hybrid methods in planetesimal dynamics: description of a

MNRAS 445, 3620–3649 (2014) doi:10.1093/mnras/stu1558

Hybrid methods in planetesimal dynamics: description of a newcomposite algorithm

P. Glaschke,1‹ P. Amaro-Seoane2‹ and R. Spurzem1,3,4‹1Astronomisches Rechen-Institut, Monchhofstraße 12-14, Zentrum fur Astronomie, Universitat Heidelberg, D-69120 Heidelberg, Germany2Max Planck Institut fur Gravitationsphysik (Albert-Einstein-Institut), D-14476 Potsdam, Germany3National Astronomical Observatories of China, Chinese Academy of Sciences, 20A Datun Lu, Chaoyang District, 100012 Beijing, China4Kavli Institute for Astronomy and Astrophysics, Peking University, 5 Yiheyuan Road, Haidian District, Beijing 100871, P. R. China

Accepted 2014 July 30. Received 2014 July 24; in original form 2013 August 29

ABSTRACTThe formation and evolution of protoplanetary systems, the breeding grounds of planet for-mation, is a complex dynamical problem that involves many orders of magnitudes. To servethis purpose, we present a new hybrid algorithm that combines a Fokker–Planck approachwith the advantages of a pure direct-summation N-body scheme, with a very accurate in-tegration of close encounters for the orbital evolution of the larger bodies with a statisticalmodel, envisaged to simulate the very large number of smaller planetesimals in the disc.Direct-summation techniques have been historically developed for the study of dense stellarsystems such as open and globular clusters and, within some limits imposed by the numberof stars, of galactic nuclei. The number of modifications to adapt direct-summation N-bodytechniques to planetary dynamics is not undemanding and requires modifications. These in-clude the way close encounters are treated, as well as the selection process for the ‘neighbourradius’ of the particles and the extended Hermite scheme, used for the very first time in thiswork, as well as the implementation of a central potential, drag forces and the adjustment ofthe regularization treatment. For the statistical description of the planetesimal disc, we employa Fokker–Planck approach. We include dynamical friction, high- and low-speed encounters,the role of distant encounters as well as gas and collisional damping and then generalize themodel to inhomogenous discs. We then describe the combination of the two techniques toaddress the whole problem of planetesimal dynamics in a realistic way via a transition massto integrate the evolution of the particles according to their masses.

Key words: methods: numerical – methods: statistical – planets and satellites: dynamical evo-lution and stability – protoplanetary discs.

1 IN T RO D U C T I O N

The formation of a planetary system is closely related to the for-mation of the host star itself. Cool molecular clouds collapse andfragment into smaller substructures which are the seeds for subse-quent star formation. Angular momentum conservation in the form-ing clumps forces the infalling matter into a disc-like structure.The subsequent viscous evolution of the disc leads to a transportof angular momentum which channels gas to the protostar in thecentre. These protoplanetary discs are the birth place of planets (fora detailed review see Armitage 2011). Embedded dust grains are

�E-mail: [email protected] (PG); [email protected] (PA-S); [email protected] (RS)

the seed for the enormous growth to bodies of planetary size. Thefirst hint to the structure of protoplanetary discs has been providedby our own Solar system. Through ‘smearing out’ all planets andadding the missing fraction of volatile elements, one can estimatethe structure and mass of the protoplanetary disc. Since the effi-ciency of planet formation is unknown, this yields only a lowerlimit – the minimum-mass solar nebula (Hayashi 1981). The in-ferred surface density decreases with the distance from the sun as∝r−3/2.

Further insight has been obtained by the detection of an infraredexcess of some stars, i.e. additional infrared radiation that does notoriginate from the star but an unresolved disc. Advancements inobservation led in the mid-1990s to the direct imaging of nearbystar-forming regions which opened a new flourishing field in as-tronomy (see O’dell, Wen & Hu 1993, for an example with HubbleSpace Telescope images). Since then a big amount of observations of

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Hybrid methods: a new composite algorithm 3621

protoplanetary discs across the electromagnetic range has been ob-tained and interpreted, both space based (ISO, Spitzer and nowHerschel) and in the ground [Very Large Telescope (VLT), VLT-I,Subaru, Keck]. We refer the reader to ‘The Star Formation Newslet-ter’ URL1 for an overview of the recent papers in star and planetformation and the book review ‘Protostars and Planets V’ (Reipurth,Jewitt & Keil 2007).

Protoplanetary disc masses cover a range from 10−3 to 0.1 M�with a peak around 0.01 M� (data from Taurus/Ophiuchus;Beckwith 1996), and see Andrews et al. (2013) for Taurus–Aurigadisc masses, in accordance with mass estimates deduced from theminimum-mass solar nebula. Since the disc lifetime cannot be mea-sured directly, it is derived indirectly from the age of young, naked(i.e. discless) stars which sets an upper limit. Pre-main-sequenceevolutionary tracks are used to gauge the stellar ages, providinglifetimes of a few 106 yr. The subsequent evolution of the discproceeds in several stages.

Two different scenarios have been proposed to explain the for-mation of kilometre-sized planetesimals.

(i) One process is the gravitational instability of the dust com-ponent in a protoplanetary disc that leads to the direct formationof larger bodies. Goldreich & Ward (1973) propose that an initialgrowth phase of dust grains leads to a thin dust disc that undergoesa gravitational collapse. As the dense dust layer decouples from thegas, it rotates with the local Keplerian velocity, whereas the gascomponent rotates slower as it is partially pressure supported. Thisgives rise to a velocity shear at the boundary, which may excite tur-bulence through the Kelvin–Helmholtz instability. Since the motionof small dust grains in the boundary layer is coupled to the gas, theturbulent velocity field could suppress the formation of a stratifieddust layer, which is a necessary prerequisite for the gravitationalinstability (Weidenschilling 1977).

(ii) The collisional agglomeration of dust particles is an opposedformation mechanism. Relative velocities are dominated by theBrownian motion in the early phases of the growth process. Thismechanism becomes increasingly inefficient with growing mass,but successively the particles decouple from the gas and settle tothe mid-plane – a process that yields even larger velocities withincreasing mass. The sedimentation initiates a growth mode that issimilar to the processes in rain clouds: larger grains drop faster, thusaccreting smaller grains on their way to the mid-plane. Turbulencemay modify this basic growth scenario by forcing the dust grains ina convection-like motion. Dust grains still grow during the settlingprocess, but the turbulent velocity field could mix up dust from themid-plane, and a new cycle begins. Each cycle adds a new layer tothe dust grains – a mechanism that also operates in hail clouds – untilthe grains are large enough to decouple from the turbulent motion.Again, turbulence plays an important role in determining the growthmode and the relative velocities. While the relative velocities arehigh enough to allow for a fast growth, it is not clear a priori thatcollisions are sticky enough to allow for a net growth. High-speedencounters lead to fragmentation, which counteracts agglomeration(e.g. Blum & Wurm 2000, and references therein). An importantbottleneck in the agglomeration process is the fast orbital decay of1 m sized boulders. Their orbital lifetime is as short as 100 yr, and

1 http://www.ifa.hawaii.edu/users/reipurth/newsletter.htm

a quick increase in size – at least over one order of magnitude – isneeded to reduce the radial drift significantly.

To overcome the difficulties associated with each of these scenarios,modifications have been proposed. Magnetohydrodynamic (MHD)simulations include electromagnetic interactions in hydrodynamicalcalculations. See the reviews of Balbus & Hawley 1998 and Bal-bus. 2003. The MHD simulations by Johansen, Klahr & Henning(2006) show that trapping of larger particles in turbulent vorticeshelps in increasing the orbital lifetime, but could also trigger localinstabilities that may lead to the direct formation of planetesimals(Inaba et al. 2005). Johansen et al. (2007) describe a gravoturbu-lence mechanism as a feasible pathway to planetesimal formationin accreting circumstellar discs.

The details of agglomeration have drawn a lot of attention andare still under question (see Kempf, Pfalzner & Henning 1999;Paszun & Dominik 2009; Wada et al. 2009), but the successiveagglomeration of planetesimals is commonly accepted.

In this paper, we present a hybrid algorithm that combines the ad-vantages of a direct-summation code, NBODY6++, with a statisticalone to treat the large number of bodies in the disc. The algorithmincludes also a collisional and fragmentation model. NBODY6++has been used to integrate dense stellar systems such as globularclusters and galactic nuclei, so that we need to modify it to suit thepurpose of a long-term integration of planetesimals. On the otherhand, a statistical approach allows us the integration of a very largenumber of bodies, but it must be able to treat the velocity dispersionconsistently, deal with spatial inhomogeneities and include a whollycollision treatment. The limitation in both schemes is the particlenumber. In the case of the direct-summation code, on the largenumber, and in the case of the statistical one, on the low one. Thismeans that NBODY6++ cannot integrate a realistically large numberof particles as it would be needed and the statistical approach cannotgive an accurate description of the evolution of the orbital elementsof the bodies, because it treats the system as a continuum. We cancombine the two approaches into a single one by addressing thisissue with the introduction of a ‘transition mass’: low-mass parti-cles are integrated with the statistical code and larger mass ones aretreated with NBODY6++. A fiducial value for this mass is of about3 × 10−11 M�.

In Section 2, we present the fundamentals of the theory of planetformation models that allows us to derive the initial models for thecalculations, as well as a consideration about the orbits and pro-toplanet growth. In Section 3, we introduce the direct-summationintegrator, as well as the required modifications for the specificproblem of protoplanet formation. This had led to the so-calledextended Hermite scheme, which is used for the first time here,the introduction of additional forces to treat correctly the dynamicsof planetesimal discs, and the optimization of the integrator to thedisc. In Section 4, we introduce the collisional and fragmentationalgorithms, with the collisional cascades of Section 5 as a direct ap-plication of the fragmentation model. Size-dependent strength, thetreatment of perturbations of equilibrium, migration and collisions,and the choice of a realistic collisional model are addressed in Sec-tions 6, 7, 8, and 9, respectively. Section 10 presents and describesin detail the statistical model, and Section 11 finally explains theway we bring together the two approaches into a single code, thehybrid algorithm, which is the core of this paper. To finish with, inSection 12, we discuss our composite algorithm. All relevant tests,as well as the main results from the simulations are presented inPaper II, Amaro-Seoane et al. (2014), companion to this one.

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3622 P. Glaschke, P. Amaro-Seoane and R. Spurzem

2 IN I T I A L M O D E L S , O R B I T S A N D G ROW T H

2.1 Initial models

The basis for all planet formation models is the structure of theprotoplanetary disc. The work of Hayashi (1981) gives us a robustinitial model. Subsequent evolution of the disc may change thissimple approach, but it is still a valuable guideline.

The total surface density is estimated through the chemical com-position of the disc, which gives the ratio of gas to solids. A fiducialvalue is 1:0.017 (see Cameron 1973). The surface density of the gascomponent is therefore (see Lodders 2003, and note metal abun-dance)

�gas(r) = 1700( r

1 au

)−3/2g cm−2. (1)

Since the dust content is rather low, the gaseous component istransparent to the visible solar radiation. Thus, the gas temperaturefollows from the radiation balance. The three-dimensional densitystructure is given by an isothermal profile with a radially changingscaleheight h. Since the density profile is related to a radially varyingpressure, the gas velocity deviates from the local Keplerian velocity.The balance of forces relates the angular velocity �g to the pressuregradient and one can derive the angular velocity of the gas (Adachi,Hayashi & Nakazawa 1976). It is more appropriate to formulatethe rotation of the gaseous disc in terms of a velocity lag �vg

normalized to the local Keplerian velocity vK:

�vg = r(�g − �)

≈ −ηgvK. (2)

A typical value of �vg for the minimum-mass solar nebula at 1 auis �vg = −60 m s−1.

This simple model provides a brief description of the initial disc.However, the subsequent evolution further modifies the structure ofthe protoplanetary disc. Since embedded dust grains are coupled tothe gas, it is likely that a global migration of solid material changesthe surface density. Moreover, the dust grains are chemically pro-cessed, depending on the local temperature and composition whichintroduces additional spatial inhomogeneities. When the growingparticles pass the critical size of ∼1 m, the strong onset of radialmigration may lead to a final reshaping of the distribution of solidmaterial. While these restrictions weaken the validity of this ap-proach as the ‘true’ initial model, it is still a robust guideline tochoose reasonable surface densities for the solid and the gaseouscomponent after the formation of planetesimals.

2.2 Kepler orbits

Planetesimals in a protoplanetary disc are subjected to various per-turbations: close encounters change their orbits, a small but steadygas drag gives rise to a radial drift and accretion changes the mass ofthe planetesimals. While all these processes drive the disc evolutionon a time-scale of at least a few thousand years, each planetesimalmoves most of the time on an orbit close to an unperturbed Keplerellipse. Though the protoplanetary disc introduces additional per-turbations, the central potential dominates for typical disc massesaround 0.01 M�. Therefore, the classical orbital elements still pro-vide a proper framework to study planetesimal dynamics.

As long as no dominant body is structuring the protoplanetarydisc, it is justified to assume axisymmetry. Hence, the argument ofthe perihelion ω, the longitude of the ascending node � and theeccentric anomaly φE are omitted in the statistical description.

The deviation of planetesimal orbits from a circle is quite small.Thus, it is appropriate to expand the orbital elements of a testparticle moving around a mass M: a planetesimal at a distance r0,in a distance z above the mid-plane and with a velocity (vr, vφ , vz)with respect to the local circular velocity vK has orbital elements[leading order only, see Hornung, Pellat & Barge (1985) and checkGoldreich & Tremaine (1978) for the origin of these ideas]:

a ≈ r0 + 2r0vφ

vK(3)

e2 ≈ v2r + 4v2

φ

v2K

(4)

i2 ≈ z2

r20

+ v2z

v2K

, (5)

where a is the semimajor axis, e is the eccentricity and i is theinclination of the orbit. These expressions allow us later a convenienttransformation between the statistical representation through orbitalelements and the utilization of a velocity distribution function.

2.3 Protoplanet growth

Our work follows the evolution of a planetesimal disc into few pro-toplanets, including the full set of interaction processes. Hence, wesummarize the main aspects of protoplanet growth first to providea robust framework.

During the initial phase, all planetesimals share the same velocitydispersion independently of their mass. The initial random velocitiesare low enough for an efficient gravitational focusing. Hence, thegrowth rate of a protoplanet with mass M and radius R can beestimated as (e.g. Ida, Kokubo & Makino 1993)

M ≈ vrel�

HπR2

(1 + 2GM

Rv2rel

). (6)

The scaleheight H (equation 165) and the relative velocity vrel

are related to the mean eccentricity em =√

〈e2〉 of the field plan-etesimals:

H ≈ vrel/� vrel ≈ ema�. (7)

Thus, the accretion rate in the limit of strong gravitational focusing(2GM/R v2

rel) is

M ≈ 2πR�GM

a2�e2m

(8)

∝ M4/3. (9)

The width �a of this sphere of influence, the heating zone, is relatedto the Hill radius RHill of the protoplanet (Ida 1993):

�a = �aRHill = 4RHill

√4

3

(e2

m + i2m

) + 12

h = 3

√M

3Mc. (10)

Here em and i are eccentricity and inclination of the field planetes-imals, scaled by the reduced Hill radius h of the protoplanet. Mc

is the mass of the central star. The condition that the protoplanetcontrols the velocity dispersion of the field planetesimals reads (Idaet al. 1993)

2M2

2πa�a> �m. (11)

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This condition is equivalent to a lower limit of the protoplanetarymass:

M

m>

(π�a

3√

3

)3/5 (�a2

Mc

)3/5 (m

Mc

)3/5

, (12)

where M/m depends on several parameters, but reasonable valuesyield M/m ≈ 50–100. The velocity dispersion in the heated regionis roughly

v ≈ RHill�, (13)

which gives an interesting relation to the condition that leads togap formation. A protoplanet can open a gap in the planetesimalcomponent if it is larger than a critical mass Mgap (Rafikov 2001)

Mgap

Mc≈

⎧⎪⎨⎪⎩

�a2

Mc

(mMc

)1/3if v � �rHill

�a2

Mc

(mMc

)1/3 (�rHill

v

)2if v �rHill

, (14)

where rHill is the Hill radius of the field planetesimals. The highervelocity dispersion of the field planetesimals (see equation 13) re-duces the growth rate given by equation (8) to

M ≈ 6π��RRHill

e2m

(15)

∝ M2/3. (16)

Different mass accretion rates imply different growth mode. If twoprotoplanets have different masses M1 and M2, their mass ratioevolves as

d

dt

M2

M1= M2

M1

(M2

M2− M1

M1

). (17)

The field planetesimals also damp the excitation due to protoplanet–protoplanet interactions and keep them on nearly circular orbits. Thebalance between these scatterings and the dynamical friction dueto smaller bodies establishes a roughly constant orbital separationb (Kokubo 1997):

b = RHill5

√7e2

mM

2π�aR(18)

b = b/RHill, (19)

where R is the radius of the protoplanet, M is its mass and e isthe reduced eccentricity of the field planetesimals. The stabilizedspacing prevents collisions between protoplanets, but it also restrictsthe feeding zone – the area from which a protoplanet accretes. If allmatter in the feeding zone is accreted by the protoplanet, it reachesits final isolation mass (Kokubo & Ida 2000):

Miso = 2πba�. (20)

Inserting equation (18) yields the isolation mass in units of the massof the host star Mc:

Miso/Mc = (112π4)3/8

(1

3

)5/8 ( 4π

3

)1/8 (e2

m

)3/8

×(

a2�

Mc

)3/2 (a3ρ

Mc

)1/8

≈ 19.67 × (e2

m

)3/8

×(

a2�

Mc

)3/2 (a3ρ

Mc

)1/8

. (21)

In Table 1 we have values for the isolation mass for differentsurface densities. As the protoplanets approach the isolation mass,

Table 1. Isolation mass for differentsurface densities at r = 1 au andMc = 1 M�.

