesimals distributed between xand 2x), (iii) gravitational interactions between plan-etesimals, and (iv) physical collisions between planetesi-mals. As long as trdz < tsyn, we are free to choose t to

be anything longer than te.6Contribution (b) is difficult to model precisely since

we do not know how orbital elements of planetesimalsare distributed. These distributions are unlikely to begoverned by simple Poisson or Gaussian statistics (see,e.g., Ida & Makino 1992; Rafikov 2003; Collins & Sari2006). Nonetheless, neglecting contribution (b) will notlead to serious error. Suppose the planetesimals pre-encounter elements are distributed such that the frac-tional variation in each element is at most of order unity(e.g., the planetesimal eccentricities span a range frome/2 to 2e at most). Then the central limit theorem en-sures that the noise introduced by random sampling oforbital elements is at most comparable to the noise in-

troduced by random fluctuations in the encounter rate.Consider, for example, noise that arises from randomsampling ofe in the case where ap = ap,nc2 (Equation

[25]). For an encounter rate fixed at N, the planets semi-major axis ap changes over time interval t by Nap,where ap is the mean of N = Nt sampled values ofap. If the dispersion in e for individual planetesimals

is e and the mean eccentricity sampled over N valuesis e, then the dispersion in the sampled mean eccentric-

ity is e e/N1/2 by the central limit theorem. Forap = ap,nc2 e, the dispersion in ap is |ap|e/e.The planets RMS speed generated purely from randomsampling of e is

a2p,rnd1/2 N

t|ap| e

e |ap|

tN

1/2 ee

, (40)

which is at most comparable to the RMS speed gener-ated purely from random sampling ofN (Equation [39]),

6 If trdz > tsyn , then t > trdz and the right-hand side of

Equation (39) is multiplied by

trdz/tsyn . The planets motionis more stochastic in this case because over trdz, correlated inter-actions with planetesimals do not cancel each other as much asuncorrelated interactions would. Later, since we will be interestedin stochastic perturbations to mean-motion resonant particles, wewill require trdz < tlib , where tlib is the libration period withinresonance.


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as desired. Our original supposition that max(e/e) 1leads to no important loss of generality; if the distri-bution in e were bi-modal, for example, we could treateach population separately and add the resultant RMSspeeds in quadrature. Similar results obtain for randomsampling of other elements such as x. For simplicity,we hereafter treat explicitly only fluctuations in the en-counter rate [(Equation (39)], knowing that the noise so

calculated will be underestimated by a factor of at mostorder unity.

Expression (39) measures the contribution to the RMSspeed from planetesimals located a distance x away.Since ap scales inversely with x to a steep power in thesub-Hill regime (see [25]), the contribution to the RMSspeed is greatest from objects at small x (for reasonablevariations of m with x). We take disk material to ex-tend to a minimum distance of |xmin| RRH (R > 1)from the planets orbit. Insertion of (25) into (39) yields

a2p,rnd

1/2

R4 1(pt)1/2

ma

2pm

M2p

1/2RHap

vH,

if 1 < R < (vH/u)1/2

;

R2 1(pt)1/2

ma2pm

M2p

1/2evH,

if R > (vH/u)1/2.(41)

The RMS speed is maximized for R = 1.2.3.2. Super-Hill Velocities (u > vH)

Next we consider the noise generated by planetesimalswith super-Hill random velocities (u > vH). We refer toclose encounters that change ap by the maximal amount,max

|ap

| (m/Mp)ape (Equation [31]), as maximal

encounters. Maximal encounters, which occur at impactparameters b GMp/u

2, make an order-unity contribu-tion to the total super-Hill stochasticity. Non-maximal(more distant) encounters contribute to the total stochas-ticity through a Coulomb-like logarithm, as we show atthe end of this sub-section.

For a maximal encounter, a planetesimal must ap-proach within distance b GMp/u2. Such encountersoccur at a mean rate

N n

GMpu2

2u m

mpR

2H

vHu

4, (42)

where n m/(mu) is the number density of plan-etesimals, and we have assumed that planetesimal incli-nations and eccentricities are of the same order. Overa time interval t, the planet encounters on average

Nt such planetesimals, each of which increases or de-creases ap by about max |ap|. Since planetesimals suf-fering maximal encounters have their orbits effectivelyrandomized relative to each other, we may choose tto be any time interval longer than an encounter timete b/u < 1/p (see related discussion in 2.3.1).Therefore the planet random walks with RMS velocity(averaged over time t)

a2p,rnd

1/2 (Nt)

1/2 max |ap|t

1

pt

1/2 ma2pmM2p

1/2vHu

RHap

vH . (43)

What about the contribution from non-maximal en-counters? For a super-Hill encounter at impact parame-ter b, the specific impulse imparted to the planetesimal isGMp/(bu). We suppose that |ap| is proportional tothis specific impulse, so that |ap| 1/b. We have con-firmed this last proportionality by numerical orbit inte-grations (not shown). Since

a2p,rnd

1/2 (N)1/2|ap|

and N b2, we have

a2p,rnd

1/2 b0, which implies

that each octave in impact parameter contributes equallyto the total stochasticity. In other words, our estimate

for

a2p,rnd

1/2in Equation (43) should be enhanced by

a logarithmic factor of ln(bmax/bmin), where bmax andbmin GMp/u2 are maximum and minimum impact pa-rameters. We estimate bmax u/, the value for whichthe relative velocity of a super-Hill encounter is domi-nated by the planetesimals random velocity rather thanby the background shear. The logarithm is not large; forexample, for e = 0.2, ln[(u/)/(GMp/u2)] 5.

