The Formation of Ice Giants in a Packed Oligarchy: Instability and Aftermath

0.87 MB·English

by  Eric B. Ford

Accepted to ApJ:Jan27,2007
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
THE FORMATION OF ICE GIANTS IN A PACKED OLIGARCHY:
INSTABILITY AND AFTERMATH
Eric B. Ford1,2 and Eugene I. Chiang3,4
Accepted toApJ: Jan 27, 2007
ABSTRACT
As many as 5 ice giants—Neptune-mass planets composed of 90% ice and rock and 10%
∼ ∼
hydrogen—arethoughttoformatheliocentricdistancesof 10–25AUoncloselypackedorbitsspaced
∼
5 Hill radii apart. Such oligarchies are ultimately unstable. Once the parent disk of planetesimals
∼
is sufficiently depleted, oligarchsperturb one anotheronto crossingorbits. We exploreboth the onset
7
and the outcome of the instability through numerical integrations, including dynamical friction cool-
0
0 ing of planets by a planetesimal disk whose properties are held fixed. To trigger instability and the
2 ejectionofthe firsticegiantinsystemshavinganoriginalsurfacedensityinoligarchsofΣ 1g/cm2,
the disk surface density σ must fall below 0.1g/cm2. Ejections are predominantly by J∼upiter and
n occur within 107yr. To eject morethan 1∼oligarchrequiresσ .0.03g/cm2. For certainchoicesofσ
a ∼
and initial semi-major axes of planets, systems starting with up to 4 oligarchs in addition to Jupiter
J
andSaturncanreadilyyieldsolar-system-likeoutcomesinwhich2survivingicegiantslieinside30AU
8
andhavetheir orbits circularizedby dynamicalfriction. Ourfindings supportthe idea thatplanetary
2
systemsbegininmorecrowdedandcompactconfigurations,likethoseofshear-dominatedoligarchies.

In contrast to previous studies, we identify σ . 0.1Σ as the regime relevant for understanding the
2
v evolution of the outer solar system, and we encourage future studies to concentrate on this regime
5 while relaxing our assumption of a fixed planetesimal disk. Whether evidence of the instability can
4 be found in Kuiper belt objects (KBOs) is unclear, since in none of our simulations do marauding
7 oligarchs excite as large a proportion of KBOs having inclinations &20◦ as is observed.
1
Subject headings: celestial mechanics—Kuiper belt—planets and satellites : formation—solar system
0
: formation
7
0
h/ 1. INTRODUCTION rates are greater. Thommes, Levison, & Duncan (1999,
2002) explore a scenario in which the two planets form
p Without gravitationalfocussing, in situ coagulationof
- Uranus and Neptune takes too long to complete. In a at distances of 5–10 AU, between the cores of Jupiter
o and Saturn. Once the gas giant cores amass their en-
minimum-mass disk at heliocentric distances of 20–30
r velopes,theyscattertheicegiantsoutwardontoeccentric
t AU, timescales to assemble the ice giants exceed the age
s orbits. These orbits subsequently circularizeby dynami-
a of the solar system by 2 orders of magnitude, if growth
cal friction with planetesimals at 15–30 AU. Tsiganis et
: is unfocussed (e.g., Goldreich, Lithwick, & Sari 2004,
v hereafterGLS04).5 N-bodycoagulationsimulationsthat al.(2005)proposeanalternativehistoryinwhichUranus
i and Neptune accrete at 12 and 17 AU, are thrown out-
X do not damp relative velocities between planetesimals,
wardby Jupiter andSaturn, andhave their orbits circu-
either by dynamical friction, inelastic collisions, or gas
r larized by dynamical friction. According to their story,
a drag,failtoformUranusandNeptune (Levison&Stew-
the outward scattering of ice giants is triggered by hav-
art 2001; see also Lissauer et al. 1995). The ice giants
ing Jupiter and Saturn divergently migrate across their
contain 10–20% hydrogen by mass, a fraction so large
mutual 2:1 resonance.
that such gas must originate from the solar nebula. The
outer planets mustthereforeformwithin a few 107yr, Another approach to solving the timescale problem is
× toconsiderhowgravitationalfocussingcanbeamplified.
beforeallofthenebularhydrogenphotoevaporates(Shu,
GLS04 adopt this route by appealing to a massive disk
Johnstone, & Hollenbach 1993; Matsuyama, Johnstone,
ofsub-km-sizedplanetesimals,similarto thoseproduced
& Hartmann 2003).
by coagulation simulations set in the outer solar system
One way to alleviate (but not necessarily eliminate)
(Kenyon&Luu1999). ThediskenvisionedbyGLS04has
the timescale problem is to form Uranus and Neptune
amassseveraltimestheminimum-massvalueinconden-
closer to the Sun, where material densities and collision
sates so that the “isolation mass”—the mass to which a
Electronicaddress: [email protected],[email protected] grows by consuming all material within its
1Harvard-Smithsonian Center for Astrophysics, Mail Stop 51, annulus of influence—equals Neptune’s mass. The small
60GardenStreet, Cambridge,MA02138, USA bodiescomprisingthediskcollidesofrequentlythattheir
2HubbleFellow
3CenterforIntegrative PlanetarySciences, AstronomyDepart- velocitydispersiondampstovalueslessthanthe Hillve-
ment, University of California at Berkeley, Berkeley, CA 94720, locity of a Neptune-mass planet. Accretion rates then
USA enjoy maximal enhancement by gravitational focussing,
4AlfredP.SloanResearchFellow and proto-Neptune can accrete the last half of its mass
5 The problem does not disappear by merely raising the disk in 105yr (see eqn. 105 of GLS04). Gas drag supplies
massabovetheminimum-massvalue,sincethegravitationallyun- ∼
focussedgrowthratescalesonlylinearlywiththedisksurfaceden- anothermeanstodampplanetesimalvelocitydispersions
sity. (Rafikov 2004; Chambers 2006; see also appendix A of
2 Ford & Chiang
GLS04). sult in the survivalof two oligarchswhose final or-
Strongly focussed, in situ assembly of planets from a bits resemble those of Uranus and Neptune?
dynamically cold disk carries, however, a potential em-
barrassment of riches: The disk can spawn more ice 3. To what degree is the Kuiper belt dynamically ex-
giants than the solar system’s current allotment of 2 cited by velocity-unstable oligarchs?
