Mars• growth stunted by an early giant planet instability · 2013-10-14 · semi-major axes via...

Icarus 311 (2018) 340–356

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Icarus

journal homepage: www.elsevier.com/locate/icarus

Mars’ growth stunted by an early giant planet instability

Matthew S. Clement a , ∗, Nathan A. Kaib

a , Sean N. Raymond

b , Kevin J. Walsh

c

a HL Dodge Department of Physics Astronomy, University of Oklahoma, Norman, OK 73019, USA b Laboratoire dAstrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, all e Geoffroy Saint-Hilaire, Pessac 33615, France c Southwest Research Institute, 1050 Walnut St. Suite 300, Boulder, CO 80302, USA

a r t i c l e i n f o

Article history:

Received 21 December 2017

Revised 10 April 2018

Accepted 11 April 2018

Available online 20 April 2018

Keywords:

Mars

Planetary formation

Terrestrial planets

Early instability

a b s t r a c t

Many dynamical aspects of the solar system can be explained by the outer planets experiencing a period

of orbital instability sometimes called the Nice Model. Though often correlated with a perceived delayed

spike in the lunar cratering record known as the Late Heavy Bombardment (LHB), recent work suggests

that this event may have occurred much earlier; perhaps during the epoch of terrestrial planet formation.

While current simulations of terrestrial accretion can reproduce many observed qualities of the solar sys-

tem, replicating the small mass of Mars requires modification to standard planet formation models. Here

we use 800 dynamical simulations to show that an early instability in the outer solar system strongly in-

fluences terrestrial planet formation and regularly yields properly sized Mars analogs. Our most successful

outcomes occur when the terrestrial planets evolve an additional 1–10 million years (Myr) following the

dispersal of the gas disk, before the onset of the giant planet instability. In these simulations, accretion

has begun in the Mars region before the instability, but the dynamical perturbation induced by the gi-

ant planets’ scattering removes large embryos from Mars’ vicinity. Large embryos are either ejected or

scattered inward toward Earth and Venus (in some cases to deliver water), and Mars is left behind as

a stranded embryo. An early giant planet instability can thus replicate both the inner and outer solar

system in a single model.

© 2018 Elsevier Inc. All rights reserved.

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1. Introduction

It is widely understood that the evolution of the solar system’s

giant planets play the most important role in shaping the dynam-

ical system of bodies we observe today. When the outer plan-

ets interact with an exterior disk of bodies, Saturn, Uranus and

Neptune tend to scatter objects inward ( Fernandez and Ip, 1984 ).

To conserve angular momentum through this process, the orbits

of these planets move outward over time ( Hahn and Malhotra,

1999; Gomes, 2003 ). Thus, as the young solar system evolved, the

three most distant planets’ orbits moved out while Jupiter (which

is more likely to eject small bodies from the system) moved in.

To explain the excitation of Pluto’s resonant orbit with Neptune,

Malhotra (1993) proposed that Uranus and Neptune must have

undergone significant orbital migration prior to arriving at their

present semi-major axes. Malhotra (1995) later expanded upon this

idea to explain the full resonant structure of the Kuiper belt. In

the same manner, an orbital instability in the outer solar system

can successfully excite Kuiper belt eccentricities and inclinations,

while simultaneously moving the giant planets to their present

∗ Corresponding author.

E-mail address: [email protected] (M.S. Clement).

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https://doi.org/10.1016/j.icarus.2018.04.008

0019-1035/© 2018 Elsevier Inc. All rights reserved.

emi-major axes via planet-planet scattering followed by dynam-

cal friction ( Thommes et al., 1999 ).

These ideas culminated in the eventual hypothesis that, as the

iant planets orbits diverged after their formation, Jupiter and Sat-

rn’s orbits would have crossed a mutual 2:1 Mean Motion Res-

nance (MMR). Known as the Nice (as in Nice, France) Model

Tsiganis et al., 2005; Gomes et al., 2005; Morbidelli et al., 2005 ),

his resonant configuration of the two most massive planets causes

solar system-wide instability, which has been shown to repro-

uce many peculiar dynamical traits of the solar system. This hy-

othesis has subsequently explained the overall structure of the

uiper belt ( Levison et al., 2008; Nesvorný, 2015a; 2015b ), the cap-

ure of trojan satellites by Jupiter ( Morbidelli et al., 2005; Nesvorný

t al., 2013 ), the orbital architecture of the asteroid belt ( Roig and

esvorný, 2015 ), and the giant planets’ irregular satellites, includ-

ng Triton ( Nesvorný et al., 2007 ).

The Nice Model itself has changed significantly since its

ntroduction. In order to keep the orbits of the terrestrial

lanets dynamically cold (low eccentricities and inclinations),

rasser et al. (2009) proposed that Jupiter “jump” over its 2:1

MR with Saturn, rather than migrate smoothly through it

Morbidelli et al., 2009a; 2010 ). Otherwise, the terrestrial planets

ere routinely excited to the point where they were ejected or


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M.S. Clement et al. / Icarus 311 (2018) 340–356 341

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ollided with one another in simulations. The probability of pro-

ucing successful jumps in these simulations is greatly increased

hen an extra primordial ice giant was added to the model

Nesvorný, 2011; Batygin et al., 2012 ). In successful simulations, the

jection of an additional ice giant rapidly forces Jupiter and Sat-

rn across the 2:1 MMR. Furthermore, hydrodynamical simulations

Snellgrove et al., 2001; Papaloizou and Nelson, 2003 ) show that a

esonant chain of giant planets is likely to emerge from the dissi-

ating gaseous circumstellar disk, with Jupiter and Saturn locked

n an initial 3:2 MMR resonance ( Masset and Snellgrove, 2001;

orbidelli and Crida, 2007; Pierens and Nelson, 2008 ). This con-

guration can produce the same results as the 2:1 MMR crossing

odel ( Morbidelli et al., 2007 ). In this scenario, the instability en-

ues when two giant planets fall out of their mutual resonant con-

guration. Such an evolutionary scheme seems consistent with the

umber of resonant giant exoplanets discovered (eg: HD 60532b,

J 876b, HD 45364b, HD 27894, Kepler 223 and HR 8799 ( Holman

t al., 2010; Fabrycky and Murray-Clay, 2010; Rivera et al., 2010;

elisle et al., 2015; Mills et al., 2016; Trifonov et al., 2017 )).

Despite the fact that 5 and 6 primordial giant planet config-

rations are quite successful at reproducing the architecture of

he outer solar system ( Nesvorný and Morbidelli, 2012 ), delay-

ng the instability ∼ 400 Myr to coincide with the lunar cata-

lysm ( Gomes et al., 2005 ) still proves problematic for the terres-

rial planets. Indeed, Kaib and Chambers (2016) find only a ∼ 1%

hance that the terrestrial planets’ orbits and the giant planets’ or-

its are reproduced simultaneously. Even in systems with an ideal

jump,” the eccentricity excitation of Jupiter and Saturn can bleed

o the terrestrial planets via stochastic diffusion, leading to the

ver-excitation or ejection of one or more inner planets ( Agnor and

in, 2012; Brasser et al., 2013; Roig and Nesvorný, 2015 ). It should

e noted, however, that Mercury’s uniquely excited orbit (largest

ean eccentricity and inclination of the planets) may be explained

y a giant planet instability ( Roig et al., 2016 ). Nevertheless, the

hances of the entire solar system emerging from a late instabil-

ty in a configuration roughly resembling its modern architecture

re very low ( Kaib and Chambers, 2016 ). This suggests that the in-

tability is more likely to have not occurred in conjunction with

he LHB, but rather before the terrestrial planets had fully formed.

ortunately, many of the dynamical constraints on the problem are

airly impartial to whether the instability happened early or late

Morbidelli et al., 2018 ). The Kuiper belt’s orbital structure ( Levison

t al., 2008; Nesvorný, 2015a; 2015b ), Jupiter’s Trojans ( Morbidelli

t al., 2005; Nesvorný et al., 2013 ), Ganymede and Callisto’s differ-

nt differentiation states ( Barr and Canup, 2010 ) and the capture of

rregular satellites in the outer solar system ( Nesvorný et al., 2007 )

re still explained well regardless of the specific timing of the Nice

odel instability. In addition to perhaps ensuring the survivability

f the terrestrial system, there are several other compelling reasons

o investigate an early instability:

1 . Uncertainties in disk properties : Since the introduction of

he Nice Model, simplifying assumptions of the unknown prop-

rties of the primordial Kuiper belt have provided initial con-

itions for N-body simulations. The actual timing of the insta-

ility is highly sensitive to the particular disk structure selected

Gomes et al., 2005 ). Furthermore, numerical studies must approx-

mate the complex disk structure with a small number of bodies

n order to optimize the computational cost of simulations. In fact,

ost N-body simulations do not account for the effects of disk self

ravity ( Nesvorný and Morbidelli, 2012 ). When the giant planets

re embedded in a disk of gravitationally self-interacting particles

sing a graphics processing unit (GPU) to perform calculations in

arallel and accelerate simulations ( Grimm and Stadel, 2014 ), in-

tabilities typically occur far earlier than what is required for a late

nstability (Quarles & Kaib, in prep).

R

2 . Highly siderophile elements ( HSE ): A late instability (the

HB) was originally favored because of the small mass accreted

y the Moon relative to the Earth after the Moon-forming im-

act (for a review of these ideas see Morbidelli et al. (2012) ).

he HSE record from lunar samples indicates that the Earth ac-

reted almost 1200 times more material, despite the fact that

ts geometric cross-section is only about 20 times that of the

oon ( Walker et al., 2004; Day et al., 20 07; Walker, 20 09 ). Thus,

he flux of objects impacting the young Earth would have had

very top-heavy size distribution ( Bottke et al. (2010) , however

inton et al. (2015) showed that the pre-bombardment impactor

ize distribution may not be as steep as originally assumed). This

istribution of impactors is greatly dissimilar from what is ob-

erved today, and favors the occurrence of a LHB. New results,

owever, indicate that the HSE disparity is actually a result of

ron and sulfur segregation in the Moon’s primordial magma ocean

ausing HSEs to drag towards the core long after the moon-forming

mpact ( Rubie et al., 2016 ). Because the crystallization of the lunar

agma took far longer than on Earth, a large disparity between

he HSE records is expected ( Morbidelli et al., 2018 ).

