arX

iv:0

905.

3741

v1 [

astro

-ph.E

P] 2

2 May

2009

ApJ Letters, in pressPreprint typeset using LATEX style emulateapj v. 04/20/08

PLANET-PLANET SCATTERING IN PLANETESIMAL DISKS

Sean N. Raymond1,2, Philip J. Armitage2,3 and Noel Gorelick4

ApJ Letters, in press

ABSTRACT

We study the final architecture of planetary systems that evolve under the combined effects ofplanet-planet and planetesimal scattering. Using N-body simulations we investigate the dynamics ofmarginally unstable systems of gas and ice giants both in isolation and when the planets form interiorto a planetesimal belt. The unstable isolated systems evolve under planet-planet scattering to yieldan eccentricity distribution that matches that observed for extrasolar planets. When planetesimalsare included the outcome depends upon the total mass of the planets. For Mtot & 1 MJ the finaleccentricity distribution remains broad, whereas for Mtot . 1 MJ a combination of divergent orbitalevolution and recircularization of scattered planets results in a preponderance of nearly circular finalorbits. We also study the fate of marginally stable multiple planet systems in the presence of planetes-imal disks, and find that for high planet masses the majority of such systems evolve into resonance. Asignificant fraction lead to resonant chains that are planetary analogs of Jupiters Galilean satellites.We predict that a transition from eccentric to near-circular orbits will be observed once extrasolarplanet surveys detect sub-Jovian mass planets at orbital radii of a 5 10 AU.Subject headings: solar system: formation planetary systems: protoplanetary disks planetary

systems: formation celestial mechanics

1. INTRODUCTION

Different dynamical mechanisms are commonly in-voked to explain the architecture of the outer SolarSystem and extrasolar planetary systems. In the So-lar System, scattering of small bodies (planetesimals)by the ice giants (Fernandez & Ip 1984; Ida et al. 2000;Kirsh et al. 2009) is thought to drive outward planetarymigration and concomitant capture of Pluto and otherKuiper Belt Objects into resonance (Malhotra 1995;Murray-Clay & Chiang 2005). The effects of planetes-imal scattering on the gas giants are smaller but still sig-nificant, for example in the Nice model (Tsiganis et al.2005; Gomes et al. 2005) where a divergent resonancecrossing between Jupiter and Saturn triggers the LateHeavy Bombardment. The presence of small bodiesaround other stars can be inferred from observations ofdebris disks (e.g. Wyatt 2008), but as yet there is noevidence for a dynamical role of planetesimals in knownextrasolar planetary systems. At radii where tidal ef-fects are negligible (roughly a & 0.1 AU) the eccentricitydistribution of extrasolar planets matches relatively sim-ple models of gravitational scattering among a systemof two or more massive planets that typically includeneither planetesimals nor residual gas (Chatterjee et al.2008; Juric & Tremaine 2008).The success of pure planet-planet scattering mod-

els does not imply that other dynamical processes canbe ignored. The observed distribution of semi-majoraxes of extrasolar planets at small orbital radii re-quires the existence of an additional dissipative process

1 Center for Astrophysics and Space Astronomy,389 UCB, University of Colorado, Boulder CO 80309;sean.raymond@colorado.edu

2 Department of Astrophysical and Planetary Sciences, Univer-sity of Colorado, Boulder CO 80309

3 JILA, 440 UCB, University of Colorado, Boulder CO 803094 Google, Inc., 1600 Amphitheatre Parkway, Mountain View,

CA 94043

(Adams & Laughlin 2003), most probably gas disk mi-gration (Lin & Papaloizou 1986), which will itself af-fect planetary eccentricity (Moorhead & Adams 2005).At larger orbital radii simple arguments suggest thata dynamically significant external reservoir of planetes-imals ought to be a common feature of young plane-tary systems. The formation of giant planets becomesincreasingly difficult at large radii (Pollack et al. 1996;Kokubo & Ida 2002), and hence it is probable that disksof leftover debris surround the zone of giant planet forma-tion in most young systems. The typical masses of plan-etesimal disks are unknown, but values of 30-50 M thatare comparable to those inferred for the early outer So-lar System are plausibly typical, since they are consistentwith disk masses estimated from astronomical observa-tions of the youngest stars (Andrews & Williams 2005).The dynamical effect of such disks on currently observedextrasolar planetary systems would be small, since radialvelocity surveys preferentially detect planets that are ei-ther massive (and hence largely immune to influence fromplanetesimal disks) or orbit at very small radii where themass of leftover debris is negligible.Pooling knowledge from the Solar System and extraso-