(Uranus and Neptune). We estimate that about 5 iso-
lation masses or “oligarchs,” each having the mass of Our methods are described and tested in 2. That
Neptune, can form between 15 and 25 AU (see eqn. 1 sectioncontainsempiricaldeterminationsofho§wrapidly
below). These planets comprise a “shear-dominated oli- 5 oligarchs viscously stir one another, with and with-
garchy,” so-called because the encounter velocities be- out the gas giants Jupiter and Saturn. Comparisons are
tween planets and planetesimals are given by their min- made with analytic theory. Results of hundreds of sim-
imum values set by Keplerian shear. Initially, the oli- ulations designed to provide statistical answers to the
garchs’ nested orbits would be stabilized by dynamical above questions are presented in 3. We summarize and
friction with the disk. GLS04 suggest that excess oli- offer an outlook in 4. §
garchswouldbepurgedfromtheoutersolarsystembyan §
eventualdynamicalinstability. Accordingtotheirorder- 2. METHODANDTESTS
of-magnitude analysis, this “velocity instability” occurs
To guide the reader, we provide a condensed descrip-
once the mass of the disk becomes less than the mass
tion of our method in 2.1. Details are elaborated upon
inoligarchs,whereupondynamicalfrictionceasesto sta- §
in 2.2–2.3.
bilize the system against mutual gravitational stirring §§
(a.k.a. “viscous stirring”). In the ensuing chaos, sev-
eraloligarchswould be ejected, either by other oligarchs 2.1. Overview
or by Jupiter or Saturn, possibly leaving two survivors Wesimulatethefinalstagesofoligarchybynumerically
whose orbits could circularize by dynamical friction at integrating the trajectories of 5 closely packedNeptune-
15–30 AU. massoligarchs,togetherwiththoseof2gasgiantsresem-
Despite their disparate perspectives on the timescale bling Jupiter and Saturn. Oligarchs and gas giants are
problem and different motivations, the scenarios of referred to as planets. We employ the hybrid orbit in-
Thommesetal.(1999),Tsiganisetal.(2005),andGLS04 tegratorMERCURY6(Chambers1999),whichcombines
share quite a few features. In their simplest forms, each a conventional Bulirsch-Stoer integrator to handle close
theory starts with a more crowded configuration for so- encounters between planets, with the fast symplectic al-
lar system planets than is observed today; each is char- gorithminventedbyWisdom&Holman(1991). Viscous
acterized by an intermediate period of dynamical chaos stirringofanoligarchbyotherplanetsissimulatedasac-
during whichUranusandNeptune execute highly eccen- curately as the orbit integrator solves the gravitational
tric orbits; and each invokes final regularization of ice equations of motion. Case studies of viscous stirring are
giant orbits by dynamical friction with an ambient disk. described in 2.2.
Thommes et al. (2002) and Chiang et al. (2006, here- To model d§ynamicalfrictionbetween a planet and the
after C06) point out that these violent histories might surroundingdiskofplanetesimals,weintroduceapertur-
be encoded in Kuiper belt objects (KBOs). In particu- bative force on each planet. The force damps the com-
lar,so-calledscatteredKBOspossesslargeeccentricities, ponentofthe planet’svelocitythatdiffersfromthelocal
inclinations,andperiheliondistanceswhichmightreflect disk (circular) velocity. For simplicity, we take the disk
gravitationalstirring by marauding ice giants. to have a constant surface density between an inner and
The notion that planets originate in compact and an outer radius. Disk parameters are held fixed. In our
crowdedconfigurationsisbolsteredbythestudyofextra- simple scheme, the planets respond to the disk through
solarsystemsaswell. Toexplainthe strikingpreponder- dynamical friction, but the disk does not respond to the
anceoflargeorbitaleccentricitiesobservedamongextra- planets. The details of the perturbation force are pro-
solar giant planets (Butler et al. 2006), multiple planets vided in 2.3. The validity of our fixed disk approach is
each having on the order of a Jupiter mass are imag- briefly co§nsidered in 4.2.2.
inedto haveonceresidedonorbitssufficiently closethat Finally, to investig§ate how oligarchs might excite the
the planets scatter one another onto elliptical trajecto- Kuiper belt, an ensemble of test particles is included in
ries(Marzari&Weidenschilling 2002;Ford,Rasio,& Yu a subset of the simulations. We use the terms “test par-
2003; Ford, Lystad, & Rasio 2005). ticle” and “Kuiper belt object (KBO)” interchangeably.
GLS04outline a possible formationhistory forUranus ThesetestparticlesareintendedtorepresentlargeKBOs
andNeptuneinapackedoligarchy,andC06expandupon likethoseobservedtoday,havingsizesontheorderof100
its consequences for the Kuiper belt, by making many km. Thissizeissmallenoughthatwecanneglectdynam-
simplifyingassumptionsandorder-of-magnitudeapprox- ical friction between KBOs and the disk, yet also large
imations. In this paper, we test some of their ideas by enough that we can ignore damping of KBOs’ velocities
numerical simulations. In particular, we seek answers to by physical collisions with the disk (C06). Thus, in our
the following questions: simulations, KBOs (test particles) feel directly only the
gravity of the Sun and of the planets.
1. For a shear-dominated oligarchy containing more
In the sub-sections below, we explore separately the
than two Neptune-mass oligarchs beyond Saturn’s
processes of viscous stirring ( 2.2) and dynamical fric-
orbit, what is the critical value of the disk surface §
tion ( 2.3), in isolation from one another. We present
densitybelowwhichthevelocityinstabilityoccurs? §
full-fledged simulations, in which the two processes are
2. What is the likelihood that the instability will re- combined, in 3.
§
Ice Giants 3
2.2. Viscous Stirring unique setof starting longitudes and eachlasting 107yr.
The timestep for the symplectic integrator is set to 130
We first study how multiple oligarchs gravitationally
days. Timesteps for the conventional Bulirsch-Stoer in-
stir one another. For this sub-section, we ignore dynam-
tegratorareasshortasnecessarytoachieveanaccuracy
ical friction with the disk, but include the gas giants,
parameter of 10−10. The changeover distance that sep-
Jupiter and Saturn. We compare our findings to those
arates the symplectic regime from the close encounter
ofGLS04. The results ofthis sub-sectionwillbe applied
regime is set to ∆r =3 Hill radii.
in 3 to understanding the thresholdconditions required crit
§ A “collision” between two bodies occurs when their
for velocity instability.
mutual separation becomes less than the sum of their
physical radii. The physical radius of each oligarch is
2.2.1. Initial Conditions
computed using Neptune’s bulk density of 1.6g/cm3.
We consider N =5 oligarchs,each having the mass
olig PhysicalradiiforJupiterandSaturnarecomputedusing
of Neptune (µ = m/m⊙ = mN/m⊙ = 5.1 10−5). The densities of 1.3 and 0.7g/cm3, respectively. We assume
×
oligarchs are initially spaced 5 Hill radii apart in semi- that bodies that collide merge completely.
majoraxis(a);thatis,thedifferencebetweensemi-major An “ejection” occurs when anoligarch’sdistance from
axes of nearest-neighboring oligarchs is the Sun exceeds 10000 AU and when its total kinetic
plus potential energy (evaluated in barycentric coordi-
a a =2.5(µ/3)1/3a +2.5(µ/3)1/3a , (1) nates with the potential energy set to zero at infinity)
j+1− j j j+1 becomes positive. Ejected planets are dropped from the
where j ranges from 1 to N . The coefficient of 2.5 is simulation.
olig
inspired by numerical studies by Greenberget al.(1991) 2.2.2. Results: Outcomes After t=107yr
of the width of a protoplanet’s feeding zone, for the case
whereKepleriansheardominatestherelativevelocitybe- At t = 107yr, the outcomes for all Nall = Nreal
×
N = 200 5 = 1000 oligarchs divide into the follow-
tweenaprotoplanetandaplanetesimal(seetheireqn.9; olig
×
ing mutually exclusive categories, in order of decreasing
seealsoIda&Makino1993). While thecoefficientof2.5
frequency of incidence:
is the standard value for our study, the coefficient in re-
alitycanbe somewhatdifferent,dependingonthe accre- 1. Ejection but no collision (463)
tion and migration histories of the planets. We explore
someconsequencesofvaryingthecoefficientin 2.2.3and 2. No collision and no ejection (439)
§
4.2.4.