3 . Updated impact data : The LHB hypothesis gained significant

omentum when none of the lunar impactites returned by the

pollo missions were older than 3.9 Gyr ( Tera et al., 1974; Zell-

er, 2017 ). However, recent 40 Ar / 39 Ar age measurements of melt

lasts in Lunar meteorites are inconsistent with the U/Pb dates

etermined in the 1970s ( Fernandes et al., 20 0 0; Chapman et al.,

007; Boehnke et al., 2016 ). These new dates cover a broader range

f lunar ages; and thus imply a smoother decline of the Moon’s

ratering rate. Furthermore, new high-resolution images from the

unar reconnaissance Orbiter (LRO) and the GRAIL spacecraft have

ignificantly increased the number of old ( > 3.9 Gyr) crater basins

sed in crater counting ( Spudis et al., 2011; Fassett et al., 2012 ).

or example, samples returned by Apollo 17 that were originally

ssumed to be from the impactor that formed the Serenitatis

asin are likely contaminated by ejecta from the Irbrium basin

Spudis et al., 2011 ). Because the Serenitatis basin is highly marred

y young craters and ring structures, it is likely older than 4 Gyr,

nd the Apollo samples are merely remnants of the 3.9 Gyr Irbrium

vent.

Here we build upon the hypothesis of an early instability by

ystematically investigating the effects of the Nice Model occur-

ing during the process of terrestrial planetary formation. Since

dvances in algorithms substantially decreased the computational

ost of N-body integrators in the 1990s ( Wisdom and Holman,

991; Duncan et al., 1998; Chambers, 1999 ), many papers have

een dedicated to modeling the late stages (giant impact phase) of

errestrial planetary formation. Observations of proto-stellar disks

Haisch et al., 2001; Pascucci et al., 2009 ) suggest that free gas dis-

ppears far quicker than the timescale radioactive dating indicates

t took the terrestrial planets to form ( Halliday, 2008; Kleine et al.,

009 ). Because the outer planets must clearly form first, the pres-

nce of Jupiter is supremely important when modeling the forma-

ion of the inner planets ( Wetherill, 1996; Chambers and Cassen,

002; Levison and Agnor, 2003a ). Early N-body integrations of

lanet formation in the inner solar system in 3 dimensions from

disk of planetary embryos and a uniformly distributed sea of

lanetesimals reproduced the general orbital spacing of our 4 ter-

estrial planets ( Chambers and Wetherill, 1998; Chambers, 2001 ).

owever, these effort s systematically failed to produce an excited

steroid belt and 4 dynamically cold planets with the correct mass

atios (Mercury and Mars are ∼ 5% and ∼ 10% the mass of Earth

espectively).

Numerous subsequent authors approached these problems us-

ng various methods and initial conditions. By accounting for the

ynamical friction of small planetesimals, O’Brien et al. (2006) and

aymond et al. (2006) more consistently replicated the low

342 M.S. Clement et al. / Icarus 311 (2018) 340–356

Table 1

Giant planet initial conditions: The columns are: (1) the name of the simulation

set, (2) the number of giant planets, (3) the mass of the planetesimal disk exte-

rior to the giant planets, (4) the distance between the outermost ice giant and the

planetesimal disks inner edge, (5) the semi-major axis of the outermost ice giant

(commonly referred to as Neptune, however not necessarily the planet which com-

pletes the simulation at Neptune’s present orbit), (6) the resonant configuration of

the giant planets starting with the Jupiter/Saturn resonance, and (7) the masses of

the ice giants from inside to outside.

Name N Pln M disk δr r out a nep Resonance chain M ice

( M �) (au) (au) (au) ( M �)

n1 5 35 1.5 30 17.4 3:2, 3:2, 3:2, 3:2 16,16,16

n2 6 20 1.0 30 20.6 3:2, 4:3, 3:2, 3:2, 3:2 8,8,16,16

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1 It should be noted that, with the additional constraint of requiring that Neptune

migrate to ∼ 28 au prior to the onset of the instability, a 3:2,3:2,2:1,3:2 resonant

configuration for 5 planet scenarios is more effective ( Deienno et al., 2017 ). Because

we are mostly interested in studying the excitation of the inner solar system, which

is largely unaffected by the particular migration of Neptune, our results should be

relatively independent of the particular ice giant resonant configuration selected.

eccentricities of the terrestrial planets. However, the so-called

“Small Mars Problem” proved to be a systematic short-coming of

N-body accretion models ( Wetherill, 1991; Raymond et al., 2009 ).

The vast majority of simulations produce Mars analogs roughly

the same mass as Earth and Venus; a full order of magnitude

too large ( Morishima et al., 2010 ). However, Mars’ mass consis-

tently stayed low when using a configuration of Jupiter and Saturn

with present day mutual inclination and eccentricities twice their

modern values ( Raymond et al., 2009 ). Because planet-disk interac-

tions systematically damp the eccentricities of growing gas planets

( Papaloizou and Larwood, 20 0 0; Tanaka and Ward, 2004 ), this re-

sult presented the problem of requiring a mechanism to adequately

excite the orbits of the giant planets prior to terrestrial planetary

formation. Hansen (2009) then demonstrated that a small Mars

could be formed if the initial disk of planetesimals was confined

to a narrow annulus between 0.7-1.0 au (interestingly, a narrow

annulus might also explain the orbital distribution of silicate rich

S-type asteroids ( Raymond and Izidoro, 2017b )). The “Grand Tack”

hypothesis provides an interesting mechanism to create these con-

ditions whereby a still-forming Jupiter migrates inward and subse-

quently “tacks” backward once it falls into resonance with Saturn

( Walsh et al., 2011; Brasser et al., 2016 ). When Jupiter “tacks” at

the correct location, the disk of planetesimals in the still-forming

inner solar system is truncated at 1.0 au, roughly replicating an an-

nulus (for a critical review of the Grand Tack consult Raymond and

Morbidelli (2014) ).

Another potential solution to the small Mars problem is lo-

cal depletion of the outer disk ( Izidoro et al., 2014; 2015 ). How-

ever, systems in these studies that placed Jupiter and Saturn on

more realistic initially circular orbits failed to produce a small

Mars. Mars’ formation could have also been affected by a secu-

lar resonance with Jupiter sweeping across the inner solar sys-

tem as the gaseous disk depletes ( Thommes et al., 2008; Brom-

ley and Kenyon, 2017 ). The degree to which this process induces

a dynamical shake-up of material in the vicinity of the forming

Mars and asteroid belt is strongly tied to speed of the resonance

sweeping. Furthermore, it is possible that Mars’ peculiar mass is

simply the result of a low probability event. Indeed, there is a low,

but non-negligible probability of forming a small Mars when us-

ing standard initial conditions and assuming no prior depletion of

the disk ( Fischer and Ciesla, 2014 ). However, on closer inspection,

many of the “successful” systems in Fischer and Ciesla (2014) are

poor solar system analogs for other reasons (such as an extra large

planet in the asteroid belt ( Jacobson and Walsh, 2015 )). Addition-

ally, the large masses of Mars analogs produced in N-body integra-

tions could be a consequence of the simplifications and assump-

tions made by such simulations. The process of how planetesimals

form out of small, pebble to meter sized bodies is still an ac-

tive field of research. Reevaluating the initial conditions used by

N-body accretion models of the giant impact phase may poten-

tially shed light on the origin of Mars’ small mass. Kenyon and

Bromley (2006) considered this problem using a multi-annulus co-

agulation code to grow kilometer scale planetesimals. Subsequent

authors ( Levison et al., 2015; Dra zkowska et al., 2016; Raymond

and Izidoro, 2017b ) modeling the accretion of meter sized objects

found that dust growth and drift cause solids in the inner disk to

be redistributed in to a much steeper radial profile. Finally, mul-

tiple studies have demonstrated that more realistic, erosive col-

lisions can significantly reduce the mass of embryos during the

late stages of terrestrial planet accretion ( Kokubo and Genda, 2010;

Kobayashi and Dauphas, 2013; Chambers, 2013 ).

Here we investigate an alternative scenario wherein the still

forming inner planets are subjected to a Nice Model instability.

Though the effect of the Nice Model on the fully formed terrestrial

planets is well studied ( Brasser et al., 2009; Agnor and Lin, 2012;

Brasser et al., 2013; Kaib and Chambers, 2016; Roig et al., 2016 ), no

nvestigation to date has performed direct numerical simulations

f the effect of the Nice Model instability on the still forming ter-

estrial planets. Furthermore, our work is motivated by simulations

rom Lykawka and Ito (2013) . The authors found that a low massed

ars could be formed when Jupiter and Saturn (with enhanced

ccentricities) were artificially migrated across their mutual 2:1

MR using fictitious forces. A downfall of this scenario, however,

s that it overexcites the orbits of the forming terrestrial planets.

dditionally, previous authors ( Walsh and Morbidelli, 2011 ) have

nvestigated whether smooth migration of Jupiter and Saturn (as

pposed to the “jumping Jupiter” model) could produce a small

ars, however the speed of the process was found to be too slow

ith respect to Mars’ formation timescale (1–10 Myr; Dauphas and

ourmand (2011) ). The work of Walsh and Morbidelli (2011) is per-

aps the most similar to that of this paper. However, our work

iffers greatly in that we consider full instabilities directly, rather

han modeling terrestrial evolution while migrating the giant plan-

ts with artificial forces. It should also be noted that we do not

nclude the “Grand Tack” hypothesis in our study (in Section 5 we

rgue that it is potentially compatible with the scenario we inves-

igate).

