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Planetary system formation in
thermally evolving viscous
protoplanetary discs
Richard P. Nelson, Phil Hellary, Stephen M. Fendyke
rsta.royalsocietypublishing.org
and Gavin Coleman
Astronomy Unit, Queen Mary, University of London, Mile End Road,
London E1 4NS, UK
Research
Cite this article: Nelson RP, Hellary P,
Fendyke SM, Coleman G. 2014 Planetary
system formation in thermally evolving
viscous protoplanetary discs. Phil. Trans. R.
Soc. A 372: 20130074.
http://dx.doi.org/10.1098/rsta.2013.0074
One contribution of 17 to a Theo Murphy
Meeting Issue ‘Characterizing exoplanets:
detection, formation, interiors, atmospheres
and habitability’.
Subject Areas:
extrasolar planets, solar system
Keywords:
extrasolar planets, planet formation,
protoplanetary discs
Author for correspondence:
Richard P. Nelson
e-mail: [email protected]
Observations of extrasolar planets are providing new
opportunities for furthering our understanding of
planetary formation processes. In this paper, we
review planet formation and migration scenarios
and describe some recent simulations that combine
planetary accretion and gas-disc-driven migration.
While the simulations are successful at forming
populations of low- and intermediate-mass planets
with short orbital periods, similar to those that are
being observed by ground- and space-based surveys,
our models fail to form any gas giant planets that
survive migration into the central star. The simulation
results are contrasted with observations, and areas of
future model development are discussed.
1. Introduction
Observations of extrasolar planets using a variety of
techniques are uncovering the extraordinary diversity
of planetary systems within our Galaxy [1–3]. This
‘planetary zoo’ consists currently of gas giant Jupiterlike and Saturn-like planets orbiting with periods
ranging from a few days to a few years, warm and
hot Neptunes and super-Earths with masses between
2 and 10 M⊕ (Earth masses). Earth-like planets (and
bodies whose sizes are apparently smaller than the
Earth’s) are just being discovered—including the recently
announced planet Kepler 37b that is inferred to have
a radius smaller than Mercury’s [4]. Examples exist of
single-planet systems with no evidence of companion
planets (e.g. 51 Peg). Numerous multi-planet systems
have been discovered. Some contain a combination of
high- and low-mass bodies (e.g. GJ 876, 55 Cancri,
HD 181433), whereas others appear to host just giant
planets (e.g. Kepler 9) or systems of lower-mass
2014 The Author(s) Published by the Royal Society. All rights reserved.
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(a) Core accretion
The most widely accepted theory of planet formation involves a multi-stage process within a
circumstellar disc consisting initially of a mixture of gas and submicrometre-sized dust grains.
Dust grains collide and stick because of the combined action of Brownian motion/smallscale turbulence and van der Waals forces, settle towards the disc midplane and continue
growing through either binary collisions [6] or collective processes, for example the streaming
instability [7], until planetesimals in the 1–100 km size range are formed. Gravitational focusing
may allow a subpopulation of these planetesimals to grow rapidly (‘runaway growth’) until
dynamical heating of the surrounding planetesimals causes growth to enter a slower phase
(‘oligarchic growth’).
For in situ terrestrial planet formation (orbital migration neglected), oligarchic growth ceases
when growing embryos deplete their feeding zones and approach their isolation masses. Further
growth requires ‘giant impacts’ between planetary embryos (expected to have masses in the Moon
to Mars range), occurring on time scales of a few 107 years [8].
Formation of giant planets is expected to begin further from the star, where condensation
of ices augments the inventory of solids that can build planetary cores. Oligarchic growth is
proposed to form cores of approximately 10 M⊕ onto which gaseous envelopes accrete. If the
planet forms early enough in the disc lifetime, it may become a gas giant planet when runaway
gas accretion occurs. Otherwise, settling of a gaseous envelope onto the core forms Neptune-like
planets consisting largely of icy and rocky material with a modest hydrogen/helium envelope [9].