2.3.3. Summary

We can neatly summarize Equations (41) and (43) bydefining the Hill eccentricity,

eH RH/ap , (44)and parameterizing m such that the disk contains massMMp in planetesimals of mass m spread uniformly fromad/2 to 3ad/2:

m =MMp2a2d

, (45)

where M is a dimensionless number of order unity. Then

a2p,rnd

1/2

CR4 (pt)1/2Mm

Mp

1/2apad

eH vH,

if e < eH/R2;

CR2 (pt)1/2Mm

Mp

1/2apad

e vH,

if eH/R2 < e < ReH;

C (pt)1/2 MmMp

1/2apad

e2He

vH,

if e > ReH,(46)

where we have introduced a constant coefficient C (thesame for each case so that the function remains continu-ous across case boundaries). The coefficient C encapsu-lates all the factors of order unity that we have droppedin our derivations. By studying N-body simulation datapertaining to the case e > ReH as recorded in the lit-erature (Hahn & Malhotra 1999; Gomes et al. 2004),we estimate that C is possibly of the order of several.That C > 1 (but not 1) is consonant with our hav-ing consistently underestimated the noise by neglecting(a) distant, non-maximal encounters in the super-Hill


7/18


regime (2.3.2), and (b) stochasticity introduced by ran-dom sampling of orbital elements of planetesimals en-countering the planet (2.3.1). We defer definitive cali-bration ofC to future study, but retain the coefficient inour expressions below to assess the degree to which ourquantitative estimates are uncertain.

Planetesimals with e = ReH (|xmin| = ape) producethe maximum possible stochasticity:

max

a2p,rnd1/2 CR (pt)1/2

MmMp

1/2apad

eH vH,

for e = ReH eH . (47)We will often adopt this case for illustration purposesbelow.

Since

a2p,rnd

1/2 (Mm)1/2, the stochasticity driven

by planetesimals having a range of sizes is dominated bythose objects (in some logarithmic size bin) having max-imal mm. For common power-law size distributions,such objects will occupy the upper end of the distribu-tion. We will explicitly consider various possible sizedistributions in

3.6.

3. APPLICATION: MAXIMUM PLANETESIMAL SIZES

Neptune is thought to have migrated outward by scat-tering planetesimals during the late stages of planet for-mation (Fernandez & Ip 1984; Hahn & Malhotra 1999).As the planet migrated, it may have captured Kuiper beltobjects into its exterior resonances (Malhotra 1995), giv-ing rise to the Resonant KBOs observed today (Chianget al. 2003; Hahn & Malhotra 2005). If Neptunes mi-gration had been too stochastic, however, resonance cap-ture could not have occurred. A planetesimal disk havingfixed surface mass density generates more stochasticitywhen composed of larger (fewer) planetesimals. There-fore, assuming that planetary migration and concomitant

resonance capture correctly explain the origin of present-day Resonant KBOs, we can rule out size distributionsthat are too top-heavy during the era of migration.

Stochasticity causes a planet to migrate both outwardand inward. In 3.1, we provide background informationregarding how resonance capture and retention dependon the sign of migration. In 3.2, we lay out general con-siderations for whether a stochastically migrating planetcan retain particles in resonance. In 3.3, we derive andevaluate analytic, order-of-magnitude expressions for themaximum planetesimal size compatible with resonanceretention, in the simple case when all planetesimals havethe same size. In 3.4, we provide an analytic formulathat details precisely how the resonance retention effi-

ciency varies with average migration speed and planetes-imal size. These analytic results are quantitatively testedby numerical integrations in 3.5. Cases where planetes-imals exhibit a wide range of sizes are examined in 3.6.

3.1. Migrating Outward and Inward

As a planet migrates smoothly outward (away from theparent star), it can capture planetesimals into its exteriormean-motion resonances. By contrast, a planet whichmigrates smoothly inward cannot capture planetesimalswhich are initially non-resonant into its exterior reso-nances (e.g., Peale 1986). But a planetesimal that startsin exterior resonance with an inwardly migrating planetcan remain in resonance for a finite time.

0.00

0.04

0.08

e

not in resonancein resonance

30.2

30.4

30.6

a(AU)

amin

not in resonancein resonance

0.0 0.2 0.4 0.6 0.8Time (Myr)

21.0

22.0

23.0

ap

(AU)

Fig. 1. Evolution of a planetesimal in external 3:2 resonance asthe planet migrates smoothly inward. The planetesimal remains inresonance until its eccentricity reaches zero, at a semi-major axisof a = amin.