lar planetary systems motivates consideration of a modelin which planet formation typically yields a marginallyunstable system of massive planets in dynamical contactwith both a residual gas disk and an exterior planetesi-mal disk. In this Letter we ignore the gas disk and studythe subsequent evolution under the combined action ofplanet-planet and planetesimal scattering. We do notattempt to model the full distribution of extrasolar plan-etary properties (which would require the inclusion ofhydrodynamic effects), but rather focus on how plan-etesimal disks affect the final eccentricity of extrasolarplanets at moderately large orbital radii.

2. METHODS

2 Raymond, Armitage & Gorelick

We assume that the gas-dominated epoch of planetformation is sufficiently distinct from the subsequentphase of planet-planet and planetesimal scattering thatit makes sense to study the latter with pure N-body sim-ulations. We focus on two large ensembles of runs. Thehighmass set comprises 1000 integrations of three planetsystems in which the masses of the planets are chosenrandomly in the range MSat < Mp < 3MJ , with a distri-bution,

dN

dM M1.1, (1)

which matches that observed (Marcy et al. 2008). Theobserved distribution is derived from an incomplete sam-ple that represents (in the context of our model) thedistribution after scattering, but these subtleties do notmatter for our purposes. The lowmass set is identical ex-cept that we sample a wider swath of the mass functionbetween 10M and 3MJ. The planets are initially placedin a marginally unstable configuration defined by circu-lar, nearly coplanar orbits with a separation of 4-5 rh,m,where the mutual Hill radius,

rh,m =1

2

(M1 +M23M

)1/3(a1 + a2) . (2)

Here a1 and a2 are the planets semi-major axes, M1and M2 their masses, and M is the stellar mass. Withthis spacing the instability timescale is relatively long(Chambers, Wetherill & Boss 1996; Chatterjee et al.2008) (the median timescale before the first planet-planetencounter was 0.3 Myr for the highmass integrationswithout disks). Our initial conditions are only a smallsubset of the architectures predicted from giant planetformation models (Thommes, Matsumura & Rasio2008; Mordasini, Alibert & Benz 2009), though broadlyconsistent with scenarios in which giant planets arecaptured into mean-motion resonances during the latestages of gas disk evolution (Thommes et al. 2008)prior to being removed from resonance by turbulentperturbations (Adams, Laughlin & Bloch 2008). Eachintegration is repeated twice, once with just the threeplanets5 and once with an external planetesimal diskwhose inner radius of ain = 10 AU is 2 Hill radii beyondthe orbit of the outermost planet6. The inner edge ofthe disk lies within the radius where a test particlein the restricted 3-body problem would be stable, sothe disk is in immediate dynamical contact with theouter planet. The planetesimal disk is represented by1000 bodies distributed between 10 and 20 AU with adisk r

1 surface density profile and a total mass of50M.We integrate these systems using the MERCURY

code (Chambers 1999) for 100 Myr. The inte-grator uses the symplectic Wisdom-Holman mapping(Wisdom & Holman 1991) for well-separated bodies, andthe Bulirsch-Stoer method when objects are within N

5 Note that the simulations without planetesimal disks are iden-tical to the mixed1 and mixed2 cases from (Raymond et al. 2008,2009).

6 In the Nice model, a separation of 3 4 Hill radii betweenNeptune and the outer planetesimal disk is needed to match thetiming of the Late Heavy Bombardment (Gomes et al. 2005). Thespacing of 2 rh,m means that our models evolve on a somewhatshorter time scale.