(b) Gravitational instability
The idea that planets form through gravitational fragmentation of protostellar discs goes back
at least to Cameron [10] and has been the subject of ongoing study since that time [11]. It
is now generally agreed that fragmentation is most likely to occur in the low-temperature
outer regions of protoplanetary discs where the cooling time is short [12]. Recent discovery
of massive gas giant planets orbiting at tens of astronomical units from their central stars
through direct imaging surveys [2] may provide evidence for planetary formation occurring
through gravitational instability, but it seems unlikely that this mode of formation explains the
numerous systems of short-period super-Earths and Neptune-like planets being discovered
in abundance.
1
Extrasolar planet data may be obtained from http://exoplanet.eu/.
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2. Planet formation
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Neptunes and super-Earths (e.g. Kepler 11). Planets are also being discovered within binary
systems, orbiting just one of the binary components (e.g. gamma Cephei) or both components
in a circumbinary configuration (e.g. Kepler 16, 34, 35).1
The above planets have been discovered largely from radial-velocity and/or transit surveys.
Microlensing discoveries suggest the existence of both high- and low-mass planets orbiting at
intermediate distance from their stars [5], whereas direct imaging surveys are beginning to
uncover a population of giant planets orbiting at 10–100 astronomical units (AU) from their
parent stars [2]. Reproducing and explaining this enormous diversity in physical and orbital
characteristics provides a rather daunting challenge for planet formation theory. The purpose
of this article is to provide a brief status report of one particular programme that aims to combine
various processes involved in planet formation into a coherent model that will eventually be able
to describe the formation histories of observed exoplanetary systems.
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3. Planetary migration
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The nature of gas-disc-driven migration depends on the planet mass mp and local scale height
of the protoplanetary disc, H, with other physical properties, such as viscosity and thermal
evolution, also being important. When the planet’s Hill sphere radius RH = a(mp /3M∗ )1/3 < H
(here a is semi-major axis and m∗ is stellar mass), then the disc response to the planet’s gravity is
expected to be quasi-linear, and the planet will undergo type I migration [16]. Migration torques
experienced by an embedded planet are composed of Lindblad and corotation torques, the former
arising because of the excitation of spiral density waves at Lindblad resonances and the latter due
to material in the vicinity of the planet semi-major axis (the corotation region) where the fluid
follows horseshoe streamlines. The Lindblad torque is always essentially present in the disc and
may be written [17] as
Γ0
[−2.5 − 1.7β + 0.1α],
(3.1)
ΓLR =
γ
where γ is the ratio of specific heats, and α and β are the exponents in the surface density and
temperature radial profiles, respectively. The most important component of the corotation torque
is often referred to as ‘horseshoe drag’ and arises because of gradients in either entropy and/or
vortensity (ratio of vorticity and surface density) across the horseshoe region. An expression for
the horseshoe drag may be written [17] as
3
ζ
Γ0
1.1
− α + 7.9
,
(3.2)
ΓVHS + ΓEHS =
γ
2
γ
where the first term in square brackets arises from the vortensity gradient, the second term is from
the entropy gradient and ζ is given by
ζ = β − (γ − 1)α.
In the above, Γ0 sets the magnitude of the torque and is given by
ap Ωp 2 2 2
mp
mp
Γ0 =
ap Ωp ,
M∗
cs
Σp a2p
(3.3)
(3.4)
where ap is the planet semi-major axis, Σp is the surface density, cs is the sound speed, Ωp is
the disc angular velocity and the subscript ‘p’ indicates that quantities should be evaluated at
the orbital location of the planet. The horseshoe drag is a nonlinear corotation torque because it
arises from the horseshoe trajectories followed by disc fluid elements (and these are an inherently
nonlinear phenomenon). Linear perturbation theory also predicts the existence of a corotation
torque due to material corotating with the planet. This is smaller than the horseshoe drag and
is important when the horseshoe drag is rendered inoperative due to viscous/thermal diffusion
processes in the disc acting on time scales that are short relative to the horseshoe U-turn time scale.
The corotation torque (horseshoe drag plus linear contribution) is prone to saturation because
material phase mixes in the horseshoe region. The effect is to erase the entropy or vortensity
gradient that feeds the corotation torque, causing the torque to disappear on the horseshoe
.........................................................
