dynamical evolution of planetesimals in protoplanetary disks

8/14/2019 Dynamical evolution of planetesimals in protoplanetary disks

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Draft version February 2, 2008Preprint typeset using LATEX style emulateapj v. 02/09/03

DYNAMICAL EVOLUTION OF PLANETESIMALS IN PROTOPLANETARY DISKS.

R. R. RafikovIAS, Einstein Dr., Princeton, NJ 08540

Draft version February 2, 2008

ABSTRACT

The current picture of terrestrial planet formation relies heavily on our understanding of the dynam-ical evolution of planetesimals asteroid-like bodies thought to be planetary building blocks. Inthis study we investigate the growth of eccentricities and inclinations of planetesimals in spatiallyhomogeneous protoplanetary disks using methods of kinetic theory. Emphasis is put on clarifying theeffect of gravitational scattering between planetesimals on the evolution of their random velocities.We explore disks with a realistic mass spectrum of planetesimals evolving in time, similar to thatobtained in self-consistent simulations of planetesimal coagulation: distribution scales as a power lawof mass for small planetesimals and is supplemented by an extended tail of bodies at large massesrepresenting the ongoing runaway growth in the system. We calculate the behavior of planetesimalrandom velocities as a function of the planetesimal mass spectrum both analytically and numerically;results obtained by the two approaches agree quite well. Scaling of random velocity with mass canalways be represented as a combination of power laws corresponding to different velocity regimes

(shear- or dispersion-dominated) of planetesimal gravitational interactions. For different mass spectrawe calculate analytically the exponents and time dependent normalizations of these power laws, aswell as the positions of the transition regions between different regimes. It is shown that randomenergy equipartition between different planetesimals can only be achieved in disks with very steepmass distributions (differential surface number density of planetesimals falling off steeper than m4),or in the runaway tails. In systems with shallow mass spectra (shallower than m3) random velocitiesof small planetesimals turn out to be independent of their masses. We also discuss the damping ef-fects of inelastic collisions between planetesimals and of gas drag, and their importance in modifyingplanetesimal random velocities.

Subject headings: planetary systems: formation solar system: formation Kuiper Belt

1. introduction.

Gravity-assisted agglomeration of planetesimals in the

protoplanetary disks around young stars is a backboneof a currently widely accepted paradigm of terrestrialplanet formation, despite many uncertainties in the de-tails of specific processes involved. In many respects ourunderstanding of the planetesimal disk evolution heavilyrelies on the numerical investigations, and these studiessignificantly grew in complexity in recent years (Wetherill& Stewart 1989, 1993; Kenyon & Luu 1998, 2002; Inabaet al. 2001). On one hand, the sheer scale of simulatedsystems has expanded considerably, which was made pos-sible by the increase in computing power; on the otherhand, a wider range of concomitant physical phenomena(fragmentation, migration, etc.) can now be routinelyfollowed simultaneously with the evolution of planetesi-mal mass and velocity distributions.

Unfortunately, very often this complexity creates aproblem when one tries to understand a relative role ofeach of the specific processes in shaping the planetesimalmass and velocity spectra. Disentangling the contribu-tions of different physical nature in the results of numer-ical simulations certainly is a tantalizing task which canoften yield very confusing conclusions. Thus it is very im-portant to be able to isolate different physical processesoperating in the planetesimal disks and to study their ef-fects on the disk evolution separately. One way to do thisis to build simple but realistic models which are easy to

Electronic address: [email protected]

analyze and at the same time are able to grasp the mainfeatures of the physics involved. Such approach wouldgrant us good analytical understanding of relevant pro-

cesses, and we are going to follow this route in our study.Behavior of planetesimal velocities is one of the cru-

cial ingredients of the terrestrial planet formation pic-ture because of the very important role played by thegravitational focusing (which depends on planetesimalvelocities) in assembly of the big bodies in planetesimaldisks. The purpose of this paper is to explore the evo-lution of inclinations and eccentricities (which representplanetesimal random velocities) caused by the mutualgravitational scattering of planetesimals. This is a cleanand well-posed problem. The effects of gas drag and in-elastic collisions are initially neglected but later on wediscuss how their inclusion affects our results. To keepthings as clear as possible we have opted not to try cou-

pling self-consistently planetesimal velocity evolution tothe evolution of the mass spectrum in this study. Instead,we follow changes of random velocities in a disk with adistribution of planetesimal sizes which evolves in someprescribed manner. However simplistic this assumptionseems to be, in adopting it we try to use mass spectrawhich are similar to those produced by self-consistentcoagulation simulations, and to study a wide variety ofsuch mass distribution models.

Another argument in favor of this approach is the factthat the dynamical relaxation time in a disk of gravita-tionally interacting planetesimals is much shorter thanthe physical collision timescale (characterizing evolution
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of the mass spectrum) if relative random velocities ofplanetesimals are smaller than the escape velocities fromtheir surfaces (which is very often the case). The dynam-ical evolution of planetesimal disk can then be approxi-mated as that of a system with a quasi-stationary plan-etesimal mass spectrum. Understanding this problem isa logical step necessary to provide a clearer perspectiveon how to build a fully self-consistent theory of planet

formation.We describe the statistical approach to the problem

of the evolution of planetesimal random velocities in 2.Our model of planetesimal mass spectrum is introduced3. In 4 we outline the results for the distribution ofplanetesimal random velocities vs. planetesimal massesas functions of the input mass distribution and time; wealso compare these analytical predictions with numer-ical results. Table C1 summarizes our major findingsin a concise form. In 5 we comment on how gas dragand inelastic collisions between planetesimals can affectplanetesimal velocity spectra. We conclude in 6 by thediscussion of our results and comparison with previousstudies.

2. velocity evolution equations.

When the number of planetesimals in the protoplane-tary disk is very large and their masses are not sufficientto induce strong local nonuniformities in the disk (Ida& Makino 1993; Rafikov 2001, 2003b, 2003c), statisticalapproach and homogeneous particle in a box assump-tion are very helpful in the treatment of planetesimalevolution (Wetherill & Stewart 1989). We assume thatplanetesimals with mass m have velocity dispersions ofeccentricity and inclination e(m, t) and i(m, t) associ-ated with them. It is natural to characterize planetes-imal disk by its mass distribution, and we assume thatN(m, t)dm is the surface number densityof planetesimals

with masses between m and m + dm.In studying the dynamics of planetesimal disks wemake the following natural assumptions:

disk is locally homogeneous in radial direction andazimuthally symmetric; it is in Keplerian rotationaround central star,

epicyclic excursions of planetesimals in horizontaland vertical directions are small compared with thedistance to the central object,

planetesimal masses are much smaller than themass of the central object Mc.

Two important consequences immediately follow fromthis set of assumptions. First, the gravitational scatter-ing of planetesimals during their encounter can be stud-ied in the framework of Hill approximation (Henon &Petit 1986; Ida 1990; Rafikov 2003a). In this approach,gravitational interaction between two bodies with massesm and m introduces a natural lengthscale Hill ra-dius1

RH a

m + m

Mc

1/31 This definition follows original work of Henon & Petit (1986)

and differs from RH a[(m + m)/3Mc]1/3 often used in the

literature (e.g. Ida 1990; Stewart and Ida 2000).

= 104 AU aAU

m + m

2 1021 g1/3

, (1)

where a is the distance from the central object. Nu-merical estimate made in (1) assumes Mc = M andaAU a/(1 AU), and is intended to illustrate that typi-cally RH a. Hill approximation yields two significantsimplifications:

The outcome of the interactions between two bod-ies depends only on their relative velocities and dis-tances.

If all relative distances are scaled by RH and rel-ative velocities of interacting bodies are scaled byRH, then the outcome of the gravitational scat-tering depends only on the initial values of scaledrelative quantities.

Second, numerical studies (Greenzweig & Lissauer1992; Ida & Makino 1992) have demonstrated that in ho-mogeneous planetesimal disks the distribution function(e, i) of absolute values of planetesimal eccentricities

e and inclinations i is well represented by the Rayleighdistribution:

(e, i)dedi =ede idi

2e2i

exp

e

2

22e i

2

22i

, (2)

where e and i are the aforementioned dispersions of ec-centricity and inclination. It also follows from azimuthalsymmetry that horizontal and vertical epicyclic phases and are distributed uniformly in the interval (0, 2).These facts have important ramifications as we demon-strate below.

Lets consider the gravitational interaction of two plan-etesimal populations: one with mass m and eccentricityand inclination dispersions e and i and the other with

mass m

and dispersions

e and

i . When the veloc-ity2 distribution function of planetesimals has the form(2) it can be shown (Rafikov 2003a) that the evolutionof e and i due to the gravitational interaction withpopulation of mass m proceeds according to

2et

= |A|Na2

m + m

Mc

4/3 mm + m

m

m + mH1 + 2

2e2e +

2e

H2

,

2it

= |A|Na2

m + m

Mc

4/3m

m + m

m

m + m K1 + 2

2i2i +

2i K

2

, (3)

where N is the surface number density of planetesimalsof mass m, and A = (r/2)d/dr is a measure of shearin the disk [in Keplerian disks A = (3/4)]. Scatteringcoefficients H1,2 are defined as

H1 = H1 (er, ir)

=

derdirr(er, ir)

dh|h| (esc)2 ,

2 Throughout the paper we often refer to eccentricities and in-clinations of planetesimals as their random velocities.


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H2 = H2 (er, ir)

=

derdirr(er, ir)

dh|h|(er esc), (4)

where er,ir = (2e,i + 2e,i)

1/2a/RH are the dispersionsof relative eccentricity and inclination normalized in Hillcoordinates [see (1)], h is a semimajor axes difference

of interacting bodies scaled by RH, and er, ir are therelative eccentricity and inclination vectors in Hill units(Goldreich & Tremaine 1980; Ida 1990). Function escrepresents a change of er produced in the course of scat-tering, which is a function of not only absolute values ofer, ir, and h, but also of the relative epicyclic phases (forwhich reason we use vector eccentricities and inclinationshere). The distribution function of relative eccentricities

and inclinations of planetesimals r(er, ir) is analogousin its functional form to (2), but with e,i replaced byrelative velocity dispersions er,ir . Expressions analogousto (4) can be written down also for inclination scatteringcoefficients K1 and K2. Note that the only assumption

used in deriving (3) is that of the specific form (2) of thedistribution function of planetesimal eccentricities andinclinations (Rafikov 2003a).