0.0 0.2 0.4 0.6 0.8 1.0Eccentricity e

0.0

0.2

0.4

0.6

0.8

1.0

Frac

tion

( 5 104 with a timestep of 10 days. A smallnumber of the re-run simulations (typically 15-35) stilldid not meet our energy criterion and were discarded.

3. RESULTS FOR MARGINALLY UNSTABLE PLANETARYSYSTEMS

In the absence of planetesimal disks our model plane-tary systems are typically unstable on Myr time scales.There are also systems that are stable over the 100 Myrduration of our runs. In our initial analysis we assumethat the typical outcome of giant planet formation is asystem that, in the absence of a disk, would be unsta-ble. We therefore analyze the subset of disk-less simula-tions that are unstable, and compare the results to thematched set of simulations that include disks. This is nota perfect one-to-one comparison, since the chaotic natureof the evolution means that disk-less planetary systemscan display different instability time scales in the pres-ence of even negligible perturbations. Nonetheless wedo observe statistical differences between the evolutionof systems (with disks) that correlate with the stabilityof the disk-free systems, and hence it makes sense as afirst approximation to separately consider the results forstable and unstable cases.The results of our disk-less simulations agree with

Planet-planet scattering in planetesimal disks 3

prior studies (Chatterjee et al. 2008; Juric & Tremaine2008; Ford & Rasio 2008). Scattering from initiallyunstable initial conditions frequently leads to the lossof one or more planets via ejection or collisions andsets up a broad eccentricity distribution (Rasio & Ford1996; Weidenschilling & Marzari 1996; Lin & Ida 1997).Scattering among equal-mass planets produces largereccentricities than scattering of unequal-mass planets(Ford, Rasio & Yu 2003; Raymond et al. 2008). Figure 1compares the final eccentricities obtained from the unsta-ble highmass simulations and the observed distribution(Butler et al. 2006; Schneider 2009) of extra-solar plan-ets. They are in good quantitative agreement. The ec-centricity distribution from the unstable lowmass sim-ulations without disks is shifted toward lower values(Raymond et al. 2008; Ford & Rasio 2008), and fits theobserved distribution of extra-solar planets with Mp 2MJ ). Each row groups simulations by outcome: unstable cases underwent closeencounters between planets, moderately stable systems underwent significant orbital changes during the simulation but the system remainedstable, and stable systems did not undergo any close encounters between planets and remained stable throughout. The evolution in thecenter and top-left panels are qualitatively similar to different models of early Solar System dynamics (Thommes, Duncan & Levison 1999;Tsiganis et al. 2005; Gomes et al. 2005).

0.00.2

0.4

0.6

0.81.0

0.00.2

0.4

0.6

0.81.0 With Disks

0.1 0.3 1 3 Total Mass (M

J)

0.00.2

0.4

0.6

0.8

Ec

cent

ricity

of I

nner

mos

t Pla

net

0.1 Total Mass (M

J)

0.00.2

0.4

0.6

0.8No Disks

Fig. 3. Final eccentricity of the innermost planet as a functionof the total mass in surviving planets for the highmass (black)and lowmass (grey) simulations. The plotted sample shows thosesystems that were unstable without disks (bottom panel), togetherwith the matched sample including disks (upper panel). Disksresult in a sharp transition to a low-eccentricity regime for systemmasses below about one Jupiter mass.

4. RESULTS FOR MARGINALLY STABLE PLANETARYSYSTEMS

Although the eccentricity distribution of extraso-lar planets is consistent with the hypothesis that allnewly formed multiple systems are unstable in theabsence of disks, this conclusion may also be biasedby selection effects. Most known extrasolar planetsprobably suffered significant gas disk migration priorto scattering (Lin, Bodenheimer & Richardson 1996;Trilling et al. 1998; Bodenheimer, Hubickyj & Lissauer2000), so the high incidence of instability may be a conse-quence of migration rather than formation. With this inmind we have separately analyzed those (previously ex-cluded) systems that were stable in the absence of disksto see what impact disks have on them. As expected,low eccentricity outcomes predominate. Resonances arealso common: about 70% of all stable highmass sim-ulations and 1/3 of stable lowmass simulations includeat least one pair of planets in the 3:2 or, more often,the 2:1 mean motion resonance. This is a much higherprobability of resonance capture than occurs for pureplanet-planet scattering without disks (Raymond et al.2008), and it also exceeds the fraction of resonant sys-tems that are expected to survive the gaseous disk phase