�(g cm−2) Miso/ M� Miso/M⊕2 3.91 × 10−8 0.01310 4.33 × 10−7 0.144100 1.37 × 10−5 4.548

interactions with the gaseous disc and neighbouring protoplanetsbecome increasingly important. We estimate the onset of orbitcrossing by a comparison of the perturbation time-scale τ pert ofprotoplanet–protoplanet interactions with the damping time-scaleτ damp due to planetesimal–protoplanet scatterings. Since the pro-toplanets are well separated (b ≈ 5 . . . 10), it is possible to applyperturbation theory (see e.g. Petit 1986):

τpert ≈ b5

7 h�. (22)

We anticipate Section 10 to derive the damping time-scale

τdamp ≈ 1

2

T 3/2r√

2πG2 ln(�)(M + m)n0 m, (23)

where Tr and Tz are the radial and vertical velocity dispersion ofthe field planetesimals. Hence, the criterion for the onset of orbitalcrossing is

τpert < τdamp. (24)

As the protoplanets control the velocity dispersion of the field plan-etesimals (see equation 13), this condition reduces to

�M > �m ln(�)72

7π

(b

e

)4

> �m × f . (25)

3 T H E D I R E C T- S U M M AT I O N PA RT O F T H EC O D E

3.1 Direct N-body

The protoplanet formation is essentially an N-body problem. Al-though we seek for a more elaborated solution to this problemwhich benefits from statistical methods, the pure N-body approachis a logical starting point. Direct calculations with a few thousandbodies have provided us with a valuable insight into the growthmode (see e.g. Ida 1992; Kokubo 1996), but they are also pow-erful guidelines that help developing other techniques. Statisti-cal calculations rely on a number of approximations and ‘exact’N-body calculations provide the necessary, unbiased validation ofthe derived formula.

The choice of the integrator is a key element in the numericalsolution of the equations of motion. Our requirements are the sta-ble long-term integration of a few ten thousand planetesimals withthe capability of treating close encounters, collisions and the per-spective to evolve it into an improved hybrid code. Approximativemethods like the Fast Multipole Method or Tree codes have a scal-ing of the computational time close to N, but the accuracy in thisregime is too poor to guarantee the stable integration of Keplerianorbits (compare the discussion in Hernquist, Hut & Makino 1993;Spurzem 1999).

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Taking all the requirements into account we have chosenNBODY6++2, an integrator which is the most recent descendantfrom the famous N-body code family from S. Aarseths’ ‘factory’.This version was parallelized by Spurzem (1999), which opened theuse of current supercomputers.

This parallel version, named NBODY6++, offers many versatilefeatures that were included over the past years and more and morerefined as time passed by. While all these elaborated tools deserveattention, we restrict ourselves for brevity to the components whichcontribute to the planetesimal problem. The main components ofthe code are as follows.

(i) Individual time steps and a block time step scheme.(ii) Ahmad–Cohen neighbour scheme.(iii) Hermite scheme.(iv) KS-regularisation for close two-body encounters.

In section 2 of Paper II, we present a series of tests of the hybridcode, including obviously a check of the direct-summation N-bodypart of it. We successfully test the energy balance, we comparethe cumulative size distribution, the mean square eccentricities andinclinations and the surface density and radial velocity dispersions.

3.2 Extended Hermite scheme

The Hermite scheme (Aarseth 1999, 2003) is an integral part ofNBODY6++ and proved its value in many applications. It wouldhave been natural to improve the performance with an additionaliteration, but our first tentative implementations showed rather neg-ative results: the iterated scheme was more unstable, slower andeven less accurate than the plain Hermite scheme. An inspectionof the code structure revealed that the Ahmad–Cohen neighbourscheme is the cause of this.

Each particle integration is composed of two parts – frequentneighbour force evaluations and less frequent total force evaluationsincluding derivative corrections. Every regular correction leads toan additional change in the position of a particle, which introducesa spurious discontinuity in the neighbour force and its derivatives.The Hermite iteration reacts to this artificial jump in two ways: itincreases the regular correction, and – what is more important – itamplifies any spurious error during the iteration which leads to anextreme unstable behaviour.

Since the Hermite iteration is a key element in the efficient inte-gration of planetesimal orbits, we sought for a modification of theHermite scheme that circumvents the depicted instability. The prob-lem gives already an indication of a possible solution. A schemewith much smaller corrections would not suffer from the feedbackof spurious errors.

NBODY6++ stores already the second and third time derivative ofthe forces for the time step calculation. A manifest application ofthese derivatives at hand is the improvement of the predictions tofourth order:

xp = x0 + v0�t + 1

2a0(�t)2 + 1

6a0(�t)3

+ 1

24a(2)

0 (�t)4 + 1

120a(3)

0 (�t)5 (26)

2 Aarseth (1999) gives a nice review on the remarkable history of the NBODY-codes, more details are given in Aarseth (2003).

vp = v0 + a0�t + 1

2a0(�t)2

+1

6a(2)

0 (�t)3 + 1

24a(3)

0 (�t)4. (27)

The prediction to fourth order was used in the iterative schemesof Kokubo, Yoshinaga & Makino (1998) and Mikkola & Aarseth(1998). Again, the new forces ap and ap are calculated to improvexp and vp – but with a modified corrector:

ap = f (xp, vp) (28)

ap = f (xp, vp) (29)

a(2)n (t) = −4a0 − 2ap

�t+ −6a0 + 6ap

�t2(30)

a(3)n (t) = 6a0 + 6ap

�t2+ 12a0 − 12ap

�t3(31)

xc = xp + 1

24(a(2)

n − a0(2))(�t)4

+ 1

120(a(3)

n − a0(3))(�t)5 (32)

vc = vp + 1

6(a(2)

n − a0(2))(�t)3

+ 1

24(a(3)

n − a0(3))(�t)4. (33)

Finally, the derivatives are updated:

a0(t + �t) = ap (34)

a0(t + �t) = ap (35)

a0(2)(t + �t) = a(2)

n + �t a(3)n (36)

a0(3)(t + �t) = a(3)

n . (37)

Our new scheme has an appealing property, which is related to theusage of the higher force derivatives: as the predictor is fourth-orderaccurate, it is equivalent to one full Hermite step. Since the correctoruses new forces to improve the two highest orders, it is equivalent toa first iteration step. Thus, we obtained a one-fold iterated Hermitescheme at no extra cost. This extended Hermite scheme reduces theenergy error by three orders of magnitude, compared to the plainNBODY6++ with the same number of force evaluations.

3.3 Additional forces for planetesimal disc dynamics

NBODY6++ only includes the gravitational interaction of all par-ticles, therefore additional forces have to be added ‘per hand’. Aplanetesimal disc requires two new forces: the presence of a centralstar introduces an additional central potential, while the gaseouscomponent of the protoplanetary disc is the source of a frictionforce. It is important that the new forces are properly included inthe neighbour scheme to assure that regular steps remain larger thanirregular steps. Since a dissipative force breaks the energy conser-vation, one has to integrate the energy loss as well to maintain avalid energy error control. In the next subsections, we describe howwe have done this.

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3.3.1 Central potential

A star is much heavier than a planetesimal. Thus, the central star isintroduced as a spatially fixed Keplerian potential:

c = −GMc

rF = −GmMc x

|x|3 . (38)

Since the orbital motion of the planetesimals sets the dominant(and largest) dynamical time-scale in the system, we included thecentral force as a component of the regular force. Moreover, thecentral potential also introduces a strong synchronization, sinceplanetesimals in a narrow ring share virtually the same regularblock time step. Keeping the star spatially fixed is, in general, avalid approximation but, however, we note that for larger – butplausible – disc masses, of the order of 0.1 M�, if a substantialfraction is incorporated into a massive planet, e.g. a 10 Jupiter-massplanet (rare, but known), then we would be running into a problem,because by holding fixed the star, the energy would alter otherelements in the system, leading to instability. We refer the reader toBromley & Kenyon (2006) for an example of code that allow thecentral star to move, and see Shu et al. (1990) for a general exampleof the issues involved in fixing the position of the star.

3.3.2 Drag force

As the whole planetesimal system is embedded in a dilute gaseousdisc, each planetesimal is subjected to a small, but noticeable dragforce. The drag regime3 depends on the gas density and the size ofthe planetesimals. Kilometre-sized planetesimals are subjected tothe deceleration

dv

dt= −πCD

2mρgasR

2|v − vg|(v − vg) (39)

CD = 0.5, (40)

which is inversely proportional to the radius R(m) of the planetesi-mal in this drag regime. vg is the rotational velocity of the gaseousdisc, which rotates slower than the planetesimal system as it is par-tially pressure supported. The drag force leads to an orbital decaya of the semimajor axis of a planetesimals:

a = −3

4CD

ρgas

ρBody

〈(�v)2〉R(m)�

. (41)

Thus, smaller particles migrate faster, with a maximum at R ≈ 1 m.Even smaller bodies couple to the gas, which reduces the effectivedrag force. The dissipation rate and its time derivative are

Wdrag = Fdrag · v

Wdrag = Fdrag · v + Fdrag · Ftot. (42)

We integrate the dissipation rate Wdrag to maintain a valid energyerror:

�E =∫ t2

t1

Wdrag dt (43)

�t = t2 − t1 (44)

3 The main drag regimes are Stokes (laminar flow), Epstein (mean freemolecular path larger than object size) and Newton’s drag law (turbulentflow). Weidenschilling (1977) provides a nice review on the different dragregimes.

�E = 1

2

(Wdrag,1 + Wdrag,2

)�t

+ 1

12

(Wdrag,1 − Wdrag,2

)�t2 + O(�t5). (45)

The expression is fourth-order accurate in accordance with the orderof the extended Hermite scheme. See Youdin & Chiang (2004) andPinte & Laibe (2014) for examples of the application of the dragforce.

3.3.3 Accurate integration of close encounters: tidalperturbations of KS-pairs and impact of the gaseous disc

Both new forces also demand a modification of the regularizationtreatment. They perturb the relative motion of a KS-pair and mod-ify the orbit of the centre of mass. While the modification of theequations of motion is rather clear, the neighbour scheme requiressome additional work.

Let r1, r2 be the positions of the two regularized particles. Theequations of motion read (G = 1)

r = r2 − r1

r1 = −Mcr1

r31

+ m2rr3

+ Fdrag,1

r2 = −Mcr2

r32

− m1rr3

+ Fdrag,2, (46)

where the perturbations by other particles have been omitted forclarity. Centre-of-mass motion and the orbital motion are sepa-rated:

r = −Mrr3

− Mcr2

r32

+ Mcr1

r31

+ Fdrag,2 − Fdrag,1 (47)

R = −Mcm1

M

r1

r31

− Mcm2

M

r2

r32

+ m1

MFdrag,1 + m2

MFdrag,2 (48)

r = r2 − r1 M = m1 + m2 (49)

R = 1

M(m1r1 + m2r2) . (50)

Two new contributions show up due to the external forces: theKS-pair is tidally perturbed by the central star and influenced bythe gaseous disc. While the aerodynamic properties of a single par-ticle are well understood, two bodies revolving about each othermay induce complex gas flows in their vicinity, which could inval-idate the linear combination of the drag forces on each component.Therefore, we drop the drag force term to avoid spurious dissi-pation. Since the dynamic environment allows virtually no stablebinaries4 in a planetesimal disc, the influence of the drag force onthe encounter dynamics is negligible.

We further decompose the additional acceleration of the centre-of-mass motion, since the neighbour scheme benefits from a clearseparation of the time-scales. Therefore, the tidal perturbation is

4 Tidal capturing of moons starts in the late stages of planet formation, but islimited to the planets or their precursors. However, the quiescent conditionsin an early Kuiper belt lead to a more prominent role of binaries. See thesummary of Astakhov, Lee & Farrelly (2005).

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split into a smooth mean force and a perturbation force:

R = Fmean + Fpert

Fmean = −McRR3

Fpert = McRR3

− Mcm1

M

r1

r31

− Mcm2

M

r2

r32

(51)

= 0 + O(r2). (52)

The mean forces varies on the orbital time-scale and is henceincluded as a regular force component, while the perturbation istreated as an irregular force as it changes with the internal orbitalperiod of the pair.

3.4 Further tuning the dial: optimizing N-body to the disc

An astrophysical simulation is a tool to analyse problems and predictdynamical systems which are not accessible to experiments. Thedesign of a new simulation tool does not only require the carefulimplementation of the invoked physics, but also an analysis of thecode performance to make best use of the available hardware.

We want to apply NBODY6++ for the first time to planet forma-tion, a subject that is quite different to stellar clusters. The centralstar forces the planetesimals on regular orbits which need higheraccuracy than the motion of stars in a cluster. In addition, the or-bital motion also introduces a strong synchronization among theplanetesimals, thus allowing a more efficient integration.

We examine the differences due to the integration of a disc sys-tem in the following sections. In particular, we will address in detailthe role of the geometry of the problem and the neighbour scheme,the prediction of the number of neighbouring particles, the com-munication, the block size distribution and the optimal neighbourparticle number for the direct summation of the massive particlesin the protoplanetary system

3.4.1 Disc geometry and neighbour scheme

The introduction of the neighbour scheme by Ahmad & Cohen(1973) has provided us with a technique to save a considerableamount of computational time in star cluster simulations. Since theaverage ratio of the regular to the irregular time step γt is of the orderof 10, the integration is speeded up by the same factor. One mayexpect a similar speedup for planetesimal systems, but in this casethe time step ratio is roughly 3. The time step is calculated withthe standard Aarseth time step criterion (it should be mentioned,however, that the relation of regular and irregular costs is morecomplicated with GPU technology)

�t =√

ηFF (2) + (F (1))2

F (1)F (3) + (F (2))2, (53)

where F(i) are the force and its time derivatives. It is applied to thecalculation of the regular step using the regular force and accord-ingly to the irregular step based on the irregular force. The regulartime step of a particle orbiting the central mass Mc at a distance r0

is

�tr = √ηr

1

�� =

√GMc

r30

. (54)

For simplicity, we introduce the scaled time step ratio γt =γt√

ηirr/ηreg. The free parameters of the problem are the mean

Figure 1. Time step ratio for Nnb = 100. Curves are plotted for differentvalues of σv/(rHill�). The dotted line is approximation (equation 59).

particle distance r , the velocity dispersion σv (additional to theKeplerian shear), the particle mass mi and the neighbour numberNnb. We employ Hill’s approximation for the central potential andobtain

�ti ≈√

ηi

�f (�, rHill, r, σv, Nnb) (55)

rHill = r03

√2mi

3Mc. (56)

Here f is a yet unknown function. Dimensional analysis leads to

�ti ≈√

ηi

�f

(σv

rHill�,

r

rHill, Nnb

)(57)

γt ≈ f

(σv

rHill�,

r

rHill, Nnb

)(58)

which shows that the time step ratio is essentially controlled bythe interparticle distance and the velocity dispersion. We generateddifferent random realizations of planetesimals discs with differentdensities and velocity dispersions to cover the range of possiblevalues. The neighbour number is fixed to Nnb = 100 to reduce thenoise due to small number statistics, but γt converges to a valueindependently of the neighbour number already for Nnb > 10. Fig. 1shows the numerical calculation of the time step ratio for variousvalues of r and σv. A good approximation to the calculated valuesof γt is

γt ≈ 1.79 ×√

1 + 1.03σ 2

v

r2�2+ 0.94

r3Hill

r3. (59)

Planetesimal discs have usually a small velocity dispersion (com-pared to the orbital velocity) and a low density in terms of the Hillradius, which leaves a major influence to the Keplerian shear. Sincethe shear motion is directly linked to the local Keplerian frequency,this synchronization reduces γt to values smaller than 10. The nu-merical calculations show larger time step ratios with increasingvelocity dispersion and for high densities5, but planetesimal discsare far from these extreme parameter values.