Equations (3) describe dynamical evolution driven onlyby gravitational scattering of planetesimals, implicitlyassuming that they are point masses. Physical size ofplanetesimal rp 3 107 AU (m/(1021 g))1/3 (forphysical density 3 g cm3) is very small compared tothe corresponding Hill radius (1), which often justifiesthe neglect of physical collisions between planetesimals.However, at high relative velocities higher than theescape speed from planetesimal surfaces one can nolonger disregard highly inelastic physical collisions whichstrongly damp planetesimal random motions. In this

study we proceed by assuming first that velocities ofplanetesimals are below their escape velocities, and thendiscussing in 5 how abandoning this assumption affectsour conclusions.

Equation (3) and definitions (4) have been previouslyderived by other authors in a slightly different form(Wetherill & Stewart 1988; Ida 1990; Tanaka & Ida 1996;Stewart & Ida 2000):

2et

= |A|Na2

m + m

Mc

4/3m

m + m

m

m + m(H1 + 2H2) + 2

m

m + m2e

2e + 2e

H2

2m

m + m2e

2e + 2e H2

. (5)In this form the first term in brackets on the r.h.s. de-scribes the so-called viscous stirringwhich is proportionalto the phase space average of (e2r) (see definitions ofH1 and H2). Depending on er and ir it can be eitherpositive or negative. Second term represents the phe-nomenon of dynamical friction well known from galacticdynamics. In the limit m m its contribution is pro-portional to the mass of particle under consideration andto the surface mass density of field particles, but is inde-pendent of individual masses of field particles (cf. Binney& Tremaine 1987). This term is negative since it repre-sents the gravitational interaction of a moving body with

the wake of field particles formed behind it as a result ofgravitational focusing; thus H2 < 0 and K2 < 0. Finally,the third term describes the increase of random velocitiesof a particular body at the expense of random motion offield planetesimals it interacts with. This effect is anal-ogous to the first order Fermi mechanism of the cosmicray acceleration via scattering of energetic particles byrandomly moving magnetic field inhomogeneities (Fermi

1949). Note that in previous studies of planetesimal scat-tering it is the combination of the second and third termson the r.h.s. of (5) that is called the dynamical friction(Stewart & Wetherill 1988; Ida 1990). The combined ef-fect of these two terms is to drive planetesimal system toequipartition of random energy between planetesimals ofdifferent mass (which would be realized in the absence ofviscous stirring).

In this work we analyze planetesimal velocity evolutionwith the aid of equation (3). We call the first (positive)term on its r.h.s. gravitational stirringor heating (differ-ent from viscous stirring), while second (negative) termis called gravitational friction or cooling (different fromdynamical friction3). In this sense terms stirring and

friction are only used to describe positive and negativecontributions to the growth rate of planetesimals randommotion. Use of equations (3) rather than (5) has two ob-vious advantages: (1) gravitational stirring and frictionhave different dependences on m/(m + m) which con-siderably simplifies analysis of velocity evolution equa-tions, and (2) stirring and friction terms have definitesigns (unlike viscous stirring and dynamical friction usedin previous studies of planetesimal dynamics).

Gravitational scattering of planetesimals can proceedin two rather different velocity regimes, shear-dominatedand dispersion-dominated. The former one is realizedwhen 2e,i +

2e,i (RH/a)2, while the latter holds when

2e,i + 2e,i

(RH/a)

2. Analytical arguments and nu-

merical calculations demonstrate that in the dispersion-dominated regime e and i are of the same order andevolve in a similar fashion (e.g. Ida 1990; Stewart & Ida2000). This happens because scattering in this regimehas a three-dimensional character making different com-ponents of velocity ellipsoid comparable to each other.Thus, rough idea of the dynamical evolution in this casecan be obtained by studying only one of the equations(3). Moreover, the behavior of the scattering coefficientsin the dispersion-dominated regime can be calculated an-alytically as a function of er and ir in two-body ap-proximation (Stewart & Wetherill 1988; Tanaka & Ida1996), and one finds that

H1K1

=

A1 (ir/er)C1 (ir/er)

ln 2er,

H2K2

=

A2 (ir/er)C2 (ir/er)

ln

2er, (6)

where A1,2, C1,2 are positive functions of inclination toeccentricity ratio ir/er, and ln is a Coulomb loga-rithm (Binney & Tremaine 1987), which in our case canbe represented as (Rafikov 2003a)

a1ir(2er + a22ir), (7)3 I am grateful to Peter Goldreich and Reem Sari for pointing

this out to me.


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with a1,2 being some constants. Explicit analytical ex-pressions for coefficients A1,2, C1,2 have been derived byStewart & Ida (2000). One can deduce a specific use-ful property of these functions by considering a single-mass planetesimal population: in dispersion-dominatedregime the distribution of random energy between thevertical and horizontal motions tends to reach a quasi-equilibrium state (see e.g. Ida & Makino 1992). Then

the ratio ir/er is almost constant and both er and irgrow with time, meaning that [see (3) & (6)]

A1 > 2A2, C1 > 2C2. (8)

This property will be used later in 4 & C.In the shear-dominated regime e and i evolve along

different routes: eccentricity is excited much strongerthan inclination because in this regime disk is geometri-cally thin and forcing in the plane of the disk is strongerthan in perpendicular direction. Eccentricity evolution isalso quite rapid in this case because of the vigorous scat-tering (relative horizontal velocity increases by RHin each synodic passage of two bodies initially separatedby

RH in semimajor axes). Simple reasoning con-

firmed by numerical experiments suggests the followingbehavior of scattering coefficients in the shear-dominatedregime:

H1(er , ir) B1, H2(er, ir) B22er,K1(er, ir) D12ir, K2(er, ir) D22ir, (9)

where B1,2, D1,2 are some positive constants, which canbe fixed using numerical orbit integrations, see AppendixA. Simple qualitative derivation of (6) and (9) can befound in Ida & Makino (1993) and Rafikov (2003c).

For further convenience, we now reduce evolution equa-tions to a dimensionless form. We relate planetesimalmasses to the smallest planetesimal mass4 m0 by usingdimensionless quantity x = m/m0. Instead of N(m) weintroduce dimensionless mass distribution function f(x)such that

N(m) =pm20

f(x) and N(m)dm =pm0

f(x)dx, (10)

where p is the total mass surface density of planetesi-mals in the disk, which is a conserved quantity. We alsorescale eccentricity and inclination dispersions:

s = e

m0Mc

1/3, sz = i

m0Mc

1/3. (11)

This is equivalent to rescaling all distances by Hill radiusof smallest planetesimals. In this notation the boundary

between the shear- and dispersion-dominated regimes isgiven by conditions

s2 + s2 (x + x)2/3, s2z + s2z (x + x)2/3. (12)We also introduce dimensionless time by

=t

t0, where t0 = |A|1 m0

pa2

m0Mc

2/3(13)

4 Protoplanetary disks should contain planetesimals of differentsizes but we assume that bodies lighter than m0 are not importantfor the disk evolution. In the case considered here the choice of m0is dictated by the mass at which distribution would depart fromthe given power law form towards shallower scaling with mass (seealso 4.1.1).

is a timescale for the stirring of planetesimal disk com-posed of bodies of mass m0 only, on the boundary be-tween the shear- and dispersion-dominated regimes. Nu-merically, for Mc = M one finds that

t0 16 yr aAU

10 g cm2

p0

m0

1021g

1/3, (14)

where we assumed that p(a) = p0a

3/2

AU (Hayashi1981). Dynamical timescale is rather short when s, sz 1, but it grows very rapidly as epicyclic velocities of plan-etesimals increase (t t0s4 in the dispersion-dominatedregime).

Random velocities of planetesimals of mass x evolvedue to the interaction with all other bodies spanning thewhole mass spectrum. Using our dimensionless notationwe can rewrite equations (3), (6), & (9) in the followinggeneral form:

s2

=

1

dxf(x)x(x + x)1/3

x

x + x H1 + 2s2

s2 + s2 H2

, (15)

s2z

=

1

dxf(x)x(x + x)1/3

x

x + xK1 + 2

s2zs2z + s

2z

K2

, (16)

where s s(x), s s(x) and similarly for sz, sz; in theshear-dominated regime [s2 +s2, s2z + s

2z (x+ x)2/3]

H1 C1, H2 C2 s2 + s2

(x + x)2/3,

K1 D1s2z + s

2z

(x + x)2/3 , K2 D2s2z + s

2z

(x + x)2/3 , (17)

and in the dispersion-dominated regime [s2 + s2, s2z +s2z (x + x)2/3]

H1K1

=

A1B1

(x + x)2/3

s2 + s2ln ,

H2K2

=

A2B2

(x + x)2/3

s2 + s2ln . (18)

We explore this system in 4 using two approaches:asymptotic analysis utilizing analytical methods, and di-rect numerical calculation of velocity evolution.