. Materials and methods

Because the parameter space of possible giant planet configu-

ations (number of planets, resonant configuration, planet spacing,

isk mass and disk spacing) emerging from the primordial gas disk

s substantial, as a starting point for this work we take two of

he most successful five and six giant planet configurations from

esvorný and Morbidelli (2012) 1 Both begin with Jupiter and Sat-

rn in their 3:2 MMR. Though placing Jupiter and Saturn in an

nitial 2:1 configuration on circular orbits can be highly success-

ul at replicating the correct planetary spacing of the outer solar

ystem, only ∼ 0.2% such simulations sufficiently excite Jupiter’s

ccentricity ( Nesvorný and Morbidelli, 2012 ). For this reason we

ocus on the scenario where Jupiter and Saturn emerge from the

as disk locked in a 3:2 MMR. Furthermore, Nesvorný and Mor-

idelli (2012) found advantages and disadvantages which are mu-

ually exclusive to both the five and six planet cases. Thus, for

ompleteness we select one five and one six planet setup for our

tudy. We summarize our chosen sets of giant planet initial condi-

ions in Table 1 .

To create a mutual resonant chain of gas giants, we apply

n external force to the giant planets which mimics the effects

f a gas disk by modifying the equations of motion with forced


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Table 2

Summary of initial conditions for terrestrial planetary formation simulations.

Run number Disk mass ( M �) Disk inner edge (au)

0–24 5.0 0.5

25–49 5.0 0.7

50–74 3.0 0.5

75–99 3.0 0.7

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igration ( a ) and eccentricity damping ( e ) terms ( Lee and

eale, 2002 ). Though the precise underlying physics of the inter-

ction between forming giant planets and a gas disk is not fully

nderstood, this method is employed by many authors studying

oth the solar system, and observed resonant exoplanets because

t consistently produces stable resonant chains ( Matsumura et al.,

010; Beaugé and Nesvorný, 2012 ). The exact functional forms of

˙ and ˙ e depend on the timescale and overall distance of migration

esired. We utilize a form of ˙ a = ka and ˙ e = ke/ 100 ( Batygin and

rown, 2010a ), where the constant k is adjusted to achieve a mi-

ration timescale τmig ∼ .1-1 Myr. Because the actual migration rate

s a complex function of the properties of the gas disk and relative

asses of the planets, achieving a specific migration timescale is

f less importance than placing the resonant planets on the proper

rbits ( Kley and Nelson, 2012; Baruteau et al., 2014 )

We first evolve the gas giants (without terrestrial planets or a

xterior disk of planetesimals) using the Mercury6 Bulirsch-Stoer

ntegrator (as opposed to the faster hybrid integrator) with a 6.0

ay timestep ( Chambers, 1999; Stoer et al., 2002 ). The Bulirish-

toer method is necessary when building resonant configurations

ecause the force on a particle is a function of both the positions

nd momenta ( Chambers, 1999; Batygin and Brown, 2010b ). The

lanets are initially placed on orbits just outside their respective

esonances, and then integrated until the outermost planet’s semi-

ajor axis is at the appropriate location ( Table 1 ). Fig. 1 shows

n example of this evolution for a simulation in the n1 batch.

o verify that the planets are in a MMR, libration about a series

f resonant angles is checked for using the method described in

lement and Kaib (2017) .

After the resonant chain is assembled, we add a disk of 10 0 0

qual massed planetesimals using the inner and outer radii that

esvorný and Morbidelli (2012) showed best meet dynamical con-

traints for the outer solar system ( Table 1 ). The orbital distribu-

ion of the planetesimals is chosen in a manner consistent with

revious authors ( Batygin and Brown, 2010b; Nesvorný and Mor-

idelli, 2012; Kaib and Chambers, 2016 ), and follows an r −1 sur-

ace density profile. Angular orbital elements (arguments of peri-

enter, longitudes of ascending node and mean anomalies) for

he planetesimals are selected randomly. Eccentricities and in-

linations are drawn from near circular and co-planar gaussian

istributions (standard deviations: σe = . 002 and σi = . 2 ◦). These

ame distributions are utilized throughout our study to maintain

ll initial eccentricities less than 0.01, and inclinations within 1 °.hese systems are integrated using the Mercury6 hybrid integra-

or ( Chambers, 1999 ) with a 20.0 day timestep up until the point

hen two giant planets first pass within 3 mutual Hill Radii. Be-

ause we wish to investigate specific instability times with respect

o terrestrial planetary formation, and our resonant configurations

an often last for tens of millions of years prior to experiencing

n instability, we integrate these systems up until the onset of the

nstability with an empty inner solar system. Only after this point

re the terrestrial planetary formation disks at various stages of

volution embedded in these systems. The method allows us to

ave computational time, and control at exactly what point the in-

tability occurs during the giant impact phase. Though the giant

lanets are already on significantly eccentric orbits at this point

and therefore affecting the terrestrial disk), we find that systems

an last for millions of years before experiencing an instability if

he simulation is stopped sooner. Moreover, the terrestrial planets

re far more sensitive to the evolution of Jupiter and Saturn than

hat of the ice giants. We find that, in the vast majority of sim-

lations, Jupiter and Saturn don’t begin evolving substantially un-

il after this first close encounter time. In the vast majority of our

imulations, the instability ensues within several thousand years of

he first close-encounter time. This is consistent with simulations

f planet-planet scattering designed to reproduce giant exoplanet

ystems ( Chatterjee et al., 2008; Juri c and Tremaine, 2008; Ray-

ond et al., 2010 ).

Because we want to embed terrestrial planetary disks at dif-

erent stages of development into a giant planet instability, we

egin by modeling terrestrial disk evolution in the presence of

static Jupiter and Saturn in a 3:2 MMR. We form 100 sys-

ems of terrestrial planets using the Mercury6 hybrid integrator

Chambers, 1999 ) and a 6.0 day timestep. Because of the integra-

or’s inability to accurately handle low pericenter passages, objects

re considered to be merged with the sun at 0.1 au ( Chambers,

999; 2001 ). We choose the simplest initial orbital distributions

or objects in the terrestrial forming disk in order to mirror pre-

ious studies which assumed no prior disk depletion ( Chambers,

0 01; 20 07 ). Half of the disk mass is in 100 equal massed embryos,

ith the remainder in 10 0 0 equal massed planetesimals. The spac-

ng of the embryos and planetesimals is selected to achieve a

urface density profile that falls off radially as r −3 / 2 . Angular or-

ital elements are selected randomly, and eccentricities and incli-

ations are drawn from near circular, co-planar gaussian distribu-

ions ( σe = . 002 and σi = . 2 ◦). The initial disk mass in simulations

umbered 0–49 is set to 5 M �. Runs numbered 50–99 begin with

M � of material. Additionally, half of the simulations begin with

he inner disk edge at 0.5 au (numbers 0–24 and 50–74) as op-

osed to 0.7 au. This gives us 25 simulations with each mass/edge

ermutation. Additionally, the embryo spacing varies between ∼ 5-

2 mutual hill radii (depending on the simulation initial condi-

ions and disk locations), and is consistent with simulations of oli-

archic growth of embryos ( Kokubo and Ida, 1998 ). Furthermore,

errestrial planet formation is a highly chaotic process in which

he stochasticity of the actual process dominates over the effects

f small changes to initial conditions such as embryo masses and

pacing ( Hoffmann et al., 2017 ). For a summary of these initial con-

itions, see Table 2 . In all simulations the outer disk edge is set at

.0 au. Furthermore, we place Jupiter and Saturn in a 3:2 MMR at

oughly the same orbital locations as in the n1 and n2 configura-

ions (ice giants are not included for this portion of the simula-

ion). As in previous studies, we evolve each system for 200 Myr

Raymond et al., 2009; Kaib and Cowan, 2015 ), outputting a snap-

hot of each system at 10 4 , 10 5 , 10 6 and 10 7 years (these times

oughly correspond to the time elapsed following gas disk disper-

al, which we loosely correlate with the onset of the giant impact

hase). These snapshots are then input into the giant planet con-

gurations n1 and n2 described above (therefore each output is

sed twice, Table 1 ), and integrated through the giant planet insta-

ility for an additional 200 Myr using the same integrator package

nd timestep.

This gives us 800 different instability simulations. We refer to

he two different giant planet configurations (n1 and n2; each con-

aining 400 individual systems) as the simulation “set.” We then

enote each subset of 100 integrations with a unique instability

elay time (10 4 , 10 5 , 10 6 and 10 7 years) as “batches.” And finally

ach unique simulation within a batch (100 each) is referred to as

“run.” The completed terrestrial formation simulations (with a

tatic Jupiter and Saturn in their pre-instability 3:2 MMR) become

ur control batch (100 runs). Thus the control batch represents

sample of terrestrial formation outcomes in a late Nice Model

cenario, where the giant planets remain locked in their mutual


Fig. 1. An example of the resonant evolution of a system of outer planets in the n1 batch. With the forcing function mimicking interactions with the gaseous disk in place,

once a set of planets falls in to a MMR, they remain locked in resonance for the remainder of the evolution.


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esonant configuration until the instability occurs ∼ 400 Myr af-

er the planets finish forming.

.1. Instability timing

We relate time zero in our simulations (the beginning of our

ontrol runs) with the epoch of gas dispersal, which loosely corre-

ates with the beginning of the giant impact phase. However, the

xact dynamical state of the terrestrial disk (for our purposes, the

elative abundance of larger planet embryos within the planetesi-

al sea) at the time of gas disk dispersal is not exactly known. For

his reason, the specific instability times we test, and draw conclu-

ions about, are of less importance than the particular dynamical

tate of the disk. In the subsequent sections of this text, we com-

are broad characteristics of our early (0.01 and 0.1 Myr) and late

1 and 10 Myr) instability delay times. Though we can confidently

ake a general connection between these specific instability times

nd the time elapsed following gas dispersal in the solar system,

elating the dynamical state of the terrestrial disk with the timing

f the instability is the more important conclusion of our work. As

e expand upon in the subsequent sections, the later two instabil-

ty delays we test tend to be more successful than the earlier ones.

herefore our work correlates the giant planet instability timing

ith a terrestrial disk at a state of evolution that is mostly de-

leted of small planetesimals, with most of the mass concentrated

n a handful of growing planet embryos. Subsequently overlaying

his timeline on that of Mars’ growth inferred from isotopic dating

Dauphas and Pourmand, 2011 ) is difficult because the relationship

etween gas disk dispersal and CAI (Calcium-Aluminum-rich inclu-

ion) formation is not well known.