5 r/rHill < 1 corresponds to an unstable self-gravitating disc.

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Figure 2. Regular steps per particle and per 1 N-body time in the inner core(Ri < 0.5) of a 5000 particle plummer model. Plotted are (1) different massexponents with velocity exponent 0 and (2) different velocity exponentswith mass exponent 1/6.

3.4.2 Optimal neighbour criterion

The standard neighbour criterion uses the geometrical distance:particles are neighbours if their distance to the reference particleis smaller than a limit Rs. This criterion is simple and probablythe best choice for an equal-mass system. However, a multimasssystem may require a different criterion, since a massive particleoutside the neighbour sphere could have a stronger influence thanlighter particles inside the neighbour sphere. Also, relative effectsare smaller at large distances. A more appropriate selection shouldrely on some ‘perturbation strength’ of a particle.

It turned out that a better criterion is the magnitude of the fourthtime derivative of the pairwise force F

(4)ij , i.e. those particles are se-

lected as neighbours which produce the largest integration error inaccordance with the Hermite scheme. F

(4)ij is a complicated expres-

sion (compare Appendix A), but the leading term can be estimatedvia dimensional analysis:

F(4)ij ∝ mjv

4ij

r6ij

. (60)

We use this expression to define a new apparent distance betweenthe integrated particle i and a neighbour j:

rapp = rij

(mi

mj

)1/6 (vs

vij

)2/3

, (61)

where vs is an arbitrary scaling velocity to obtain a distance withdimension length. This new distance definition moves massive orfast neighbours to an apparently smaller distance, thus enforcingthat these particles are preferentially included in the neighbour list.In addition, the modified distance is readily included in the con-ventional neighbour scheme. We tested different mass and velocityexponents to verify that equation (61) is the optimal choice. Fig. 2shows that these exponents are indeed the optimal choice for aPlummer model (Plummer 1911) with mass spectrum. The newscheme saves 25 per cent of the force evaluations in the core, butthe impact on a planetesimals system is smaller, as it is the casefor the neighbour scheme. While a velocity-dependent distance re-duces the number of necessary full force evaluations, it introducesa distance changing with time which destabilizes the integration.The result is a much larger energy error compared to the achieved

speedup. Therefore, we only recommend the mass modification ofthe apparent distance.

3.4.3 Neighbour changes

The rate at which the neighbours of a given particle change has anoticeable influence on the accuracy of the code. During the courseof an integration, the second and third time derivative of the regularand irregular force are calculated from an interpolation formula.Whenever a particle leaves (or enters) the neighbour sphere, thesederivatives are corrected by analytic expressions6. Hence, manyneighbour changes lead to a pronounced spurious difference.

We estimated the rate at which particles cross the neighboursphere boundary to quantify this effect. Neighbour changes are dueto the Keplerian shear and the superimposed random velocities ofthe particles. The two effects lead to

N+/− = �trNnb

TorbShear

N+/− = 3

2�trNnb

σv

Rnb

= �trNnb

Torb

3πσv

Rnb�Dispersion, (62)

where Rnb and Nnb are the neighbour sphere radius and the neighbournumber, respectively. In practice, the neighbour changes due to theshear account for up to 80 per cent of the total neighbour changes.The standard regular time step �tr = 2−5 and 50 neighbours yielda change of one particle per regular step, which is fairly safe.

3.4.4 Neighbour prediction

Each integration step is preceded by the prediction of all neighboursof the particles that are due. A regular step requires the full predic-tion of all particles, so there is no possibility to save computingtime. In contrast, an irregular step calculates only neighbour forces,which requires the prediction of less particles. Thus, the predictionof all particles to prepare an irregular step is a simple, but, depend-ing on the block size, computational costly solution. It seems to bemore efficient to predict only the required particles, but random ac-cess to the particle data and the complete check of all neighbour listentries introduces an additional overhead. Therefore, large blocksizes should favour the first approach, whereas the second approachis more suitable for small block sizes.

Both regimes are separated by a critical block size N∗irr. If Nirr

particles with 〈Nnb〉 neighbours are due, then only Nmerge particlesneed to be predicted:

Nmerge ≈ Ntot

(1 − exp

(−Nirr〈Nnb〉

Ntot

))≤ Nirr〈Nnb〉. (63)

The size Nmerge of the merged neighbour lists is smaller thanthe total number of neighbour list entries, since some particles areby chance members of more than one neighbour list. Performancemeasurements show that the prediction of the merged neighbourlists is 10 per cent more costly (per particle) than the full prediction,mainly due to additional sorting and a random memory access. ThusN∗

irr satisfies

Ntot = 1.1 × Nmerge. (64)

6 Appendix A gives a complete set of the force derivatives up to third order.

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Table 2. Ring Communication. Communication part-ners are fixed, while the exchanged data varies. np − 1cycles are needed.

Process 0 1 2 3 4 5 6 7

Send to 1 2 3 4 5 6 7 0Receive from 7 0 1 2 3 4 5 6

Table 3. Hierarchical Communication. Communication partnerschange after every cycle. The exchanged data amount doubleswith every new cycle, hence only ln2(np) cycles are needed.

Cycle Process 0 1 2 3 4 5 6 7

1 Exchange with 1 0 3 2 5 4 7 62 Exchange with 2 3 0 1 6 7 4 53 Exchange with 4 5 6 7 0 1 2 3

Inserting equation (63) yields the critical block size:

N∗irr ≈ 2.4

Ntot

〈Nnb〉 . (65)

The prediction mode is chosen according to the actual block size.

3.4.5 Communication scheme

NBODY6++ is parallelized using a copy algorithm. A complete copyof the particle data is located on each node, so the integration stepof one particle does not need any communication. Therefore, ablock of Nbl particles is divided in np parts (np is the processornumber), which are integrated by different processors in parallel.The integration step is completed by an all-to-all communication ofthe different subblocks to synchronize the particle data on all nodes.Hence, the amount of communicated data is proportional to Nbl ×np. A communication in a ring-like fashion (see Table 2) needs np −1 communication cycles, but a hierarchical scheme (see Table 3)sends the same amount of data with only ln2(np) communicationcycles. The difference between the two approaches remains small, aslong as the communication is bandwidth limited, i.e. the blocks arelarge. Small block sizes shift the bottleneck to the latency, which issignificantly reduced by the second scheme – especially if the coderuns on many processors. See Table 5 for details on computers.

A hierarchical scheme reduces the latency, but nevertheless it ispossible that the parallel integration is actually slower than a singleCPU integration. We estimated both the runtime on one CPU andon a parallel machine to explore the transition between these tworegimes. The latency time τl per communication is included in thewallclock time expressions for one regular/irregular step:

τl = αA

tsingle = αNblNnb (66)

tpar = α

(NblNnb

np︸︷︷︸Arithmetic

+ A ln2(np)︸︷︷︸Latency

+ BNbl︸︷︷︸Communication

). (67)

If tsingle (runtime on a single CPU) is equal to tpar (parallel compu-tation), one can deduce the critical block size Nmin which gives the

Table 4. Timings on a Beowulf cluster (Hydra, seeTable 5). See text for an explanation of the vari-ables. Timings are obtained for a maximal neigh-bour number LMAX=64. In practice, B is twice aslarge due to storage rearrangements in NBODY6++.See Appendix 5 for details on the computers.

Block np α(μs) τl(μs) A B

Irregular 10 0.35 51 145 4.5Regular 10 0.22 113 512 40Irregular 20 0.35 308 877 8.8Regular 20 0.22 368 1668 75

minimal block size for efficient parallelization:

tsingle = tpar

Nmin = A ln2(np)np

Nnb(np − 1) − Bnp. (68)

The hierarchical communication gives a minimal block size thatincreases logarithmically with the processor number. Equation (67)gives immediately the speedup S and the optimal processor numberfor a certain block size Nbl:

S = np

1 + Anpln2(np)NblNnb

+ Bnp

Nnb

np,opt(Nbl) = ln(2)Nbl × Nnb

A− Hierarchical. (69)

A comparison to the optimal processor number for a ring commu-nication

np,opt(Nbl) =√

Nbl × Nnb

A− Ring

tpar = α

(NblNnb

np+ Anp + BNbl

)(70)

stresses the efficiency of the hierarchical communication, since itallows a much larger processor number for a given problem size.Equations (66) and (67) are also useful to derive the total wallclocktime, since the total runtime scales with the number of regular andirregular blocks:

Nreg ≈ TN1/3

N1/3nb

√ηreg

(71)

Nirr ≈ TN1/3

√ηirr

. (72)

These equations are only approximate expressions, but they give theright order of magnitude without detailed calculations that need aprecise knowledge of the N-body model. Tables 4 and 6 summarizethe timing parameters drawn from our experience with the Hydraand JUMP parallel supercomputers.

3.4.6 Block size distribution

The preceding section showed that the block size is closely related tothe efficiency of the parallelization. Small blocks are dominated bythe latency and the parallelization could be even slower than a singleCPU calculation. Therefore, we derive the block size distribution forthe block time step scheme to assess its influence on the efficiency.

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Table 5. Specs of the different supercomputers used in running the algorithm.

Name Beowulf (Hydra) Titan JUMP

Institute ARI/ZAH ARI/ZAH Forschungszentrum JulichLocation Heidelberg Heidelberg JulichProcessors 20 64 1248Speed 2.2 GHz 3.2 GHz 1.7 GHzProcessors/Node 2 2 32Network Myrinet Infiniband Gigabit–EthernetBandwidth 2 Gbit s−1 20 Gbit s−1 10 Gbit s−1

Table 6. Timings on the IBM. More than 32processors require more than one node.

Block np α(μs) τl(μs) A B

Irregular 8 0.29 255 837 1.7Regular 8 0.60 981 1763 4.1Irregular 16 0.28 188 700 1.7Regular 16 0.60 306 561 6.7Irregular 64 0.27 241 887 7.7Regular 64 0.46 401 871 21.7

Figure 3. Time step distribution f = dp/dh. The short-dashed line on theleft indicates approximation equation (74), whereas the dashed line on theright defines a reasonable upper limit hmax .

Suppose that the time steps7 h of all N particles in the model aredistributed according to some known function f:

dp = f (N, h) dh, (73)

where f is in most cases a complicated function. It involvesspatial averaging and integration over the velocity distribution,which could be quite complicated even for simple time step formu-lae. Nevertheless, there is a constraint on the time step distribution,simply because every particle has a neighbour within a finite dis-tance: there is some upper limit hmax that restricts the major fractionof the time steps to a finite interval. Thus, it is possible to capturethe main features of the time step distribution with an expansionaround h = 0 (Fig. 3 sketches this approximation):

7 We use h instead of �t in this section to avoid unclear notation.

f ≈ C(N )ha h ≤ hmax, (74)

where a is the lowest non-vanishing order of the expansion. Now,we consider a block level with the largest possible time step hk. Thenumber of particles Nbl in this block is

Nbl = N

∫ hk

0f dh ≈ C(N )

a + 1(hk)a+1. (75)

According to the block time step scheme, the number of blocksper time with the largest possible time step hk is proportional to(hk)−1. Therefore, the probability that a block size is in the range[Nbl, Nbl + dNbl] is

dp ∝∑

k

δ

(Nbl − C(N )

a + 1ha+1

k

)1

hk

dNbl, (76)

where δ is Dirac’s delta function. The sum over the logarithmicallyequidistant time steps hk is approximated by an integral

dp ∝∫ ∞

0δ

(Nbl − C(N )

a + 1ha+1

)d ln(h)

hdNbl

≈ 1

a + 1N

−(a+2)/(a+1)bl dNbl. (77)

Thus, the average block size and the median of the block sizedistribution are

〈Nbl〉 ≈ 1

aNa/(a+1)

median(Nbl) ≈ 2a+1. (78)

Special expressions for the average block size were already derivedby Makino (1988), but the general relation of the time step distri-bution to the block size distribution is a new result. The median issurprisingly independent of the particle number, i.e. 50 per cent ofall blocks are always smaller than a fixed value. It seems that thisis a threat to the efficiency of the method, but the median of thewallclock time

median(N2bl) ≈ N

2(a+1)/a(79)

demonstrates that these small blocks account only for a small frac-tion of the total CPU time. We confirmed the derived block sizedistribution (equation 77) by numerical calculations (see Fig. 4).The order parameter a is roughly two in (at least locally) homoge-nous systems, while an additional Keplerian potential reduces theorder to a = 1. A planetesimal disc – or more precisely, a nar-row ring of planetesimals – has a very narrow distribution of timesteps since all particles share nearly the same orbital period. Thus,the regular block size is always equal to the total particle numbermaking the parallelization very efficient.

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Figure 4. Cumulative irregular block size distribution for an N = 2500particle Plummer model.

3.4.7 Optimal neighbour number

We treated the mean neighbour number Nnb so far as some fixedvalue. But it is also a mean to optimize the speed of the integration.Large neighbour spheres reduce fluctuations in the regular forces al-lowing larger regular steps, which reduces the total number of forceevaluations. But larger neighbour lists also imply a larger commu-nication overhead, as all the neighbour lists have to be broadcast tosynchronize the different nodes. The best choice balances these twoextremes, thus maximizing the speed.

Before we derive the optimal neighbour number on a parallel ma-chine, we briefly summarize the known solution for a single CPUrun (Makino 1988) for an extensive derivation. The computationaleffort of the irregular steps is proportional to the neighbour number,while the number of force evaluations for the regular steps is pro-portional to the total number of particles, reduced by the time stepratio γt:

γ t : = �treg

�tirr

TCPU = f (N )

(Nnb + N

γ t(Nnb)

)(80)

γ t(Nnb) ≈ N1/3nb , (81)

where f(N) collects all factors depending only on the total numberof particles. Optimization with respect to the neighbour number Nnb

yields the well-known result:

0 = d

dNnbTCPU

Nnb,opt ∝ N3/4. (82)

The calculation of the elapsed time for NBODY6++ on a PC clusterincludes more terms. For clearness, we restrict ourselves to a rathersimple model that involves only the dominant terms to show howparallelization influences the optimal neighbour number. We makethe following approximations.

(i) We only take the force calculation and communication intoaccount.

(ii) We use the same time constants for regular and irregularexpressions.

(iii) We neglect all numerical factors that are comparable to unity.

The total CPU time is an extension of equation (70), which isapplied to the regular and the irregular step. A new constant Bn

includes the neighbour list communication separately, while allfactors depending on N are represented by f(N):

Nbl ≈ N2/3

γ t ≈ N1/3nb

Tirr = f (N )

(NblNnb

p+ Ap + BNbl

)

Treg = 1

γ tf (N )

(NblN

p+ Ap + (B + BnNnb)Nbl

)Ttot = Tirr + Treg. (83)

Optimization with respect to the processor number p leads to

0 = ∂

∂pTtot

popt =√

N2/3N4/3nb + N5/3

A(N1/3nb + 1)

≈ N5/6

√AN

1/6nb

. (84)

Further optimization with respect to the neighbour number givesthe expression:

0 = ∂

∂NnbTtot (85)

0 = N4/3nb − 1

3N − 1

3AN−2/3p2 − 1

3Bp

+ 2

3NnbBnp. (86)

For a fixed p or Bn = 0 (very fast neighbour list communication),we recover for large N:

Nnb,opt ∝ N3/4. (87)

In general, one cannot neglect the neighbour list communication.Therefore, we seek for the optimal choice of p and Nnb, thus com-bining equations (84) and (86):

AN5/3nb +

(2

3Nnb − B

3Bn

)Bn

√AN

1/6nb N5/6

= 1

3(N1/3

nb + 1)NA. (88)

Since this equation has no closed solution, we identify the domi-nant terms in equation (88) to calculate the asymptotic solution forlarge N:

Nnb,opt ≈(

A

4B2n

)3/5

N1/5 N (

3A

4B2n

)3/2

. (89)

For small N, we get the approximated solution

Nnb,opt ≈(

N

3

)3/5

N <

(3A

4B2n

)3/2

. (90)

Fig. 5 compares the approximate expressions with the numericalsolution of equation (88). In spite of the complicated structure ofequation (88), both approximate expressions are reliable solutions.