3. Collisionwithanotheroligarchbutnoejection(42;
§
We assume the semi-major axis of the innermost oli-
i.e.,21oligarchsremain,eachwithmasstwicethat
garch a = 15 AU. Then according to eqn. (1), semi-
1 of an original oligarch)
major axes of the next four oligarchs equal 17.1, 19.4,
22.1,and25.1AU. Initialeccentricities, andinitial incli- 4. Collision with Sun only (19)
nationsrelativetoanarbitraryreference(x-y)plane,are
5. Collision with Jupiter only (14)
suchthat e =sini =10−4. All orbitalelements in this
j j
paper are osculating and referred to a barycentric coor- 6. Collision with another oligarch, and subsequent
dinate system. That is, when computing the osculating ejectionorsubsequentcollisionwithJupiterorSat-
Kepler elements for a given body, the position and ve- urn (13)
locity of the body are referred to the system barycenter
7. Collision with Saturn only (10)
(calculated using all massive bodies), while the central
massofthe assumedKepler potentialequalsthe massof The dominant outcome is ejection. In 50% of the re-
theSunplusthatofthegivenbody(alone). Foreacholi- alizations (i.e., 100 out of N = 200), at least one
real
garch, the initial longitude of ascending node, longitude oligarch is ejected before t = 1.6 106yr. By t =
ofpericenter,andmeananomalyarerandomlygenerated 3.2 106yr,85%ofallrealizationsexp×erienceafirstejec-
×
from uniform distributions between 0 and 2π. tion. Allbut5outof200realizationsexperienceatleast
Jupiter and Saturn are assigned initial masses and one ejection of an oligarch by t=107yr.
semi-major axes equal to their current values: µ = Jupiter and Saturn are responsible for the preponder-
J
mJ/m⊙ = 9.5 10−4, µS = mS/m⊙ = 2.9 10−4, ance of ejections. When we repeat the experiment with
a = 5.18 AU, a×nd a = 9.54 AU. Initial eccen×tricities Jupiter and Saturn omitted, outcomes at t = 107yr are
J S
equale =e =0.05,andinclinationsrelativetothe ref- as follows: 870 out of 1000 oligarchs experience neither
J S
erence plane are such that sini =sini =0.01. Orbital an ejection nor a collision, 118 collide with another oli-
J S
longitudes are randomly generated, just as they are for garch(so that 59 remain), and 12 collide with two other
the oligarchs. oligarchs (so that 4 remain). No ejection is observed to
These initial conditions, particularly our choices for occur by t=107yr when the gas giants are absent.
a =15AUandN =5,aresomewhatarbitrary. They These outcomes are consistent with timescale esti-
1 olig
are intended to represent qualitatively the final stages matesbyGLS04. NeglectingJupiterandSaturn,GLS04
of shear-dominated oligarchy in the outer solar system. predictthattheoligarchejectiontimescaleis 109yr(see
∼
We will adjust starting parameters (e.g., a , N ) in their eqn. 114). This is consistent with our finding that
1 olig
later sections to achieve simulation outcomes in better no ejection occurs within t = 107yr in the absence of
agreement with observed properties of the solar system. gas giants. GLS04 mention the possibility that Jupiter
A total of N = 200 orbital integrations (“re- and Saturn hasten ejections. We confirm this possibil-
real
alizations”) are performed with the hybrid integrator ity. When gas giants are present, we find the ejection
MERCURY6 (Chambers 1999), each characterized by a timescale is 106yr.
∼
4 Ford & Chiang
themasssurfacedensityofoligarchs. Conjunctions
with these oligarchs occur over the synodic period
a
t t , (3)
syn orb
∼ x
where t is the orbital period. Then
orb
de ∆e Σa2 1
N constant (4)
dt ∼ tsyn ∼ m⊙ torb ∼
as roughly observed in Fig. 1. As time elapses,
ever closer neighbors at smaller x drive the stir-
ring. This reasoningmatches that givenby GLS04
in their treatment of viscous stirring in the shear-
dominated regime; their eqn. (33) is identical in
form to our eqn. (4). The t1 scaling is also derived
by Collins & Sari (2006) and Collins, Schlichting,
& Sari (2007). These latter studies concentrate on
the limit N 1.
Fig. 1.— Viscous stirring of oligarchs, in the absence of dy- olig
≫
namical friction, for an oligarch separation of 5RH. We show the
median(solidcurve),85thpercentile(longdashedcurve),and15th 2. Conjunctions with Nearest Neighbors: At inter-
percentile(dotted curve)oftheoligarchs’eccentricitydistribution mediate times 500 . t(yr) . 1.5 104, plane-
versustime,basedonN-bodyintegrationswithnodynamicalfric- tary orbits remain non-crossing but×the eccentric-
tion. In panel (a), each integration includes Jupiter, Saturn, and
ity distribution hardly changes. Since the syn-
five oligarchs. In panel (b), only the five oligarchs are integrated.
Arrows in panel (a) mark the times when 15%, 50%, and 85% of odic period between nearest neighboring oligarchs
Nreal=200realizations(integrations)experiencethefirstejection is tsyn 500yr, a given oligarch during this phase
of an oligarch. Vertical lines separate various phases of evolution experie∼nces repeated conjunctions with its nearest
discussedin§2.2.3.
neighbor. Suchrepeatedcloseencountersmightbe
2.2.3. Results: Eccentricity and Inclination Growth expected to produce chaotic motion and to cause
(“Stirring Curves”) eccentricities to random walk, in which case e
50
∝
Fig. 1 tracks the median eccentricity, e (t), of all oli- t1/2. Thatthis scaling is not observedimplies that
50
garchs. The sample from which the median is drawn al- our 5 oligarchsdo not behave in a strongly chaotic
wayscontainsN =1000objects,regardlessofwhether manner despite their close spacing. Indeed, we ob-
all
oligarchs collide or are ejected. When computing the serve in our simulations that the epicyclic phases
median, we adopt the following rules: ejected oligarchs (true anomalies) of a given oligarch at successive
have their eccentricities set equal to 1 (but remain part conjunctions with its nearestneighbor do not vary
of the sample); an oligarch that collides with either the completelyrandomly. Perturbationsfromconjunc-
Sun, Jupiter, or Saturn has its eccentricity set equal to tions with a nearest neighbor apparently tend to
1;andoligarchsthatcollide withotheroligarchsarestill cancel out during this second phase.