(a) Type I migration
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130074
The above descriptions of planet formation clearly do not tell the full story. Observations of
extrasolar planets indicate substantial mobility: the existence of short-period Jupiter-, Saturnand Neptune-like planets indicates that either the planets themselves or their building blocks
must have experienced substantial inward orbital migration. Systems of closely packed low-mass
bodies detected by Kepler (e.g. Kepler 11) clearly support a disc-driven scenario. Planets whose
orbits are significantly eccentric [13] or whose orbit planes are significantly inclined relative to
the equatorial plane of the host star [14] may have experienced gravitational scattering with
additional planets in the system, with tidal interaction with the central star also playing a role
for the short-period circular systems [15].
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Migration transitions from type I to type II when a planet opens a gap in the disc. The conditions
for this are that the disc response to the planet perturbation must be nonlinear and the tidal torque
due to the planet must exceed the viscous torque in the disc [20]. Inward migration under these
conditions then occurs on the viscous evolution time scale of the disc [21,22].
It is noteworthy that planets formed through either core accretion or gravitational
fragmentation may undergo substantial migration. The dense gaseous clumps that form through
gravitational instablity are prone to extremely rapid inward migration [23]. Realization that
this rapid inward migration may occur has led to suggestions that these gaseous protoplanets
may experience tidal downsizing, as they migrate inwards and fill their Hill spheres [24]. This,
however, is not a scenario that we consider further in this article.
4. Corotation torques experienced by eccentric low-mass planets
Planetary orbital eccentricity can strongly influence Lindblad migration torques [25] and
eccentricity/inclination damping rates of embedded planets [26]. In a recent study, Bitsch & Kley
[18] showed that corotation torques decrease significantly with modest growth of eccentricity. We
present here a brief summary of recent calculations that explore this issue further, motivated by
the simple observation that planet formation in the oligarchic growth scenario necessarily leads
to eccentricity excitation through mutual encounters between embryos.
Hydrodynamic simulations were performed to examine how eccentricity influences corotation
torques. Details about the simulation set-up and results are given in Fendyke & Nelson [27]. These
two-dimensional simulations were performed for 5 M⊕ planets embedded in discs with scale
height h = 0.03, 0.05, 0.07 and 0.1. Additional simulations for 10 M⊕ planets embedded in a disc
with h = 0.1 were also performed. Eccentricities in the range 0 ≤ e ≤ 0.3 were considered.
Figure 1a shows the time-averaged surface density perturbation for the h = 0.07 disc with
mp = 5 M⊕ for different eccentricities. Thermal relaxation and viscosity were optimized to prevent
entropy/vortensity corotation torque saturation. The density within the horseshoe region has
the characteristic structure for a disc that exerts a positive unsaturated corotation torque: a
positive density perturbation leading the planet and a negative perturbation trailing it. The black
lines delineate the horseshoe region, and we see that the width of this region diminishes as
eccentricity increases, owing to the material in this region experiencing an effective softening
of the gravitational acceleration due to the planet.
The corotation torque experienced by the planet is shown in figure 1b, as a function of orbital
eccentricity. As reported by Bitsch & Kley [18], the torque drops off with increasing eccentricity.
We ascribe this effect as being due in large part to the decrease in the horseshoe width displayed
in figure 1a. Fendyke & Nelson [27] provide a fit to the results from their full suite of simulations
.........................................................
(b) Type II migration
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libration time scale, τlib . Saturation can be prevented if viscosity and thermal diffusion are
present and able to re-establish the vortensity and entropy gradients. This occurs when the
thermal diffusion time scale, τtherm , and viscous time scale, τvisc , across the horseshoe region
are approximately equal to the libration time scale, τlib . If, on the other hand, τtherm τlib ,
then the entropy contribution to the horseshoe drag is erased and only the linear contribution
remains. If τvisc τlib , then the vortensity contribution to the horseshoe drag is erased and only
the linear contribution remains. If τtherm and τvisc τlib , then the corotation torque saturates
completely.
The corotation torque can also be substantially reduced in the presence of planetary
eccentricity [18]. In a recent study, Hellary & Nelson [19] suggested that this could strongly
influence the evolution of planetary swarms undergoing oligarchic growth. Results from a
detailed study of this effect are discussed briefly in §4.