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Figure 5. Optimal neighbour number as function of particle number N. Theplot includes the numerical solution of equation (88) and the two asymptoticsolutions. Timing constants are taken from a Beowulf cluster.

The example uses timing constants derived from our local Beowulfcluster:

A ≈ 200 B ≈ 5 Bn ≈ 0.5. (91)

If we compare the new optimal neighbour number to the single CPUexpression (equation 82), we find that the influence of the neigh-bour list communication favours much smaller neighbour numbers.Nnb increases so slowly with the particle number that a neighbournumber around 100 is a safe choice.

4 C O L L I S I O NA L A N D F R AG M E N TAT I O NM O D E L

Some examples of fragmentation and accretion in the literatureare Cazenave, Lago & Dominh (1982), Beauge & Aarseth (1990),Spaute et al. (1991), Wetherill & Stewart (1993) and Kenyon &Bromley (2004) for examples of coagulation as applied to planetformation. In our own algorithm, we use the approach of Glaschke(2003), which was applied to asteroid families. The following sec-tion describes the approach to fragmentation that is implemented inthe code.

4.1 Handy quantities for quantifying the models

Although the whole scenario describing the process is quite com-plex, there are a few measures that describe the most importantaspects.

(i) Mass of the largest fragment ML, or dimensionless fl =ML/M where M is the combined mass of the two colliding bodies.

(ii) fl < 12 refers to fragmentation, whereas fl > 1

2 is denoted ascratering.

(iii) Energy per volume S that yields fl = 12 is denoted as impact

strength.(iv) fKE := 2E

fragkin /Ekin: fraction of the impact energy that is con-

verted into kinetic energy of the fragments.

Different fragment sizes and velocities are summarized by ap-propriate distribution functions. mi, Di and vi are mass, diameterand modulus of the velocity of a given fragment, respectively.

(i) Fragment size distribution:

(a) Nm(m) : number of all fragments with a mass mi ≥ m,

(b) M(m) : mass of all fragments with a mass mi ≥ m,(c) ND(D) : number of all fragments with a diameter Di ≥ D.

The distribution functions are related to each other:

Nm(m) = ND(D(m))

M(m) =∫ ∞

m

x

∣∣∣∣dNm(x)

dx

∣∣∣∣ dx

Nm(m) =∫ ∞

m

1

x

∣∣∣∣dM(x)

dx

∣∣∣∣ dx.

D(m) is the size–mass relation.

(ii) Velocity distribution:

(a) v(m): mean velocity as a function of mass.

4.2 Prediction of collisional outcomes: derivationfrom a Voronoy tessellation

Any theoretical or empirical prescription of a collision has to relatethe aforementioned parameters, namely the impact energy, to thesizes and velocities of the produced fragments. The central quantityis the impact strength, which is a measure for the overall stabilityof a body. Objects smaller than 1 m are accessible to laboratoryexperiments, while collisions of larger bodies up to asteroid sizehave to be analysed by complex computer simulations. Asteroidfamilies, which are remnants of giant collisions in the asteroidbelt, provide independent insight, although the data is difficult tointerpret.

We selected two different impact strength models as referencefor our work. The first was obtained by Housen & Holsapple (1990)through the combination of asteroid family data and laboratoryexperiments via scaling laws:

S = S0

(R

1 m

)−0.24[

1 + 1.6612 × 10−7

(R

1 m

)1.89]

fKE = 0.1 (92)

S0 = 1.726 × 106 J m−3 = 1.726 × 107erg cm−3. (93)

Later, Benz & Asphaug (1999) obtained another resultthrough smoothed particle hydrodynamics simulations (for basalt,v =3 km s−1):

S = S0

(R

1 m

)−0.38[

1 + 6.989 × 10−5

(R

1 m

)1.74]

fKE ≈ 0.01 (94)

S0 = 6.082 × 105 J m−3 (95)

ρ = 2.7 g cm−3. (96)

fKE is a measure of the kinetic energy that is transferred to thefragments:

Efragkin = fKE

2Ekin. (97)

See Fig. 7 for a depiction, we introduce a dimensionless measure γ

of the relative importance of gravity for the result of a collision. It

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Figure 6. Section for fl = 0.04 and n = 3. The largest fragment is colouredin dark grey. In this calculation, 60 × 60 × 60 grid cells are used. Note thedecomposition in grid cells and the Voronoy polyhedra which form thefragments.

is defined as the ratio of the energy per volume SG that is necessaryto disperse the fragments to the impact strength S0:

SG = 2π4 − 2 3

√2

5fKEGR2ρ2

γ := SG/S. (98)

The first step towards the prediction of a collisional outcome is torelate the impact energy and the impact strength to ascertain the sizeof the largest fragment fl. Laboratory experiments and simulationsindicate the functional form

ε(fl) ={

2(1 − fl) for fl > 12

(2fl)−1K otherwise

.

ε = Ekinρ

2SM, (99)

which is both valid in the fragmentation regime and the crater-ing limit. The size of the largest fragment is used to derive thefull size distribution. To accomplish the decomposition ‘seed frag-ments’ are distributed inside the target according to the largest de-sired fragment. The full set of fragment is derived from a Voronoytessellation8 using these seed points. Fig. 6 depicts the result of sucha decomposition. The fragment velocities are calculated from thetotal kinetic energy after the collision to initiate a post-collisionalN-body calculation to treat reaccumulation.

We conducted a large set of such calculations to cover a suffi-cient range in f i

l (i.e. impact energy) and γ (i.e. body size). Table 7summarizes the derived values of the largest and second largest frag-ment including reaccumulation. For the strength of planetesimalsand large objects, see Leinhardt & Stewart (2012).

8 The Voronoy tessellation assigns every volume element to the closest seedpoint. First applications date back to the 17th century, but the Russianmathematician of Ukrainian origin Georgy Feodosevich Voronoy put it ona general base in 1908.

Figure 7. Impact strength according to equations (92) and (94). The rightaxis gives the corresponding impact velocity according to S = 1/2ρv2 withρ = 2.7 g cm−3.

We note that the connection to the collision energy needed tofragment an object and eject half of the mass of the combined pairto infinity, usually referred to as Q, is that this quantity is the totalenergy, and S the energy per volume.

5 C O L L I S I O NA L C A S C A D E S

Although the formation of planets requires a net growth due tocollisions, this destructive process plays a role in the formation oflarger bodies as the overall size distribution controls the accretionrate of the protoplanets. While this has been addressed in detail byKobayashi et al. (2010) and Belyaev & Rafikov (2011), we deemit worth to have a closer look into this mechanism now within ourapproach.

5.1 Self-similar collisions

The first step is to decompose an inhomogeneous system intosmaller subvolumes which are locally homogenous. Furthermore,it is assumed that these subvolumes hardly interact with eachother. Hence, it is possible to apply the particle-in-a-box-method(Safronov 1972) to analyse collisions within the small subvolumes.

The distribution function is evolved by the coagulation equation.We modified the equation given by Tanaka, Inaba & Nakazawa(1996) by introducing a new function Mred to arrive at a moreconcise expression:

0 = ∂

∂tmn(t, m) + ∂

∂mFm(t, m), (100)

where n(m) is the distribution function. The mass flux Fm is givenby

Fm = −∫ ∫

n(t, m1)n(t, m2)ξdm1dm2

ξ ≡ σ (m1, m2)vrelMred(m,m1, m2) (101)

Mtot =∫

n(t, m)mdm (102)

∂

∂tMtot = Fm(mmin), (103)

where ξ is the coagulation kernel, n is the already introduced sizedistribution,vrel is the mean relative velocity, σ is the cross-sectionfor colliding bodies (m1, m2) and Mred is the newly introduced

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Table 7. Data compilation of the fragmentation calculations.

γ f il 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Largest fragment

0.025 52 0.100 00 0.200 00 0.300 00 0.400 00 0.500 00 0.600 00 0.700 00 0.800 000.198 97 0.100 00 0.200 00 0.300 00 0.400 00 0.500 00 0.600 00 0.700 00 0.800 000.679 85 0.100 00 0.200 00 0.300 00 0.400 00 0.576 12 0.673 15 0.820 06 0.960 731.140 50 0.100 00 0.315 26 0.357 08 0.613 62 0.835 11 0.928 32 0.943 80 0.975 721.770 57 0.108 84 0.588 83 0.759 74 0.879 22 0.927 55 0.936 62 0.971 07 0.979 242.260 21 0.158 95 0.688 91 0.872 17 0.895 92 0.929 65 0.960 89 0.967 01 0.987 273.116 26 0.309 54 0.837 74 0.902 72 0.926 82 0.959 43 0.955 65 0.976 77 0.987 91

Second largest fragment

0.025 52 0.081 71 0.107 70 0.064 71 0.081 07 0.049 76 0.039 82 0.031 71 0.021 550.198 97 0.091 74 0.095 33 0.084 10 0.069 67 0.049 30 0.048 25 0.035 10 0.020 770.679 85 0.077 13 0.083 65 0.077 91 0.083 87 0.070 26 0.061 47 0.060 82 0.008 471.140 50 0.086 21 0.072 56 0.103 31 0.096 40 0.024 67 0.006 75 0.007 83 0.002 651.770 57 0.079 09 0.055 49 0.069 61 0.020 35 0.003 29 0.007 19 0.002 73 0.001 612.260 21 0.066 93 0.022 88 0.005 28 0.008 82 0.005 84 0.002 68 0.006 64 0.001 263.116 26 0.069 40 0.008 84 0.003 84 0.004 88 0.000 64 0.010 07 0.002 25 0.001 62

fragment redistribution function. Mred contains all information onthe fragments arising from the breakup of body m1 due to the im-pact of body m2. Its definition avoids double counting of collisionsin the above integral. The redistribution function is related to thedifferential number distribution function ncoll(m1, m2, m), i.e. thenumber of fragments produced by a collision per mass interval.Since the target m1 formally disappears, it is included as a negativecontribution:

Mred(m, m1, m2) :=∫ m

0

(ncoll(m1,m2, m) − δ(m − m1)

)m dm. (104)

Mass conservation in each collision is reflected by Mred(0, m1,m2) = Mred(∞, m1, m2) = 0. The cross-section σ depends on thevelocities and radii Ri of the particles. A simple approach is thegeometric cross-section:

σ (m1,m2) = π (R1 + R2)2. (105)

If gravity plays an important role during encounters, we have totake into account gravitational focusing:

σ (m1,m2) = π (R1 + R2)2

[1 + 2G(m1 + m2)

v2rel(R1 + R2)

]. (106)

A special class of collisional models are self-similar collisions. Self-similarity implies an invariance of the collisional outcome withrespect to the scale of the colliding bodies. If the target mass aswell as the projectile mass are enlarged by a factor of 2, then onlythe masses of all fragments doubles without further changes inthe collisional outcome. They allow us to introduce the convenientdimensionless fragment redistribution function fm:

Mred(m, m1, m2) = mfm(m1/m, m2/m). (107)

We follow Tanaka et al. (1996) and employ the substitution9

m1 = mx1, m2 = mx2 to simplify equation (101):

Fm = −∫ ∫

n(t, mx1)n(t, mx2)m11/3

×σ (x1, x2)vrelfm(x1, x2) dx1 dx2. (108)

9 A similar approach to the solution of the coagulation equation is theZakharov transformation, see Connaughton, Rajesh & Zaboronski (2004).

A simple solution is a steady-state cascade with Fm = const.The loss of bodies of a given size is balanced by the fragmentsupply from larger bodies, hence the system maintains a steadystate ∂

∂tn(t, m) = 0. Equation (108) inspires the ansatz n(m) ∝

m−k, which yields k = 11/6. This is the well-known equilibriumslope in self-similar collisional cascades, which was already foundby Dohnanyi (1969). Strong gravitational focusing changes theexponent10 to k = 13/6. Both steady-state solutions seem to berather artificial, as they contain an infinite amount of mass and re-quire a steady mass influx from infinity. However, they provide anappropriate description for the relaxed fragment tail of a size distri-bution, as long as the largest bodies provide a sufficient flux of newfragments. Once the largest bodies start to decay, the finite amountof mass in the system leads to an overall decay of the collisionalcascade. Thus, we seek for a more general solution to equation (100)using the ansatz n(t, m) = a(t)n0(m):

∂

∂ta(t) = −Ca(t)2 (109)

mn0(m) = 1

C

∂

∂mFm, (110)

where C is determined by fixing n0 at an arbitrary value m∗. a(t) isindependent of the collision model:

a(t) = 1

1 + Ct(111)

C ∝ n(m∗). (112)

A power-law solution is n0(m) ∝ − Cm−k+1 which is only valid forC < 0 (agglomeration dominates). To examine C > 0, we perturbthe already known equilibrium solution:

n0(m) = N0 m−k − CN1 m−2k+2 + O(C2) (113)

1

N1= (2 − k)

∫ ∫x−k

1 x−2k+22 σ (x1, x2)

×vrel (fm(x1, x2) + fm(x2, x1)) dx1dx2, (114)

10 Tanaka et al. (1996) state that k < 2 is a necessary condition for a finitemass flux. However, their analysis is not valid for all possible collisionalmodels.

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where N1 is small if the integral on the right-hand side is large. Thisis the case for a sufficiently large impact strength. Equation (113)has the interesting property that n(m′) = 0 for some mass m′, giventhat k < 2. This mass m′ represents the largest body in the system,e. g. the largest asteroid in a fictitious asteroid belt.

6 SIZE-DEPENDENT STRENGTH

We model the size-dependent strength S with a power law to exam-ine the influence on the equilibrium solution. The velocity disper-sion v and the collisional cross-section σ are also modelled withpower laws to account for relaxation processes:

v = v0

(m

m0

)w

(115)

σ = σ0

(m

m0

)s

(116)

S = S0

(m

m0

)α

. (117)

The subscript ‘0’ denotes values for an arbitrarily chosen scalingmass. Since smaller bodies are more abundant than larger ones, wesafely assume that most collisions involve a large mass ratio. Inaddition, we assume w < 0, since we expect energy equipartition tosome degree in most cases. These restrictions lead to the followingsimplifications (m1 > m2):

σ (m1,m2) ≈ σ (m1) (118)

vrel ≈ v(m2) (119)

ε ≈ 1

2

m2ρv2rel

m1S1. (120)

Therefore, the smaller body m2 enters only through the specificenergy ε:

Fm ≈ −∫ ∫

n(t, m1)n(t, m2)σ (m1)vrel(m2)m1

×fm(m1/m, ε)dm1dm2. (121)

We introduce new dimensionless quantities with the help of equa-tions (118)–(120) to simplify the integral:

m1 = mx1

m2 = m0

(m1

m0

) 1+α1+2w

(2S0

ρv20

) 11+2w

ε1

1+2w . (122)

Again, we assume a power law for the density n ∝ m−k and changethe integration parameters to (x1, ε). Applying the constant-fluxcondition yields the equilibrium exponent

k ≈ s + 3 + α + w(2s + α + 5)

2 + α + 2w(123)

and the scaling exponent k′ of the total mass-loss

k′ ≈ s − w + 1

2 + α + 2w

∂

∂tMtot ∝ −S−k′

S = 2S0

ρv20

. (124)

The exponent k′ in equation (124) is close to unity for realistic valuesof the free parameters. Thus, the mass-loss is roughly inverselyproportional to the strength of the bodies. The general formula

equation (123) contains the special solution of O’Brien (2003),who concentrated on the parameters s = 2/3, w = 0 and a specialcollisional model. In fact, the derivation applies to a much widerclass of collisional models that we denote as scalable collisionalmodels. Scalable indicates that the model is self-similar except ascaling of the impactor mass.

7 PE RT U R BAT I O N O F E QU I L I B R I U M

The derived scaling relations provide insight into the overall prop-erties of a collisional cascade, which is in (or close to) equilibrium.However, they do not provide information on how the equilibriumis attained or how the system responds to various external pertur-bations. A rigorous approach would be the approximate solution ofthe coagulation equation11, which is by no means simple since itrequires a careful analysis of the collision model.

Hence, we turn to perturbations of the equilibrium size distribu-tion, as it is easier to assess the quality of the derived expression fora variety of collision models. In addition, all equations are linear inthe perturbation, allowing the detailed analysis of the solution.