countedasseparateobjects andareeachassignedanec-
centricityequaltothe currenteccentricityofthe merged 3. Onset of Orbit Crossing: Thecancellationscharac-
body. Experiments with alternative sets of rules pro- terizing the preceding phase are not perfect, how-
duced no qualitative changes to our results. ever. Eventually, from t 1.5 104yr to t
Theresultant“stirringcurves”ofFig.1exhibitavari- 5 104yr, eccentricities su∼rge as×oligarchs star∼t
×
ety of behaviors. We first discuss the case when Jupiter crossing orbits. The median eccentricity e50 sur-
and Saturn are omitted from the integrations (Fig. 1b). passes the orbit-crossing value, e 0.06, during
≈
As annotated in Fig. 1b, we distinguish four phases of this third phase.
viscous stirring: Our finding that orbits cross in a few 104yr is
×
consistent with numerical experiments by Cham-
1. Distant Conjunctions: At early times t . 500yr, bers,Wetherill, &Boss(1996),who measuretimes
planetaryorbitsdonotcrossande50growsroughly required for close encounters to occur in initially
linearlywithtime. Alineardependenceisexpected circular,co-planar,multi-planetsystemsasa func-
from viscous stirring by distant conjunctions, i.e., tion of planet mass and orbital spacing. For refer-
conjunctionsbetweenoligarchsthatarenotnearest ence,ourspacingof5R correspondsto 4“mutual
H
neighbors. Toderivethet1 scaling,weestimateus- Hill sphere radii” as defined by those authors.
ing the impulse approximation that a conjunction
between two oligarchsseparatedby distance x<a 4. Orbit Crossing: At late times t & 5 104yr, oli-
×
imparts eccentricities on the order of garchs routinely cross orbits and we observe e
50
t0.25. We can reproduce this scaling using the fo∝l-
a 2
∆e µ (2) lowing particle-in-a-box argument. An oligarch’s
∼ (cid:16)x(cid:17) random velocity v at time t is determined largely
to both bodies, provided they have eccentricities by its closest encounter with another oligarch up
less than ∆e prior to conjunction. For a given until that time. We call the impact parameter
oligarch, a total of N Σax/m oligarchs all re- characterizingthisclosestencounterb . Fromki-
min
side about the same di∼stance x away, where Σ is netictheory,nb2 vt 1,wheren ΣΩ/mvisthe
min ∼ ∼
Ice Giants 5
number density of oligarchs and we have assumed
that the random velocity distribution of oligarchs
is isotropic. It follows that b 1/t1/2 and
min
∝
v (Gm/b2 )1/2 t1/4. This scaling agrees with
∼ min ∝
that of eqn.(49)of GLS04 andthat ofeqn. (14)of
C06.
WhenwerestoreJupiterandSaturntotheintegrations
(Fig.1a),wecanstilldiscernthefourphasesenumerated
above. However, compared to the case without giants,
some phase boundaries are shifted to earlier times, and
e risesmorequicklyduringsomephases. Phase1tran-
50
sitionstophase2aatt 30yr;atthistime,alloligarchs
≈
haveundergonetheir firstconjunctionswith Jupiterand
Saturn. Phase2atransitionstophase2batt 500yr;as
≈
inthecasewithoutgiants,thistransitionmarksthetime
when every oligarchhas experienced about one conjunc-
tion with its nearestneighboring oligarch. During phase
2b, we witness the same remarkable near-constancy of Fig. 2.— Viscous stirring of oligarchs, in the absence of dy-
e 0.01. Finally, during phase 4 at t & 3 104yr, namicalfriction,forassumedoligarchicspacingsof3RH and7RH.
w5h0en≈oligarchs are on crossing orbits, e50 t0.×38. Such Dbuattaonalryemgeendeiraanteedccaenntdripcirteiseesnaterde dinispthlaeyesda.mWe witahyouatstfhoer Fgaigs.g1i-,
∝
growth outpaces that observed in the absence of the gas ants, the same 4 phases characterizing viscous stirring for 5RH
giants. (Fig.1)areevident for7RH. Aspacingof 3RH produces systems
sufficiently chaotic that phase 2b is indiscernible. Adding the gas
Whataboutoligarchinclinations? Insimulationswith-
giants, however, reduces the sensitivity of the results to the oli-
outgasgiants,weobservethatthemedianinclinationi50 garchicspacing.
remains fairly constant at the initial value of 10−4 until
phase 3. As orbits cross, sini surges up to 0.05, and migration. We adopt in this paper a standard value of
50
thereaftergrowsast0.25 duringphase4,justa∼se does. 5R ,identicaltothatassumedbyGLS04,andmotivated
50 H
By t = 107yr, i 10◦. When Jupiter and Saturn are bystudiesofshear-dominatedaccretionbyGreenberget
50
included, sini ≈t0.28 during phase 4. By t = 107yr, al. (1991). Shorter spacings seem less attractive insofar
50
i 10◦, as was∝the case without the gas giants. The as they will produce smaller isolation masses for a given
50
≈
modestgrowthofoligarchs’inclinationswilllimitthede- disk surface density.
gree to which inclinations of KBOs are stirred ( 3.3.2). The intermediate phase of slow-to-nogrowthofeccen-
§
Not all of the different phases of viscous stirring that tricity that characterizes oligarchic spacings 5R will
H
≥
we observe are anticipated from the study of GLS04, prove important in determining the threshold disk sur-
whichdocumentsonlythet1scalingcharacterizingshear- face density below which dynamicalfrictioncooling can-
dominated oligarchy(phase 1)and the t1/4 scaling char- not balance viscous heating, i.e., the threshold surface
acterizing the super-Hill, orbit-crossingregime (phase 4, density for the velocity instability ( 3.2).
§
no giants). Their analysis misses the intermediate phase
2 of slow-to-no growth just prior to orbit crossing, and 2.3. Dynamical Friction
the significant roles that Jupiter and Saturn play in ac- Oligarchs grow from a disk of planetesimals. Those
celerating viscous stirring during phase 4 (t0.38 vs.t1/4). planetesimalsthatarenotaccretedexertdynamicalfric-
Thatdifferencesexistisnottoosurprising,astheiranal- tion on oligarchs. The conditions for velocity instability,
ysisisrootedinthelargeNolig 1limit,whereasforour andtheeasewithwhichsurvivorsoftheinstabilityreturn
≫
system, Nolig = 5. More importantly, we take nearest- to low-eccentricity,low-inclinationorbits, depend on the
neighboring oligarchs to be separated by 5 Hill sphere strength of dynamical friction. We describe how we im-
radii, as dictated by the extent of an oligarch’s feeding plement dynamical friction in our simulations in 2.3.1,
zone in a shear-dominated disk (Greenberg et al. 1991), present a test case in 2.3.2, and show that our §imple-
whereastheorder-of-magnitudeequationsofGLS04gov- mentation is compatibl§e with the formulae of GLS04 in
erning shear-dominated oligarchyassume the separation 2.3.3.
is closer to 1 Hill sphere radius. In this regard, we §
∼
presentinFigure2viscousstirringcurvesforcaseswhere 2.3.1. Prescription
the oligarch separation is 3R and 7R (corresponding
H H Consider a planet having an eccentricity and an in-
to coefficients in eqn. (1) of 1.5 and 3.5, respectively).
clination much greater than those of disk planetesimals.