Furthermore, Marty et al. (2017) presented evidence that

ometary bombardment accounted for ∼ 22% of the noble gas

oncentration in Earth’s atmosphere. At first glance, this constraint

ppears to be slightly at odds with the various delay times we ex-

mine in this paper (0.01–10 Myr). We choose to discuss this here

n detail because it can potentially be construed to undermine the

erit of our study. Because the noble gas makeup of the mantle is

o different from that of the atmosphere, and that of comet 67P,

his seems to imply that the onslaught of comets (the timing of

hich would correlate with the giant planet instability) occurred

fter the moon forming impact. Because the moon forming impact

ccurred after Mars had completed forming ( Kleine et al., 2009;

auphas and Pourmand, 2011 ), the giant planet instability could

ot be the mass-depleting event in the Mars forming region. How-

ver this argument does not take in to account the timing of im-

acts with respect to the Earth’s magma ocean phase. It is reason-

ble to assume that some cometary delivery must have occurred

rior to core closure. If fractionalization of Xenon occurred dur-

ng the magma ocean phase, the preserved signature in the mantle

ould very well be different from that in the atmosphere. Because

either the distribution of impact times of primordial Kuiper belt

bjects (KBOs), nor the fractionalization of Xenon in the magma

cean are well known, drawing a broad conclusion of the timing

f the instability from this constraint is difficult. Additionally, de-

ivery itself may have been stochastic; in that an early instability

ight set the Xenon content of the mantle by delivering many

mall comets, and then a later impact of a large comet could boost

he Xenon fraction in the atmosphere. Finally, Xenon isotope trends

re not well known over a large enough sample of comets. It is rea-

onable to expect that comet 67P’s specific Xenon concentration

ould fall somewhere on a continuum when compared to other

imilar comets. A larger sample of such measurements must be

ade before these conclusions can be applied to the giant planet

nstability timeline. Therefore, as a starting point for our study, we

rgue that the most important constraint for instability timing is

he survivability of the terrestrial planets, which no scenario to

ate can ensure.

. Success criteria

When analyzing the results of our simulations, the parameter

pace for comparison to the actual solar system is extensive. Fur-

hermore, for many metrics our accuracy is strongly limited by the

esolution of our simulations. For example, the planetary embryos

sed in the majority of our simulations begin the integration with

fourth of Mars’ present mass. Therefore, a “successful” Mars ana-

og could be formed from as few as 2–3 impacts. For these reasons

ur criteria must be broad, because we are more interested in look-

ng at statistical consistencies and order of magnitude agreements

han perfectly replicating every nuance of the actual solar system.

hus, we focus on 10 broad criteria for replication of both the in-

er and outer solar systems, which we summarize in Table 3 .

.1. The inner solar system

Because our goal is to look for systems like our own, with par-

icular emphasis on forming Mars analogs, we employ an analysis

etric similar to Chambers (2001) . A system is considered to meet

riterion A if it forms a Mars sized body in the vicinity of Mars’

emi-major axis, exterior to two Earth sized bodies. We first check

or any planets formed in the region of 1.3-2.0 au, where the in-

er edge of this region is roughly equal to Mars’ current pericen-

er ( ∼ 1.38 au) and the outer limit lies at the inner edge of the

steroid belt. If this planet has a mass less than 0.3 M �, is imme-

iately exterior to two planets each with masses greater than 0.6

�, and the system contains no planets greater than 0.3 M � in

he asteroid belt, criterion A is satisfied. A separate success crite-

ia (criterion D, Section 3.3 ) filters out systems that finish with an

mbryo in the Asteroid Belt as unsuccessful. While some authors

Hansen, 2009 ) select 0.2 M � as the upper mass limit for Mars

nalogs, due to the previously discussed resolution limitation en-

ountered when using 0.025 M � embryos, we follow the prescrip-

ion in Raymond et al. (2009) and use 0.3 M � as our limit. Addi-

ionally, because our simulations do not take collisional fragmenta-

ion in to account (for further discussion of this phenomenon, see

ection 5.6), it is possible that the masses of our Mars analogs are

omewhat over-estimated. Because we set the Venus/Earth mini-

um mass to 0.6 M � (approximately 75% that of Venus’ present

ass), this provides an adequate mass disparity of a factor of 2

etween the Venus/Earth and Mars analogs. We also look at sys-

ems which form three planets of the correct mass (criterion A1),

ut do not have the correct semi major axes (eg: Mars formed at a

emi-major axis greater than 2.0 au). For this criterion, we include

ystems which form no Mars, but do accrete appropriately sized

arth and Venus analogs in the correct locations as being success-

ul.

.2. The formation timescales of Earth and Mars

Mars is often thought to have been left behind as a “stranded

mbryo” ( Morbidelli et al., 20 0 0 ) during the process of planetary

ormation because the timescale for its accretion inferred from

f/W dating (.1–10 Myr) is so quick ( Nimmo and Agnor, 2006;

auphas and Pourmand, 2011 ). Contrarily, Earth is believed to have

ormed much slower; of order 50–150 Myr ( Touboul et al., 2007;

leine et al., 2009 ). There is a significant amount of uncertainty in

oth of these timescales. The specific timing of the moon forming

mpact, which is thought to correlate with the last major accretion

vent on Earth, is still not well known. Unfortunately, these metrics

re quite difficult to meet when using standard embryo accretion

umerical models. In fact, planets with semi-major axes greater


Table 3

Summary of success criteria for the solar system. The columns are: (1) the semi-major axis of Mars, (2–4)

The masses of Mars, Venus and Earth, (5–6) the time for Mars and Earth to accrete 90% of their mass,

(7) the final mass of the asteroid belt, (8) the ratio of asteroids above to below the ν6 secular resonance

between 2.05-2.8 au, (8) the water mass fraction of Earth, (9) the angular momentum deficit (AMD)

of the inner solar system, (10) the final number of giant planets, (11–13) the semi-major axes, time-

averaged eccentricities and inclination of the giant planets, (14) the orbital period ratio of Jupiter and

Saturn.

Code Criterion Actual value Accepted value Justification

A a Mars 1.52 au 1.3-2.0 au Inside AB

A, A1 M Mars 0.107 M � > 0.025, < .3 M � Raymond et al. (2009)

A, A1 M Venus 0.815 M � > 0.6 M � Within ∼ 25%

A, A1 M Earth 1.0 M � > 0.6 M � Match Venus

B τ Mars 1–10 Myr < 10 Myr

C τ� 50–150 Myr > 50 Myr

D M AB ∼ 0.0 0 04 M � No embryos Chambers (2001)

E ν6 ∼ 0.09 < 1.0

F WMF � ∼ 10 −3 > 10 −4 Order of magnitude

G AMD 0.0018 < 0.0036 Raymond et al. (2009)

H N GP 4 4 Nesvorný and Morbidelli (2012)

I a GP 5.2/9.6/19/30au 20% Nesvorný and Morbidelli (2012)

I e GP 0.046/0.054/0.044/0.01 < .11 Nesvorný and Morbidelli (2012)

I i GP 0.37/0.90/1.02/.67 ° < 2 ° Nesvorný and Morbidelli (2012)

J P Sat / P Jup 2.49 < 2.8 Nesvorný and Morbidelli (2012)

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than 1.3 au in our control simulation (which assume no gas gi-

ant evolution) almost always form far slower than interior planets.

With these metrics in mind, we require our Mars analogs accrete

90% of their mass within 10 Myr of the beginning of our terrestrial

planetary formation simulations (not the onset of the instability,

criterion B). Additionally, we require our Earth analogs take at least

50 Myr to accrete 90% of their mass (criterion C)

3.3. The asteroid belt

Imposing strict constraints on the asteroid belt is difficult be-

cause, of the 10 0 0 planetesimals that begin a given simulation,

typically only 10 to 30 complete the integration in the asteroid belt

region. Furthermore, because the smallest objects in our simula-

tions have masses ∼ 16 times greater than Ceres, our initial condi-

tions are quite unrealistic for an appropriate study of the asteroid

belt. Our ability to model the depletion of the asteroid belt is thus

limited. Therefore, we cannot draw any conclusions about the to-

tal mass of the asteroid belt as all the particles in our simulation

are simply too large. However, the dynamical behavior of our small

planetesimals should be roughly similar to that of the larger aster-

oids in the belt (such as Ceres). Because there are only a few such

large asteroids in the actual asteroid belt, it is important that we

heavily deplete the region of such objects in our simulations. Sev-

eral studies have already investigated the effects of the Nice Model

on the asteroid belt ( O’Brien et al., 2007; Walsh and Morbidelli,

2011; Roig and Nesvorný, 2015; Deienno et al., 2016 ). A similar

study, using tens of thousands of smaller particles in the asteroid

belt region will be required to study the particular dynamical con-

straints our scenario places on the asteroid belt. Furthermore, the

most successful models for the asteroid belt ( Walsh et al., 2011;

Raymond and Izidoro, 2017b ) successfully reproduce the composi-

tional dichotomy between “S-types” (Silicate rich, moderate albedo

asteroids) and “C-types” (low albedo, carbonaceous asteroids mak-

ing up about 75% of the belt) ( Gradie and Tedesco, 1982 ). Improv-

ing our mass resolution within the asteroid belt in the future will

allow us to test this constraint as well.

For our purposes, we simply require that no embryos remain

in the region (a > 2.0 au). This method (criterion D) is similar to

that employed in Chambers (2001) and Raymond et al. (2009) . The

fact that there are no significant gaps between observed mean mo-

tion and secular resonances in the actual asteroid belt implies an

upper limit (about a Mercury mass) for the mass of the largest ob-

ect in the belt that could survive terrestrial planetary formation

O’Brien and Sykes, 2011 ). Because the total mass of the asteroid

elt is thought to have depleted by about a factor of ∼ 10 4 over

he life of the solar system ( Petit et al., 2001 ), a detailed calcula-

ion of the actual numerical value of what is left over is far beyond

he scope of this paper. Moreover, because Gyr timescale modeling

f test particles in the asteroid belt indicates that depletion is loga-

ithmic over the life of the solar system, the majority of this deple-

ion must happen during the first ∼ 200 Myr of evolution because

oss in the next 4 Gyr is only of order ∼ 50% ( Minton and Malho-

ra, 2010 ).