Goldreich (1965) demonstrates how an outwardly mi-grating body adds angular momentum to a test particlein exterior resonance at just the right rate to keep theparticle in resonance. Reversing the signs in his proof im-plies that an inwardly migrating body removes angular

momentum from a particle in exterior resonance, pullingit inward while preserving the resonant lock. A planetes-imals eccentricity decreases as it is pulled inward. Theadiabatic invariant,

N=

GMa(p

1 e2 q) , (48)which is preserved for migration timescales long com-pared to the synodic time (e.g., Murray-Clay & Chiang2005), implies that a planetesimal in p : q exterior reso-nance (p > q) cannot be pulled inward to a semi-majoraxis less than

amin =1

GM N

p

q

2

= a0 p

1 e20 qp

q

2

,

(49)the value for which e = 0. Here a0 and e0 are the ini-tial semi-major axis and eccentricity of the planetesimal,respectively.

Thus an exterior particle follows an inwardly migratingplanet in resonant lockstep until it either reaches zeroeccentricity (view Figure 4 of Peale 1986 or Figure 8.22of Murray & Dermott 1999 in reverse) or until it crossesthe separatrix (view Figure 5 of Peale 1986 or Figure 8.23of Murray & Dermott 1999 in reverse), whichever comesfirst. We illustrate the former possibility in Figure 1 andthe latter possibility in Figure 2, using our own orbitintegrations. The value ofamin is annotated for reference.


8/18


9/18


for the maximum planetesimal sizes compatible with res-onance retention, for the simple case when the disk iscomposed of ob jects of a single size. The assumption ofa single size is relaxed in 3.6.

Say the planet takes time T to migrate at speed ap,avgfrom its initial to its final semi-major axis. Over thistime, ap random walks an expected distance of

ap,T a2p,rnd1/2 T

CR4Mm

Mp

1/2apad

eH vH

T

p

1/2,

if e < eH/R2;

CR2Mm

Mp

1/2apad

e vH

T

p

1/2,

if eH/R2 < e < ReH;

C MmMp

1/2 apad

e2He

vH T

p1/2

,

if e > ReH,

(51)

where we have set t = T in evaluating

a2p,rnd

1/2.

Ifap,T < ap,lib/2, the planet can keep a large fractionof planetesimals in resonance. That is, most particlesare retained in resonance when the disk mass comprisesplanetesimals of mass

m mcrit

R8C2libC2M

adap

21

pT

ereseH

Mp,

if e < eH/R2;

R4C2lib

C2M adap2

1

pT

ereseH

e2 Mp,

if eH/R2 < e < ReH;

C2libC2M

adap

21

pT

erese2

e3HMp,

if e > ReH.(52)

Equation (52) can be equivalently interpreted as an up-per limit on T for planetesimals of given mass m. For afixed degree of noise, resonant objects are more difficultto retain if the average migration is slow.

We evaluate (52) to estimate the maximum planetes-

imal radius, s = (3m/4)1/3

, compatible with reso-nant capture of KBOs by Neptune. For an internaldensity = 2 g/cm3, Mp = MN = 17M, eH = 0.03,ap = ad = 26.6 AU, eres = 0.25, M = 2 (so thatm = 0.2 g c m

2), and T = 3 107 yr, resonant cap-ture and retention require

s

km

scritkm

700R8/3C2/3, if e < eH/R2;70e2/3R4/3C2/3, if eH/R2 < e < ReH;7000e2/3C2/3, if e > ReH.

(53)For example, ife = 0.1 and R = 1, then line 3 of (53) ob-tains and scrit = 1500C2/3 km. Maximum stochasticity

results when R = 1 and e eH (see also Equation [47]);either of lines 1 or 2 then yield scrit 700C2/3 km.These size estimates decrease by about 20% when cor-rected to reflect the fact that the width of the 3:2 res-onance is somewhat smaller than the pendulum modelimplies (see the discussion following Equation [50]).

3.4. Analytical Formula for the Retention Fraction

As defined in 3.2, Pkeep is the resonance retention frac-tion, or the probability that a typical resonant particleis retained in resonance over some duration of migration.We calculate Pkeep by modelling the random componentof the planets migration as a diffusive continuum pro-cess. In the limit that the planet encounters a largenumber N 1 of planetesimals, the Poisson distribu-tion (Equation [37]) is well-approximated by a Gaussiandistribution with mean N and variance N. Thus, over

a time interval t N1

, the random displacement ofthe planet, Srnd = ap(N N), has the probabilitydensity distribution

f(Srnd, t) =1

2Dt exp((Srnd)2

/(2Dt)) ,

(54)

where D = (ap)2N is the diffusion coefficient and werecall that ap is the change in ap due to an encounterwith a single planetesimal. The evolution ofap,rnd with tis continuous and the distribution f is independent overany two non-overlapping intervals t (the random walkhas no memory). In other words, Srnd evolves as aWiener process, or equivalently according to the rules ofBrownian motion (e.g., Grimmett & Stirzaker 2001a).From Equation (54), it follows that over time T, theprobability that |Srnd| does not exceed ap,lib/2 equals