Planet-planet scattering in planetesimal disks 5

in the presence of turbulence (Adams, Laughlin & Bloch2008). Most surprisingly within the highmass set asubstantial fraction (about 1/3) of stable systems be-come locked into mean-motion resonances that involveall three of the planets analogs of the Laplace reso-nance among Jupiters Galilean satellites. Chains of res-onances7 arise preferentially in higher-mass systems andin systems where planetesimal-driven migration causescompression rather than divergent migration. Detectionof high mass planets at the relevant radii (between 5-10 AU) should soon be possible via astrometric or di-rect imaging techniques, and observation of resonantchains would be consistent with our model. Intrigu-ingly, the recently-discovered triple planet system HR8799 may be in a 4:2:1 resonant chain (Marois et al. 2008;Fabrycky & Murray-Clay 2009). Determining whethercapture into resonance was initiated by a gas or plan-etesimal disk may be possible via detailed compari-son of the outcome of resonant capture in planetes-imal (Murray-Clay & Chiang 2005) versus gas disks(Lee & Peale 2002; Adams, Laughlin & Bloch 2008).

5. CONCLUSIONS

Circumstantial evidence suggests that many observedproperties of the outer Solar System (Malhotra 1995)and of extrasolar planetary systems (Rasio & Ford 1996;Weidenschilling & Marzari 1996; Lin & Ida 1997) maybe attributable to the dynamical effects of planetesimalscattering and planet-planet scattering. Here, we havestudied the predicted architecture of planetary systemsthat results from the joint action of both mechanisms.We have argued that this regime will be relevant oncelower mass extrasolar planets are discovered at largerorbital radii than those currently known. Genericallywe predict that a transition to Solar-System-like archi-tectures, characterized by near-circular orbits and rela-tively stable planetary separations, will be observed once

surveys detect planets in the regime where planetesimaldisks play a dynamical role. Our simulations suggestthat the transition is a surprisingly sharp function of to-tal planetary system mass, and that it occurs for systemmasses a factor of several larger than the initial planetes-imal disk mass.Our initial conditions do not sample anything ap-

proaching the full range of initial planetary system ar-chitectures. We believe that the existence of a tran-sition between typically eccentric and near-circular or-bits is a general feature of joint models of planet-planetand planetesimal scattering, but the transition mass andminimum orbital radii at which planetesimal effects be-come manifest is of course a function of the poorly knownmasses and radial extent of planetesimal disks. Our re-sults suggest that the transition might be seen for sub-Jovian mass planets at orbital radii of 5-10 AU, but thetransition would be pushed to greater orbital radii if giantplanet formation consumes planetesimals across a widerextent of the disk. We also find that the final systemarchitecture varies substantially depending on the initialseparation of the planets. In particular, if planet forma-tion yields a mixture of massive systems in initially stableorbits, interaction with planetesimals drives a large frac-tion of systems into resonance. Whether such systemsexist should be testable in the near future.

We thank Google for the large amount of com-puter time needed for these simulations. S.N.R. ac-knowledges support from NASAs Astrobiology Institutethrough the Virtual Planetary Laboratory lead team,and from NASAs Origins of Solar Systems program(NNX09AB84G). P.J.A. acknowledges support from theNSF (AST-0807471), from NASAs Origins of Solar Sys-tems program (NNX09AB90G), and from NASAs As-trophysics Theory program (NNX07AH08G).

7 Our definition of resonance requires one resonant argument tolibrate with an amplitude A < 150. Roughly half of the resonantsystems were deep in the resonance, with A < 60. This fractionappears to be independent of the number of planets in resonance,

as about 1/4 of the highmass resonant chains had A < 60 for bothpairs of planets.