If the equilibrium solution n(m) = n0(m/m0)−k is perturbed witha small deviation �n(m), we get to first order:

0 = ∂

∂tm�n(m) + ∂

∂mF p(t, m)

F p = −∫ ∫

�n(m1)n(t, m2)σ (m1, m2)vrel

× (Mred(m,m1, m2) + Mred(m,m2, m1)) dm1dm2. (125)

Despite of the expansion in �n, equation (125) is still a compli-cated integro-differential equation. Thus, it is not possible to obtaina solution without further information about the problem. Whilethere is no general solution, we restrict our attention to self-similarcollisional processes. In virtue of this assumption, it is possible tosimplify equation (125), as we can see in equations (126) and (126).In those expressions

0 = ∂

∂tm�n(m) − n0m

30σ0v0

∂

∂m

×∫

�n(t, mx1)F (x1)(mx1/m0)kdx1 (126)

F (x1) =∫

m2k−30 x−k

1 x−k2

σ (x1, x2)

σ0

vrel

v0

×(fm(x1, x2) + fm(x2, x1)) dx2, (127)

where σ 0 and v0 are velocity and cross-section of an arbitrarilychosen scaling mass m0. F(x1) contains all information about thecollisional process. If collisions do not result in extreme outcomes,like cratering or a complete destruction of the target, most of thefragment mass is contained in bodies with similar size as the parentbody. Hence, we expect that F(x1) peaks around x1 ≈ 1 and drops tozero as x1 gets larger (or smaller). We introduce the dimensionlessrelative perturbation g(m):

g(m) = �n(m)

n(m)= �n(m)mk

n0mk0

. (128)

11 Appendix C highlights a possible approach.

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Figure 8. Scaled fragmentation kernel G(u) for a simple fragmentationmodel (see equation 160) and different scaled impact strength S.

Thus, the new differential equation reads:

0 = ∂

∂t(m/m0)1−k�g(m)

− n0m20σ0v0

∂

∂m

∫�g(t, mx1)F (x1) dx1. (129)

We change to logarithmic coordinates to arrive at a convolutionintegral:

u = ln(m/m0) u1 = ln(x1). (130)

Furthermore, we define a collisional time-scale τ 0

τ0 = (n0m0σ0v0)−1 (131)

to obtain a more concise expression. The transformed equation is

0 = ∂

∂tg(t, u)eu(2−k) − 1

τ0

∂

∂u

∫g(t, u + u1)G(u1)du1 (132)

G(u) = F (eu)eu. (133)

If g(u) is varying on a scale larger than the width of the kernel G(u)(compare Fig. 8), it is justified to expand g(u) under the integral.We retain the first two moments of G(u):

0 = ∂

∂tg(t, u)eu(2−k) − G0

τ0

∂

∂ug(t, u) − G1

τ0

∂2

∂u2g(t, u) (134)

Gk = ∫ukG(u) du. (135)

The first-order moment G1, which introduces a diffusive term, isomitted in the following for clarity12. We introduce a fragmentationtime τ frag(u) and transform equation (134) back to m:

0 = ∂

∂tg(t, m) − m

τfrag(m)

∂

∂mg(t, m) (136)

τfrag = τ0

G0eu(2−k)

= τ0

G0(m/m0)2−k. (137)

12 The study of wave-like structures in the size distribution (see e.g. Bagatinet al. 1994) requires even the second-order moment G2.

Equation (136) is a modified advection equation, which conservesthe total mass. It is possible to derive equations similar to equation(136) for any collisional model. However, the general approachis less fruitful, as it lacks a robust frame of a known equilibriumsolution and reliable scaling relations. Therefore, we provide onlythe extension to scalable collisional models in Appendix B. Wereadily obtain the general solution

g(t, m) = f

(t + τ0

(m/m0)(2−k)

G0(2 − k)

)

�n(t, m) = n(m)f

(t + τ0

(m/m0)(2−k)

G0(2 − k)

). (138)

The function f is determined by the initial value g(0, m) of the per-turbation. As the collisional cascade evolves, the initial perturbationfunction is shifted as a whole to smaller masses. This evolution be-comes clearer if we attach labels M(0) to the initial perturbationfunction and follow the time evolution of these tags. The functionsM(t) are the characteristics13 of the differential equation (136):

M(t) = m0

[(M(0)/m0)(2−k) − t/τ0G0(2 − k)

]1/(2−k). (139)

The meaning of the fragmentation time τ frag becomes clear by therelation

M

M= −τfrag (140)

which is the time until a body has lost a significant fraction of itsmass due to destructive collisions. A comparison of the perturbationequation (136) with the scaling relations from the previous sectiongives the scaling of the zeroth-order moment G0 with respect to theimpact strength:

G0 = G′0S

−k′. (141)

G′0 should only depend on the fragmentation model (i.e. fragment

size distribution as a function of the largest fragment fl) within thelimits of this approximation. Fig. 8 shows that the scaling with theimpact strength works quite well, except slight variations which aresmall compared to the covered range of impact strengths. Likewise,it is possible to restate the total equilibrium flux Feq in terms of G′

0:

Feq(m) ≈ −G′0

2n(m)2σ (m)m3vrelS

−k′. (142)

The fragmentation time-scale τ frag(m) allows a more intuitive ex-pression:

Feq(m) ≈ −1

2

n(m)m2

τfrag(m). (143)

Our simple collisional model (see Fig. 8 and equation 160) refersto

Feq(m) = −(1 . . . 30) × n(m)2σ (m)m3vrelS−k′

. (144)

8 M I G R AT I O N A N D C O L L I S I O N S

The local perturbation analysis is only applicable to a planetesimaldisc if the migration velocity of the planetesimals is negligibly small.This assures that collisional cascades at different radial distancesdo not couple to each other, so that the whole disc is composed ofmany local cascades. While this assumption is justified for larger

13 In general, characteristics of a partial differential equation are paths alongwhich the solution is constant.

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bodies, migration strongly influences bodies below 1 km in size.Hence, we must extend our analysis to examine the influence ofmigration on the (no longer) local collisional processes.

We assume that the collisional evolution of the system leads toan equilibrium planetesimal distribution everywhere in the disc:

�0(r,m) = �r,0(r)C0(m), (145)

where �r(r) is the total surface density at a given distance r, whileC0(m) is the universal equilibrium distribution. Though the plan-etesimal distribution at larger sizes is likely different at different lo-cations in the disc, we only demand a universal function at smallersizes, where migration is important. The power-law exponent kdepends on the details of the invoked physics, but numerical simu-lations show that k ≈ 2 is a fiducial value. Equation (145) does notyet include migration effects. If we include migration, the surfacedensity is modified to

�(r,m) = g(r,m)�0(r,m), (146)

where the dimensionless function g contains the changes due tomigration. The collisional evolution is governed by the continuityequation with an additional collisional term

∂�(r,m)

∂t− 1

r

∂

∂r(v(r,m)r�(r,m)) = �coll, (147)

where v(r, m) is the migration velocity (see equation 41), definedsuch that positive v imply an inward migration. We express thecollisional term with the help of equation (136) and seek for asteady-state solution � = 0:

1

τfrag(m, r)

∂g

∂mm�r,0(r) + 1

r

∂

∂r

(gvr�r,0(r)

) = 0, (148)

where τ frag(m, r) is the fragmentation time-scale of a mass m at adistance r. Since the surface density � and the various contributionsto the drag force are well described by a power law (with respect toradius), equation (148) further simplifies to

1

τfrag(m, r)

∂g

∂mm + ∂g

∂rv − b

rgv = 0, (149)

where b is a combination of the various invoked power-law ex-ponents. As the surface density � and the gas density drop withincreasing radius in any realistic disc model, it is safe to assumeb > 0. We choose a self-similar ansatz for g:

g(r,m) = g(ζ ) , ζ = mgm(r). (150)

The new differential equation is

1

τfrag(m, r)

dg

dζmgm(r) + m

dg

dζ

dgm

drv − b

rgv = 0 (151)

which is equivalent to the more concise expression:

d ln(g)

d ln(ζ )

(r

vτfrag+ d ln(gm)

d ln(r)

)= b. (152)

We assume a power-law dependence for the time-scale ratioτmig/τ frag:

r

vτfrag= τmig

τfrag= (m/m0)km (r/r0)kr . (153)

The cut-off mass m0 at a distance r0 has a time-scale ratioτmig/τ frag = 1, which defines a proper lower cut-off within thiscontext. Hence, the solution is

b = d ln(g)

d ln(ζ )

(ζ km + kr

km

)(154)

gm(r) = (r/r0)kr /km

m0(155)

g(ζ ) =(

1 + kr

kmζ km

)−b/kr

. (156)

Though the analytical solution equation (156) provides a completedescription of the lower cut-off of the size distribution, it is moreappropriate within the frame of this discussion to translate the equi-librium solution to an equilibrium mass-loss due to migration:

�mig(r,m) = −bv

r� + bv

r�

kr/km

ζ km + kr/km

= − b�

τmig + kr/kmτfrag. (157)

An inspection of the time-scale ratio shows that the mass exponentkm should be positive, whereas simple estimations of kr on the basisof the minimum-mass solar nebula are somewhat inconclusive. Thevalue of kr is so close to zero that any change in the assumedequilibrium slope or the impact strength scaling gives easily bothpositive and negative values. Moreover, equation (157) requiresa globally relaxed planetesimal disc, but the huge spread in thevarious involved time-scales at different radii inhibits any significantrelaxation in the early stages.

However, it is possible to gain valuable information from thetwo limiting cases kr > 0 and kr < 0. Both values of kr give theproper limit g → 1 at large masses, where the migration time-scaleis much larger than the fragmentation time-scale and we recover thesteady-state collisional cascade.

A positive exponent kr reduces the effective mass-loss due tomigration, as fragments from the outer part of the disc replenishthe local mass-loss. Hence, the fragmentation time-scale controlsthe net loss of smaller planetesimals. In contrast, a negative expo-nent kr leads to a pronounced cut-off in the size distribution, sinceonly larger planetesimals are replenished through inward migration.Though the mass-loss rate is singular at some mass m′, this sharpcut-off is an artefact due to the perturbation approximation.

Our analysis is subjected to several restrictions. We applied theperturbation equation to values of g that exceed the limit for a safeapplication (i.e. g �≈ 1) of the perturbation expansion. Furthermore,the steady-state solution requires a global relaxation of the col-lisional processes, which is practically never obtained during thedisc evolution. Despite of these restrictions, we gained insight on amore qualitative level. Numerical calculations indicate that the per-turbation approximation is inappropriate close to the lower cut-offof the size distribution. However, a comparison of different expo-nents kr (see Fig. 9) attributes only a minor role to the replenishmentof fragments due to inward migration. Only unrealistic small slopesb of the migrational mass influx would strengthen the importanceof this process. Though temporally non-equilibrium phenomena arenot ruled out by the previous derivation, their study would requirethe global simulation of the system.

9 C H O I C E F O R TH E C O L L I S I O NA L M O D E L

Any detailed study of a collisional system requires the specifica-tion of a realistic collisional model. We first subtract the well-known perfect accretion model. While it is an oversimplificationfor collisions among kilometre-sized planetesimals, its simplicity

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Figure 9. Cut-off function g according to equation (156). The mass expo-nent is km = 1/3, while the mass influx exponent is b = 1.75 according tothe minimum solar nebula.

allows a reliable code testing and eases the comparison with otherworks:

Mred(m, m1, m2) = −m1�(m − m1) − m2�(m − m2)

+ (m1 + m2)�(m − m1 − m2). (158)

Although our fragmentation model (see Section 4) provides avery detailed description of the outcome of a collision, we abandonmost of the details for the following reasons. The computationaleffort of the numerical solution of the coagulation equation scaleswith the third power of the number of mass bins. Hence, we choosea mass grid whose resolution is by far smaller than the informationprovided by the detailed collisional model. As a mismatch of themass resolution could produce undesired artefacts, a lower resolu-tion of the collisional model is needed for consistence. Thus, onlythe largest fragment fl(f i

l , γ ) and the second fragment f(2)l (f i

l , γ )(which contains information on reaccumulation) enter the fragmentsize distribution:

Mred(xM)

M=

⎧⎪⎪⎨⎪⎪⎩

1 if x ≥ fl

1 − fl if fl > x ≥ f(2)l

(1 − fl − f(2)l )(x/f

(2)l )fl otherwise

.

(159)

Both values fl and f(2)l are interpolated from Table 7, where the

initial fragment size f il is calculated from the dimensionless impact

energy ε. We used a reduced fragmentation model for test purposes:

Mred(xM)/M ={

1 if x ≥ fl

(1 − fl)(x/fl)fl otherwise. (160)

Table 8 summarizes the most important model parameters.

Table 8. Main parameters ofthe collisional model.

ρ 2700 kg m−3

k 1/6Model Gaussian scatterfKE 0.1K 1.24

We note that Mred is defined in such a way that the equationsconserve the mass (see equation 100). Also, by using fl and f

(2)l ,

we avoid the problem that is inherent to using only fl: namely thatthe exponent of the distribution of fragmentation is 1 − fl, whichgoes to 0 in the cratering limit.

10 STATISTICAL MODEL

We work out a scheme to integrate the evolution of particle numberand velocities in our coagulation code. Our implementation followsthe standard procedure of former works, such as Hornung, Pellat& Barge (1985). The direct approach to the integration of an N-body system is, in principle, possible for any particle number. Thisprocedure becomes computationally too expensive for very largeparticle numbers. In the following of this section, we employ theusual notation as followed by e.g. Binney & Tremaine (2008), intheir description of the Fokker–Planck equation, and will avoidusual equations that can be found in that one or any other text book.

Instead of tracking all particle orbits, we can define a distributionfunction f (a phase-space density), which gives the probability tofind a particle at a position x with a velocity v, i.e. the state ofthe system. As long as only dynamical interactions are taken intoaccount, the number of all particles (e. g. stars, planetesimals) isconserved. f is a function of six variables, so an exact solution isusually very complicated or even impossible. However, it is possibleto gain valuable insight into the problem by taking the moments ofthe distribution function (see Schneider, Amaro-Seoane & Spurzem2011):

〈xni vm

j 〉 =∫

f (x, v)xni vm

j d3xd3v n, m > 0. (161)

The first-order moment with respect to velocity gives the time evo-lution of the mean velocity v :

ν∂vj

∂t+ νvi

∂vj

∂xi

= −ν∂

∂xj

− ∂(νσ 2ij )

∂xi

+ ν

(∂vj

∂t

)coll

(162)

vi = 1

ν

∫f (x, v)vid

3v

σ 2ij = vivj − vi vj , (163)

where σ ij is the anisotropic velocity dispersion and the continuityequation was used to arrive at a more concise formulation. While thestructure of the moment equations is already familiar from hydro-dynamics, they do not provide a closed set of differential equations,since each differential equation of a given moment is related to (yetunknown) higher order moments. Hence, any finite set of momentaneeds a closure relation – additional constraints that relate the high-est order moments to known quantities. The choice of this relationis a key element in the validity of the equations, but it is not uniqueand depends well on the problem at hand (compare e.g. Larson1970).

10.1 Distribution function

Any statistical description of a planetesimal disc requires the knowl-edge of the distribution function. Since the full problem includingcollisions, encounters and gas drag has no analytic solution, a col-lisionless planetesimal disc (i.e. no perturbations) is a natural basisfor further investigations. The distribution function that describessuch a simplified system is a solution of the Boltzmann equation.

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A special solution to the Boltzmann equation is a thin homogenousplanetesimal disc

f (z, v) = ��

2π2TrTzmexp

(−v2

r + 4v2φ

2Tr− v2

z + �2z2

2Tz

)(164)

provided that the radial velocity dispersion Tr and the vertical dis-persion Tz are small compared the mean orbital velocity vK. Theazimuthal velocity dispersion Tφ is locked to Tr by the local epicyclicfrequency κ in a central potential, where the ratio 1 : 4 is a specialsolution of (see e.g Binney & Tremaine 2008) κ2Tr = 4�2Tφ . Allvelocities vr, vφ and vz refer to the local Keplerian velocity. Thenormalization is the same as in Stewart & Ida (2000). A planetesi-mal disc is a slowly evolving system compared to the orbital time,hence it is reasonable to use equation (164) as a general solutionof the perturbed problem. �, Tz and Tr are now functions of timeand of the radial distance to the star. All information on the systemis contained in these three momenta of the distribution function,where higher order moments can be deduced from equation (164).Thus, the functional form of the distribution function represents animplicit closure relation.