Pkeep =

n=1

4

n sin 3n

2

e

n

T , (55)

where n = (n)2D/(2a2p,lib) (see Appendix A for a

derivation).Suppose migration occurs in a disk of planetesimals

having a single size s. Figure 3 displays Pkeep as a func-tion ofs and of exponential migration timescale definedaccording to

ap,avg(t) = ap,f (ap,f ap,i)et/ , (56)where ap,i and ap,f are the planets initial and final aver-age semi-major axes, respectively. In Equation (55), wetake T = 2.6, and evaluate remaining quantities for thecase of maximum stochasticity: e

eH and

R= 1. Then

ap = ap,nc1 (Equation [25]) and N = 2mR2H/m

(Equation [36], with a factor of 2 inserted to account fordisk material both inside and outside the planets orbit).As in 3.2, we take = 2 g/cm3, Mp = MN = 17M,eH = 0.03, ap = ad = 26.6 AU, RH = eHap, and M = 2.To evaluate ap,lib, we take eres = 0.25 for a particle in3:2 resonance. Figure 3 describes how for a given sizes, the retention fraction decreases with increasing ; thelonger the duration of migration, the more chance a par-ticle has of being jostled out of resonance. For = 10Myr, planetesimals must have sizes s 500km for theretention fraction to remain greater than 1/2. These re-sults confirm and refine our order-of-magnitude estimates


10/18


made in 3.3. Similar results were obtained for the 2:1resonance.

The continuum limit is valid as long as the expectationvalue of the time required for a resonant particle to es-

cape, tescape N1

[(ap,lib/2)/ap]2

, greatly exceedsthe time for the planet to encounter one planetesimal,

N1

. This criterion is satisfied for the full range of pa-

rameters adopted in Figure 3.

3.5. Numerical Results for the Retention Fraction

To explore how a stochastically migrating planet cap-tures and retains test particles into its exterior reso-nances, and to test the analytic considerations of 3.23.4, we perform a series of numerical integrations. Wefocus as before on the 3:2 (Plutino) resonance with Nep-tune.

Following Murray-Clay & Chiang (2005, hereafterMC05), we employ a series expansion for the time-dependent Hamiltonian,

H=

(GM)2

2(3 +N)2 GM

ap(t)31/2 (2 +N)

GMpap(t)

(f1 + f2e

2 + f31e cos )

, (57)

where = ap/a 0.76, the fis are given in Murray &Dermott (1999), and N (Equation [48]) is a constant ofthe motion determined by initial conditions. The reso-nance angle,

= 3res 2p res , (58)is defined by the mean longitude res and longitude ofperiastron res of the resonant particle, and the meanlongitude p of the planet. The resonance angle libratesabout for particles in resonance. The momentum con-

jugate to is . We integrate the equations of motion,

=H

, = H

, (59)

using the Bulirsch-Stoer algorithm (Press et al. 1992) forfixed and fis.

The Hamiltonian in Equation (57) faithfully repro-duces the main features of the resonance potential; seeBeauge (1994, his Figures 12a and 12c) for a direct com-parison between such a truncated Hamiltonian and theexact Hamiltonian, averaged over the synodic period (seealso that paper and Murray-Clay & Chiang 2005 for adiscussion of the pitfalls of keeping one too many a termin the expansion). Of course, even the exact Hamilto-

nian, because it is time-averaged and neglects chaoticzones, is inaccurate with regards to details such as thelibration width, but these inaccuracies are slight; see thediscussions following Equations (50) and (53).

To compute ap(t), we specify separately the averageand random components of the migration velocity, ap,avgand ap,rnd. For ap,avg, we adopt the prescription (equiv-alent to Equation [56])

ap,avg =1

(ap,f ap,i)et/ , (60)

where ap,i and ap,f are the planets initial and final av-erage semi-major axes, respectively, and is a time con-stant. To compute ap,rnd, we divide the integration into

time intervals of length 1/p. The only requirement forthe time interval is that it be less than the libration pe-riod tlib 400/p (see 3.2). Over each interval, werandomly generate

ap,rnd = pap(N N1p ) . (61)We focus on the case of maximum stochasticity, so that

ap = ap,nc1 (Equation [25]) and N = 2mR2Hp/m(Equation [36] with R = 1 and an extra factor of 2 in-serted to account for disk material on both sides of theplanets orbit). We assume that the entirety of the diskmass is in planetesimals of a single mass m. Each Nis a random deviate drawn from a Poisson distributionhaving mean N1p .

Figure 4 displays the sample evolution of a test particledriven into 3:2 resonance by a stochastically migratingplanet. For this integration, Mp = MN, ap,i = 23.1AU,ap,f = 30.1AU, = 107 yr, R = 1, M = 2 (so thatm = 0.2 g c m

2), and s = 150km (m = 3 1022 g).This choice for s is sufficiently small that the particle issuccessfully captured and retained by the planet.