3. mass spectrum.

Starting from (15)-(18) one would like to obtain thebehavior of s and sz as functions of x and , given someplanetesimal mass spectrum f(x) as an input. In gen-eral, to do this one has to evolve equations (16)-(18) nu-merically. However, it is usually true that the planetes-imal size distribution spans many orders of magnitudein mass; thus one would expect that some general pre-dictions can be made analytically about the asymptoticproperties of planetesimal velocity spectrum.

In this study we assume that planetesimal mass distri-bution has a self-similar form; specifically we take

f(x, ) = ()

x

xc()

, (19)


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Fig. 1. Schematic representation of the shape of planetesimalmass spectrum and its evolution in time. Mass distribution has apower law form exponentially cut off at xc() with a tail of runawaybodies for x xc().

where xc() 1 is some fiducial planetesimal masswhich steadily grows in time as a result of coagulation,() is a temporal modulation, and function repre-sents a self-similar shape of the mass spectrum. We alsoassume in this study that asymptotically

(y)

y, y 1,exp(y), y 1, (20)

where > 0. Such scaling behavior is often found incoagulation simulations. We will see in 4 that for clari-fying the asymptotic properties of planetesimal velocityspectrum it is enough to know only the asymptotic be-havior of (y) and not its exact shape. Moreover, in 6we demonstrate how our results for power-law size dis-tributions can be generalized for other mass spectra.

Normalization () is not an independent function. Itis related to xc() because of the conservation of the total(dimensionless) planetesimal surface density

M1

1

xf(x)dx. (21)

Taking this constraint into account one finds that

() =

x2c (), < 2,xc (), > 2.

(22)

In the case > 2 mass spectrum behaves as f(x, ) =x for x xc(); only the position of the high masscutoffxc() shifts towards higher and higher masses withtime.

We define M() to be the -th order moment of themass distribution:

M()

1

(x)f(x, )dx. (23)

Integral in (23) is dominated by the upper end of thepower-law part of the mass spectrum (i.e. by x xc) if > 1; in this case we can write [using (19)]

M() = M [xc()]+1()

= M

x1c , < 2,x+1c , > 2,

, M

0

y(y)dy,(24)

where M reduced moment of order is a time-independent constant for a given mass distribution.

One of the interesting features exhibited by self-consistent coagulation simulations is the development ofthe tail of high mass bodies beyond the cutoff of thebulk distribution of planetesimals (Wetherill & Stewart1989; Inaba et al. 2001). This is interpreted as the man-ifestation of the runaway phenomenon taking place in acoagulating system. To explore the possibility of the run-away scenario we add to the mass spectrum (20) a tail ofhigh mass planetesimals extending beyond the exponen-tial cutoff xc (see Appendix A). In doing this we alwaysmake sure that runaway tail contains a negligible part

of the systems mass and does not affect its dynamicalstate (i.e. stirring and friction caused by these high massplanetesimals are small) which is true if biggest bodiesare not too massive (Rafikov 2003c). Under these as-sumptions the explicit functional form of the runawaytail is completely unimportant. Throughout this study,we use the term planetesimals for bodies belonging tothe power-law part of the spectrum (x xc), and mas-sive or runaway bodies to denote the constituents ofthe tail (x xc).

We do not restrict the variety of possible functionaldependences ofxc on time . It will turn out that all ourresults can be expressed as some functions of xc whichleads to the time-dependence of planetesimal velocities

in a very general form. In some cases it will be impor-tant that xc() does not grow too fast, which, however,we believe is a fairly weak constraint (see discussion in4.1.2).

4. velocity scaling laws.

To facilitate our treatment of planetesimal velocitieswe introduce a set of simplifications into our considera-tion:

We split mass spectrum into several regions, suchthat in each of them planetesimal interactionscan be considered as dominated by a single dy-namical regime (shear-dominated or dispersion-dominated). Transitions between such regions are

not considered but in principle might be treated byinterpolation.

When studying the dispersion-dominated regimewe neglect the difference between the vertical andhorizontal velocity dispersions, and treat them bya single equation. We also set Coulomb logarithmequal to constant, because of its rather weak de-pendence on s, sz or x.

The validity and impact of these assumptions on the ve-locity spectrum are further checked using numerical tech-niques. Since we are mostly interested in qualitative be-havior of solutions, we do not expect numerical constants


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to be correctly reproduced by our analysis; they can betrivially fixed using numerical calculations.

Mass distributions are parametrized by the value ofpower law exponent [see (20)]. We split our investi-gation according to the value of into 3 cases: shal-low mass spectrum, < 2, intermediate mass spectrum,2 < < 3, and steep spectrum, > 3 (this regime splitsinto two more important subcases, see

4.3). Note that

for the sake of avoiding additional complications we donot consider borderline cases = 2, 3; they can be eas-ily studied in the framework of our approach if the needarises.

In our analytical work planetesimals are started withlarge enough s and sz so that they interact with eachother in the dispersion-dominated regime. We thenalso assume that planetesimals with masses x xc()(containing most of the mass) stay in the dispersion-dominated regime w.r.t. each other also at later time,i.e.

s(x, ), sz(x, ) x1/3 for x xc(). (25)This is a reasonable assumption for all mass spectra atthe beginning of evolution, although for steep size distri-

butions it may break down when maximum planetesimalmass becomes very large (see 4.3).

4.1. Shallow mass spectrum.

Planetesimal mass distributions shallower than m2

result from coagulation of high-velocity planetesimals,when gravitational focusing is unimportant and collisionrate is determined by geometrical cross-section of collid-ing bodies (Wetherill & Stewart 1993; Kenyon & Luu1998). It is worth remembering however that in highlydynamically excited disks (1) energy dissipation in in-elastic collisions must be important, see 5, and (2) plan-etesimal fragmentation cannot be ignored. Despite that,the case of shallow mass spectrum is interesting because

it facilitates understanding of the disks with other plan-etesimal size distributions.

4.1.1. Velocities of planetesimals.

We start by considering planetesimals, x xc() part of the size distribution containing most of themass. Regarding them as interacting in the dispersion-dominated regime [i.e. the condition (25) is fulfilled] wemay write using (16) & (18) that

s2

=

1

dxf(x)x(x + x)

s2 + s2

x

x + x

A1

2

s2

s2

+ s2

A2 (26)for x xc(). Since we use a single equation to describethe evolution of both s and sz, the values of constantsA1,2 are not well defined but this is not important forderiving general properties of the velocity spectrum.

We can represent (26) asymptotically in the followingform:

s2

A1

x1

dxx2f(x)

s2 + s2 2A2s2x

x1

dxxf(x)

(s2 + s2)2

+

x

dxx2f(x)

s2 + s2

A1 2 s

2

s2 + s2A2

. (27)

Fig. 2. Sketch of the behavior of the scaled planetesimal eccen-tricity and inclination s = ea/rH and sz = ia/rH as functionsof their dimensionless mass x = m/m0, predicted by asymptoticanalysis of 4.1. Solid line displays s(x) while dashed line is forsz(x) (for x < xshear their behaviors are the same). Dotted linedelineates shear- and dispersion-dominated regimes of planetesimalinteraction with bodies of comparable mass. The rest of notationand theoretical predictions for the temporal behavior of s(x) andsz(x) can be found in the text.

The first two terms on the r.h.s. represent correspond-ingly the gravitational stirring and friction by bodiessmaller than x. The last term is responsible for the com-bined effect of stirring and friction produced by bodiesbigger than x. From this equation it is easy to see thats() independent of x is a legitimate solution of (27) inthe case < 2. Indeed, assuming s to be independent ofx we find that all integrals over x in (27) are dominatedby their upper limits when mass spectrum is shallow.Using (19)-(24) we can estimate that

s2

A1 xc

2s2

x

xc

3 2A2 xc

4s2

x

xc

3

+(A1 A2) M2()2s2

. (28)

Note that in deriving this expression we have extendedthe integration range of the last term in (27) from 1 to

infinity (not from x). This is legitimate because thisintegral is dominated by the contribution coming from xc x and, thus, adding interval from 1 to x tointegration range would introduce only a subdominantcontribution to the final answer. Clearly, two first termsin (28) are negligible compared to the third one for x xc, and third term is positive [see (8)] and independentof x, meaning that s() is also independent of mass, inaccord with our assumption. One can then easily findthat5 [dropping constant factors A1,2 but bearing in mind

5 Throughout the paper we imply by s0 the velocity dispersion ofsmallest planetesimals and s0 is different for different mass spectra,see (40) and (53) below.


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7

condition (8)]

s(x, ) s0 M2()d

1/4

= M2

xc()d1/4

, < 2, x xc (29)

[if s(x, ) s(x, 0)]. The second equality holds onlyfor < 2, according to (22) & (24). Thus, in the caseof mass spectrum shallower than x2 random velocitiesare independent of planetesimal masses and uniformlygrow with time. Both gravitational stirring and frictionare dominated by biggest planetesimals (x xc), withstirring being more important; friction by small bodies iscompletely negligible and energy equipartition betweenplanetesimals of different mass is not reached. For thisreason, planetesimal velocity expressed in physical unitsusing (11), (13), & (29) is independent of the choice ofm0. A schematic representation of velocity spectrum for

the shallow mass distribution is displayed in Figure 2.