The validity of this approximation can be further assessed bya closer examination of the Boltzmann equation. We summarizeall perturbations in an evolution time-scale Tevol and reduce theradial structure to some typical length-scale �r to estimate thedeviation from the functional form equation (164). A comparisonwith the Boltzmann equation shows that the difference is smallif the migration time-scale and the evolution time-scale are largecompared to the orbital time T0, T0 � �r/〈vr〉 and T0 � Tevol. Anorder-of-magnitude estimate of the evolution time supports theserequirements. Furthermore, numerical calculations confirm that thevelocity distribution stays triaxial Gaussian (see Ida 1992).

The distribution function is equivalent to an isothermal verticaldensity structure with scaleheight h:

h =√

Tz

�2(165)

ρ(z) = ρ0 exp

(− z2

2h2

). (166)

Thus, the central density ρ0 and the mean density 〈ρ〉 are related tothe surface density in a simple way:

ρ0 = �√2πh

〈ρ〉 = ρ0√2. (167)

The triaxial Gaussian velocity distribution is equivalent to aRayleigh distribution of the orbital elements e and i14:

dn(e2, i2) = 1

〈e2〉〈i2〉 exp

(− e2

〈e2〉 − i2

〈i2〉)

de2di2

〈e2〉 = 2Tr

(�r0)2〈i2〉 = 2Tz

(�r0)2. (168)

Planetesimal encounters couple the time evolution of eccentricityand inclination, so that the ratio i2/e2 tends to an equilibrium valueafter a few relaxation times. It is close to 1/4 in a Keplerian potential,but the precise value also depends on the potential itself (Ida et al.1993).

14 Equations (3)–(5) provide the coordinate transformation.

10.2 Dynamical friction

The Coulomb logarithm in the standard formulation ofChandrasekhar (1942), �, arises from an integration over all impactparameters smaller than an upper limit lmax , given by

� ≈ σ 2v lmax

G(m + M). (169)

Although encounters in the gravitational field of the sun deviatefrom pure two-body scatterings, it is safe to neglect the presenceof the sun if the encounter velocity is large compared to the Hillvelocity15 �RHill. Thus, the classical dynamical friction formulais also applicable to planetesimal encounters in the high-velocityregime, though a generalization to triaxial velocity distributions σ i

is necessary (see e.g. Binney 1977). An additional complication isthe choice of lmax (i.e. the choice of the Coulomb logarithm). Thereare several scalelengths, which could determine the largest impactparameter lmax: the scaleheight of the planetesimal disc, the radialexcursion due to the eccentric motion of the planetesimals and theHill radius of the planetesimals. As it is not possible to derive aunique expression for lmax from first principles, a proper formulais often fitted to N-body calculations (compare equation 173). Thevelocity dispersion of a planetesimal disc is triaxial with Tφ/Tr =1/4 and Tz/Tr ≈ 1/4. We take these values and expand the set ofequations of Binney (1977) for small velocities vM:

dvM,r

dt≈ −1.389 vM,r

√2πG2 ln(�)(M + m)n0 m

T3/2

r

dvM,φ

dt≈ −3.306 vM,φ

√2πG2 ln(�)(M + m)n0 m

T3/2

r

dvM,z

dt≈ −3.306 vM,z

√2πG2 ln(�)(M + m)n0 m

T3/2

r

. (170)

The derived expressions provide a compact tool to analyse dynam-ical friction in disc systems. However, the involved approximationsare too severe compared to the needs of an accurate description.While these concise expressions are valuable for basic estimations,the following sections derive viscous stirring and dynamical frictionformulae for a planetesimal system in a rigorous way.

10.3 High-speed encounters

We return to the Boltzmann equation as a starting point for thederivation of the scattering coefficients. Again, we omit the basicformulae that can be found in any textbook and employ the usualnotation and terminology. In virtue of the ansatz for the distribu-tion function (see equation 164), it is sufficient to derive the timederivative of the second-order velocity moments Tr and Tz. Sincethe distribution function is time independent in the absence of en-counters, only the collisional term contributes to the time derivativeof the velocity dispersions Tk (k ∈ (r, z, φ) in the following:

dρTk

dt=∫

d3vmv2k

(∂f

∂t

)coll

. (171)

The collisional term invokes the averaging over many differentscattering trajectories and is, given that the underlying encountermodel is analytically solvable, still too complex to derive an ex-act expression. If most of the encounters are weak – a realistic

15 Whenever relative velocities are classified as ‘high’ or ‘low’ in the fol-lowing sections, a comparison with the Hill velocity is implied.

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assumption in a planetesimal disc – it is possible to expand the col-lisional contribution in terms of the velocity change �vi. This is theusual Fokker–Planck approximation (see e.g. Binney & Tremaine2008). The diffusion coefficients D contain all information on theunderlying scattering process. The usual next step is to considertwo interacting planetesimal populations m, m∗ with distributionfunctions

f = ��

2π2TrTzmexp

(−v2

r + 4v2φ

2Tr− v2

z + �2z2

2Tz

)

f ∗ = ��∗

2π2T ∗r T ∗

z m∗ exp

(−v2

r + 4v2φ

2T ∗r

− v2z + �2z2

2T ∗z

)(172)

to be able to evaluate the terms in the Fokker–Planck approximation.In this regard, we follow in our approach the scheme of Stewart &Ida (2000) except some minor changes in the notation, and refer thereader to their work for more details.

The determination of a proper Coulomb logarithm � leaves roomfor further optimization. A careful comparison with N-body modelsgives rise to the empirical choice (Ohtsuki, Stewart & Ida 2002):

� = 1

12(〈e2〉 + 〈i2〉)〈i2〉1/2

(173)

e =√

2Tr

�RHilli =

√2Tz

�RHill. (174)

Ohtsuki et al. (2002) also report a further improvement by settingB ≡ A.

10.4 Low-speed encounters

Encounters in the low-velocity regime exhibit a wealth of differentorbits, as the solar gravity field perturbs the two-body scattering.Only a small subset of the trajectories represents simple, regularorbits like Tadpole or Horseshoe orbits16. Hence, an examinationof this velocity regime is best done by integrating the equations ofmotions numerically.

Ohtsuki et al. (2002) integrated a large set of planetesimal en-counters and extracted fitting formulae that cover the low-velocityregime. Their expressions for viscous stirring are given in theirwork. The stirring rate of the radial velocity dispersion approachesa finite value for very low velocity dispersions, while the stirring ratefor the vertical velocity dispersion drops to zero as the velocity dis-persion decreases. This different behaviour of the two limits is dueto the encounter geometry: if two planetesimals have zero inclina-tion, they may still excite higher eccentricities during an encounter,but they remain confined to the initial orbital plane preventing anyexcitation of inclinations.

As the stirring rates are only valid in the low-velocity regime,Ohtsuki et al. (2002) introduced special interpolation coefficients Ci.These coefficients tend to unity for very small velocity dispersions,and drop to zero in the high-velocity regime. Thus, the interpolationformulae are properly ‘switched off’ in the high-velocity regime,so they do not interfere with the known high-velocity stirring rates.

10.5 Distant encounters

All formulae include only the stirring rates due to close encoun-ters, but non-crossing orbits also contribute to the overall change of

16 The most famous example of such a regular orbit are the two Saturnianmoons Janus and Epimetheus which share nearly the same orbit.

the velocity distribution. As these distant encounters lead to smallchanges of the orbital elements, the problem is accessible to per-turbation theory; see Hasegawa (1990) for a detailed treatment.Stewart & Ida (2000) integrated the perturbation solution over allimpact parameters to derive the collective effect of all distant en-counters:

d〈e2〉dt

= �m∗�∗r20

(m + m∗)2〈PVS,dist〉

〈PVS,dist〉 = 7.6α(m + m∗)2

M2c

×EXINT

(α h2

(〈e2〉+〈e∗2〉)

)− EXINT

(α h2

(〈i2〉+〈i∗2〉)

)〈e2〉 + 〈e∗2〉 − 〈i2〉 − 〈i∗2〉

EXINT(x) := exp(x)�(0, x) h = 3

√m + m∗

3Mc

α ≈ 1, (175)

where α accounts for the uncertainty in the smallest impact param-eter that is regarded as a distant encounter. While distant encountersare already included in the interpolation formula of the low-velocityregime, we use the modified expression:(

dTr

dt

)dist

= 1

2(�r0)2

(d〈e2〉

dt

)dist

(1 − C1)

= GMcr0�m∗�∗

2(m + m∗)2〈PVS,dist〉(1 − C1). (176)

Stewart & Ida (2000) omitted the change in the inclination, as it issmall due to the encounter geometry. We derive here the integratedstirring rate for completeness, in the lengthy equation (177).

d〈i2〉dt

= �m∗�∗r20

(m + m∗)2〈QVS,dist〉〈QVS,dist〉 = 0.4

α2(m + m∗)2

M2c

× 1

〈e2〉 + 〈e∗2〉 − 〈i2〉 − 〈i∗2〉

×⎡⎣1 − αh2

〈i2〉 + 〈i∗2〉EXINT

(α

h2

(〈i2〉 + 〈i∗2〉))

− (〈i2〉+〈i∗2〉)EXINT

(α h2

(〈e2〉+〈e∗2〉)

)− EXINT

(α h2

(〈i2〉+〈i∗2〉)

)〈e2〉 + 〈e∗2〉 − 〈i2〉 − 〈i∗2〉

⎤⎦.

(177)

A close inspection of the integrated perturbation shows that theabove formula is roughly a factor 〈i2〉+ 〈i∗2〉 smaller than the cor-responding changes in the eccentricity.

10.6 Gas damping

The presence of a gaseous disc damps the velocity dispersion ofthe planetesimals and introduces a slow inward migration. Adachiet al. (1976) used the drag law equation (39) to approximate17 the

17 A formal expansion at e = 0, i = 0, ηg = 0 is not possible, since thedrag law involves the modulus of the relative velocity. Kary, Lissauer &Greenzweig (1993) corrected a missing factor 3/2 in equation (178).

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average change of the orbital elements:

τ0 = 2m

πCDρgR2vKηg = |vK − vg|

vK

d

dte2 ≈ −2e2

τ0

(0.77 e + 0.64 i + 3

2ηg

)(178)

d

dti2 ≈ −2i2

τ0

(0.39 e + 0.43 i + 1

2ηg

)d

dta ≈ −2a

τ0ηg

(0.97 e + 0.64 i + ηg

)(179)

ηg is the dimensionless velocity lag of the sub-Keplerian rotatinggaseous disc.

10.7 Unified expressions

All expressions for the different velocity regimes are constructedsuch that a smooth transition between the different regimes is as-sured. Thus, a simple addition of all contributions yields alreadythe unified expressions

dTr

dt=(

dTr

dt

)high

+(

dTr

dt

)low

+(

dTr

dt

)gas

+(

dTr

dt

)dist

(180)

dTz

dt=(

dTz

dt

)high

+(

dTz

dt

)low

+(

dTz

dt

)gas

+(

dTz

dt

)dist

(181)

which cover the full range of relative velocities. Although only twopopulations m and m� were assumed, equations (180) and (181) arereadily generalized to a multimass system by adding a summationover all masses.

10.8 Inhomogeneous disc

The preceding derivations assumed a homogeneous disc, whichsimplified the calculation, since the integration over all impact pa-rameters needed no special precaution. A more sophisticated con-sequence is that the spatial density and the density in semimajoraxis space are equal:

�(r) = �(a) = �0. (182)

Density inhomogeneities break this simple relation, as particlesat the same radial distance could have different semimajor axes,and particles with the same semimajor axis are located at differentpositions at a given time. While both representations are equivalent(i.e. describe the same system in different ways), we choose thedensity in semimajor axis space as the primary density18. The spatialdensity is derived as

�(r) =∫

1√2πa2〈e2(a)〉 exp

(− (a − r)2

2a2〈e2(a)〉)

�(a) da. (183)

Likewise, Tr and Tz are also functions of the semimajor axis.Furthermore, an inhomogeneous surface density invalidates the

averaging over all impact parameters. Planetesimal encounter aremost efficient for impact parameters smaller than a few Hill radii, sothe derivation is still valid if the surface density is roughly constant

18 We denote �(a) also as ‘surface density’ and refer to a as a radial coor-dinate. However, all formulae are precise in discriminating both representa-tions in r and a.

on that length-scale. However, a planetesimal that is large enoughwill ‘feel’ the spatial inhomogeneities or even generate densityfluctuations. Hence, it is essential to extend the validity of the aver-aged expressions to inhomogeneous systems. We use the averagedexpressions⟨

dTr,z

dt

⟩= �(a)

∫ ∞

−∞

dTr,z(b)

dtdb (184)

as a starting point (dTr,z/dt excludes the surface density, as opposedto the averaged expressions). The (yet unknown) scattering contri-bution dTr,z/dt as a function of the impact parameter b is our startingpoint for a general expression for a varying surface density:

dT (a0)r,z

dt=∫ ∞

−∞�(a0 + b)

dTr,z(b)

dtdb. (185)

We restate equation (185) in terms of a weight function w(b):

dT (a0)r,z

dt=⟨

dTr,z

dt

⟩1

�(a0)

∫ ∞

−∞�(a0 + b)w(b) db (186)

w(b) = �(a0)dTr,z(b)

dt

⟨dTr,z

dt

⟩−1

. (187)

The numerical solution of the Hill problem gives some insightinto how the weight function w(b) changes with the impact parame-ter. We follow a similar approach as in Petit (1986) for the evolutionof particle densities in the Hill problem.

10.9 Diffusion coefficient

We concentrated on the evolution of the velocity dispersion so far,but scatterings among planetesimals also change the semimajor axisof the disc particles, inducing a diffusive evolution of the surfacedensity:

∂�

∂t= �a(D�). (188)

The diffusion coefficient D is related to the typical change in semi-major axis �a and the time-scale T2Body on which planetesimalencounters operate:

D ≈ (�a)2

T2Body. (189)

If we neglect the radial displacement during an encounter, thechange in semimajor axis is solely due to the change of thevelocity:

− GM

2a= −GM

r+ 1

2v2

�a ≈ 2a2

GMv · �v. (190)

An average over all orientations of the velocity v and the velocitychange �v yields the mean square change in semimajor axis:

〈(�a)2〉 ≈ 4a3

3GM〈(�v)2〉

= 4

3�2(�Tr + �Tφ + �Tz). (191)

This yields the mean diffusion coefficient

D ≈ 4

3�2

(5

4

d

dtTr + d

dtTz

), (192)

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where the time derivatives of the velocity dispersions Tr and Tz aretaken with respect to encounters. We note that this approach does notreplace large angle scattering by close approaches of planetesimalsto large planets. However, Tr and Tz do contain large angle scatter-ing. We note that it is important to maintain an integral number ofparticles as they migrate radially inward.

10.10 Coagulation equation

We treat collisions with the particle in a box method, where thecollision rate is a number density times a cross-section times arelative velocity. In our scheme, we use numerical calculations fromGreenberg (1991) and Greenzweig (1992)19 as a basis for a unifiedfitting formula

σ = σ × 0.572 (1 + 3.67vHill/vrel)

(1 + 1.0

σ�2

Tz

)−1/2

(193)

σ = σgeom

(1 + v2

∞v2

rel + 1.8v2Hill

)(194)

v2rel = 1

2(Tr + Tφ + Tz) vHill = �rHill (195)

which gives an effective cross-section σ for planetesimal–planetesimal encounters. Equation (193) reduces to the well-knowngravitational focusing formula in the limit of high velocities:

σ ∝ σgeom

(1 + v2

∞v2

rel

)(196)

v2∞ = 2G(m1 + m2)

R1 + R2. (197)

If the vertical velocity dispersion is small, the disc becomes two-dimensional and the cross-section is proportional to R. The maindifferences to the two-body cross-section in equation (196) is a finitegravitational focusing factor, since the Keplerian shear inhibits azero relative velocity, and a finite collisional probability for verysmall velocities, again due to the shear which provides a finiteinflux of particles. At the low-velocity limit, our results comparewith those of Goldreich, Lithwick & Sari (2004), which have beenexplored by N-body simulations.

The precise calculation of the coagulation kernel should includean integration over all semimajor axes with a proper weightingkernel. As collisions among particles in the statistical model playonly a major role when the system is still homogenous, we omit-ted this contribution. In addition, this helps saving computationaltime, since the solution of the coagulation equation is very costly.However, interactions between N-body particles and the statisticalmodel include spatial inhomogeneities properly (see Section 11).