Contrast Figure 4 with Figure 5, in which all modelparameters are the same except for a larger s = 700 km.In this case the planet eventually loses the test particlebecause the migration is too noisy.

Figure 6 displays the fraction of particles caught andkept in resonance as a function of and s. For each datapoint in Figure 6, we follow the evolution of 200 parti-cles initialized with eccentricities of approximately 0.01and semi-major axes that lie outside the initial positionof resonance by about 1 AU.8 Figure 6 is the numericalcounterpart of Figure 3; the agreement between the twois excellent and validates our analytic considerations. If = 107 yr (consistent with findings by MC05), then thecapture fraction rises above 0.5 for s 500km. Since

these results pertain to the case {R = 1, e eH} whichyields the largest amount of noise for given m and s,we conclude that s 500 km is the lowest, and thus themost conservative, estimate we can make for the maxi-mum planetesimal size compatible with resonant captureof KBOs by a migrating Neptune, assuming that the en-tire disk is composed of planetesimals of a single size(this assumption is relaxed in the next section). In otherwords, if Neptunes migration were driven by planetesi-mals all having s 500 km, stochasticity would not haveimpeded the trapping of Resonant KBOs. Of course,our numerical estimate of 500 km is uncertain insofar aswe have not kept track of order-unity constants in ourderivations. We suspect a more careful analysis will re-

vise our size estimate downwards by a factor of a few (seethe discussion of C in 2.3.3).3.6. Planetesimal Size Distributions

Actual disks comprise planetesimals with a range ofsizes. From Equation (46), the stochasticity in theplanets migration is dominated by those planetesimalshaving maximal mm. What was the distribution ofsizes during the era of Neptunes migration? A possi-ble answer is provided by the coagulation simulations

8 The particles do not all have the same initial eccentricities andsemi-major axes. This is because they occupy the same Hamilto-nian level curve; see section 3.5 of MC05.


11/18


0 2 4 6 8 10Timescale of Migration (Myr)

0.0

0.2

0.4

0.6

0.8

1.0

A

nalyticRetentionFra

ctionPkeep

200km400km

500km

600km900km

1600km

Fig. 3. Fraction of particles retained in external 3:2 resonance by a stochastically migrating planet as a function of migration timescale,calculated according to Equation (55). The entire disk mass is assumed to be in planetesimals of a single size s, and a range of choices fors are shown. The diffusivity D is evaluated at its maximum value, appropriate for the case e eH and R = 1. We set ap = ap,nc1(Equation [25]), N = 2mR2H/m, (Equation [36]), = 2 g/cm

3, Mp = MN = 17M, eH = 0.03, ap = ad = 26.6 AU, RH = eHap,M = 2, eres = 0.25, and T = 2.6. Planetesimals having sizes smaller than 200 km produce so little noise in the planets migration thatno object is lost from the 3:2 resonance. Compare this Figure with its numerical counterpart, Figure 6. In calculating Pkeep , we assume

C = 1; probably C is of order several, in which case the sizes indicated in the Figure should be revised downward by a factor of a few (C2/3;see Equation [53]).

of Kenyon & Luu (1999, hereafter KL99). The left-hand panel of their Figure 8 portrays the evolution ofthe size distribution, starting with a disk of seed bodieshaving sizes up to 100 m and a total surface density of = 0.2 g c m2. After t = 11 Myr, the size bin for whichmm is maximal is centered at s 4 km; for this bin atthat time, m = 10

3 g cm2 (evaluated within a loga-rithmic size interval 0.3 dex wide). After t = 37 Myr,the planetesimals generating the most stochasticity haves 750 km and m = 2 103 g cm2. Note that att = 37 Myr, the stochasticity is dominated by the largestplanetesimals formed, but they do not contain the bulkof the total disk mass; the lions share of the mass isinstead sequestered into km-sized objects.

In Figure 7, we plot the resonance retention fractionPkeep (Equation [55]) for the KL99 size distribution att = 11 and 37 Myr, using the values of s(m) and mcited above. The remaining parameters that enter intoPkeep are chosen to be the same as those employed forFigure 3; i.e., we adopt the case of maximum stochastic-ity. Evidently, Pkeep = 1 for the KL99 size distributions;stochasticity is negligible.

For comparison, we also plot in Figure 7 the retentionfraction for pure power-law size distributions: d/ds sq, where d is the differential number of planetesimalshaving sizes between s and s + ds. Since mm s7q,stochasticity is dominated by the upper end of the size

distribution for q < 7. We fix the maximal radius to bethat of Pluto (supper = 1200 km), set the total surfacedensity = 0.2 g c m2, and calculate Pkeep for threechoices of q = 3.5, 4, and 4.5. For q 4, the lowerlimit of the size distribution significantly influences thenormalization of d/ds; for q = 4 and 4.5, we exper-iment with two choices for the minimum planetesimalradius, slower = 1 km and 1 m. We equate m with theintegrated surface density between supper/2 and supper.According to Figure 7, steep size distributions q 4are characterized by order-unity retention efficiencies. Incontrast, shallow size distributions q < 4 for which thebulk of the mass is concentrated towards supper can in-troduce significant stochasticity.