4.1.2. Velocities of runaway bodies.

Now we turn our attention to the runaway bodies, x xc. By assumption, this tail contains so little mass, thatit cannot affect velocities not only of planetesimals butalso of runaway bodies themselves.

Because of our assumption of the dispersion-dominatedinteraction between all planetesimals with x xc itis natural to expect that at least those runaway bod-ies which are not too heavy still experience dispersion-dominated scattering by planetesimals with masses xc.Assuming that the velocity dispersion of these runawaybodies is much smaller than that of the bulk of planetes-

imals natural consequence of the tendency to equipar-tition of random epicyclic energy between bodies of dif-ferent mass we can rewrite (26) as

s2

A1

1

dxx2f(x)

s2 2A2s2x

1

dxxf(x)

s4.(30)

Although the integration range of all terms in the r.h.s.is extended to infinity, the exponential cutoff of f(x) at xc effectively restricts the integration range to be from1 to xc.

One can easily see that s(x, ) = const(x) cannot be asolution of(30) because dynamical friction is too strong.

Indeed, if we were to assume the opposite, we would findthat s(x, ) is still given by (29). However, direct substi-tution of (29) into (30) shows that the gravitational fric-tion term exceeds other contributions for x xc. Thismight tempt one to suggest that heating (first) term inr.h.s. (30) should be neglected compared to the friction(second) term. In this case one would find that s(x, )exponentially decays in time and very soon heating termbecomes the dominant one, contrary to initial assump-tion. These results suggest the only remaining possibil-ity that heating of massive bodies by planetesimalsalmost exactly balances friction by planetesimals, andminute difference between them is accounted for by thetime dependence ofs(x, ) embodied in the l.h.s. of (30).

Fig. 3. Numerical results in the case of = 1.5 for (a) ec-centricity dispersion (dimensionless) s and (b) inclination disper-sion sz. In the bottom panels the velocity spectrum is plotted as afunction of dimensionless mass x, while in the top panels the powerlaw index = d ln s/d ln x of the spectrum is displayed. Differentcurves correspond to increasing values of maximum planetesimalmass xc (and, consequently, time) indicated by arrows of corre-sponding line type on the upper panels. Dotted line in the lowerpanels is s = x1/3 (boundary between different velocity regimes).Dotted curve in the upper panels is the theoretical prediction forthe run of the spectral index corresponding to the largest xc dis-

played (for which numerical result is always shown by thick solidline). These results are to be compared with theoretical predictionsshown in Figure 2.

Given that this assumption is correct we immediatelyfind that

s(x, )

1x

1

dxx2f(x)

s2

1/2

1

dxxf(x)

s4

1/2

.(31)

It is easy to see that scaling s x1/2 follows di-


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8

rectly from assuming that (1) the interaction is in thedispersion-dominated regime, and (2) heating of runawaybodies by planetesimals is balanced by friction also dueto planetesimals. This specific result is completely inde-pendent of the mass or velocity spectra of small bodies.We will highlight this again when we obtain similar re-sults for intermediate and steep mass spectra in 4.2.2and

C.2. For a specific case < 2 considered in this

section s = s0() and one finds that

s(x, ) s0()x

M2()

M1

1/2

= s0

xcx

1/2 M2M1

1/2, xc x xshear (32)

[xshear is defined below in equation (33)]. At x xc thisformula is consistent with s0 extrapolation of smallmass result (29) up to x xc.

Time dependence enters (32) eventually through xc().Thus, it is the behavior of xc() that determines s

2/in (30). The timescale on which xc varies is usually much

longer than the dynamical relaxation time of the system,thus one would expect l.h.s. of (30) to be small in ac-cordance with the assumptions which led to (31). Therequirement of the negligible effect of time derivatives insituations like the one described here is the only con-straint which must be imposed on the possible behaviorof xc().

As we consider even bigger x we find that at some massrunaway bodies start interacting with bodies of similarmass in the shear-dominated regime; this happens when

x1/3 s0(xc/x)1/2, or x xc(s0/x1/3c )6/5. This, how-ever does not affect the dynamics of these bodies, be-cause by our assumption runaway tail contains too littlemass to dynamically affect its own constituents. It is

only after the runaway bodies start interacting in theshear-dominated regime with planetesimals (i.e. bod-ies lighter than xc), that the velocity evolution of thetail gets affected. In our case this clearly happens whenx1/3 s0(). Thus, runaway bodies with masses

x xshear s30 (33)interact with all planetesimals in the shear-dominatedregime.

As mentioned in 2, in the shear-dominated case ec-centricities and inclinations may no longer evolve alongsimilar routes and we have to consider them separately.Using (16) & (17) we write

s2

=

1

dxf(x)x(x + x)1/3

C1x

x + x 2C2 s

2

(x + x)2/3

, (34)

s2z

=

1

dxf(x)xs2z + s

2z

(x + x)1/3

D1x

x + x 2D2 s

2z

s2z + s2z

. (35)

Assuming that s, sz s sz (because of the gravi-tational friction) and x x (both heating and cooling

Fig. 4. Comparison of numerical (solid curves) and analytical(dotted curves) behaviors of s0 as a function of dimensionless time. We display cases: (a) = 1.5, (b) = 2.3, (c) = 2.7, (d) = 3.2 and = 4.3.

are dominated by planetesimals, not runaway bodies),

and neglecting l.h.s. of both equations (similar to thepreviously discussed situation for xc x xshear), weobtain the following result:

s(x, ) x1/6

M2()

M1

1/2,

sz(x, ) 1x

1

M1

1

dxx2s2f(x)

1/2

. (36)

Finally, setting s s0() as appropriate for < 2 wefind that

s(x, ) x1/6 M2

M1 xc

1/2

,

sz(x, ) s0xc

x

1/2 M2M1

1/2, x xshear. (37)

These expressions demonstrate that transition to theshear-dominated regime is accompanied by a change ofthe power-law index of eccentricity scaling with mass,while the power-law slope of inclination scaling staysthe same. This happens because in the case of inclina-tion evolution heating term is functionally similar to thefriction term (exactly like in the dispersion-dominatedcase), which is not the case for eccentricity evolution, see


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9

(17). As a result, for x xshear one finds that s sz planetesimal velocity ellipsoid flattens considerably inthe vertical direction, which is very unlike the dispersion-dominated case (see Figure 2). This conclusion is verygeneral and should be found whenever runaway bodiesinteract with all planetesimals in the shear-dominatedregime.

4.1.3. Comparison with numerical results.To check the prediction of our asymptotic analysis

we have performed numerical calculations of planetesi-mal velocity evolution for a distribution of planetesimalmasses which was artificially evolved in time accordingto prescriptions outlined in 3. Details of our numericalprocedure can be found in Appendix A.

In Figure 3 we show the evolution of planetesimal ec-centricity and inclination dispersions for a shallow massspectrum with = 1.5. For each curve displaying s orsz run with x at a specific moment of time we also com-puted power law spectral index d ln s/d ln x. Wethen compare power law indices and z with analyt-ical predictions given by equations (29), (32) , & (37).

Dotted line in the upper panels of Figure 3 representsanalytical predictions for and z at the moment whenthe snapshot of velocity spectrum displayed by the solidline (corresponding to the highest xc shown) was taken.One can see that agreement between two approaches isquite good. Power law slopes of both eccentricity andinclination z exhibit a well defined plateau at massesconsiderably smaller than xc which is predicted by (29).Above upper mass cutoff and z plunge towards 1/2which is in agreement with (32), although they do not fol-low this value very closely. The reason is most likely thelack of dynamical range transitions between differentregimes typically occupy substantial intervals in mass.Finally, at x above xshear, where runaway bodies inter-

act with all planetesimals in shear-dominated regime, goes up to 1/6 while z stays at a value of 1/2, ex-actly like equation (37) predicts. Glitches in the curvesof and z in the vicinity ofxshear are the artifacts of in-terpolation of scattering coefficients between shear- anddispersion-dominated regimes.

In Figure 4a we display the time evolution of s0 ec-centricity dispersion of smallest planetesimals versus theprediction of our analysis given by equation (29). Onecan see that the agreement between two approaches isexcellent analytical and numerical results closely fol-low each other as time increases. Vertical offcet betweenthem amounts to a constant factor by which numericallydetermined s0() differs from analytical s0(). This, of

course, is expected because we completely ignored con-stant coefficients in our asymptotic analysis. Numericalresults shown in Figure 4a allow us to fix constant coef-ficient in (29), and we do this when we calculate xshearin Figure 3 using equation (33). Note that the rapidgrowth of s0 with time should certainly increase veloc-ities of planetesimals beyond their escape speeds (if itwere not the case initially). Dynamical evolution in thiscase is discussed in more details in 5.

An apparent break in the behavior of s0() at 105can be seen in Figure 4a. It occurs because this is thetime at which mass scale xc starts to grow. The pre-scription for xc() which we use (described in AppendixA) is such that xc is almost constant for 10

5; as a

Fig. 5. Same as Figure 2 but for the case of intermediate massspectrum. Specific case of 2 < < 5/2 is displayed. Notation isexplained in the text. Analogous plot for 5/2 < 3 wouldexhibit different behavior of sz in the high-mass tail and can beeasily constructed using the results of 4.2.2.

result, s0 1/4 according to equation (29). At 105scale factor starts to increase rapidly and this causes s0to grow faster. To conclude, the results of this sectionshow that the overall agreement between numerical cal-culations and analytical theory of planetesimal velocityevolution is rather good.