10.11 Discretization

All involved quantities are only functions of a and m. Therefore,we introduce a two-dimensional grid, where �, Tr and Tz are cell-centred quantities. Fig. 10 summarizes the definition of the two-dimensional grid. Since the full planetesimal size range coversseveral orders of magnitude in mass, we chose a logarithmicallyequidistant discretization in mass to cover the necessary mass rangein a reliable way. The radial spacing of the grid cells is equidistant.

19 Their work includes an averaging over the Rayleigh distributed inclina-tions and eccentricities of the colliding planetesimals.

Figure 10. Numerical grid. The arrows indicate transport of kinetic en-ergy (red), spatial transport of mass(green) and accretion (black). Non-neighbouring cells are coupled by the coagulation kernel and the radialinterpolation kernel.

Thus, the grid setup for the mass discretization reads (N grid cellsfrom mmin . . . mmax ):

mi = mminδ−i(1/2 + δ/2) i = 1, . . . , N

�mi = mminδ−i(1 − δ)

�i = d�

dm�mi

δ =(

mmin

mmax

)1/N

. (198)

The grid spacing δ controls the number of cells which are necessaryto cover a specified mass range. As the evaluation of the coagulationequation scales with the third power of the number of grid cells, δ

should be as large as possible. If the flux integral is approximatedin a standard way

Fi = −N∑

j=1

N∑k=1

F(jk)i

F(jk)i = 1√

2π(h2j + h2

k)

�j

mj

�k

mk

× σ (mj, mk)

× vrelMred(mi − �mi/2, mj , mk), (199)

a spacing δ much smaller than 2 is required to guarantee a sufficientaccuracy20. However, it is possible to use a spacing of 2 if specialprecaution is taken. Spaute et al. (1991) approximated the surfacedensity with a power law, thus taking the gradient with respect tomass into account. While they reached only a sufficient accuracywith further special adaptations, we use a more rigorous approach.A large spacing δ reduces the accuracy, since the partial flux (equa-tion 199) is strongly varying even inside one grid cell. Thus, we

20 Ohtsuki, Nakagawa & Nakazawa (1990) give a thorough analysis of theimportance of the resolution.

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rearrange this expression to identify the most important terms, aswe can see in equation (200):

F(jk)i = vrel√

2π(h2j + h2

k)

�k

mk

�j

mj

× σ (mj ,mk)mj︸︷︷︸FV (mj )

1

mj

Mred

× (mi − �mi/2, mj , mk)

j ≥ k. (200)

The strongest varying contribution FV is now approximated by apower law with respect to mj:

FV (m) ≈ FV (mj )

(m

mj

)q

. (201)

Thus, equation (201) is used to provide improved partial fluxes F(jk)i

and thus we come to equation (202).

F(jk)i = vrel√

2π(h2j + h2

k)

�k

mk

×∫ mj +�mj /2

mj −�mj /2

FV (m)

m�mj

Mred(mi − �mi/2, m, mk) dm

j ≥ k. (202)

Since the fragment redistribution function is a piecewise power law,an analytical solution of the integral is possible. Equation (202)gives reliable results even with a spacing δ = 2. The time derivativeof the surface density reads

�im = −Fi+1 + Fi (203)

which should assure the conservation of mass within numericalaccuracy.

We note that for calculations with a fixed mass grid (where theaverage mass of a bin is constant in time), δ < 2; otherwise, spu-rious solutions result (Ohtsuki et al. 1990). Other schemes use theapproach of Wetherill (1989), the mass ‘batches’, where the aver-age mass of a bin varies with time. In this case, δ can be any value.However, the latency between the actual time a particle reaches aspecific size and the calculated time grows with δ. It has been notedthat lags of a few per cent require δ ≤ 1.25 (Kenyon & Luu 1998).Nevertheless, a δ ∼ 2 yields lags of about 10 per cent, acceptablein view of the uncertainties in the whole approach.

10.12 Integrator

All contributions to the evolution of the surface density � and thevelocity dispersions Tr and Tz are summarized by the following setof differential equations:

D = 4

3�2

(5

4Tr,enc + Tz,enc

)d�

dt= �a(D�) + �coll

d�Tr

dt= �a(D�Tr) + �Tr + d

dt(�Tr)coll

d�Tz

dt= �a(D�Tz) + �Tz + d

dt(�Tz)coll. (204)

The Laplace operator is approximated in accordance with theequidistant radial grid:

�af = 1

a

∂

∂a

(a

∂

∂af

)

�af ≈ fi+1(1 + �a/(2ai)) − 2fi + (1 − �a/(2ai))fi−1

(�a)2. (205)

We chose the Heun method 21 as the basic integrator for the statisticalmodel. It is a second-order accurate predictor–corrector scheme(X is a vector containing all the above quantities):

dX

dt= f (X)

Xp = Xn + �tf (Xn)

Xn+1 = Xp + 1

2�t(f (Xp) − f (Xn)) + O(�t3). (206)

The Heun method is readily extended to an iterate scheme, whichis equivalent to the implicit expression:

Xn+1 = Xn + 1

2�t(f (Xn+1) + f (Xn)). (207)

This adds stability to the method and allows the secure integrationof stiff configurations that may appear during the runaway accretionphase. In practice, three iterations are sufficient to guarantee a stableintegration. As the diffusive part is discretized with a first-orderaccurate formula (see equation 205), the whole iterated scheme isequivalent to the Crank–Nicolsen method. We choose a global timestep for the statistical model according to the expression

�t = min

(ηDisc

X

X, X ∈ {�, Tr, Tz}

), (208)

where the hybrid code (see next section) applies an additional dis-cretization in powers of two to achieve a better synchronization withthe N-body code component.

Since we use the continuous form of the coagulation equation(see Sections 5 and 5.1), we also have ‘half bodies’. In the massregime of relevance, these are so copious that non-integer collisionrates do not need to be treated with a random number generator. Thisis where the direct-summation N-body integrator of the scheme isrelevant.

1 1 B R I N G I N G T H E T WO S C H E M E STO G E T H E R : TH E H Y B R I D C O D E

We introduced two different methods to solve the planetesimalgrowth problem. On the one hand, we modified NBODY6++, whichhas been used so far mainly for the simulation of stellar clusters,to adapt it to the special requirements of a long-term integrationof planetesimal orbits. On the other hand, we developed a new sta-tistical code with a consistent evolution of the velocity dispersion,the capability to treat spatial inhomogeneities and a thoroughlyconstructed collision treatment. Neither of the two approaches ispowerful enough to provide a complete and accurate descriptionof the planetesimal problem, since each method is confined to acertain range of the particle number. However, these restrictionsare complementary in the sense that each method covers a regime

21 The name of this method is not unique. Some texts denote it as themodified Euler method. The Heun method is a special case of the Runge–Kutta methods.

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Figure 11. Interplay between the N-body component and the statisticalcomponent of the hybrid code. Black arrows indicate mass transfer, redarrows exchange of kinetic energy and green arrows indicate spatial struc-turing, respectively.

where the other method fails. This intriguing relation stimulated theconstruction of a hybrid code which combines the benefits of bothmethods.

The basic idea is to introduce a transition mass mtrans, whichseparates the two mass regimes. Particles with a lower mass aretreated by the statistical model, whereas larger particles belong tothe N-body model. Though both parts are clearly divided in differentmass ranges, they are connected by various interdependences.

(i) Direct collisions between particles lead to a mass exchange.One process is the accretion of small particles by N-body particles,but agglomeration within the statistical model can also produce par-ticles larger than the transition mass. This requires the generation ofnew N-body particles. Energetic impacts may erode larger particles,so a corresponding particle removal is also required for consistency.

(ii) Mutual scatterings among N-body particles and smaller plan-etesimals transfer kinetic energy. While energy equipartition leadsto a systematic heating of the smaller field planetesimals, a consis-tent treatment has to include both transfer directions.

(iii) Accretion and scattering by the N-body particles induce spa-tial inhomogeneities or even gaps in the planetesimal component, ifthe particles have grown massive enough. Likewise, the small par-ticles could induce some structure in the distribution of the N-bodyparticles. Since the spatial structure is dominated by the stirringfrom few protoplanets, we neglect the latter process.

Fig. 11 summarizes this brief overview of the interactions betweenthe two code components in a schematic diagram. The followingsections explain the implementation of each interaction term inmore detail.

11.1 Mass transfer

An N-body particle accretes smaller particles in its vicinity. Wealready derived expressions which describe agglomeration withinthe statistical model, so it is manifest to apply these formulae toderive the accretion rate of an N-body particle.

Most of the material is accreted within the cross-sectional areaσ (see equation 194), but the finite eccentricity of an orbit extends

the accessible radial feeding zone. Thus, we assign the followingsurface density to each particle

�(a) = M

2πa√

2πlexp

(− (a − a0)2

2l2

)(209)

l2 = σ/π + 1

2a2e2 + Tr/�2 (210)

by smearing it out over its feeding zone. Tr is the radial velocitydispersion of particles in the statistical model with semimajor axisa. This density distribution is projected on to the radial grid tocalculate the accretion rate. As the time step of the statistical modelis much larger than the regular step of an N-body particle, theparticle mass update is synchronized with the statistical integration.The projection technique allows the calculation of the accretionrates in a simple way, which gives the right size of the feeding zoneand the proper total accretion rate.

Particle generation is included in the following way: a ‘virtual’mass bin is introduced as the boundary between the statistical grid(denoted by the dashed area in Fig. 11) and the N-body component.Its sole task is to store mass and kinetic energy that drives thestatistical model towards higher masses. If the mass content exceedsone mtrans, a new particle is created with inclination and eccentricityaccording to the stored velocity dispersions.

The masses of the N-body particles are regularly checked to de-tect any particle which dropped below the transition mass. Whilethis procedure would remove the particle and transfer the associ-ated quantities back to the grid, we never observed such a particleerosion.

11.2 Disc excitation

The projection of an N-body particle on to the grid with the helpof a proper weight function is also useful for the calculation of thedisc excitation due to stirring by the larger particles. Since the Hillradius sets the proper length-scale for planetesimal encounters, theweight function is modified to

�(a) = M

2πa√

2πlexp

(− (a − a0)2

2l2

)

l2 = R2Hill + 1

2a2e2 + Tr/�2, (211)

where Tr is the radial velocity dispersion of the heated planetesimalcomponent. The velocity dispersion of the stirring N-body particleis [in accordance with equations (4) and (5)]:

Tr,0 = 1

2(�a0)2e2Tz,0 = 1

2(�a0)2i2. (212)

We employ the orbital elements as mediators between the fast vary-ing instantaneous position and velocity of a particle and the slowevolution of the statistical model, which operates on a longer relax-ation time-scale. In virtue of the projection of the particle, we readilyapply the standard interaction terms (see Section 10) to evaluate theadditional heating due to the presence of N-body particles.

11.3 Pseudo-force

While an N-body particle is moving through the disc, it also inter-acts gravitationally with the particles in the statistical model. Thecollective effect of all these encounters leads to a change in the or-bital elements of the N-body particle. Again, we project the N-body

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particle on to the grid and evaluate the stirring rates Tr and Tz, whichcorrespond to a change in the orbital elements:

d

dte2 = 2Tr

(�a0)2

d

dti2 = 2Tz

(�a0)2. (213)

These time derivatives of eccentricity and inclination are translatedto a pseudo-force that effects the desired change of the orbitalelements. We chose the ansatz

Fx,y = Cr(vx,y − (vK)x,y)

Fz = Czvz, (214)

where vK is the local Keplerian velocity. In addition, we tried asimpler expression

Fx,y = 2Crrx,y

r · v

r2

Fz = Czvz (215)

without any significant differences in the accuracy or the simulationoutcome. The proper friction coefficients are

Cr = Tr

2Tr

Cz = Tz

2Tz

. (216)

Since the relevant quantities are the time derivatives of the orbitalelements, any other pseudo-force is also applicable. Though thisapproach yields the right mean change of the orbital elements,it lacks the statistical fluctuations from the particle disc. Hence,the distribution of the orbital elements of the N-body particles isartificially narrowed, which is especially important when the N-body particles and the statistical particles have a comparable mass.As the mass contrast between the two code parts is quite significantin planet formation simulations, it is safe to neglect the fluctuatingpart without major restrictions on the realism of the simulations.

The friction coefficients Ci are kept constant between two inte-gration steps of the statistical model. While a more frequent updateof the coefficients would be easily possible, a regular update on thebasis of the statistical time step is accurate enough. Moreover, eachupdate poses a considerable computational effort (roughly equiva-lent to 1000 force evaluations), so our approach also saves valuablecomputational time.

11.4 Spatial structure

The first insight into planetesimal formation was obtained by theparticle-in-a-box method, which invokes the underlying assumptionthat the planetesimal disc stays homogeneous throughout the pro-toplanet growth (see e.g. Greenberg et al. 1978). While few largebodies introduce some coarse-graininess of the surface density, allsmaller bodies are assumed to be evenly spread in the disc. Researchon the interaction of protoplanets showed that this is an oversim-plification, as bodies that are massive enough could open gaps intheir vicinity (see e.g. Lin & Papaloizou 1979; Rafikov 2001). Gapformation does not only change the overall surface density, but alsocontrols the accretion on to the protoplanet through the amount ofplanetesimals in the feeding zone. If gap formation is too effective,the growth of the protoplanet may well stop before the isolation mass

Figure 12. Gap opening in a planetesimal disc. The gap is fully developedafter 2000 yr.

is reached. Hence, any hybrid code should provide a framework thatallows this mechanism to operate. A necessary condition is a radialdensity grid with a sufficient resolution to describe possibly emerg-ing gaps. A too low resolution suppresses local perturbations fromthe protoplanets by a simple averaging, thus inhibiting the forma-tion of any spatial inhomogeneities. A second requirement is thatthe interaction terms relating statistical model and N-body modelinclude the local interaction between particles and the statisticalcomponent in a proper way.

Our hybrid approach includes gap formation implicitly throughthe diffusive terms. A protoplanet heats only the planetesimals in itsvicinity (defined by the heating kernel), thus also increasing locallythe diffusion coefficient. Hence, the surface density drops due tooutward diffusion of the planetesimals, given that the protoplanetis massive enough. The minimum gap opening mass is set by thecondition that the protoplanet controls the random velocities of thefield planetesimals in its heating zone (see equation 12), whichis equivalent to the independently derived gap formation criterion(compare equation 14).

Although our algorithm invokes a simplified picture of theprotoplanet–planetesimal interaction, it is surprisingly accurate withrespect to the width of the forming gap and the opening criterion.Fig. 12 shows a simulation which examines the accuracy of ourapproach, and see Table 9 for a summary of the initial conditionsfor the comparative runs. The overall performance of the statisti-cal code is quite remarkable, except a significant overestimation ofthe surface density at the gap boundary compared to the N-bodymodel. This deviation is due to the improper treatment of strongplanetesimal–protoplanet encounters, which exceed the diffusiveapproximation. Moreover, the higher concentration of planetesi-mals near the gap boundary leads to an additional overestimationof the velocity dispersion of the smaller planetesimals in the sta-tistical calculation (see Fig. 13). While the comparison with theN-body calculation clearly indicates a necessary improvement ofthe treatment of spatial inhomogeneities, our approach catches themain features of gap formation.

11.5 Transition mass

Since the inventory of the new hybrid code is now completed, weturn to the specification of the transition mass mtrans. The massboundary between statistical and N-body part has a major influence

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Table 9. Parameters of the statistical and the N-body gap simulation. The perturber is placed at the centre ofthe ring.

No. � �a N Nrad e2/h2 i2/h2 m Type

G1 1.1251 × 10−6 0.2 1406 – 0.001 35 0.001 35 1 × 10−9 N-bodyPerturber – 1 – e = 6.1 × 10−5 i = 3.2 × 10−5 1 × 10−7 –

G2 1.1251 × 10−6 0.2 – 201 0.001 35 0.001 35 1 × 10−9 StatisticPerturber – 1 – e = 6.1 × 10−5 i = 3.2 × 10−5 1 × 10−7 –

Figure 13. Mean square eccentricity and inclination of the smaller plan-etesimals in terms of the reduced Hill radius H of the protoplanet accordingto simulation G1 (N-body) and G2 (Statistic).

on the realism and the speed of the simulation. On the one hand,optimization with respect to speed favours a large transition mass,whereas a reasonable resolution of the transition between the twocomponents introduces some upper limit.