4. CONCLUDING REMARKS

We summarize our findings in 4.1 and discuss quanti-tatively some remaining issues in 4.2.

4.1. Summary

Newly formed planets likely occupy remnant planetes-imal disks. Planets migrate as they exchange energy andangular momentum with planetesimals. Driven by dis-crete scattering events, migration is stochastic.

In our solar system, Neptune may have migrated out-ward by several AU and thereby captured the manyKuiper belt objects (KBOs) found today in mean-motion


12/18


0

/2

3/2

2

0.0

0.1

0.2

e

32

34

36

38

a(AU)

0 5 10 15 20 25Time (Myr)

23

25

27

29

ap

(AU)

Fig. 4. Evolution of a particle caught into 3:2 resonance with astochastically migrating planet. Stochasticity is driven by a disk of

surface density m = 0.2 g c m

2

, all in planetesimals having sizess = 150 km and sub-Hill random velocities. The random walk inthe planets semi-major axis causes the libration amplitude of theresonant particle to undergo a corresponding random walk. Thenoise in this example is too mild to prevent the planet from bothcapturing and retaining the particle in resonance.

resonance with the planet. While resonance capture isefficient when migration is smooth, a longstanding is-sue has been whether Neptunes actual migration wastoo noisy to permit capture. Our work addressesanddispelsthis concern by supplying a first-principles the-ory for how a planets semi-major axis fluctuates in re-

sponse to intrinsic granularity in the gravitational poten-tial. We apply our theory to identify the environmentalconditions under which resonance capture remains vi-able.

Stochasticity results from random variations in thenumbers and orbital properties of planetesimals encoun-tering the planet. The degree of stochasticity (as mea-sured, say, by ap,T, the typical distance that the planetssemi-major axis random walks away from its averagevalue) depends on how planetesimal semi-major axes aand random velocities u are distributed. We have param-eterized a by its difference from the planets semi-majoraxis: x a ap RRH, where RH is the Hill sphereradius and R 1. In the case of high dispersion when

0

/2

3/2

2

not in resonance

0.0

0.1

0.2

e

3132

33

34

35

a(AU)

0 5 10 15 20 25Time (Myr)

23

25

27

29

ap

(AU)

Fig. 5. Evolution of a particle caught into, but even-tually lost from, 3:2 resonance with a stochastically migrating

planet. Stochasticity is driven by a disk of surface density m =0.2 g c m2, all in planetesimals having sizes s = 700 km and sub-Hill random velocities. The particle is expelled from resonancehaving had its eccentricity raised to 0.2 during its time in resonantlock.

u > RvH (where vH pRH is the Hill velocity andp is the planets orbital angular velocity), planetesimalorbits cross that of the planet. Stochasticity increaseswith decreasing u in the high-dispersion case becausethe cross-section for strong scatterings increases steeplywith decreasing velocity dispersion (as 1/u4). In theintermediate-dispersion case when vH/R2 < u < RvH,planetesimal and planet orbits do not cross, and stochas-ticity decreases with decreasing u. In the low-dispersioncase when u < vH/R2, the amount of stochasticity isinsensitive to u.

The values ofu and R which actually characterize disksare unknown. The random velocity u, for example, isexpected to be set by a balance between excitation bygravitational scatterings and damping by inelastic colli-sions between planetesimals and/or gas drag. Dampingdepends, in turn, on the size distribution of planetesi-mals. These considerations are often absent from currentN-body simulations of planetary migration in planetesi-mal disks. Despite such uncertainty, we can still identifythe circumstances under which stochasticity is maximal.


13/18



0.0

0.2

0.4

0.6

0.8

1.0

NumericalCapture

andRetentionFra

ction

Smooth

200km

400km

500km

600km900km

1600km

Fig. 6. Fraction of particles caught into, and retained within, external 3:2 resonance by a stochastically migrating planet. For every and s, we numerically integrate the trajectories of 200 test particles with initial eccentricities of 0.01 and semi-major axes that lie 1 AUoutside of nominal resonance. These particles respond to the time-averaged potential of a Neptune-mass planet which migrates outwardfrom 23.1AU to 30.1 AU within a disk of fixed surface density m = 0.2 g c m2 in planetesimals of a single size s. The planetesimalshave sub-Hill random velocities and semi-major axes that lie within R = 1 Hill radius of the planets; these choices maximize the amountof stochasticity in the planets migration. Compare this Figure with its analytic counterpart, Figure 3; the agreement is excellent. Thesolid curve labelled Smooth corresponds to the case when all noise is eliminated from the planets migration. Planetesimals having sizessmaller than 200 km yield an essentially smooth migration. For 105 yr, capture is not possible even if migration were smooth, sincethe migration is too fast to be adiabatic. These results are calculated for C = 1; probably C is of order several and so the sizes indicated inthe Figure should be revised downward by a factor of a few (C2/3; see Equation [53]).