4.2. Intermediate mass spectrum.Self-consistent coagulation calculations (Kenyon &

Luu 1998) and N-body simulations of planet formation(Kokubo & Ida 1996, 2000) often yield intermediate massdistributions of planetesimals (2 < < 3). Kokubo &Ida (1996) have found, for example 2.5 0.4 intheir N-body calculation of collisional evolution of severalthousand massive (m = 1023 g) planetesimals. Such out-come seems to be ubiquitous for planetesimals which ag-glomerate under the action of strong gravitational focus-ing (planetesimal velocities are well below the escape ve-locities from their surfaces) in the dispersion-dominatedregime, which makes the case of intermediate mass spec-tra very important.

4.2.1. Velocities of planetesimals.

As in 4.1 we start our consideration of mass spectrawith 2 < < 3 by concentrating on planetesimals withmasses xc(), assuming that all these bodies interactin the dispersion-dominated regime (see 4). Then equa-tions (26) and (27) continue to hold.

It is natural to start by assuming again that s is in-dependent of x. Performing analysis similar to that of4.1.2 we find that stirring is still dominated by largeplanetesimals ( xc), but the biggest contribution to thegravitational friction is produced by smallest planetes-imals (with m m0, or x 1 in our dimensionless


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10

notation):

s2

A1 x

3

2s2 2A2 x

4s2M1 + (A1 A2) M2()

2s2.(38)

Clearly, the first term on the r.h.s. which is responsi-ble for the stirring by small planetesimals is negligiblecompared to the last one, representing stirring by plan-etesimals bigger than x [see (24)]. Also, by definition,

M1 is just a dimensionless total surface density of plan-etesimal disk, and thus has to be constant in time. Thenour assumption of s independent of x is self-consistentand s is given by the first equality in (29) only if frictionby small planetesimals [ second term in (38)] is smallcompared to stirring by big ones (third term), which istrue provided that

x xk() M2()M1

= xcM2M1

x2c xc. (39)The last equality follows from the fact that M2 =M2x3c when 2 < < 3. Thus, for small planetesimals(x xk) friction by even smaller ones is again unim-portant; their velocities are excited by big planetesimals

(x xc), and as a result [cf. (29)]

s(x, ) s0() =M2

[xc(

)]3d

1/4

= const(x), 2 < < 3, x xk. (40)

However, for x xk, gravitational friction by smallplanetesimals is no longer negligible compared to the ve-locity excitation by big ones. One would then expectfriction to lower the velocities of big bodies below thoseof small planetesimals (because of the tendency to en-ergy equipartition associated with friction term). Start-ing with (27) and using (40) we obtain the following equa-

tion for x

xk:s2

=

A1s20

xk1

dx(x)2f(x) + A1

xxk

dx(x)2f(x)

s2

2A2 s2x

s40

xk1

dxxf(x) 2A2s2xx

xk

dxxf(x)

s4

+A1 2A2

s2

x

dx(x)2f(x). (41)

In this equation different terms represent stirring or fric-tion by different parts of planetesimal mass spectrum,

below and above xk. We now take an educated guessthat the solution for s can be obtained by equating thirdand fifth terms in the r.h.s. of (41). Physically this meansthat velocity spectrum results from the balance of veloc-ity stirring by biggest planetesimals (masses xc x,similar to the case of shallow mass spectrum) and frictionwhich is mainly contributed by smallest planetesimals.One then finds that [see (39)]

s(x, ) s0()

M2()

M1

1/4x1/4

s0()

xk()

x

1/4, xk x xc. (42)

Fig. 6. The same as Figure 3 but for the case of intermediatemass spectrum with = 2.3 (cf. Fig. 5).

Substituting this solution into (41) we easily verify thatour neglect of all other terms in the r.h.s. of (41) isindeed justifiable. The l.h.s. of (41) can be neglected forxk x xc on the basis of arguments identical to those

advanced in 4.1.2. Thus, s given by (42) is indeed alegitimate solution in this mass range.Equations (40) & (42) highlight an interesting feature

of the case 2 < < 3: although the mass spectrumhas a power law form with a continuous slope all theway up to xc, velocity spectrum exhibits a knee at someintermediate mass xk (see Figure 5). The resemblanceof the velocity spectrum for x xk to the spectrum for < 2 is due to the fact that in both cases stirring is doneby biggest planetesimals and friction has a small effect.However, for x xk effect of friction becomes important,and now it is dominated by the part of mass spectrum(smallest planetesimals) different from that responsiblefor stirring (biggest planetesimals). This causes velocity


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11

spectrum to change its slope from 0 to 1/4 in this massrange (see Figure 5).

4.2.2. Velocities of runaway bodies.

Similar to the case of shallow mass spectrum, some(lightest) runaway bodies interact with the bulk of plan-etesimals in the dispersion-dominated regime. In this

case all the considerations of 4.1.2 hold, i.e. evolutionequation (30) stays unchanged and after the same line ofarguments we arrive at the expression (31) for the plan-etesimal velocity dispersion. Integrals entering (31) mustbe reevaluated anew using equations (40) & (42) appro-priate for 2 < < 3. Performing this procedure andsubstituting resulting expressions into (31) we find that

s(x, ) s0x

M5/2()

M1x1/2k ()

1/2

= s0

xcx

1/2 M5/2M2

xkxc

1/2, xc x xh. (43)

This velocity distribution is valid only for runaway bodieslighter than

xh [s(xc)]3 s30

xkxc

3/4, (44)

because only such bodies interact with all small massplanetesimals (x xc) in the dispersion-dominatedregime. In agreement with what we found earlier in4.1.2, velocity dispersion decreases with mass as x1/2.At x xc this expression matches the low mass result(42).

Runaway bodies heavier than xh but still lighterthan xshear s30 are in a mixed state: they in-teract with the most massive planetesimals in theshear-dominated regime, and with lighter ones in thedispersion-dominated regime. This, of course, is a di-rect consequence of the fact that the velocity spectrumof planetesimals has a knee at xk, see 4.2.1. Introducing

xs(x) xk s0

x1/3

4. (45)

we find that runaway bodies with masses x betweenxh and xshear interact with planetesimals less massivethan xs(x) in the dispersion-dominated regime, and withplanetesimals heavier than xs(x) in the shear-dominatedregime. Self-consistent analysis of planetesimal velocityspectrum for xh x xshear is described in detail inAppendix B, and only final results are shown here. Itturns out that eccentricity scales as

s(x, ) s0x5/6

s20

M2()

M1

1/2= s0

s0x1/2k

x5/6,

xh x xshear, (46)

while inclination behaves in a different manner:

sz(x, ) s0x7/6

s40x

1/2k

M3/2()

M1

1/2

=s0

x7/6

s40xk

xkxc

1/2 M3/2M2

1/2, < 5/2, (47)

Fig. 7. The same as Figure 3 but for the case of intermediatemass spectrum with = 2.7 (cf. Fig. 5).

sz(x, ) s0

(xs(x))7/2

x1/2k xM1

1/2

x(417)/6, > 5/2, (48)In the case of = 5/2 a more general expression for sz

following directly from (B4) should be used.Finally, bodies heavier than xshear interact with all

planetesimals in the shear-dominated regime. Expres-sions (36) adequately describe velocity behavior in thiscase. Manipulating them with the aid of equations (40)and (42) we find that

s(x, ) x1/2k

x1/6, x xshear. (49)

while

sz(x, ) s0xk

x

1/2 xkxc

1/2 M3/2M2

1/2,


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12

< 5/2, (50)

sz(x, ) s0xk

x

1/2 xkxc

(3)/2, > 5/2. (51)

Scalings of velocity dispersions with mass x in (49)-(51)are the same as in the case of shallow mass spectrum, butthe time dependences are different. Velocity behavior in

the case of intermediate mass spectrum is displayed inFigure 5.

4.2.3. Comparison with numerical results.

We calculated numerically the evolution ofs and sz fortwo intermediate mass spectra: = 2.3 and = 2.7. InFigure 4b,c we compare numerically calculated s0() eccentricity dispersion of smallest planetesimals withpredictions of our asymptotic theory. As in the case ofshallow mass spectrum discussed in 4.1.3, agreement be-tween the two approaches is very nice. This allows us tofix coefficient in (40), absence of which in our asymptoticanalysis causes a constant offset between numerical andanalytical curves in Figure 4b,c. Results for the velocity

dependencies on mass at different moments of time andtheir comparison with analytical theory of intermediatemass spectra outlined above are shown in Figures 6 and7.

In the case = 2.3 one can clearly see the appearanceof all features we described in 4.2.1 and 4.2.2. Indeed,in Figure 6 one can observe both the zero-slope part ofthe spectrum at x xk and the trend of reaching theslope 1/4 for xk x xc; the last feature is notfully developed when we stop our calculation but it isalmost certain that and z would reach slope 1/4given enough time and mass range. Calculation has tobe stopped when x reaches 108 because later on biggestplanetesimals start interacting with bodies of similar sizein the shear-dominated regime. Such a possibility wasnot explicitly considered in the present study (but it canbe easily dealt with using analytical apparatus developedin this work).

Results for biggest runaway bodies (x xshear) arein nice concordance with analytical predictions given byequations (51). In the range xc x sshear velocities ofthe runaway bodies clearly have not yet converged to asteady state but are evolving in the right direction. Therange of masses where velocity spectrum should exhibita slope 1/2 [xc x xh, see (43)] is so narrow6 for thelast (solid) curve (because planetesimals with mass xcare already very close to shear-dominated regime) thatwe do not see this regime realized. At the same time, itis conceivable that runaway bodies which interact with

planetesimals in the mixed regime (xh x xshear, seeAppendix B) will finally reach their asymptotic velocitystate with = 5/6 and z = 7/6 [see equation (48)],although it will take very long time for them to get there.