Without gas giants, for a separation of 3R , phases 2
H Dynamical friction reduces the planet’s random (pecu-
and 3 disappear, leaving only phases 1 and 4 as origi-
liar) velocity: the difference ~v vvˆ between the orbital
nally envisioned by GLS04. The time to orbit crossing ≡
velocity of the planet and that of the mean disk flow.
variesfrom 300yrto 2 106yrasthespacingchanges
∼ ∼ × From Binney & Tremaine (1987, their eqn. 7-17),
from 3 to 7 Hill radii. This extreme sensitivity to spac-
ing wasalsofound byChamberset al. (1996). Including d~v 2πG2mρ
gas giants, however, reduces this sensitivity, as Fig. 2a = ln(1+Λ2)vˆ, (5)
dt − v2
shows.
Theactualoligarchicspacingmightonlybedetermined where ρ is the local mass density of the disk, m is the
by carefulnumericalsimulationsofaccretionandorbital mass of the planet, G is the gravitationalconstant, and
6 Ford & Chiang
strengthofdynamicalfrictioninoursimulationsdepends
b (v2+2 sin2i v2 ) muchmorestronglyontheplanet’srandomspeedvthan
Λ= max h diski circ (6)
Gm onthemuchsmallerrandomspeedsofplanetesimals(see
eqns. 6–7). Planetesimals can maintain low velocity dis-
istheCoulombparameterappropriatefordynamicalfric-
persions by inelastic collisions or by gas drag.
tionin a Kepleriandisk (Stewart& Ida 2000). For b ,
max Our scheme for dynamical friction damps orbital in-
the maximumimpactparameterbetweenthe planetand
clinations relative to the x-y (disk) plane. The inclina-
a disk planetesimal, we adapt the expression of Stewart
tion may become so small that ∆t 1/sini exceeds
&Ida(2000;seethediscussionfollowingtheireqn.2-17): ∝
t , at which point the planet is immersed within the
orb
disk and our impulse approximation breaks down. To
bmax =RH+r sin2idisk +sin2i 1/2 , (7) account for this possibility, we arbitrarily set ∆t =
h i
(cid:0) (cid:1) min(h/v ,0.025/Ω). Our softening prescription slows
wherer istheinstantaneousdistancebetweentheplanet | z|
but does not stop the damping of inclination and eccen-
and the system barycenter, R = (µ/3)1/3r is the Hill
H tricity for sini . 0.004. The softening might represent
sphere radius, and sin2i 1/2 1 is the inclination
disk slight misalignments between the planet’s orbital plane
h i ≪
dispersion of disk planetesimals, held constant for each
and the disk midplane, which in reality will be warped.
simulation (more on its precise value later). The term
Wehaveverifiedthatourprincipalfindings,describedin
t2hhseinv2eliodcisiktiyvcd2iirscpienrseioqnn.of(6d)isakppplraonxeimteasitmesaltsh,ewshqeuraerveciorcf §p3r,esdcroipntoiotnd.epWenhdilesetnhseitipvreelycisuepovanlutehseodfettahielsinocflitnhais-
is the local mean disk speed. We take vcirc to equal tions that we compute are clearly not very meaningful,
the speed that the planet would have on a circular orbit we expect that our results are still qualitatively correct,
about the Sun. i.e., the code correctly identifies when mutual inclina-
Usually it is assumed in writing eqn. (5) that Λ 1. tions between planetary orbits are large (> 1rad) and
≫
We do not make this assumption. In fact, we use small ( 1rad).
eqns. (5)–(7) regardless of the magnitude of Λ. When Them≪ainvirtueofourprescriptionfordynamicalfric-
Λ 1, dynamical friction is in the shear-dominated tion is its simplicity. We need only specify the disk sur-
≪
regime. In 2.3.3, we justify our universal application face density σ(r), notthe volumetricdensity ρ(r,z), and
§
of (5)–(7) by showing that these equations correctly re- we need only apply dynamical friction at disk crossings.
duce to forms appropriate to the shear-dominated case The main shortcomingofour prescriptionis thatit does
when Λ 1. not account for the response of the disk to the plan-
≪
We implement dynamical friction as follows. We are ets. The clearing of gaps and the generation of time-
interestedinthe casewherethe oligarchsaresodynami- dependent, non-axisymmetric structures (see, e.g., Gol-
callyexcitedthateachplungesthroughaverticallythin, dreich&Tremaine1982)willalterthegravitationalback-
dynamically cold disk of planetesimals twice per orbit. reactionof the disk onto embedded planets in ways that
Specifically, we assume that the time a planet spends couldbesignificantbutthatwe(andGLS04)donotcap-
immersed in the disk, ∆t h/vz , where h is the full ture. Forwaystomodeltheresponseoftheplanetesimal
≈ | |
verticalthicknessofthediskand vz istheverticalcom- diskmorerealistically,seeLithwick&Chiang(2007)and
| |
ponentof~v atthemomentofdiskcrossing,isshortcom- Levison & Morbidelli (2007).
pared to the orbital period, t = 2π/Ω. Equivalently,
orb
sini sin2i 1/2. At every disk crossing, a planet
disk
≫ h i 2.3.2. Test with Single Planet
receives a specific impulse of
We test our prescription for dynamical friction in the
d~v 2πG2m case of a single planet interacting with a disk. The disk
∆~v ∆t ln(1+Λ2)ρ∆tvˆ has constant surface density σ = 1g/cm2. A Neptune-
≈ dt ≈− v2
mass oligarch is placed on an orbit having initial semi-
2πG2m σ
ln(1+Λ2) vˆ, (8) major axis ainit = 20 AU and initial eccentricity and
≈− v2 |vz| inclinationsuchthateinit =siniinit =0.3. Twocasesare
considered, one where the initial argument of perihelion
whereσ isthe disk surfacedensity (height-integratedρ).
ω = 0 and another where ω = π/2. The evolution
At every timestep of the integration, we check whether init init
of e(t) and i(t) depends on ω modπ.
the planet crosses through the disk, which is fixed to lie init
Fig. 3 displays the resultant evolution. The planet’s
in the x-y plane. At moments of disk crossing,we apply
eccentricity and inclination both drop precipitously to-
akickaccordingtoeqn.(8): weincrementthevelocityof
ward zero after a time on the order of 106yr; the exact
theplanetby∆~vbutdonotchangetheplanet’sposition.
time variesby a factor of3between ourchoicesfor ω .
We compute the difference velocity~v by subtracting the init
Thesemi-majoraxiscanincreaseordecrease. Itchanges
barycentric velocity of the planet from v φˆ, where φˆ
circ by 3–13%, on the order of but less than the starting ec-
is the unit vector in the azimuthal direction. The kick
centricity.
isappliedinthesubroutineMDT HY.FORintheMER-
We check our numerical results by comparing them to
CURY6code,afterthepositionsareadvancedbutbefore
the following approximate analytic solution. Since the
the velocities are updated for the second time by the in-
kick ∆~v is applied twice per orbit, we write
teraction Hamiltonian.