Maximum stochasticity obtains when R 1 and u vH,that is, when planetesimals have semi-major axes withina Hill radius of the planets and when their velocity dis-persion is no greater than the Hill velocity.

A stochastically migrating planet cannot retain objectsin a given resonance if the planets semi-major axis ran-dom walks away from its average value by a distancegreater than the maximum libration width of the reso-nance. This simple criterion is validated by numericalexperiments and enables analytic calculation of the res-onance retention efficiency as a function of disk param-eters. A disk of given surface density generates morenoise when composed of fewer, larger planetesimals. In

the context of Neptunes migration, we estimate that ifthe bulk of the minimum-mass disk resided in bodieshaving sizes smaller than O(100) km and if the fractionof the disk mass in larger bodies was not too large (a few percent for planetesimals having sizes of 1000 km,for example), then the retention efficiency of Neptunesfirst-order resonances would have been of order unity( 0.1). Such order-unity efficiencies seem required byobservations, which prima facie place 122/474 26% ofwell-observed KBOs (excluding Centaurs) inside mean-motion resonances (Chiang et al. 2006). Drawing conclu-sions based on a comparison between this observed per-centage and our theoretical retention percentage Pkeep

is a task fraught with caveatsa more fair comparisonwould require, e.g., disentangling the observational biasagainst discovering Resonant vs. non-Resonant objects;account of the attrition of the Resonant population dueto weak chaos over the four-billion-year age of the so-lar system; and knowledge of the initial eccentricity andsemi-major axis distributions of objects prior to reso-nance sweeping, as these distributions impact captureprobabilities in different ways for different resonances(Chiang et al. 2003; Hahn & Malhotra 2005; Chiang etal. 2006). But each of these caveats alters the relevantpercentages only by factors of a few, and when combined,their effects tend to cancel. Therefore we feel comfort-

able in our assessment that Pkeep must have been of or-der unity to explain the current Resonant population. Inthat case, O(100 km) is a conservative estimate for themaximum allowed size of planetesimals comprising thebulk of the disk mass, derived for the case of maximumstochasticity.

How does an upper limit of O(100) km compare withthe actual size distribution of the planetesimal disk?While todays Kuiper belt places most of its mass inobjects having sizes of 100 km, this total mass istinyonly 0.1M (Bernstein et al. 2004; see Chiang etal. 2006 for a synopsis). The current belt is therefore 23orders of magnitude too low in mass to have driven Nep-


14/18



0.0

0.2

0.4

0.6

0.8

1.0

A

nalyticRetentionFra

ctionPkeep

KL99 t=11 Myr; KL99 t=37 Myr; q=4.5, slower=1 m

q=4.5, slower=1 km

q=4, slower=1 m

q=4, slower=1 km

q=3.5

Fig. 7. Fraction of particles retained in external 3:2 resonance by a stochastically migrating planet for various planetesimal sizedistributions. The retention efficiency is calculated analytically using Equation (55), with parameters the same as those for Figure 3except for m m; that parameter is evaluated at its maximum value within a logarithmic size bin spanning a factor of 2 for a given sizedistribution. The size distributions considered include two from Kenyon & Luu (1999; their Figure 8), evaluated at times t = 11 Myr and37 Myr; and five different power-law distributions, each characterized by a total integrated surface density = 0.2 g c m2, an upper sizelimit supper = 1200 km, a differential size index q (such that d/ds sq), and a lower size limit slower as indicated (the curve for q = 3.5is insensitive to slower since the bulk of the mass is concentrated towards supper). The three curves for the size distributions of KL99 andfor {q = 4.5, slower = 1 m} overlap at Pkeep = 1.

tunes migration. The current size distribution is suchthat bodies having radii 40 km are collisionless over theage of the solar system and might therefore represent adirect remnant, unadulterated by erosive collisions, of theplanetesimal disk during the era of migration (Pan & Sari2005). If so, the bulk of the primordial disk mass musthave resided in bodies having sizes 40 km. Theoreti-cal calculations of the coagulation history of the Kuiperbelt are so far consistent with this expectation. Kenyon& Luu (1999) find, for their primordial trans-Neptuniandisk of 10M, that 99% of the mass failed to coagu-late into bodies larger than O(1) km, because the forma-tion of several Pluto-sized objects (comprising 0.1% ofthe total mass) excited velocity dispersions so much that

planetesimal collisions became destructive rather thanagglomerative. The average-mass planetesimals in theirsimulation have sizes O(1) km, much smaller than evenour most conservative estimate of the maximum allowedsize ofO(100) km.