Comparison for the case = 2.7 (Figure 7) is verysimilar. All major features corresponding to the interme-diate mass spectrum can be traced in this case as well.Unfortunately, the comparison with analytical results iscomplicated by the fact that in this case we have to stop

6 Position ofxh in Figure 6 is determined by equation (44) which

assumes that between xk and xc velocity behaves x1/4. In

reality, as we discussed above, velocity slope in that mass region issomewhat steeper causing (44) to overestimate xh.

Fig. 8. (a) Same as Figure 2 but for the Case 1 of steep massspectrum (3 < < 4). Note that 1/4 < < 1/2 [see equation(55)]. (b) Same as Figure 2 but for steep mass spectrum with > 4.

numerical calculation even earlier than in = 2.3 caseto avoid bringing most massive planetesimals into theshear-dominated regime (for this reason mass scale inFigure 7 does not go as far as in Figure 6). As a re-sult, even a velocity plateau at small masses does nothave time to fully develop, not speaking of runaway bod-ies in mixed interaction state. Despite this, the overallagreement between the numerical results and asymptotictheory of velocity evolution of intermediate mass distri-butions is rather good, especially if we make allowancefor the short duration of our calculation and small massrange of planetesimals.

4.3. Steep mass spectrum.

Planetesimal spectra decreasing with mass steeperthan m3 are usually not found in calculations of plan-etesimal agglomeration. Still, this case is quite interest-


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13

Fig. 9. The same as Figure 3 but for the case of steep massspectrum with = 3.2 (cf. Fig. 8a).

ing and we briefly discuss it here.As demonstrated in 4.1 & 4.2 stirring of planetesi-

mals is determined solely by biggest bodies with masses xc when < 3. This is no longer true when the massspectrum becomes steeper than x3. Indeed, let us con-

sider the evolution ofs0 velocity dispersion ofsmallestplanetesimals having dimensionless mass x = 1. Usingequation (26) we find that

s20

1s20

1

dxf(x)x(x + x)

1 + s2/s20

x

x + xA1 2 1

1 + s2/s20A2

(52)

Assuming that s/s0 0 when x/x as a re-sult of gravitational friction we find that integral in (52)is dominated by its lower integration limit for > 3.This means that the velocity of smallest planetesimals is

now mediated by smallest planetesimals themselves, un-like the case of < 3. We also find that the temporalbehavior of s0 is given simply by

s0() 1/4, > 3. (53)We now turn our attention to studying planetesimal

velocities in the mass range 1 x xc. We make an

a priori assumption that planetesimal velocity spectrumhas a form of a simple power law (with constant powerlaw index)

s(x, ) s0()x , x xc, (54)and verify later if it is correct. In Appendix C we showthat whenever 3 < < 4, both heating and friction ofplanetesimals of a particular mass x are determined bybodies of similar mass, x x. This is somewhat unusualin view of our previous results when only the extrema ofpower-law mass spectrum were important (see 4.1 and4.2. In this case power law index can only be foundby numerically solving equation (C2); there is no simpleexpression for but its value has to satisfy condition

24

< < 12

. (55)

For even steeper mass distributions, > 4, we demon-strate in Appendix C that smallest planetesimals domi-nate both stirring and friction not only of themselves,but also of bigger bodies. This occurs because massspectrum is very steep and high mass planetesimals con-tain too little mass to be of any dynamical importance.This is very similar to the velocity evolution of the run-away tail in the dispersion-dominated regime described in4.1.2, and, not surprisingly, we find that = 1/2 when > 4. Thus, a solution leading to a complete energyequipartition between planetesimals (s x1/2) is onlypossible for very steep mass spectra. Velocity behaviorof runaway bodies in the case of steep mass spectrum isdescribed in detail in Appendix C. Theoretical spectraof planetesimal velocities are schematically illustrated inFigures 8a (for 3 < < 4) and 8b (for > 4).

We verify the accuracy of these predictions by numer-ically calculating the velocity evolution of planetesimaldisks with = 3.2 and = 4.3. In the first case, shownin Figure 9, the slope of planetesimal velocity spectrumfor x xc is evidently shallower than 1/2 but steeperthan ( 2)/4 = 0.3, in agreement with constraint(55). Analytical prediction for the behavior of (x) intop panels of Figure 9 is computed using the numeri-cally determined value of and equations (C4), (C5),and (C6). It agrees with numerical result (solid curvescorresponding to the largest xc displayed) rather well,especially for x xshear.

In the case = 4.3 (Figure 10) the slope of plan-etesimal velocity scaling with mass is essentially indis-tinguishable from 1/2 for x xc, in accordance withour asymptotic prediction. Velocity evolution of runawaytail also agrees with the discussion in Appendix C.2 quitewell. In both = 3.2 and = 4.3 cases the final velocitycurves displayed are at the limit where we can still apply

our theory: s(xc) x1/3c ; this washes out some details ofvelocity spectra of runaway tails predicted in AppendixC.2. Differences of and z from their predicted valuesat smallest planetesimal masses are caused by boundary


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14

Fig. 10. The same as Figure 3 but for the case of steep massspectrum with = 4.3 (cf. Fig. 8b).

effects. Note that the eccentricity dispersion s is virtu-ally independent of time above xshear for both = 3.2and 4.3 (while inclination dispersion sz slowly grows as increases), which is in complete agreement with equa-tion (C6). Finally, numerically computed s0() followsreasonably well analytical result represented by formula(53), which is demonstrated in Figure 4d.

5. velocity scaling in the presence of dissipativeeffects.

Until now we have been assuming that gravity is theonly force acting on planetesimals. In reality there shouldbe other factors such as gas drag and inelastic collisionsbetween planetesimals which might influence their eccen-tricities and inclinations. Here we briefly comment on thepossible changes these effects can give rise to.

In the presence of gas drag eccentricity and inclinationof a particular planetesimal with mass m and physical

size rp tend to decrease with time. Adachi et al. (1976)investigated the damping of planetesimal eccentricitiesand inclinations assuming gas drag force proportional tothe square of planetesimal velocity relative to the gasflow. They came out with orbit-averaged prescription forthe evolution of random velocity of planetesimals whichcan be written at the level of accuracy we are pursuingin this study (dropping all possible constant factors and

assuming that eccentricity is of the same order as incli-nation) as

e2

t e2(e + ) ga

prp, (56)

where g is the mass density of nebular gas, p is thephysical density of planetesimal, and (cs/a)2 1,with cs being the sound speed of nebular gas. In ournotation (see 2) this equation translates into

s2

s2x1/3

s +

Mcm0

1/3,

gp acs

m0Mc(pa3)21/3

106a5/2AU

g/p100

1 km s1

cs

m0

1018g

1/3,(57)

where g is the surface mass density of the gas disk,and numerical estimate is made for p = 3 g cm3 andMc = M. For protoplanetary nebula consisting of so-lar metallicity gas one would expect g/p 50 100,but during the late stages of nebula evolution this ratioshould be greatly reduced by gas removal7.

We use the results of4 to discuss the effect of the gasdrag on the planetesimal random velocities for mass dis-tributions with < 3. We know from

4.1 and

4.2 that

stirring in this case is dominated by biggest planetesimalsand stirring rate is M2()/s2 [see equations (28), (38),and (41)]. Comparing this heating rate with dampingdue to gas drag (57) we immediately see that gas dragcan become important for small planetesimals if planetes-imal velocities are large (s 1). In this case dynamicalexcitation by biggest planetesimals is balanced by gaddrag. Very interestingly, it turns out that such a balanceleads to s x1/12 or x1/15 (depending on whether s isbigger or smaller than (Mc/m0)

1/3), i.e. smaller plan-etesimals have smaller random velocities. Such a behav-ior has been previously seen in coagulation simulationsincluding effects of gas drag on small planetesimals (e.g.Wetherill & Stewart 1993). The time dependence of ve-

locities of small planetesimals affected by the gas dragshould also be different from that given by (29) or (40).In Figure 11 we schematically display the scaling of s asa function ofx in the presence of gas drag for 2 < < 3and different values of .

Inelastic collisions between planetesimals become im-portant for their dynamical evolution roughly when rel-ative velocity at infinity with which two planetesimalsapproach each other becomes comparable to the escapespeed from the biggest of them; in this case gravitational

7 Note that for planet formation in some exotic environmentssuch as early Universe prior to metal enrichment or metal-poorglobular clusters this ratio can be much larger than 100.


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Fig. 11. Schematic representation of the effect of gas drag onplanetesimal velocity spectrum in the case 2 < < 3 (cf. Figure5). Velocity dependence on planetesimal mass is displayed for twovalues of gas drag parameter defined by equation (57): 1 (dashedcurve) and 2 (solid curve), with 1 < 2. Velocity spectra overlapat high masses for which gas drag is no longer important. Inelasticcollisions between planetesimals would affect their velocity spec-trum in a similar manner.

focusing is inefficient. The escape speed vesc 1 ms1rp/(1 km) (for physical density = 3 g cm3) is muchlarger than the Hill velocity for the same mass RH 5cm s

1

a1/2

AU rp/(1 km) (for Mc = M), which justifiesinitially our previous neglect of inelastic collisions (es-pecially far from the Sun where RH is small). Lateron, however, continuing dynamical heating in the diskwould certainly bring velocities of small planetesimalsabove their escape speeds. When this happens, collisionswith bodies of similar size lead to strong velocity dissi-pation. This damping should produce similar effect onplanetesimal random velocities as gas drag does. Indeed,kinetic energy losses in physical collisions between highvelocity planetesimals are roughly proportional to theirinitial kinetic energies, which is similar to the behaviorof the gas drag losses.