Additionally, we look at the number of remaining planetesimals

bove and below the ν6 secular resonance between 2.05-2.8 au.

n the actual solar system, this ratio is about ∼ .09. Again, due to

he small number statistics, the ratio inferred from an individual

imulation will be very imprecise. Therefore we only require this

atio to be less than one (criterion E).

Furthermore, resonance sweeping during giant planet migra-

ion and evolution can drastically effect the dynamical structure

f the asteroid belt ( Walsh and Morbidelli, 2011; Minton and Mal-

otra, 2011; Roig and Nesvorný, 2015 ). In a slow migration sce-

ario, as Saturn moves outward towards its current semi-major

xis, the ν16 secular resonance excites inclinations as it sweeps

hrough the asteroid belt. As Saturn continues to migrate, the

6 resonance erodes the remaining low inclination, low eccen-

ricity component. Though the process of resonance sweeping

an undoubtedly have an effect on the resulting mass of Mars

Bromley and Kenyon, 2017 ), the mechanism can also remove low

nclination asteroids which are common ( Fig. 6 ) in today’s asteroid

elt ( Walsh and Morbidelli, 2011 ). To preserve the structure of the

steroid belt from the effects of resonance dragging, previous au-

hors ( Roig and Nesvorný, 2015 ) utilized a “jumping Jupiter” model

nstability (wherein Jupiter and Saturn “jump” toward their present

rbital locations, ideally preserving the fragile terrestrial planets

nd asteroid belt structure). In order for our model to be success-

ul, it must heavily deplete the asteroid belt while still maintaining

he low inclination component.

.4. Water delivery to Earth

Many models trace the origin of Earth’s water to the early

epletion of the primordial asteroid belt ( Raymond et al.,

0 07; 20 09 ). The actual topic of water delivery to Earth is


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Table 4

Summary of percentages of systems which meet the various terrestrial planet suc-

cess criteria established in Table 3 . It should be noted that, because runs beginning

with a disk mass of 3 M � were not successful at producing appropriately massed

Earth and Venus analogs, criterion A and A1 are only calculated for 5 M � systems.

The subscripts TP and AB indicate the terrestrial planets and asteroid belt respec-

tively.

Set A A1 B C D E F G

a, m TP m TP τ mars τ� M AB ν6 WMF AMD

Control 0 0 9 86 2 53 87 8

n1/.01Myr 3 15 31 84 41 35 40 14

n2/.01Myr 2 6 15 75 48 46 34 9

n1/.1Myr 0 2 16 79 38 39 27 12

n2/.1Myr 6 10 20 69 50 39 38 12

n1/1Myr 13 13 6 90 38 31 47 13

n2/1Myr 8 14 11 87 54 42 44 7

n1/10Myr 12 20 3 73 48 26 47 16

n2/10Myr 8 20 19 80 54 31 62 8

Table 5

Summary of percentages of systems which meet the various giant planet success

criteria established in Table 3 . The subscript GP indicates the giant planets.

Set H I J

N GP a,e, i GP S:J

n1/.01Myr 27 14 47

n2/.01Myr 18 6 41

n1/.1Myr 14 7 42

n2/.1Myr 18 3 38

n1/1Myr 23 15 57

n2/1Myr 22 9 38

n1/10Myr 17 9 36

n2/10Myr 16 8 36

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xtensive, with many competing models, and far beyond the

cope of this paper (for a more complete discussion of various

deas see Morbidelli et al. (20 0 0) , Morbidelli et al. (2012) and

arty et al. (2016) ). Uncertainties in the initial disk properties and

ocations of various snow lines make it challenging for embryo ac-

retion models like our own to confidently quantify the water mass

raction (WMF) of Earth analogs. In addition, the actual bulk water

ontent of Earth is extremely uncertain. Estimates of the mantle’s

ater content range between 0 to tens of oceans ( Lécuyer et al.,

998; Marty, 2012; Halliday, 2013 ), awhile the core may contain 0

o nearly 100 ( Badro, 2014; Nomura et al., 2014 ). See the review

y Hirschmann (2006) for a discussion of the difference between

he capacity of Earth’s water reservoirs and geochemical evidence

or the actual water contained in Earth’s interior. Furthermore,

iven the amount of planetesimal scattering which occurs when

he giant planets grew and migrated during the gas disk phase,

he material from which Earth formed during the giant impact

hase may have already been sufficiently water rich ( Raymond and

zidoro, 2017a ). For our simulations, we first look at the bulk WMF

f Earth analogs calculated using an initial water radial distribution

imilar to that used in Raymond et al. (2009) ( Eq. 1 , this assumes

hat the primordial asteroid belt region was populated by water-

ich objects from the outer solar system during the gas disk phase

Raymond and Izidoro, 2017a )). Any system which boosts Earth’s

MF to greater than 10 −4 is considered to satisfy criterion F. We

lso analyze the percentage of objects Earth analogs accrete from

ifferent sections of the disk.

MF =

{

10

−5 , r < 2 au

10

−3 , 2 au < r < 2 . 5 au

10% , r > 2 . 5 au

(1)

.5. Angular momentum deficit

One defining aspect about our solar system is the remarkably

ow eccentricities and inclinations of the terrestrial planets. Over

engthy integrations, the orbits of all the inner planets but Mer-

ury typically stay extremely low ( Quinn et al., 1991; Laskar and

astineau, 2009 ). This orbital constraint was very difficult for early

ccretion models to meet ( Chambers and Wetherill, 1998; Cham-

ers, 2001 ). O’Brien et al. (2006) used dynamical friction to explain

ow the orbits could stay cold through the standard process of

lanetary formation. However, it has proven even more challenging

o keep eccentricities low when a giant planet instability is con-

idered ( Brasser et al., 2009; Kaib and Chambers, 2016 ). Any suc-

essful model of terrestrial planetary formation must maintain low

rbital excitation in the inner solar system. To measure this in our

ystems, we measure the angular momentum deficit (AMD, crite-

ion F) of each system ( Laskar, 1997 ). AMD ( Eq. (2) ) quantifies the

eviation of the orbits in a system from perfectly coplanar, circular

rbits. We follow the same procedure as Raymond et al. (2009) and

equire our systems maintain an AMD less than twice the value of

he modern inner solar system ( ∼ .0018).

MD =

∑

i m i

√

a i [1 −√

(1 − e 2 i ) cos i i ] ∑

i m i

√

a i (2)

.6. The outer solar system

When analyzing the success of our terrestrial planetary forma-

ion simulations, it is important to consider how dependent our re-

ults are on the fate of the outer solar system. Indeed, the chance

f our chosen resonant chains reproducing all the important traits

f the outer solar system after undergoing an instability is often

ow. For example, Nesvorný and Morbidelli (2012) report only a

3% chance of our n1 configurations finishing the integration with

he correct number of outer planets. When all four success criteria

n that work are considered, n1 resonant chains only successfully

atch the outer solar system ∼ 4% of the time. Given the computa-

ional cost of our integrations, we are less interested in how of ten

e correctly replicate the orbital architecture of the giant planets,

nd more concerned with how d epend ent our results are on the

ate of the outer solar system.

To quantify the outer solar system, we adopt the same success

riteria as Nesvorný and Morbidelli (2012) . First, criterion H re-

uires that the simulation finish with 4 giant planets. If this is sat-

sfied, criterion I stipulates that the final semi-major axis of each

lanet be within 20% of the modern location, and the time aver-

ged eccentricity and inclination of each planet be less than twice

he largest current value in the outer solar system. Finally, criterion

states that the period ratio of Jupiter and Saturn stay less than

.8. It should be noted that, unlike Nesvorný and Morbidelli (2012) ,

e check for criterion J independently of whether the other two

tandards are met. The dynamics of the forming terrestrial planets

re far less sensitive to the behavior of the ice giants than they

re to that of Jupiter and Saturn. Therefore, we are nearly just as

nterested in systems that correctly produce Jupiter and Saturn but

ject too many ice giants as we are in those that replicate the outer

olar system perfectly.

. Results and discussion

We provide complete summaries of our results in Tables 4 , 5

nd 6 . It should be noted that a small number of our instabili-

ies fail to properly eject an ice giant, and complete the integra-

ion with greater than four giant planets. Because we are only in-

erested in systems similar to the solar system, we do not include

hese few outliers in any of our analyses. Table 4 shows the to-

al percentage of systems in each simulation batch which meet

ur success criteria for the inner solar system. Table 5 summarizes

he percentage of systems satisfying our giant planet success crite-

ia. We find that our systems adequately replicate the outer solar


Table 6

Summary of percentages of systems which meet the various terrestrial planet suc-

cess criteria established in Table 3 AND finish with Jupiter and Saturn’s period ratio

less than 2.8 (criterion J). It should be noted that, because runs beginning with a

disk mass of 3 M � were not successful at producing appropriately massed Earth

and Venus analogs, criterion A and A1 are only calculated for 5 M � systems. The

subscripts TP and AB indicate the terrestrial planets and asteroid belt respectively.

Set A A1 B C D E F G

a, m TP m TP τ mars τ� M AB ν6 WMF AMD

Control 0 0 9 86 2 53 87 8

n1/.01Myr 0 15 33 94 24 33 61 15

n2/.01Myr 5 5 17 80 30 60 56 10

n1/.1Myr 0 0 14 78 13 36 42 5

n2/.1Myr 12 18 7 84 32 59 64 11

n1/1Myr 26 26 12 95 20 29 65 14

n2/1Myr 11 27 12 92 42 44 69 2

n1/10Myr 9 18 7 92 16 29 53 25

n2/10Myr 20 33 27 87 25 52 77 2

Fig. 2. Cumulative distribution of Mars analog masses formed in instability systems

and our control batch (note that some systems form multiple planets in this region,

here we only plot the largest planet). The vertical line corresponds to Mars’ actual

mass. All control runs with a Mars analog smaller than 0.3 M � ( ∼ 20% of the batch)

were unsuccessful in that they also formed a large planet in the asteroid belt.