Hence, we identify first the set of large masses, which controls thevelocity dispersion of the disc, since these objects are also possiblecandidates for gap opening. The inspection of all involved stirringterms gives approximately the inequality:∫ mtrans

0

d�

dmm dm <

∫ ∞

mtrans

d�

dmm dm. (217)

While this is a necessary condition to select all potential majorperturbers, criterion (217) does not imply that all particles in theselected mass range exert indeed a strong influence on the disc. Thenumber of possible gaps – and therefore the number of perturbersassociated with them – is ultimately limited by the available space.Thus, we integrate the area of all potential gaps (width ≈f�RHill)and normalize it to the total disc area:

fC ≈∫ ∞

mtrans

f�

d�

dm

2πaRHill

mdm. (218)

If the covered fraction fC is much larger than one, it is possible toincrease the transition mass until the condition

fC � 1 (219)

is fulfilled. Of course, conditions (217) and (219) defined only anupper limit of the transition mass, so the adaptation of a lower valueis also possible. Though there are two reliable conditions at hand, thetransition mass is still a function of time owing to the time evolutionof the density �(m). Therefore, we chose a priori a fiducial value ofthe transition mass, run the simulation and conduct an a posteriori

check, whether the initial choice matches our requirements at anyevolutionary stage of the disc. A reliable value for a Solar systemanalogue at 1 au is

mtrans ≈ 3 × 10−11 M� (220)

which restricts the number of N-body particles to a tractable amount.Later stages would allow an even larger transition mass, but thecurrent hybrid code does not include any dynamical adjustment ofthe transition mass at runtime.

11.6 Boundary conditions

Any numerical simulation is limited to a finite simulation volumeand a finite time interval. Therefore, it is mandatory to introduceproper boundary conditions which provide a reasonable closure ofthe simulation volume.

While boundary conditions with respect to time are the familiarinitial conditions, the choice of the spatial boundary conditions forthe various involved quantities depends on the problem at hand andthe type of the boundary. A simulation boundary can be due tophysical reasons (like walls of a concert hall, surface of a terrestrialplanet) or simply due to a limitation in computational power thatinhibits the complete numerical coverage of the problem.

The current capability of the hybrid code sets limits on the radialrange as well as on the covered mass range, which a simulation canhandle in a reasonable time. Hence, we have to introduce artificialboundaries in radius, and a lower limit for the mass grid.

Any migration process couples the evolution of a local ring areain the planetesimal disc to the evolution of the whole disc. Inward(or outward) migrating material also transports information on theradial zone where the material originated from. As this informationis not available within the frame of a local simulation, any choiceof the boundary condition alters the evolution to some extent.

However, we focus on a formation stage where migration is nota dominant process, but provides only removal of the smaller col-lisional fragments. Thus, we apply closed boundary conditions forthe outer and inner radius of the ring area (i.e. all fluxes vanishat the boundary), and an open boundary for the lower end of themass range. While these conditions exclude the study of migrationalprocesses, we gain clearer insight into the protoplanet growth.

1 2 D I S C U S S I O N A N D C O N C L U S I O N S

The formation of planetary systems represents a challenge froma numerical standpoint. The dynamical problem spans over manyorders of magnitudes in length and demands the combination ofdifferent techniques. We have presented a composite algorithm thatbrings together the advantages of direct-summation tools and statis-tics for the description of the planetesimal disc. Direct-summationN-body techniques have been around for some decades and haveproven their accuracy in a very large number of studies of stellar

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clusters such as galactic nuclei and globular and open clusters. Wedeem it to be the numerical tool to integrate the motion of the bod-ies for the very precise integration of the orbits and treatment ofclose encounters. Typically, in a simulation of a stellar system, theenergy is conserved in each time step by E/�E ∼ 10−11 (where Eis the total energy and �E the difference between the former andcurrent total energy for a specific time), so that even if we integratefor a long time the cluster, the accumulated energy error is negli-gible. Nevertheless, porting the numerical tool to the problem ofplanetary dynamics is not straightforward and requires importantmodifications and additions. In this work, we present them in de-tail: the neighbour radius selection for the protoplanets, the Hermiteiteration and we introduce for the very first time the new extendedHermite scheme, since the usual Hermite scheme is not sufficientto integrate planetesimal orbits accurately enough. Then, we bringin new forces to the problem, namely the introduction of the centralpotential of the star, as well as the drag forces, which depend on thegas density and size of the planetesimals. Hence, the regularizationscheme, crucial to exactly integrate the close encounters, has to beaccordingly modified. We then introduce the disc geometry and dis-cuss the required changes to the neighbour scheme and prediction,as well as the communication algorithm and block size distribution.

For the statistical description of the planetesimal disc, we employa Fokker–Planck approach. We include dynamical friction, high-and low-speed encounters, the role of distant encounters as wellas gas and collisional damping and then generalize the model toinhomogenous discs. We then describe the combination of the twotechniques to address the whole problem of planetesimal dynamicsin a realistic way via a transition mass to integrate the evolution ofthe particles according to their masses.

In particular, we introduce and describe the extended Hermitescheme, which reduces the energy error by three orders of magni-tude with the same number of force evaluations, compared to thestandard version of NBODY6++.

While the implementation and some code details are newly intro-duced to the field of planet formation simulations, the first hybridapproach was developed in the early 1990’s. Spaute et al. (1991, fur-ther improved in Weidenschilling et al. 1997) constructed a hybridcode with a statistical component to treat the smaller particles and aspecial treatment for the larger particles. A statistical model coversthe field planetesimals with the help of a distribution function (sim-ilar to Wetherill 1989), whereas the larger particles are individuallystored and characterized by mass, semimajor axis, eccentricity andinclination. While the interaction between these single particles andthe statistical component is expressed by standard viscous stirringand dynamical friction terms, perturbations among the single par-ticles are equated in a different way. First, the probability of anencounter of two neighbouring particles is calculated. This prob-ability is used in a second step to decide whether a (numericallyintegrated) two-body encounter of the neighbouring particles is car-ried out to derive the change in the orbital elements. Though thesetwo well-defined code components justify to speak about a hybridapproach, the Monte Carlo like integration of the largest particlesis still closely related to a statistical treatment.

A modified N-body approach is used in the work of Levison &Morbidelli (2007). Their method covers the largest particles by adirect N-body code, which includes the smaller particles as ‘tracer’particles. The term ‘tracer’ indicates that each particle represents awhole ensemble of planetesimals. In a similar line of approach andinspired by this idea, Levison, Thommes & Duncan (2010) modi-fied a symplectic algorithm, SYMBA, to study the formation of giantplanet cores. However, they made some assumptions in order to

calculate the gravitational interaction between the planetesimals. Inparticular, they ignored totally close encounters between planetes-imals (but see Levison, Duncan & Thommes 2012, for an updatebuilt on SYMBA), although they use their code in situations whereinteractions between planetesimals are small.

Ormel & Spaans (2008) present in their work a scheme based onMonte Carlo techniques to cover the vast range of sizes. For this,they assign more resolution to those particles that are more relevantto the interactions, typically the largest bodies. Smaller particles aregrouped and treated collectively, which means that they all sharethe same mass and structural parameters. This classification is donein accordance to the ‘zoom factor’, a free parameter. Later, Ormel,Dullemond & Spaans (2010a) presented an detailed comparison oftheir Monte Carlo code with other techniques, in particular with puredirect-summation N-body results and other statistical studies andfound that system leaves the runaway at a larger radius, in particularat the outer disc. With their simulations, the authors propose a newcriterion for the runaway growth-oligarchy transition: from severalhundreds of km in the inner disc regions up to a thousand km forthe outer disc (Ormel, Dullemond & Spaans 2010b).

Bromley & Kenyon (2006) published a description of a hybridmethod with a basic approach similar to our work. They employtwo velocity dispersions and the surface density of the planetesi-mals to describe the planetesimal system. The statistical componentincludes migration of the planetesimals and dust particles due to gasdrag and Pointing–Robertson drag. In contrast to our approach, theydid not include mass transport due to the diffusion of the planetes-imals, which precludes the study of spatial structures induced bythe protoplanets. This issue was addressed later in their later workof Bromley & Kenyon (2011a,b), in which planets open up gaps indiscs. One must note also that their method uses the standard dis-cretization of the collisional flux (see equation 199) and thus restrictthe spacing factor to δ � 1.25 (Kenyon & Luu 1998). Bromley &Kenyon (2006) chose a set of test calculations which focused less onthe technical aspects of their method, but on an overall comparisonwith a selected set of standard works on planet formation. Their testsimulations are in good agreement with the references simulations,thus indicating a comparable quality of the method. Four years later,the authors presented an updated version of their code for planetformation. The new characteristics of the code included 1D evolu-tion of the viscous disc, gas accretion on to massive cores, as wellas accretion of small particles in planetary atmospheres (Bromley& Kenyon 2011a).

While a variety of hybrid approaches emerged over the past years,this technique is still far from a routinely application and is stillchallenged by many open issues. Hybrid codes bear the potentialto address the dynamical evolution of a whole planetary system,the later stages of protoplanet formation initiate a strong interactionwith the gaseous disc, which may require more diligence than theinclusion of a few additional interaction terms. However, the de-velopment is picking up speed, which places our work in a goodposition for further research.

AC K N OW L E D G E M E N T S

PAS thanks the referee, Scott Kenyon, for his patience and time ona detailed and helpful refereeing of the paper. It is a pleasure tothank Sverre Aarseth, Cornelis Dullemond and Phil Armitage forcomments on the manuscript. PAS thanks the National Astronomi-cal Observatories of China, the Chinese Academy of Sciences andthe Kavli Institute for Astronomy and Astrophysics in Beijing, foran extended visit, as well as the Aspen Center of Physics and the

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organizers of the summer meeting, where this work was finished.PAS expresses his utmost gratitude to Hong Qi, Wenhua Ju andXian Chen for their hospitality during his stay in Beijing and to theKavli Institute for Theoretical Physics where one part of this workhas been completed. This research was supported in part by theNational Science Foundation under Grant No. NSF PHY11-25915and supported by the Transregio 7 ‘Gravitational Wave Astronomy’,financed by the Deutsche Forschungsgemeinschaft DFG (GermanResearch Foundation). PG and RS acknowledge support by theCollaborative Research Group FOR 759 (Project C3) of GermanScience Foundation (DFG) ‘The formation of Planets: The Criti-cal First Growth Phase’. RS acknowledges support by the ChineseAcademy of Sciences Visiting Professorship for Senior Interna-tional Scientists, Grant Number 2009S1-5 (The Silk Road Project).The special supercomputer Laohu at the High Performance Com-puting Center at National Astronomical Observatories, funded byMinistry of Finance under the grant ZDYZ2008-2, has been used.Simulations were also performed on the GRACE supercomputer(grants I/80 041-043 and I/84 678-680 of the Volkswagen Foun-dation and 823.219-439/30 and /36 of the Ministry of Science,Research and the Arts of Baden-urttemberg). Computing time onthe IBM Jump Supercomputer at FZ Julich is acknowledged. PASand RS thank the Aspen Center for Physics and the NSF Grant#1066293 for hospitality while part of the work for this paper wasdone.

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A P P E N D I X A : C E N T R A LFORCE-DERIVATIVES

Central force F per mass (i.e. acceleration) and its time derivativesare

F = − xM

x3

F(1) = −vM

x3− 3AF

F(2) = − aM

x3− 6AF − 3B F

F(3) = − aM

x3− 9AF(2) − 9B F(1) − 3C F

a = v

A = x · v

x2

B = v2

x2+ x · a

x2+ A2 = A + 3A2

C = 3v · ax2

+ x · ax2

+ A(3B − 4A2). (A1)

The F(i) denote the central force and its time derivatives, whereasa and a refer to the total acceleration of the particle. The assumptionthat x, v, a and a are independent of each other allows the derivationof averaged expressions for particle–particle interactions:

〈(F)2〉 = m2 1

x4

〈(F(1))2〉 = m2 2v2

x6

〈(F(2))2〉 = m2

(12

v4

x8+ 2

a2

x6

)

〈(F(3))2〉 = m2

(144

v6

x10+ 126

a2v2

x8+ 2

a2

x6

). (A2)

We combine these expressions with Aarseth’s time step formula toderive the regular time step as a function of the neighbour sphereradius Rs:

�treg ≈ √ηreg

Rs

v

1

1 + √Rs/R0

R0 = 4v2

a≈ 4v2r2

Gm= 4

r2

rclose, (A3)

where r is the average particle distance and rclose is the impactparameter for a 90-degree deflection.

A P P E N D I X B : SC A L A B L E C O L L I S I O N S FL U X

The mass flux according to the perturbation equation (125) is

F p = −∫ ∫

(n(m2)�n(m1) + n(m1)�n(m2))

× σ (m1)v(m2)m1fm(m1/m, ε)dm1dm2

= F (1) + F (2). (B1)

First, we employ the substitution

m1 = mx1

m2 = m0

(m1

m0

) 1+α1+2w (

S) 1

1+2w ε1

1+2w (B2)

to solve for the partial flux F(1):

F (1) = −n20m

30σ0v0

∫g(mx1)F1(x1) dx1

F1(x1) = S−k′∫

ε− w+s+3+α2+α+2w

fm(x1, ε)

x1(1 + 2w)dε. (B3)

The second contribution F(2) requires a slightly different transfor-mation:

m1 = mx1ε−1/(1+α)

m2 = m0

(mx1

m0

) 1+α1+2w (

S) 1

1+2w . (B4)

Thus, the partial flux F(2) is

F (2) = −n20m

30σ0v0

∫g(m2)F2(x1) dx1

F2(x1) = S−k′∫

ε− w+s+3+α2+α+2w

fm(x1ε−1/(1+α), ε)

x1(1 + 2w)dε. (B5)

We change to a new set of logarithmic coordinates

u = ln(m/m0) u1 = ln(x1) s = ln(S)

1 + α(B6)

which transforms the total flux Fm to a convolution integral:

F p = −n20m

30σ0v0

∫[g(u + u1)G1(u1)

+ g (p(u + u1 + s1)) G2(u1)] du1 (B7)

p = 1 + α

1 + 2w, (B8)

where p = 1 refers to the already derived solution for self-similarcollisions. Hence, we expand equation (B7) at p = 1 and retain onlythe zeroth-order moment of the fragmentation kernel:

F p = −n20m

30σ0v0

[g(u)G1,0 +

(g(u) + u(p − 1)

∂g

∂u

)G2,0

].

(B9)

This expression is equivalent to

F p = −n20m

30σ0v0(g(u)G1,0

+ [g(u) + (p − 1)(g(u) − g(0))]G2,0), (B10)

where higher derivatives of g(u) are neglected. Hence, we recoverthe same functional form of the perturbed mass flux Fp as for self-similar collisions:

F p = −n20m

30σ0v0 g(u)

(G1,0 + p G2,0

) + const.

× ∝ S−k′. (B11)

MNRAS 445, 3620–3649 (2014)

at MPI G



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A P P E N D I X C : C OAG U L AT I O N E QUAT I O N

While the success of a general approximation of the coagulationequation depends heavily on the used coagulation kernel, we nev-ertheless provide a more general approach to embed Section 5 in abroader context. The standard coagulation equation is

0 = ∂

∂tmn(t, m) + ∂

∂mFm(t, m)

Fm = −∫ ∫

n(t, m1)n(t, m2)σ (m1, m2)

× vrelMred(m,m1, m2) dm1dm2. (C1)

In virtue of our experience drawn from the perturbation expansion,we transform the coagulation equation to logarithmic coordinates

u = ln(m) (C2)

and employ the size distribution g(u) relative to the steady-statesolution neq(m):

0 = ∂

∂tg(u, t)neq(u)e2u + ∂

∂uFu(t, m)

Fu = −∫ ∫

g(t, u1)g(t, u2)K(u, u1, u2) du1 du2, (C3)

where K(u, u1, u2) is the properly transformed new coagulationkernel. g(u) is expanded under the integral to arrive at a momentexpansion of the flux Fu:

Fu = −K00(u)g(u)2 − (K10(u) + K01(u))g(u)∂g

∂u+ · · ·

Kij =∫ ∫

K(u, u1, u2)ui1u

j2du1du2. (C4)

Retaining only the leading-order terms, we recover an approximatecoagulation equation which is similar to the inviscid Burgers’ equa-tion 22:

0 = ∂

∂tg(u, t)neq(u)e2u − ∂

∂u

(K00(u)g(u)2

). (C5)

22 This notion goes back to Burgers (1948), but the equation was alreadyintroduced by Bateman (1915).

This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 445, 3620–3649 (2014)

at MPI G



ownloaded from


hybrid methods in planetesimal dynamics: description of a

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