For a given size distribution of planetesimals, moststochasticity is produced by the size bin having maxi-mal m2, which need not be the size bin containing themajority of the mass. Here, and m are the numberof planetesimals and the mass of an individual planetes-imal in a logarithmic size bin. For power-law size dis-tributions d/ds sq such that q < 7, stochasticity isdominated by the largest planetesimals. For disks having

as much mass as the minimum-mass disk of solids andwhose largest members are Pluto-sized, size distributionswith q 4 enjoy order-unity efficiencies for resonance re-tention. The size distributions of Kenyon & Luu (1999)resemble q = 4 power laws, but with a large overabun-dance of planetesimals having sizes of O(1) km. Thissequestration of mass dramatically reduces the stochas-ticity generated by the largest bodies, which have sizesofO(1000) km.

We conclude that Neptunes Brownian motion did notimpede in any substantive way the planets capture andretention of Resonant KBOs.

4.2. Extensions

4.2.1. Single Kick to PlanetOur focus thus far has been on the regime in which

many stochastic kicks to the planet are required for res-onant particles to escape. Of course, a single kick from aplanetesimal having sufficiently large mass m1 could flushparticles from resonance. To estimate m1, we equate thechange in the planets semi-major axis from a single en-counter, ap, to the maximum half-width of the reso-nance, ap,lib/2 (see Equation [50] and related discus-sion). In the likely event that the perturbers eccentric-ity e is of order unity, then max(ap) (m1/Mp)ape(Equation [31]) and therefore m1 0.6 (0.5/e) M forPlutinos to escape resonance. Our estimate for m1 agrees


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with that of Malhotra (1993). Why such enormous per-turbers have not been observed today is unclear andcasts doubt on their existence (Morbidelli, Jacob, & Petit2002). If such an Earth-mass planetesimal were presentover the duration T of Neptunes migration, then thelikelihood of a resonance-destabilizing encounter would

be P1 N T 102(0.5/e)4, where the encounter rate

N is given by Equation (42) with m m1/(2a2p), and

we have set T 2.7 107 yr.4.2.2. Kicks to Resonant Planetesimals

Finally, we have ignored in this work how disk planetes-imals directly perturb the semi-major axis of a resonantparticle. This neglect does not significantly alter our con-clusions. Take the resonant planetesimal to resemble atypical Resonant KBO observed today, having size sres 100 km. Then its Hill velocity is eH,resa 104a. Therelative velocity between the resonant planetesimal andan ambient, perturbing planetesimal greatly exceeds thisHill velocity, if only because migration in resonant lockquickly raises the eccentricity of the resonant planetes-

imal above eH,res. Equation (43), appropriate for the

super-Hill regime, implies that a2p,rnd1/2 M0p theRMS random velocity does not depend on the mass of theobject being perturbed! Therefore when both the reso-nant planetesimal and the planet are scattering planetes-imals in the super-Hill regime, their random walks arecomparable in vigor. The conservative limit of O(100)km on the planetesimal size which allows resonance re-tention is derived, by contrast, for the sub-Hill, maxi-mum stochasticity regime, and is therefore little affectedby these considerations.9

This work was made possible by grants from theNational Science Foundation, NASA, and the AlfredP. Sloan Foundation. We thank B. Collins, J. Hahn,A. Morbidelli, I. Shapiro, J. Wisdom, and an anonymousProtostars and Planets V referee for encouraging and in-formative exchanges. Section 3.6 was written in responseto insightful comments by Mike Brown and Reem Sari.An anonymous referee helped to improve the presenta-tion of this paper.

9 The probability that a planetesimal will be ejected from reso-nance by a planetesimal of comparable mass in a single encounter

is negligibly small.

APPENDIX

ABSORPTION PROBABILITY FOR BROWNIAN MOTION WITH A DOUBLE BOUNDARY

Here we derive Equation (55), the probability that a particle experiencing Brownian motion between two absorbingboundaries has not been absorbed by time t (e.g., Grimmett and Stirzaker 2001b). Consider Brownian motion alonga path x(t) with x(0) = 0 and absorbing boundaries at x = b, b 0. The probability density distribution f(x, t)satisfies the diffusion equation

f

t=

1

2D

2f

x2, (A1)

where D is the diffusion coefficient. The absorbing boundaries generate the boundary conditions

f(b, t) = 0 (A2)for all time,10 and the initial condition is

f(x, 0) = (x) . (A3)

To solve for f(x, t), we expand f in a Fourier series, keeping only terms that satisfy ( A2):

f(x, t) =

n=1

kn(t)sin

n(x + b)

2b

. (A4)

Plugging (A4) into (A1), we find

kn(t) = cnent , (A5)

where the cns are constants and

n n22

8b2D . (A6)

The cns must satisfy (A3). From Fourier analysis at time t = 0, we find

cn =1

2b

4b0

(y b) (y 3b)

sin

ny2b

dy (A7)

=1

bsinn

2

. (A8)

10 Equation (A2) holds as long as D is non-zero. Particles near the boundary are carried across by fluctuations too quickly to maintaina non-zero density f at x = b (see Grimmett and Stirzaker 2001a for a proof).


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