Fig. 3. Cumulative mass distribution of embryos and planetesimals at the beginning

of the control simulations (black line), 10 Myr after the early instability delay times

(.01 and 0.1 Myr; red line), 100 Myr after the early instability delay times (orange

line), 10 Myr after the late instability delay times (1 and 10 Myr; cyan line) and

100 Myr after the late instability delay times (blue line). The green line represents

the current mass distribution of the inner solar system. (For interpretation of the

references to colour in this figure legend, the reader is referred to the web version

of this article.)

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system with frequencies consistent with those reported in

Nesvorný and Morbidelli (2012) . In Table 6 , we look at our success

rates for systems with Jupiter/Saturn configurations most similar

to the actual solar system (period ratios less than 2.8). Clearly, the

fate of the terrestrial planets is highly dependent on the evolution

of the solar system’s two giant planets. This is largely due to strong

secular perturbations which result from the post-instability excita-

tion of the giant planet orbits. Indeed, when we look at systems

which eject all ice giants and finish with a highly eccentric Jupiter

and Saturn outside a period ratio of 2.8, we find terrestrial planets

which are too few in number, on excited orbits and systematically

under-massed. Though we still see these symptoms in some of the

systems summarized in Table 6 , they are noticeably less frequent.

4.1. Formation of a small Mars

An early instability is highly successful at producing a small

Mars, regardless of instability timing and the particular evolution

of the giant planets. 75% of all our instability systems form either

no Mars or a small Mars (less than 0.3 M �), as opposed to none

of our control runs. Additionally, as shown in Table 4 , most of our

control systems leave at least one embryo in the asteroid belt. In

fact, many of these systems even form multiple small planets, or

an Earth massed planet in the asteroid belt. Only 9% of our insta-

bility simulations form a planet more massive than Mars in the

asteroid belt, as opposed to 65% of our control runs. Clearly a Nice

Model instability is a highly efficient means of depleting the plan-

etesimal disk region of material outside of 1.3 au.

Fig. 2 shows the cumulative distribution of the largest planets

in each system formed between 1.3 and 2.0 au for our instability

sets versus the control batch. The solar system fits in well with this

distribution, with slightly greater than half of our systems forming

Mars analogs larger than the actual planet. Indeed, the instability

consistently starves this region of material and produces a small

planet. In fact, 22% of our systems produce no planet in the Mars

region whatsoever. This is slightly lower for the two earliest insta-

bility delays (0.01 and 0.1 Myr) which we test, with 18% of such

systems forming no Mars in the 1.3 to 2.0 au region. This is due to

the fact that, when the instability occurs, the ratio of the number

of planetesimals to embryos, and that of total planetesimal mass

to total embryo mass is much higher. In the late instability cases,

the majority of the system mass is trapped in several large em-

bryos. The dynamical excitement of the additional planetesimals in

the early instability delay cases allows the disk mass to disperse,

thereby enhancing the mass of Mars analogs. We find that planets

in the outer disk (a > 1.3 au) from early instability delay systems

(0.01 and 0.1 Myr) where the outer planets meet criterion J accrete

∼ 6 times more material from the inner disk than in late insta-

ilities (1 and 10 Myr). Furthermore, dynamical friction from the

igher number of planetesimals de-excites material in the outer

isk. Indeed, our early instability delay systems which satisfy crite-

ion J lose an average of 0.25 M � less mass in the outer disk (a >

.3 au) to ejection or collisions with the Sun than the later delays.

ig. 3 shows how a late instability delay results in a dramatically

teeper mass distribution profile.

In Fig. 4 we show the distributions of semi-major axes and

asses for the planets we form, compared with our control sim-

lations. Regardless of the instability delay time, there is a stark

ontrast between our simulations and the control set. Earth mass

lanets in the Mars region and beyond are very common in the

ontrol simulations, and rarely occur when the system undergoes


Fig. 4. Distribution of semi-major axes and masses for all planets formed using 5 M � massed planetesimal disks. The red squares denote the actual solar system values for

Mercury, Venus, Earth and Mars. The vertical dashed line separates the Earth and Venus analogs (left side of the line) and the Mars analogs (right side). The top panel shows

our control runs and each of the 4 lower plots depict a different instability delay time.


Fig. 5. Semi-Major Axis/Eccentricity plot depicting the evolution of a successful sys-

tem in the n1/10Myr batch. The size of each point corresponding to the mass of the

particle (because Jupiter and Saturn are hundreds of times more massive than the

terrestrial planets, we use separate mass scales for the inner and outer planets). The

final planet masses are 0.37, 1.0, 0.69 and 0.15 M � respectively.

Fig. 6. The upper plot shows the inclination distribution of the modern asteroid

belt (only bright objects with absolute magnitude H < 9.7, approximately corre-

sponding to D > 50 km, are plotted). The bottom plot combines all planetesimals

remaining in the asteroid belt region from all instability simulations that form a

Mars analog less massive than 3 times Mars’ actual mass, and finish with Jupiter

and Saturn’s period ratio less than 2.8. Grey points correspond to high-eccentricity

asteroids on Mars crossing orbits which will be naturally removed during subse-

quent evolution up to the solar system’s present epoch. The vertical dashed lines

represent the locations of the important mean motion resonances with Jupiter. The

bold dashed lines indicate the current location of the ν6 secular resonance.

a Nice Model instability. Though the general trends for all four

plots are quite similar, we note that our distributions for late in-

stabilities are slightly better matches to the actual solar system for

two reasons. First, the number of outlying Mars analogs which are

larger than ∼ .6 M � is substantially less for the later instability de-

lay times. As discussed previously, because these simulations be-

gin with fewer planetesimals, the lack of disk dispersal and de-

excitation via dynamical friction between small bodies makes it

difficult for the system to accrete a large Mars over the next ∼200 Myrs of evolution. Next, these systems tend to form more ac-

curate Earth and Venus analogs. Many of the failed early instability

delay simulations are clear examples of the instability’s tendency

to hinder the formation of Earth and Venus. The larger dispersal

of the disk mass profile in these delays consistently deprives the

Earth and Venus forming regions of material. In these simulations,

we often form systems with small (less than ∼ .6 M �) Venus and

Earth analogs, just one total terrestrial planet, or no inner planets

at all.

4.2. Strengths of an early instability

Tables 4 and 6 show the percentage of systems in each batch

that meet our success criteria for the inner solar system. Because

the solar system very well could have been formed in a low like-

lihood scenario, it is important not to place too much weight on

meeting specific numerical values and exactly replicating every

particular dynamical trait of the actual system. For this reason, we

try to keep our success criteria as broad as possible.

We consider our systems roughly successful at meeting our es-

tablished criteria. Generally, our systems perform better than our

control runs in almost all categories. Because the unique dynami-

cal state of the solar system represents just one point in a broad

spectrum of possible outcomes, it is unreasonable to expect that

our simulations meet every single success criterion exactly, every

time. By these standards, our simulations are successful on most

accounts. In particular, an instability is very successful at meeting

the requirements for the asteroid belt (criterion D) and the forma-

tion timescale of Earth (criterion C).

Given the large number of constraints involved in criterion A,

our success rates of ∼ 5 − 20% are still very encouraging. For this

reason, we also use the broader criterion A1 for the orbital ar-

chitecture of the inner solar system. This metric considers sys-

tems that form planets of the correct mass ratios, but incorrect or-

bital locations, and those which form no Mars but a proper Earth

and Venus pair to be successful. When we look at our rates of

success for meeting this criterion when Jupiter and Saturn finish

the integration within a period ratio of 2.8 ( Table 6 ), we find our

later instability delay times are remarkably successful with values

closer to ∼ 30%. In these successful scenarios, Mars often forms

as a stranded embryo. The simulation begins with multiple bod-

ies of order 0.25-2.5 M Mars in the vicinity of Mars’ present orbit.

When the instability ensues, most of these bodies are ejected. 40%

of the time, “Mars” undergoes no further major accretion events

with other embryos after the instability simulation begins. Fig. 5

shows an example of such an evolution scheme. Notice that after

the instability ensues the proto-Venus and proto-Earth continue to

accrete material while objects in the Mars forming region do not.

4.2.1. The asteroid belt

Our simulations are successful at depleting the asteroid belt

because of the dynamical excitation provided by the embryos we

place in the belt. The embryos pre-excite the asteroid belt during

the evolution leading up to the instability. When the instability

ensues, excited planetesimals in the belt scatter off the embryos,

leading to high mass loss. To test this, we performed a follow-

on suite of integrations using our 1 Myr instability control disks,


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nd the Mercury6 hybrid integrator. We place Jupiter and Saturn

n orbits corresponding to a period ratio of 1.6, and set an extra

ce giant immediately exterior to Saturn. When the ice giant scat-

ers and is ejected, Jupiter and Saturn jump. Next, systems where

he post-jump period ratio of Jupiter and Saturn is between 2.1

nd 2.4 are selected, and integrated for an additional 10 Myr with

code that mimics smooth migration and eccentricity damping

n ∼ 3 Myr e-folding timescales using fictitious forces ( Lee and

eale, 2002 ). To attain final states similar to Jupiter and Saturn, we

hut off migration and eccentricity damping when the two gas gi-

nts attain a period ratio above 2.45 and eccentricities below 0.06.

hrough this process, we create a sample of asteroid belts ( ∼ 20)

hich experience a pre and post-instability evolution broadly sim-

lar to the runs from our original simulations that best matched

he currently observed orbital architecture of Jupiter and Saturn.

y performing 2 sets of runs (embryos and planetesimals and plan-

tesimals only), we are able to test the effects of embryo excita-

ion. Planetesimal only simulations are created by converting all

mbryos with a > 1.5 au in a given system in to an appropri-

te number of equal-mass planetesimals with similar semi-major

xes, eccentricities and inclinations, and random angular orbital el-

ments. Simulations using embryos and planetesimals lost about

wice as much mass beyond 1.5 au over just 10 Myr of evolution as

he planetesimal only systems. This is consistent with the idea that

he dynamical excitement of embryos leads to significantly more

ass loss in the asteroid belt.