REFERENCES

Adams, F. C., & Laughlin, G. 2003, Icarus, 163, 290Adams, F. C., Laughlin, G., & Bloch, A. M. 2008, ApJ, 683, 1117Andrews, S. M., & Williams, J. P. 2005, ApJ, 631, 1134Bodenheimer, P., Hubickyj, O., & Lissauer, J. J. 2000, Icarus,143, 2

Butler, R. P., et al. 2006, ApJ, 646, 505Chambers, J. E. 1999, MNRAS, 304, 793Chambers, J. E., Wetherill, G. W., & Boss, A. P. 1996, Icarus,119, 261

Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008,ApJ, 686, 580

Fabrycky, D. C., & Murray-Clay, R. A. 2009, ApJ, submitted(arXiv:0812.0011v1)

Fernandez, J. A., & Ip, W.-H. 1984, Icarus, 58, 109Ford, E. B., & Chiang, E. I. 2007, ApJ, 661, 602Ford, E. B., & Rasio, F. A. 2008, ApJ, 686, 621Ford, E. B., Rasio, F. A., & Yu, K. 2003, in Scientific Frontiersin Research on Extrasolar Planets, ASP Conference Series Vol294, Eds. Drake Deming and Sara Seager (ASP: SanFrancisco), 181

Goldreich, P., Lithwick, Y., & Sari, R. 2004, ARA&A, 42, 549Gomes, R., Levison, H. F., Tsiganis, K., & Morbidelli, A. 2005,Nature, 435, 466

Ida, S., Bryden, G., Lin, D. N. C., & Tanaka, H. 2000, ApJ, 534,428

Juric, M., & Tremaine, S. 2008, ApJ, 686, 603Kirsh, D. R., Duncan, M., Brasser, R., & Levison, H. F. 2009,Icarus, 199, 197

Kokubo, E., & Ida, S. 2002, ApJ, 581, 666Lee, M. H., & Peale, S. J. 2002, ApJ, 567, 596Lin, D. N. C., Bodenheimer, P., & Richardson, D. C. 1996,Nature, 380, 606

Lin, D. N. C., & Ida, S. 1997, ApJ, 477, 781Lin, D. N. C., & Papaloizou, J. 1986, ApJ, 309, 846Malhotra, R. 1995, AJ, 110, 420Marcy, G. W., et al. 2008, Physica Scripta, 130, 014001Marois, C., Macintosh, B., Barman, T., Zuckerman, B., Song, I.,Patience, J. Lafreniere, D., & Doyon, R. 2008, Science, 322,1348

Moorhead, A. V., & Adams, F. C. 2005, Icarus, 178, 517Mordasini, C., Alibert, Y., & Benz, W. 2009, A&A, in press(arXiv:0904.2524v1)

Murray, N., Hansen, B., Holman, M., & Tremaine, S. 1998,Science, 279, 69

Murray-Clay, R. A., & Chiang, E. I. 2005, ApJ, 619, 623Pollack, J. B., et al. 1996, Icarus, 124, 62Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954

6 Raymond, Armitage & Gorelick

Raymond, S. N., Barnes, R., Armitage, P. J., & Gorelick, N.2008, ApJ, 687, L107

Raymond, S. N., Barnes, R., Veras, D., Armitage, P. J., Gorelick,N., & Greenberg, R. 2009, ApJ, 696, L98

Schneider, J. 2009, http://exoplanet.euThommes, E. W., Duncan, M. J., & Levison, H. F. 1999, Nature,402, 635

Thommes, E. W., Matsumura, S., & Rasio, F. A. 2008, Science,321, 814

Thommes, E. W., Bryden, G., Wu, Y., & Rasio, F. A. 2008, ApJ,675, 1538

Trilling, D. E., et al. 1998, ApJ, 500, 428Tsiganis, K., Gomes, R., Morbidelli, A., & Levison, H. F. 2005,Nature, 435, 459

Weidenschilling, S. J., & Marzari, F. 1996, Nature, 384, 619Wisdom, J., & Holman, M. 1991, AJ, 102, 1528Wright, J. T., Upadhyay, S., Marcy, G. W., Fischer, D. A., Ford,E. B., & Johnson, J. A. 2009, ApJ, 693, 1084

Wyatt, M. C. 2008, ARA&A, 46, 339