For all our simulations, we fix sin2i 1/2 = 10−3,
disk
h i
a value sufficiently small that sini sin2i 1/2 for d~v 2∆~v
disk
≫ h i
all but a tiny fraction of the time. In other words, the dt ≈−t
orb
Ice Giants 7
We next recognize that 4πG2ρ2R4/3 = 3v4 /16π,
p p esc,p
where v is the escape velocity from the surface of
esc,p
the planet. Then eqn. (13) simplifies to
1dv 3√2 σΩ v 4
= ln(1+Λ2) esc,p . (14)
v dt − 8π ρpRp (cid:16) v (cid:17)
When ln(1+Λ2) is a constant of order unity, eqn. (14)
matchesthe formofeqn.(45)ofGLS04,evaluatedusing
the first line of their eqn. (46), with their planetesimal
random velocity u replaced by v (since v > u; see their
section 5.5, end of first paragraph). This formula de-
scribes dynamical friction in the dispersion-dominated
regime, where v exceeds the Hill velocity v =ΩR .
H H
On the other hand, it is possible for Λ 1. This
≪
happens, according to (6)–(7), when v √2iΩ
∼ ≪
Gm/R v (terms proportional to sin2i 1/2 =
H H disk
Fig. 3.— Test of our prescription for dynamical friction. We p10−3 are ne∼gligible). In this shear-dominhated regimi e,
calculatetheorbitalevolutionofasingleNeptune-massplanetin-
teracting withadiskofsurfacedensity σ=1g/cm2. Solidcurves
denote the evolution when the planet’s initial argument of peri- RHv2
Λ 1, (15)
centerωinit=0(sothattheorbitintersectsthediskatpericenter ≈ Gm ≪
and apoocenter), while dashed curves correspond to ωinit = π/2
(sothattheorbitintersectsthediskatquadrature). Dottedcurves ln(1+Λ2) Λ2, and eqn. (14) reduces to
representouranalyticsolution(11)whichassumesaconstantsemi- ≈
majoraxisaandCoulombparameter Λ.
2
1dv 3 σΩ R
H
G2σm ~v =
4πln(1+Λ2) . (9) v dt −π√2ρpRp (cid:18)Rp(cid:19)
≈− t v3 v
orb z 3 σΩ 1
| |
= , (16)
We set e=sini and make the following approximations: −π√2ρpRpα2
~v =(epˆ+izˆ)Ωa,v =√2eΩa,and v =iΩa. Herepˆand
zˆ are unit vectors that lie paralle|l azn|d perpendicular to where we have defined, following GLS04, α ≡ Rp/RH.
Eqn. (16) matches, to within a numerical constant,
the disk, respectively. Then (9) simplifies to
eqn. (45) of GLS04, evaluated using the second line of
their eqn. (46).
de ln(1+Λ2)Gσ m 1
= , (10) We conclude that our treatment of dynamical friction
dt − √2 Ωa m⊙e3 is compatible with that of GLS04.
with an analogous equation for i. For fixed Λ and a,
3. RESULTS
eqn. (10) integrates to
Wepresenttheresultsofsimulationsthatcombinevis-
4ln(1+Λ2) m Gσ 1/4 cous stirring due to multiple oligarchs, with dynamical
e= e4 t . (11) friction due to a planetesimal disk.
(cid:18) init− √2 m⊙ Ωa (cid:19)
3.1. Initial Conditions and Integration Times
The eccentricity (equivalently, inclination) vanishes in a
finite time Each system begins with either Nolig = 5, 4, or 3 oli-
garchs, together with the gas giants, Jupiter and Sat-
tvanish = 4ln(1√+2 Λ2)mm⊙GΩσae4init. (12) tuinironn§.s2.I2on.f1it,oialeilxgccaoernpchdtsittihaoarnetsisanuricethiatlhtehecasctaemneteriac=sititeshsinosaiendd=eisncc1rli0ibn−ea2d-.
j j
We overlay eqn. (11) in Fig. 3, taking as representative Initial inclinations are therefore ten times larger than
valuesa=a =20AUandln(1+Λ2)=ln(1+Λ2 )= the assumed inclination dispersion of disk planetesimals
init init
13. Theanalyticsolutionlies betweenthe twonumerical ( sin2i 1/2 = 10−3). Initial eccentricities are the
disk
h i
solutions. We consider the agreement acceptable. same as those that characterize phase 2b of the viscous
stirringcurves(Fig.1a). Planetaryorbitsinitiallydonot
2.3.3. Connecting Our Prescription to GLS04 cross.
Dynamicalfrictionisexertedbyadiskofconstantsur-
Our equations can be re-cast into the same forms as
face density σ which extends from a barycentric radius
those of GLS04, under the assumption e = i. We start
of12.5AUto45AU.Theouterboundarycoincideswith
with eqn. (9) and substitute v = v/√2, t = 2π/Ω,
and m=(4π/3)ρ R3, where ρ| z|and R areotrhbe internal the location of the classical Kuiper belt (C06). The in-
p p p p ner boundary is less well motivated. It is chosen so that
density and physical radius of the planet:
the oligarchsreside initially inside the disk while the gas
giants do not. We explore values for σ ranging from 0.4
1dv = 23/2ln(1+Λ2)σΩ4πG2ρpRp3 . (13) tinoo0l.i0g0a1rcgh/scmis2Σ. For1.r5efge/rcemnc2e., the initial surface density
v dt − 3v4 ≈
8 Ford & Chiang
Inasubsetofruns,weinclude400testparticlesrepre-
senting large KBOs. These feel the gravity of the plan-
ets but do not feel dynamical friction from the disk (see
2.1). Initialsemi-majoraxesoftestparticlesrangefrom
§
a = 40 to 45 AU, and initial eccentricities and inclina-
tions are such that e=sini=10−2. For all planets and
KBOs, initial longitudes of ascending node, longitudes
of pericenter, and mean anomalies are randomly chosen
from uniform distributions between 0 and 2π.