Our simulations’ ability to replicate the asteroid belt population

bout the ν6 resonance is subject to numerical limitations. Our

imulations start with 0.0025 and 0.0015 M � planetesimals, both

f which are more massive than the entire present contents of the

steroid belt. Furthermore, most simulations finish with between

0 and 30 bodies in the main belt. Such small numbers makes it

ifficult to discern subtle dynamical features within our individ-

al simulated asteroid belts. Although statistics can be improved

y co-adding many simulations to examine the general effects of

iant planet instabilities ( Fig. 6 ), every instability is unique at some

evel, and dynamical sculpting processes occurring during some in-

tabilities may not operate in others. When we co-add the aster-

ids from all of our criterion J satisfying simulations (those which

nish with Jupiter and Saturn’s period ratio less than 2.8) and re-

ove objects on planet-crossing orbits, we find a ratio of bodies

bove to below the ν6 resonance (between 2.05 and 2.8 au) to be

0.71. However, when we only consider asteroids between 2.05

nd 2.5 au, the ratio is a poorer match (2.24). Though neither num-

er is close to the actual ratio ( ∼ 0.09), the first is quite promising

ith respect to other numerical modeling attempts. For example,

eienno et al. (2016) imposed a “Grand-Tack” style migration on

he asteroid belt and found a ratio of ∼ 1.2. In a similar manner,

alsh and Morbidelli (2011) reported a ratio of ∼ 5.2 in a smooth

igration scenario.

Due to numerical limitations, further simulations, involving tens

f thousands of smaller bodies in the asteroid belt region are re-

uired to comprehensively study the detailed effect of an early in-

tability on the asteroid belt. However, an early instability seems to

enerate an asteroid belt similar to the actual belt in broad strokes.

he presence of embryos appears to provide sufficient dynamical

xcitation to substantially deplete the mass in the region (most

imulations deplete more than 95% of belt material in 200 Myr).

hough this does fall short of the required depletion of a factor

f ∼ 10 4 ( Petit et al., 2001 ), our mechanism does produce substan-

ial depletion in the asteroid belt when compared with our con-

rol runs. More realistic initial conditions and handling of collisions

ill be required to more accurately model depletion in the Aster-

id Belt in our model. Nevertheless, embryos remaining in the as-

eroid belt are extremely rare in our instability systems. By using a

ull instability, rather than a smooth migration scenario, we avoid

ragging resonances across the belt, thus broadly preserving its or-

ital structure.

.3. Weaknesses of an early instability

On average, our instability simulations are less successful at

eeting the success criteria for the formation timescale of Mars,

he WMF of Earth and the AMD of the terrestrial planets. Repro-

ucing the formation timescale of Mars is a difficult constraint for

-body accretion models of terrestrial planetary formation. A suc-

essful Mars analog in our simulations need only be composed

f 4 embryos. Meanwhile, the real Mars formed from millions of

maller objects that accreted prior to and during the giant impact

hase. This difference must be weighed when considering moder-

te discrepancies between the formation timescales of simulated

ars analogs and the real planet. In fact, ∼ 40% of all our Mars

nalogs undergo no impacts with other embryos following the in-

tability, and Mars’ form on average ∼ 39 Myr faster than their

arth counterparts. Additionally, 6 of the 7 Mars analogs in the

riterion A1 satisfying [n2/10 Myr] batch (our most successful sim-

lations), form in under 10 Myr. Because Mars’ growth only con-

inued at the ∼ 10% level after ∼ 2–4 Myr ( Dauphas and Pour-

and, 2011 ), our 1 Myr instability delays are the most successful

t simultaneously matching the mass distribution of the terrestrial

ystem and the proposed accretion history of Mars. However, the

eological accretion history of Mars is inferred relative to CAI for-

ation; the timing relative to gas disk dissipation of which is not

ully understood.

Providing a means of water delivery to Earth is not a strict re-

uirement for the success of an embryo accretion model. It should

e noted that many ideas for how Earth was populated with wa-

er exist, several of which have nothing to do with delivery via

odies from the outer solar system ( Morbidelli et al., 20 0 0; 2012 ).

n fact, water delivering planetesimals may have been scattered

n to Earth-crossing orbits during the giant planets’ growth and

igration phase ( Raymond and Izidoro, 2017a ). Despite the small

umber statistics involved with using only 10 0 0 initial particles in

he Kuiper belt, about half of which typically deplete in the ini-

ial phase of integration (before the terrestrial disks are imbedded),

e do find 12 instances of Earth analogs accreting objects from

his region in our simulations. Interestingly, Earth’s noble gases

re thought to come primarily from comets, despite the fact that

omets are likely a minor source of water ( Marty et al., 2016 ). We

nd that a late instability delay time (1 and 10 Myr) systematically

tretches the feeding zone of Earth analogs further in to the ter-

estrial disk. This broader feeding zone is basically a result of ec-

entricity excitation of planetesimals ( Levison and Agnor, 2003b ).

n these cases, mass from the outer disk is able to “leap-frog” its

ay towards the proto-Earth. In the first phase of evolution (before

he instability), forming embryos in the middle part of the disk

∼ 2.0–3.0 au) accrete material from the outermost section of the

isk ( ∼ 3.0–4.0 au). When the instability ensues, these embryos are

estabilized, and occasionally scattered inward towards the form-

ng “Earth”.

Our simulations often leave the inner planets with too large of

n AMD. Though most runs only exceed the actual AMD of the

olar system by a factor of 2–3, some systems occasionally reach

MDs as high as 10 times the value of the current solar system.

any of these outliers are from integrations where particularity

iolent instabilities leave behind a system of overly excited giant

lanets. Even when we remove these instances which are not anal-

gous to the actual solar system, our “successful” simulations still

end to possess high AMD values. Often, an overly excited Mars is

he source of this orbital excitation (values of e Mars ∼ . 1 − . 25 are

ypical for these systems). The obvious source of this excitation is

ecular interactions with the excited giant planets. One potential


Fig. 7. Values of the amplitudes of Jupiter’s g 5 mode versus the mass of Mars

analogs formed for 10 Myr delayed instability systems where the orbital period

ratio of Saturn to Jupiter completing the integration less than 2.8. The red stars

correspond with the present solar system values.

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solution to this problem might be accounting for collisional frag-

mentation. Chambers (2013) showed that angular momentum ex-

change resulting from hit-and-run collisions noticeably reduces the

eccentricity of planets formed in embryo accretion models. Addi-

tionally, Jacobson and Morbidelli (2014) showed that the AMD of

systems increases as the total amount of initial mass placed in em-

bryos instead of planetesimals increases, and as individual embryo

mass decreases. We observe a similar relationship in overall disk

mass loss (section 5.1). Moreover, because of the chaotic nature of

the actual solar system, its AMD can evolve by as much as a factor

of 2 in either direction over Gyr timescales ( Laskar, 1997 ).

4.4. Varied initial conditions

Our simulations are broken up into 4 different sets of 25 runs

with unique inner disk edge and initial disk mass combinations

( Table 2 ). In half of our simulations, we use a disk mass of 3 M �

rather than a more typical choice of ∼ 5 M � ( Chambers, 2001; Ray-

mond et al., 20 09 ). 10 0% of these systems with lower mass disks

fail to meet criterion A for correctly replicating the semi-major

axes and masses of the terrestrial planets. Using a lower overall

disk mass leads to less dynamical friction available to save bodies

from loss after the instability. We find that by far the most likely

final configuration for these simulations is a single Venus analog,

occasionally accompanied by a Mars analog. However, we note that

the percentages of systems that meet the other 6 success criteria

(criterion B through G) are roughly similar (within ∼ 5%) for sys-

tems of either initial disk mass.

We see no noticeable differences between the sets of simula-

tions which truncate the inner planetesimal disk at 0.5 au and

those with an inner edge at 0.7 au. Both batches are roughly

equally likely (9% and 10% of the time, respectively) to form a Mer-

cury analog (we define this as any planet smaller than 0.2 M � in-

terior to an Earth and a Venus analog). For more discussion on the

formation of Mercury, see Section 6 . Finally, our rates for meet-

ing all success criteria for the inner planets are roughly the same

(within ∼ 5%), regardless of the selected inner disk edge location.

4.5. Excitation of Jupiter’s g 5 mode.

The sufficient excitation of Jupiter’s g 5 mode is another impor-

tant constraint on the evolution of the giant planets. The current

amplitude of the mode, e 55 = 0 . 044 , is very important in driv-

ing the secular evolution of the solar system ( Morbidelli et al.,

2009b ). Additionally, the amplitude of Saturn’s forcing on Jupiter’s

eccentricity, e 56 = 0 . 016 , is important for the long term evolu-

tion of Mars and the asteroid belt. Overexciting e 56 might lead

to a small Mars and a depleted asteroid belt. However, this sce-

nario is not akin to the actual evolution of the solar system. To

evaluate the relationship between the g 5 mode and the mass of

Mars, we integrate all systems which finish with Jupiter and Sat-

urn within a period ratio of 2.8 for an additional 10 Myr, and per-

form a Fourier analysis of the additional evolution ( Šidlichovský

and Nesvorný, 1996 ). In Fig. 7 , we plot the values of e 55 and e 56

against the masses of Mars analogs produced for these systems

in our 10 Myr delayed instabilities. We find that systems with

an e 55 amplitude greater than that of actual solar system never

produce a large Mars analog (greater than 0.3 M �). The average

Mars analog mass in systems with e 55 less than half the solar

system value (0.022) is 1.96 times Mars’ mass, compared to 1.03

for systems with e 55 > 0.022. Additionally, we see multiple exam-

ples of systems where e 56 is close to the solar system value, that

produce a small Mars. Clearly, the complete excitation of the g 5 mode is linked to reducing the mass of planets in the Mars form-

ing region. Additionally, in Fig. 8 , we plot the normalized AMD of

upiter and Saturn versus the mass of Mars analogs. It is very ap-

arent that the range of possible values is extensive, with the so-

ar system falling well within the range of our results. Therefore,

he actual solar system is consistent with our dynamical evolu-

ion model. Furthermore, Jupiter’s excitation also effects Earth and

enus. When we plot the cumulative mass of Earth and Venus

gainst the mass of Mars ( Fig. 9 ), we find that many of the sys-

ems with similar values to the solar system have correspondingly

imilar values of e 55 .