At the same time, velocities of light bodies should still

be smaller than the escape velocity from the biggest plan-etesimals, since otherwise these planetesimals would notbe able to dynamically heat the disk and it would coolbelow their escape velocity. Thus, planetesimal veloci-ties cannot exceed the escape speed from the surfaces ofbodies doing most of the stirring, result dating back toa classical study by Safronov (1972). This implies thatgravitational stirring must still be done by the highestmass planetesimals (if < 3) in accordance with what weassumed in the case of gas drag. As a result, one wouldagain expect s to exhibits a power law dependence onmass with positive (but small) slope. Exact value of thepower law index of this dependence is determined by thescaling of energy losses with the mass of planetesimals

involved in collision8. Effect of the collisions should bemost pronounced for small planetesimals for which rela-tive encounter velocities can be much higher than theirvesc. Threshold mass at which highly inelastic collisionsbecome important must constantly increase in time asplanetesimal disk is heated up by massive planetesimals.One should also remember that planetesimal fragmenta-tion in high velocity encounters may become important

in this regime which would significantly complicate sim-ple picture we described here.

In summary, dissipative effects are unlikely to affectthe dynamics of massive planetesimals. However, theybecome important for small mass planetesimals and leadto positive correlation between random velocities of plan-etesimals and their masses (unlike the case of pure grav-ity studied in 4), which has been previously observedin self-consistent coagulation simulations (Wetherill &Stewart 1993). It is thus fairly easy to modify our simplepicture developed in 4 to incorporate the effects of gasdrag and inelastic collisions.

6. discussion.

Although our study has concentrated on a particularcase of power-law size distributions (most of the massis locked up in the power-law part of the planetesimalmass spectrum) its results have a wider range of appli-cability. The reason for this is that the velocity scalingsderived in 4 depend only on what part of planetesimalspectrum dominates stirring and friction and not on theexact shape of mass distribution. For example, supposethat f(x, ) does not have a self-similar power law formof the kind studied in 4.2 (intermediate mass spectrum)but its first moment M1 is still dominated by the small-est masses (most of the mass is locked up in smallestplanetesimals, which dominate friction), while the sec-ond moment M2 is mainly determined by highest masses

(heating is determined by biggest planetesimals xc).Then it is easy to see from our discussion in 4.2.1 thatthe velocity distribution has a form of a broken power-law in mass, changing its slope from 0 at small massesto 1/4 at high masses [see (40) and (42)]. If, on thecontrary, most of the mass is not in smallest bodies butin largest ones (M1 and M2 converge near the cutoff ofplanetesimal spectrum, xc), then both friction andstirring are dominated by biggest planetesimals and ve-locity spectrum must have zero slope for the entire rangeof planetesimal masses, in accord with our considerationof shallow mass distributions (4.1.1). Whenever bothstirring and cooling of planetesimals of mass m are dom-inated by planetesimals of similar mass, we go back tothe Case 1 of

4.3. Finally, in all cases when stirring

and friction are produced by planetesimals much smallerthan those under consideration, velocity profile scales likem1/2 (energy equipartition, see Case 2 in 4.3). Simi-lar rules can be derived on the basis of the results of 4for more complicated mass spectra, and this makes ourapproach very versatile.

One of the interesting features found in our study ofpurely gravitational scattering (see 4) is the develop-

8 Note that in the crude picture of collision energy losses men-tioned above d2e/dt is independent of planetesimal mass. Thenthe power law index of velocity spectrum of small planetesimals isexactly zero. More accurate treatment of planetesimal collisionsmight result in some mass dependence of random velocities.


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ment of a plateau at the low mass end of planetesimalsvelocity distribution for shallow and intermediate massspectra ( < 3). These mass spectra are among themost often realized in the coagulation simulations whichmakes this prediction quite important for interpretationof numerical results. One should be cautious enough notto confuse these plateaus with the manifestations of dis-sipative processes such as gas drag or inelastic collisions

which tend to endow velocity distribution with only veryweak (positive) dependence on mass, see 5.

Another observation which can be made based on ourresults is that the bulk of planetesimals (power law partwhere most of the mass is locked up) rarely reaches acomplete equipartition in energy. Equipartition is real-ized when e,i m1/2, or in our notation s, sz x1/2.Our calculations (4.3) show that when x xc such avelocity spectrum is only possible for very steep massdistributions, > 4, when both dispersion-dominatedstirring and friction are due to the smallest planetesi-mals with masses m m0 (x 1). Mass distributionsthat steep are not often encountered in self-consistentcoagulation or N-body simulations (Wetherill & Stewart

1993; Kenyon 2002; Kokubo & Ida 1996, 2000) and yetthe equipartition argument is very often used to draw im-portant conclusions about the bulk of planetesimal pop-ulation (see below).

At the same time, random energy equipartition is ubiq-uitous for runaway bodies in the high mass tail (x xc)if they still interact with the rest of planetesimals in thedispersion-dominated regime. Since, by our assumption,runaway tail contains very little mass, this equipartitionis very similar to the case of > 4, because in both situ-ations stirring as well as friction are driven by planetes-imals much lighter than the bodies under consideration.It is also worth mentioning that this conclusion changeswhen runaway bodies start interacting with small plan-

etesimals in the shear-dominated regime (see e.g. 4.1.2).Although we did not follow the evolution of the plan-etesimal mass spectrum self-consistently we can alreadyconstrain some of the theories explaining mass distribu-tions in coagulation cascades. For example, N-body sim-ulations of gravitational agglomeration of 103 104 mas-sive planetesimals by Kokubo & Ida (1996, 2000) pro-duce roughly power law mass spectrum with an index of 2.5 within some range of masses. To explain thisresult Makino et al. (1998) explored planetesimal growthin the dispersion-dominated regime with strong gravita-tional focusing. They postulated epicyclic energy to be inequipartition and found that planetesimal mass distribu-tion has a power-law form with a slope of 8/3 2.7.However, using the results of4.2.1 one can immediatelypinpoint the contradiction with the basic assumptionswhich lead to this conclusion. Indeed, slope of8/3 cor-responds to the intermediate mass spectrum (see 4.2)which can never reach energy equipartition. Moreover,this mass spectrum cannot even have a power-law ve-locity distribution with a continuous slope necessary fora theory of Makino et al. (1998) to work. As a result,the explanation provided by these authors for the massspectrum found in N-body simulations cannot be self-consistent. More work in this direction has to be done.

Although theory developed in 4 is in good agreementwith our numerical calculations, one should bear in mind

that our analytical results are asymptotic by construc-tion, i.e. they are most accurate when planetesimal diskhas evolved for a long time and when planetesimal sizedistribution spans a wide range in mass (this can be eas-ily seen by comparing Figures 6 and 7). Real protoplan-etary disks during their evolution might not enjoy theluxury of having such conditions been fully realized; thiscan complicate quantitative applications of our results

to real systems. We believe however that this does notdevalue our analytical results because they provide basicunderstanding of processes involved in planet formation,allow one to clarify important trends seen in simulations,and enable quick and efficient classification of possibleevolutionary outcomes for a wide range of parameters ofprotoplanetary disks.

Several previous studies have concentrated on explor-ing velocity equilibria in systems with nonevolving massspectra in which inelastic collisions between planetesi-mals acted as damping mechanism necessary to balancegravitational stirring (Kaula 1979; Hornung, Pellat, &Barge 1985; Stewart & Wetherill 1988). We have out-lined a qualitative picture of the effects of dissipative

processes in 5 and it agrees rather well with these pre-vious investigations. N-body calculations and particlein a box coagulation simulations represent another classof studies in which velocity distributions have been rou-tinely computed. As we already commented in 1 thesecalculations usually absorb a lot of diverse physical phe-nomena which makes them not easy targets for compar-isons. Still, there are several generic features almost allsimulations agree upon such as (1) roughly power-lawmass spectra typically with slopes < 3 within somemass range, (2) velocity dispersion is constant or slowlyincreases with mass for small planetesimals, and (3) it de-creases with mass for large bodies. This general pictureis in good agreement with our analytical predictions.

7. conclusions.

We have carried out an exhaustive census of the dy-namical properties of planetesimal disks characterized bya variety of mass distributions. Here we briefly summa-rize our results for the case velocity evolution driven bygravitational scattering of planetesimals.

Whenever planetesimal mass distribution in a disk hasa power law slope shallower than 2, we find that plan-etesimal velocity is constant up to the upper mass cut-off. For slopes between 2 and 3 velocity dispersionsare constant at small masses but switch to m1/4 de-pendence above some intermediate mass. As a result,mass spectrum exhibits a pronounced knee at thismass. Distributions with slopes between

3 and

4

have purely power-law velocity spectra with slope satisfying 1/4 < < 1/2, which can be determined nu-merically using equation (C2). Finally, planetesimals indisks characterized by mass spectra steeper than m4

are in energy equipartition, i.e. velocity dispersions scaleas m1/2. Such a difference in behaviors is caused bythe fact that gravitational stirring and dynamical fric-tion receive major contributions from different parts ofplanetesimal mass spectra in different cases.