Settings for the MERCURY6 code are the same as
those given in 2.2.1, except for the duration of inte-
§
gration. The integration automatically halts when there
are a catastrophic number of ejections, i.e., when the
only massive bodies remaining include the Sun, Jupiter,
Saturn, and one oligarch (which may have collided with
other oligarchs). In the absence of such an event, each
systemis integratedfirstto t=2 107yr. If the planets
×
seemtohavestabilizedatthattime—i.e.,theireccentric-
Fig. 4.— Branching ratios for the outcome of the velocity in-
ities no longer grow—then we stop the integration and
stability, for systems that include dynamical friction with a disk
record the outcome as final. Otherwise, we repeat this
ofsurfacedensityσ. Forcomparison,theinitialsurfacedensityin
test as necessary at 5 107yr and 1 108yr. oligarchs is Σ ≈ 1.5g/cm2. Systems begin with Jupiter, Saturn,
Byt=1 108yr,mo×stbutnotallre×alizationsstabilize. and either Nolig = 5 (top panel), 4 (middle panel), or 3 (bottom
Excluding×systems that are stopped abruptly once only panel) oligarchs. Different symbols denote the fraction of realiza-
tions that result in 0, 1, 2, 3, or 4 ejections of oligarchs, whether
three planets remain, we find that . 10% of systems
collisionallymerged or not. Few oligarchs collidein these simula-
have undergone a close encounter (here defined to occur tions; the fraction of realizations that result in 1 collision is less
when the distance between any two planets is less than than20%forallvaluesofσtested,andthefractionofrealizations
1 Hill radius) within the last 107yr of the integration, thatresultin>1collisionis<1%. Toproduce≥1ejectionwith
>20% probabilityrequiresσ.Σ/10. Standarderrorbarsdueto
for Nolig = 5 and all values of σ tested. For Nolig = 4 counting statistics aregiven forcurves corresponding to Nolig−2
and 3, the corresponding fractions are . 10% and . ejections.
1%. Those realizations that do not stabilize by t = 1
108yr are typically characterized by small values of σ ×. for oligarchs to cross orbits, eccentricities must not be
0.006g/cm2 (i.e.,relativelyweakdynamicalfriction)and allowed to vanish by dynamical friction before tunstable:
oneoligarchremainingonaneccentricorbitthatextends
t >t . (17)
wellpasttheouteredgeofthe disk. Fortheselowvalues vanish unstable
of σ, the number of ejected oligarchs that we report will Using (12) for t , we find that (17) translates into
vanish
be underestimated, but not in a way that changes our σ <σ , where
crit
qualitative conclusions.
The initial conditions just summarizedapply to allre-
nsualttisveinsetthoeffionliltoiwalincgontdwiotiosnecs,timonost,iv§a§t3e.d2–b3y.3t.heAfinnadlitnegrs- σcrit∼4ln(1√+2 Λ2)mm⊙GΩtae4init (18)
unstable
in those sections, and results pertaining thereto are pre-
sented in 3.4. 0.2g/cm2 0.1Σ. (19)
§ ∼ ∼
Thenumericalevaluationtakesa=19.4AU,e =0.01
3.2. Threshold Disk Surface Densities for Instability init
(thevalueappropriatetophase2boftheviscousstirring
Oligarchs cross orbits when the disk surface density
curves in Fig. 1a; the median eccentricity does not rise
σ is so low that dynamical friction cooling cannot com- above 0.01 for t < t ), and ln(1+Λ2) = ln(1+
unstable
pete with viscous stirring. GLS04 estimate the critical Λ2 )=0.02.
surface density for instability to be on the order of the init
Our semi-empirical estimate for σ finds support in
crit
surface density of oligarchs: σ Σ (see also Chiang
crit Fig. 4a, which documents, for all runs starting with
∼
&Lithwick2005foracorrectioninthe derivationofthis
N = 5 oligarchs, the frequency of incidence of out-
olig
result). How well does this criterion predict the onset of
comes(“branchingratios”)asafunctionofσ. Instability
instability for our system of N =5 oligarchs?
olig and the subsequent ejection of 1 and only 1 oligarch is
As noted in 2.2, the rates of eccentricity growth(vis- the dominant outcome for σ = 0.1g/cm2 0.07Σ. For
§
cous stirring) exhibited in our N-body integrations dif- ≈
σ &0.3Σ,morethan90%ofrealizationsproducenoejec-
fer from those estimated by GLS04. Specifically, our
tion. Fig. 5 displays a sample simulation for σ 0.07Σ
rates are slower, as evidenced by the period of slow-to- ≈
in which 1 oligarch escapes before the system stabilizes.
no growth of eccentricity (phase 2b) in Fig. 1a. Over-
Figs. 4b and 4c supply branching ratios for N = 4
olig
estimating the vigor of viscous stirring leads to overes-
and 3. To produce the same number of ejections with
timates for σ . We try to predict σ ourselves by
crit crit smaller N (less viscous stirring) requires smaller σ
olig
drawing from the numerical results of 2.2. We observe
(less dynamical friction). For example, for N = 3,
in Fig. 1a that after a time t § 5 103yr, ec- olig
unstable the ejection of 1 and only 1 oligarch is the dominant
centricities surge rapidly to crossing va∼lues×.6 Therefore
asfunctions ofoligarchmassandorbitalspacing,butonlyforthe
6 Chambers et al. (1996) provide fitting formulae for tunstable casewithoutthegasgiantsJupiterandSaturn.
Ice Giants 9
Fig. 5.—Typicalrealizationforσ=0.1g/cm2≈0.07Σinwhich Fig. 6.—Cumulativedistributionsoffinalsemi-majoraxesofsys-
1oligarchescapesbeforethesystemstabilizes. Thetoppanelplots temsthatcontain, inorderofincreasingsemi-majoraxis,Jupiter,
semi-major axis (solid curves), and periastron and apastron dis- Saturn, and two oligarchs (Uranus and Neptune), each with their
tances(dottedcurves)foreachofthesevenplanets. Eccentricities originalmass. These4-planetsystemsresultfromintegrationsorig-
(middle panel) and inclinations (bottom panel) are plotted using inallycontaining7planets(Nolig=5). Eachpanelcorrespondsto
a mixed log-linear ordinate. Black curves refer to Jupiter, gray simulations performed with the disk surface density σ indicated.
curves refer to Saturn, and remaining colored curves refer to the Verticallinesmarkthecurrentsemi-majoraxesofthegiant plan-
fiveoligarchs. ets in our solar system. Apart from a tendency for Saturn and
Neptune to be pulled toward the inner (a = 12.5AU) and outer
outcome for σ 0.03Σ, occurring in about 60% of real- (a = 45AU) edges of the disk with increasing σ (increasing dy-
≈
izations. The corresponding σ for N =4 is 0.05Σ. namical friction), the characteristic final semi-major axes do not
olig
depend on σ. Saturn, Uranus, and Neptune have final orbits too
3.3. Runs with Two Surviving Oligarchs largecomparedtotheiractualcounterparts inthesolarsystem.
(Solar-System-Like Outcomes)
N =5. Increasing σ increases dynamical friction and
olig
While σcrit 0.1Σ roughly characterizes the onset of therefore tends to pull Saturn and Neptune, whose or-
∼
instability and the subsequent ejection of a single oli- bitslie neardiskboundaries,intothe disk. Forexample,
garch, the disk surface density must be reduced below if Saturn’s orbital apocenter intersects the disk while its
σcrit to produce more than 1 ejection in a large fraction pericenterremainsoutsidethe disk,thendynamicalfric-
ofruns. Togenerateanoutcomereminiscentofoursolar tion will circularize the orbit by raising the pericenter
system starting with Nolig = 5 requires 3 ejections and closer to apocenter. The kinks in the distribution func-
thesurvivalof2oligarchs. AccordingtoFig.4a,suchan tionsforSaturninFigs.6aand6barelocatedata=12.5
outcome occurs with a maximum probability of 50% AU,exactlyatthe inner disk edge. The kink vanishes in
for σ 0.01g/cm2 0.007Σ. The probability ex∼ceeds Fig.6c. ForthesimulationsinFig.6c,dynamicalfriction
≈ ≈
20% for all values of σ .0.03Σ that we tested. is strong enough to pull Saturn’s orbit wholly into the
Figs. 4b and 4c indicate that for Nolig = 4 and 3, disk at a 12.5AU.