.6. Impact velocities

Our simulations use an integration scheme where all collisions

re assumed to be perfectly accretionary ( Chambers, 1999 ). This

rovides a decent approximation of the final outcome of terres-

rial planet formation for low relative-velocity collisions between

bjects with a large mass disparity. However, higher velocity col-

isions can often be erosive ( Genda et al., 2012 ), particularly when

he projectile to target mass ratio is closer to unity. Additionally,

epending on the parameters of the impact, glancing blows can

ead to the re-accretion of either all, some or none of the original

rojectile ( Asphaug et al., 2006; Asphaug, 2010; Stewart and Lein-

ardt, 2012 ). Because of the instability’s tendency to excite small

lanetesimals on to high-eccentricity orbits, the collisional veloc-

ties in our simulations are often quite large (occasionally in ex-

ess of 10 times the mutual escape velocity). Because of this, it is

ery important to consider the effects of collisional fragmentation

f bodies when analyzing our results.


Fig. 8. The AMD of Jupiter and Saturn (normalized to the actual solar system value)

versus the mass of Mars analogs formed for all instability systems. The red star

denotes the solar system values.

Fig. 9. Values of the total mass of Earth and Venus analogs versus the mass of Mars

analogs formed in instability systems where the value of Jupiter and Saturn’s period

ratio finished the simulation less than 2.8. The color of each point corresponds to

the amplitude of Jupiter’s g 5 mode. The blue star denotes actual solar system values.

(For interpretation of the references to colour in this figure legend, the reader is

directed to the version of this article)

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To check our simulations for erosive collisions, we use a code

hich determines the collision type from the collision speed and

mpact angle by following the parameter space of gravity domi-

ated impacts mapped by Stewart and Leinhardt (2012) . We find

hat erosive collisions do occur with the forming planets in our

imulations. However, they are infrequent, and comprise less than

5% of all collisions and less than ∼ 1% by mass. Erosive collisions

ccur at similar rates for Earth, Venus and Mars analogs, and are

lmost always planetesimal-on-embryo impacts. We do note, how-

ver, that our Mars analogs undergo a significantly higher number

f hit and run collisions (36% by mass as opposed to less than 20%

or Earth and Venus analogs). This indicates that our Mars analog

asses are most likely over-estimated. Additionally, because the

ffects of hit and run collisions have been shown to reduce the

MD of planets produced ( Chambers, 2013 ), it is possible that the

esulting orbital eccentricities and inclinations of our Mars analogs

re similarly over-excited. This is encouraging because the exci-

ation of Mars significantly contributes to our systematically high

MDs.

. Conclusions

In this paper, we have presented 800 direct numerical simula-

ions of a giant planet instability occurring in conjunction with the

rocess of terrestrial planet formation. By timing this violent event

ithin the first ∼ 100 Myr following the dispersion of the gas disk,

he instability scenario no longer requires a mechanism to prevent

he destabilization and loss of the fully formed terrestrial planets

errestrial planets (such as the “Jumping Jupiter” model). When we

crutinize our fully formed systems against a wide range of suc-

ess criteria, we note multiple statistical consistencies between our

imulated planets and the actual terrestrial system. First, our Mars

nalogs are more likely to form small and quickly. In fact, 75% of

ll our instabilities form either no planet in the Mars region what-

oever, or an appropriately sized Mars. Additionally, cases where

he instability is delayed 1–10 Myr after the beginning of the gi-

nt impact phase tend to be more successful than earlier timings

< 1 Myr). In many of these runs, the instability itself sets the ge-

logical formation timescale of Mars. Thus, an early giant planet

nstability provides a natural explanation for how Mars survived

he process of planet formation as a “stranded embryo.”

We find that our simulated asteroid belts are largely depleted

f mass when compared with our control set of simulations, and

eldom form a planet in the belt region. Furthermore, the broad

rbital distribution of the asteroid belt seems to be well matched

hen we co-add the remaining asteroids from all of our simula-

ions. Because the instability itself is inherently chaotic, each re-

ulting system of giant planets has slightly different orbital charac-

eristics. When we filter out systems where the giant planets or-

its most closely resemble those in the actual solar system, we

nd higher rates of success among the corresponding terrestrial

ystems.

At first glance, certain geochemical and dynamical constraints

omewhat conflict with an instability occurring 1–10 Myr after

as disk dispersal. New isotopic data from comet 67P suggests

hat ∼ 22% of Earth’s atmospheric noble gases were delivered via

ometary impacts after the Earth had fully formed ( Marty et al.,

017 ). Because the giant planet instability is the most likely source

f such a cometary onslaught, this seems to suggest that the insta-

ility occurred after the conclusion of terrestrial planet formation.

owever, considerable uncertainty remains in the interpretation of

his noble gas signature. Another potential conflict with our result

s related to the modern Kuiper belt. The classical Kuiper belt pop-

lation on high inclination orbits can be explained if Neptune ini-

ially migrated in a slow, smooth fashion for at least 10 Myrs af-

er gas disk dispersal before being interrupted by the giant planet

nstability ( Nesvorný, 2015a ). Though such a timing matches our

ongest delay, pushing the instability time later leads to a conflict

ith constraints on Mars’ accretion history ( Dauphas and Pour-

and, 2011 ). However, our understanding of the Kuiper belt’s ori-

ins and Mars’ formation timescale are subjects of ongoing study

nd continually evolving. Given this, we believe this is not enough

o rule out the premise of our model, especially because it is able

o replicate so many features of the inner solar system.

The Nice Model explains a number of aspects of the outer solar

ystem. We have shown that, if it occurred within 10 Myr of the

issipation of the gaseous disk, the instability produces inner solar

ystem analogues that match many important observational con-

traints regarding the formation of Mars and the asteroid belt. In

ontrast, simulations lacking an instability consistently yield Mars

nalogs that are too massive, form too slowly, and are surrounded

y over-developed asteroid belts. By including a giant planet in-

tability, these same simulations show a dramatic decrease in the


R

A

A

A

B

B

B

B

B

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B

B

B

B

B

B

B

C

C

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C

C

C

C

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mass and formation timescale of Mars, and adequate depletion in

the asteroid belt.

6. Future work

Our simulations are insufficient to study the large scale struc-

ture of the asteroid belt. Simulations utilizing tens of thousands of

smaller particles in the asteroid belt are necessary to test if this

model can correctly match the orbital distribution of known as-

teroids. Additionally, the large scale distribution of “S-types” and

“C-types” is an important dynamical constraint which must be ac-

counted for Gradie and Tedesco (1982) ; Bus and Binzel (2002) ;

DeMeo and Carry (2013) . Moreover, as our knowledge of asteroids

continues to undoubtedly expand through further missions such as

Asteroid Redirect, Dawn, OSIRIS-Rex and Lucy, so will the number

of constraints.

Another major shortcoming of our project is that we do

not account for the collisional fragmentation of colliding bod-

ies. Higher velocity collisions can be erosive, rather than ac-

cretionary. Given the highly excited orbits which are produced

by the Nice Model instability in our simulations, accounting for

the fragmentation of bodies is supremely important. Furthermore,

Chambers (2013) showed that accounting for hit-and-run collisions

in embryo accretion models can result in producing planets on

colder orbits than when using standard integration schemes. Con-

sistent with most previous N-body simulations of the late stages

of terrestrial planetary formation, our integrations fail to pro-

duce Mercury analogs with any reasonable consistency ( Chambers,

2001; O’Brien et al., 2006; Raymond et al., 2009; Kaib and Cowan,

2015 ). Perhaps accounting for the fragments of material ejected in

high velocity collisions with Venus analogs might provide insight

into understanding this problem.

Finally, the capture of Mars’ trojan satellites (some of which are

rare, olivine-rich “A-type” asteroids) is undoubtedly affected by the

strong excitation of orbits we see in the proto-Mars region of our

simulations ( Tabachnik and Evans, 1999 ). The trojans of Mars are

the only such objects in the inner solar system with orbits stable

over Gyr timescales. Their unique compositions represent a poten-

tial observational constraint for N-body integrations of terrestrial

planetary formation ( Polishook et al., 2017 ).

Acknowledgments

M.S.C. and N.A.K. thank the National Science Foundation for

support under award AST-1615975 . S.N.R. thanks the Agence Na-

tionale pour la Recherche for support via grant ANR-13-BS05-

0 0 03-0 02 (grant MOJO). K.J.W. thanks NASA ’s SSERVI pro-

gram (Institute of the Science of Exploration Targets) through in-

stitute grant number NNA14AB03A . The majority of computing

for this project was performed at the OU Supercomputing Cen-

ter for Education and Research (OSCER) at the University of Ok-

lahoma (OU). Additional analyses and simulations were done us-

ing resources provided by the Open Science Grid ( Pordes et al.,

2007 ; Sfiligoi et al., 2009 ), which is supported by the National

Science Foundation award 1148698, and the U.S. Department of

Energy’s Office of Science. Control simulations were managed on

the Nielsen Hall Network using the HTCondor software package:

https://research.cs.wisc.edu/htcondor/. This research is part of the

Blue Waters sustained-petascale computing project, which is sup-

ported by the National Science Foundation (awards OCI-0725070

and ACI-1238993) and the state of Illinois. Blue Waters is a joint

effort of the University of Illinois at Urbana-Champaign and its Na-

tional Center for Supercomputing Applications ( Bode et al., 2013 ;

Kramer et al., 2015 ).

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