We also consider a possibility that mass distributionhas a tail of massive runaway bodies sticking out be-yond the exponential cutoff of the power-law mass spec-trum. We have found that the low mass end of the tail


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experiencing dispersion-dominated scattering by all plan-etesimals exhibits complete energy equipartition: e,i m1/2. Most massive runaway bodies interact with allplanetesimals in the shear-dominated regime, which leadsto highly anisotropic velocity distributions of these bod-ies: e m1/6 while i m1/2. Time dependence ofvelocities is determined mainly by the evolution of thecutoff of the power law mass spectrum m

0xc

(). Theseasymptotic results derived using analytical means are ingood agreement with numerical calculations of planetes-imal dynamical evolution presented in 4. They can beeasily generalized to cover more complicated planetesi-mal mass distributions.

Finally, we investigate qualitatively the impact ofdamping processes such as gas drag or inelastic collisionson the planetesimal disk dynamics. We find that manifes-tations of these effects are only important for small massplanetesimals; they exhibit themselves in the form of gen-

tle increase of planetesimal random velocities with masswhich has been previously observed in particle in a boxcoagulation simulations (Wetherill & Stewart 1993).

Future work in this direction should target the self-consistent coupling of the evolution of planetesimal ve-locities to the evolution of mass spectrum of planetesi-mals due to their coagulation. Successful solution of thisproblem would greatly contribute to our understanding

of how terrestrial planets grow out of swarms of plan-etesimals in protoplanetary disks.

I am indebted to Peter Goldreich for inspiration andvaluable advice I have been receiving during all stages ofthis work. I am also grateful to him and Reem Sari forcareful reading of this manuscript and making a lot ofuseful suggestions. Financial support of this work by W.M. Keck Foundation is thankfully acknowledged.

APPENDIX

numerical procedure.

Evolution of planetesimal eccentricities and inclinations is performed by numerically evolving in time the full system(3) with scattering coefficients smoothly interpolated between shear- and dispersion-dominated regimes. Planetesimalsof different masses are distributed in logarithmically spaced mass bins, with bodies in each bin being more massive thanplanetesimals in previous bin by 1.2. Using numerical orbit integrations we have determined the values of constantcoefficients in (9) relevant for the shear-dominated scattering to be

B1 11.4, B2 4.0, D1 11.4, D2 4.2. (A1)Scattering coefficients in the dispersion-dominated regime are taken from Stewart & Ida (2000) who derived analyticalexpressions for viscous stirring and dynamical friction rates in the two-body approximation. We translate their resultsfor our gravitational stirring and friction functions H1,2 and K1,2. This provides us with the dependence of coefficientsA1,2 and C1,2 in (6) on the inclination to eccentricity ratio i/e. Constants a1,2 in (7) are fixed to be a1 = 1.7, a2 = 1.

We assume that mass scale evolves as xc() = F + (), where F 40, = 104, and = 4 in the case ofshallow mass spectrum with = 1.5 and = 1.5 in all other cases. Small parameter is introduced to mimic thedifference between the timescale of dynamical evolution of planetesimal disk and the timescale of planetesimal growth

(which is usually much longer). Besides this, such a form of xc() is chosen arbitrarily. Runaway tail is assumed tohave a power-law form with the same slope but much smaller normalization than planetesimal mass distribution.Tail always extends 8 orders of magnitudes beyond xc to allow interesting velocity regimes to fully develop. Initialdistribution of planetesimal eccentricities and inclinations is set to s(x, 0) = sz(x, 0) = 3x

1/2.

velocity evolution in a mixed state for intermediate mass spectrum.

In the case of intermediate mass spectrum runaway bodies with masses x between xh [defined in (44)] and xshearinteract with small planetesimals [less massive than xs(x) defined by (45)] in the dispersion-dominated regime, butwith large ones (heavier than xs(x)) in the shear-dominated regime. As a result, the equations for s and sz are nowhybrid versions of (30) and (35):

s2

= A1

xs(x)1

dxx2f(x)

s2 2A2s2x

xs(x)1

dxxf(x)

s4+

C1x2/3

xs(x)

dxx2f(x) 2C2s2

x1/3

xs(x)

dxxf(x), (B1)

s2z

= B1

xs(x)1

dxx2f(x)

s2 2B2s2zx

xs(x)1

dxxf(x)

s4+

D1x4/3

xs(x)

dxx2s2f(x) 2D2s2z

x1/3

xs(x)

dxxf(x).(B2)

Following the approach taken in 4.1.2 we neglect l.h.s. of both equations and use (40) and (42) to evaluate integralsin evolution equations. After careful comparison of different contributions in the r.h.s. we find that

C1M2

x2/3 2A2M1 s

2x

s40, (B3)

B1x

7/2s

s20x1/2k

+ D1s20x

1/2k

x4/3

xs

dx(x)3/2f(x) 2B2M1 s2zx

s40. (B4)


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Equation (B3) implies that eccentricity stirring of runaway bodies in the mass range xh x xshear is done mainlyby planetesimals with masses xc, for which the interaction proceeds in the shear-dominated regime. This heating isbalanced by dynamical friction due to the smallest planetesimals, lighter than xk, which interact with these massivebodies in the dispersion-dominated regime.

Vertical heating is somewhat different. Gravitational friction is again dominated by smallest planetesimals whichare in the dispersion-dominated mode. Stirring, however, depends on shape of the mass distribution. If < 5/2,second term in the l.h.s. of (B4) shear-dominated heating by planetesimals with masses between xs and xc dominates9 over the first term which represents the effect of dispersion-dominated heating by planetesimals with

masses xs. In the opposite case, when > 5/2 both terms in the l.h.s. of (B4) produce roughly equal contributions.Using (B3) & (B4) one can easily derive expressions (46)-(48).

details of velocity evolution for the case of steep mass spectrum.

Substituting our guess (54) into (26) and going through all possibilities appropriate for > 3 we find that self-consistent solutions of power-law type can exist in only two cases:

Case 1: 3 + 2 > 0 & 2 + 4 > 0. Case 2: 3 + 2 < 0 & 2 + 4 < 0,

We now consider separately these two possibilities.

Planetesimal velocities.

Case 1.In this case one can easily see that integrals in the r.h.s. of (26) are mostly contributed by x x. This allows usto rewrite (26) in the following form:

x2s20

=x3+2

s20

0

dtt1(1 + t)

1 + t2

t

1 + tA1 2 1

1 + t2A2

. (C1)

In arriving at this expression we have used the fact that f(x, ) = x for > 3 and 1 x xc, and extended theintegration range over t x/x from 0 to (because only local region t 1 matters). From this equation we canreadily see that the l.h.s. of (C1) can be neglected for x 1. As a result, a self-consistent power-law solution for sexists only if integral in (C1) is equal to zero. This leads to the following constraint on the required value of :

I1(, )

I2(, )=

A12A2

, I1(, )

0

dtt1(1 + t)

(1 + t2)2, I2(, )

0

dtt2

1 + t2. (C2)

Solving this equation for given and A1/A2 one can find the slope of the mass spectrum (, A1/A2). At a first sightit is not at all clear that equation (C2) should in general possess a solution for . However, closer look at the problemreveals some interesting patterns.

First of all, it follows from (8) that r.h.s. of (C2) has to be bigger than 1. Second, in the case of energy equipartitionI1(, 1/2)/I2(, 1/2) = 1. Third, I1(, ) as ( 2)/4, while I2 stays finite. These observations prove that(C2) always has a solution for satisfying the constraint posed by equation (55). Combining restriction (55) withinitial constraints of Case 1 we find that Case 1 is only possible for mass spectra with power law slope 3 < < 4.

In fact, one can do things even better by considering separately eccentricity and inclination scalings with mass, i.e.using both equations (16). Assuming that the ratio of inclination to eccentricity sz/s is still constant for x xc, onewould obtain in addition to (C2) another equation dictated by the inclination evolution. It would be identical to ( C2),but with A1,2 replaced by B1,2. Since both A1,2 and B1,2 depend only on sz/s (see 2) one would have two equationsfor two unknowns: and sz/s. Solving them one can uniquely fix both the power law index of planetesimal velocitydependence on mass and the ratio of inclination to eccentricity of planetesimals (which is left undetermined in our

simplified analysis of4).Case 2.Restrictions imposed on and by the conditions of Case 2 imply that all integrals in the r.h.s. of (26) are

dominated by the lower end of their integration range, i.e. by masses 1. Neglecting time derivative in (26) andbalancing contributions in the r.h.s. we find that = 1/2 and

s(x, ) s0()x1/2, x xc. (C3)This solution demonstrates that for x 1 our omission of the l.h.s. of (26) is justified for arbitrary behavior of xc().It also imposes important restriction on the mass spectra for which this scaling can be realized: mus be bigger than4. This means that we have found solutions for all possible positive values of and our study is complete in this sense.

9 For < 5/2 integral in (B4) converges at the upper end of its range, near xc. When = 5/2 this integral is contributed roughlyequally by equal logarithmic intervals in mass between xs and xc; then the second term in the l.h.s. of (B4) dominates over the firstone by ln(xc/xs).


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Velocities of runaway bodies.

Case 1 (3 < < 4).Velocity evolution of runaway bodies in the Case 1 is similar to the situation with the intermediate mass spectrum.

One finds that when mass x satisfies xc x xh s30x3c velocity profile is shaped by the dispersion-dominatedinteraction with all small mass planetesimals (x xc):

s(x, )

sz(x, )

s0()

x1/2x12

c

M2+2

M1+4

1/2

, 3 < < 4, xc x xh. (C4)

The biggest contribution to both stirring and dynamical friction comes the most massive planetesimals

dynamical evolution of planetesimals in protoplanetary disks

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