the values of σ most likely to produce 4-planet sys- Toimpr≥ovestatistics,weignorethesesmallartifactsof
tems (Jupiter, Saturn, plus 2 oligarchs) are 0.01Σ and our disk boundary conditions and pool together N =
∼ real
0.03Σ,respectively. The probabilitiesforgenerating4- 438 realizations, all of which start with N = 5 and
∼ olig
planet systems starting with Nolig = 4 or 3 reach large produce 4-planet systems, but have a variety of σ’s be-
maximum values of about 50%, and remain above 20% tween 0.001and 0.1g/cm2. From this pool we construct
overa considerable range in σ, up to 0.07Σin the case thedistributionoffinalsemi-majoraxesshowninFig.7a.
∼
Nolig =3. Clearly, most realizations end with Saturn, Uranus, and
The vast majority of the resultant 4-planet systems Neptune on orbits too large compared to their actual
are correctly ordered; they contain, in order of increas- counterparts in the solar system. Furthermore, Jupiter
ing semi-major axis, Jupiter, Saturn, and 2 oligarchs. typically migrates inward as a consequence of ejecting
Moreover, in most of these systems, the surviving plan- severaloligarchsoutward. Theexcessivelylargeorbitsof
ets have not experienced a collision. In the following Saturn,Uranus,andNeptune areaconsequenceofthose
sub-sections we further quantify the properties of these planets having scattered oligarchs inward to Jupiter.
correctly ordered, collisionally unmodified, 4-planet sys- Though they only comprise (given our assumed ini-
tems, comparing them to those of the solar system. We tial conditions) a few percent of outcomes for σ =
refertothe2survivingoligarchsineachofthesesystems 0.02g/cm2, some realizations better resemble the solar
as Uranus and Neptune. system insofar as Uranus and Neptune have final semi-
majoraxeslessthan30AU.Fig.8showcasesanexample.
3.3.1. Final Semi-Major Axis Distributions
Even for this simulation, however,Saturn’s final orbit is
Because packed oligarchies evolve chaotically, we can 3 AU larger than its actual one.
only meaningfully compute probability distributions for Reducingthenumberofstartingoligarchssignificantly
their final semi-major axes. Fig. 6 illustrates how these lessens the problem of excessive migration. Figs. 7b and
distributions depend on σ, for realizations starting with 7cshowfinalsemi-majoraxisdistributionscorresponding
10 Ford & Chiang
Fig. 7.— Cumulative distributions of final semi-major axes of
correctly ordered, collision-free, 4-planet systems, shown for dif-
ferent starting values of Nolig. Each panel combines integrations Fig. 9.—OrbitalelementsofsimulatedKBOs(blackcrosses)at
performedwithavarietyofσ’sbetween0.001and0.1g/cm2. Kinks t = 5×107yr, for the same simulation shown in Fig. 8. Eccen-
inthedistributionsforSaturnandNeptuneareartifactsofouras-
tricities and inclinations are displayed against either semi-major
sumed disk boundaries at a = 12.5AU and 45AU. Vertical lines
markthecurrentsemi-majoraxesofsolarsystemgiants. Decreas- axis a or perihelion distance rp. For comparison we show orbital
elements of known classical and scattered (non-resonant) KBOs
ingNolig shrinksthefinalorbitsforSaturn,Uranus,andNeptune. (redinvertedY’s)fromC06. Orbitalelementsofsurvivingplanets
Finalorbitsresemblingthoseofthecurrentsolarsystemaremost
(Jupiter,Saturn,andthetwooligarchs)areshownasbluecircles.
easilyproducedforNolig=3. SimulatedKBOsforthisrundonotexhibitthelargeeccentricities
andinclinationscharacterizing actual KBOs,butseeFigs.11and
14forotherexamples.
for actual KBOs that do not reside in any strong mean-
motion resonance. These objects, taken from C06, com-
prise both low-e classical and high-e scattered KBOs as
classifiedbytheDeepEclipticSurvey(Elliotetal.2005).
Thesimulatedtestparticleshavetheireccentricitiesand
inclinationsexcitedupto 0.1,valuesthatmatchactual
∼
classical KBOs. But the simulated particles fail to em-
body the extreme degree ofdynamicalheating exhibited
by scattered KBOs. Marauding oligarchs in this simula-
tion stir planetesimals at 40–45 AU too briefly.
Since simulations with N = 3 more efficiently gen-
olig
erate solar-system-like planetary spacings than do sim-
ulations with N = 5, we can more thoroughly map
olig
out the possible extents to which KBOs are stirred for
N = 3. Figs. 10 and 11 document one simulation,
olig
representative of several percent of the solar-system-like
Fig. 8.—Examplerealizationforσ=0.02g/cm2 andNolig=5 realizations generated using Nolig = 3, in which KBOs
inwhich3oligarchs(denotedbymagenta,green,andcyancurves) are stirred considerably. Even here, however, the pro-
escape before the system stabilizes. The remaining 2 oligarchs portion of simulated KBOs that simultaneously attain
(blueandred)havefinal,nearlycircularandco-planarorbitsinside inclinations i & 10◦ and perihelion distances r & 35
30AU,similartothoseofUranusandNeptune. Suchanoutcome p
occurswithafrequencyofafewpercentforσ=0.02g/cm2. AU is less than observed. The proportion of simulated
KBOs having eccentricities e & 0.3 also seems under-
to N =4 and 3, respectively. Outcomes for N =3 represented.
olig olig
are most solar-system-like. In 3.4 we will experiment
with initial conditions to demo§nstrate that a level of 3.4. More Compact Initial Conditions
agreement comparable to that displayed in Fig. 7c for As described in 3.2–3.3, the simulations that begin
Nolig =3 can also be obtained for Nolig =4. with Nolig =4 or 5§o§ligarchsoften do yield 4-planet sys-
tems, but the orbital spacings of the resultant systems
3.3.2. Stirring of KBOs
do not match those of the solar system. Multiple ejec-
Fig. 9 describes how the test particles (KBOs), ini- tions displace Jupiter too far inward and displace Sat-
tially distributed in a dynamically cold ring at a = 40– urn, Uranus, and Neptune too far outward. The prob-
45 AU, are stirred by oligarchs, for the same N = 5 lem of excessive spreading is exacerbatedby the need to
olig
simulation(Fig.8)whichplacesUranusandNeptune on haveNeptuneconcludeitsorbitalevolutionbymigrating
final orbits inside 30 AU. Simulation data are collected smoothly and slowly outward from a 23 to 30 AU to
at t=5 107yr. For comparison, Fig. 9 also plots data producethe populationofresonantKB≈Os(Murray-Clay
×