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(a)
e=0
e = 0.02
e = 0.04
e = 0.06
e = 0.08
e = 0.10
e = 0.12
e = 0.14
e = 0.18
e = 0.24
5
1.1
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1.0
0.9
1.1
1.0
0.9
1.1
1.0
0.9
1.1
1.0
0.9
1.1
1.0
0.9
0
corotation torque
(b)
2
4
6
0
2
4
6
H/r = 0.07
15
10
5
0
0.05
0.10
0.15
eccentricity
0.20
0.25
Figure 1. (a) Time-averaged surface density for simulations with increasing values of eccentricity in a disc with h = 0.07 and
mp = 5 M⊕ . The superimposed black lines show the circulating streamlines that lie closest to the planet, and thus delineate
the horseshoe region. (c) Corotation torque measured for a 5 M⊕ planet embedded in a disc with h = 0.07 as a function of
eccentricity. (Online version in colour.)
covering different values of h, e and mp , and find that the fully unsaturated corotation torque can
be fitted by
e
,
Γc (e) = Γc,0 exp −
Δ
(4.1)
where Δ = h/2 + 0.01 and Γc,0 is the corotation torque at zero eccentricity. An important result
is that the parameter that determines the magnitude of Δ is the disc scale height h and not the
horseshoe width, as has been suggested in previous work. Given that the horseshoe width varies
inversely with h, the realization that Δ scales with h will be particularly important where largescale migration occurs in discs where h varies significantly with orbital radius.
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5. Planet formation simulations
In previous work, a disc model was adopted with fixed power laws for the radial surface density
and temperature distributions. This has been replaced with a thermodynamically evolving
viscous disc model that photoevaporates. We solve the standard surface density diffusion
equation [28] using explicit finite differences, including tidal torques from embedded planets, so
that gaps may form self-consistently [21]. The equilibrium temperature is calculated iteratively
through a balance between radiative cooling and heating due to stellar radiation, accretion
luminosity and viscous dissipation [29]. Rosseland mean opacities from Bell & Lin [30] are
employed. The prescription from Dullemond et al. [31] leads to photoevaporation. This removes
gas primarily from beyond the radius where it becomes unbound from the star (rg = 10 AU here).
The flux of EUV ionizing photons is f41 × 1041 s−1 , and we consider values f41 = 1, 10, 100. The gas
disc runs between 0.1 and 40 AU.
(b) The solid component
Disc solids consist of planetary embryos with mass 0.2 M⊕ immersed within a large
number of ‘super-planetesimals’. Protoplanets interact gravitationally with each other and
with planetesimals. Planetesimals interact only with protoplanets. Protoplanets accrete through
collisions with other protoplanets and planetesimals. Planetesimals experience gas drag, with
an acceleration appropriate for bodies of radius 1 or 10 km (assuming an internal density
of 3 g cm−3 ). Super-planetesimal masses equal one-tenth of the initial protoplanet masses.
Simulations are initiated with approximately 40 protoplanets distributed between 1 and 20 AU,
and between approximately 2000 and 12 000 planetesimals, depending on the particular run
(higher-mass discs include more planetesimals).
The surface density of solids falls initially as r−3/2 . For a disc with solar metallicity, the surface
density of solids equals 7.1 g cm−3 at 1 AU. unit and increases by a factor 30/7.1 beyond the snow
line, where the initial steady disc temperature falls below 160 K. The N-body components of our
simulations were conducted using MERCURY-6 [32].
(c) Gas-disc-driven migration
We incorporate migration of low-mass planets using the prescriptions for type I migration
presented in Paardekooper et al. [17]:
Γtot = ΓLR + ΓVHS Fv Gv + ΓEHS Fv Fd Gv Gd + ΓLVCT (1 − Kv ) + ΓLECT (1 − Kv )(1 − Kd ).
The total torque Γtot is composed of the Lindblad torque, ΓLR , vortensity and entropy
contributions to the horseshoe drag (ΓVHS and ΓEHS , respectively) and vortensity and entropy
contributions to the linear corotation torque (ΓLVCT and ΓLECT , respectively). The factors Fv , Gv ,
Fd , Gd , Kv and Kd determine the level of saturation of the corotation torque contributions through
estimates of the relative time scales between thermal/viscous diffusion across the horseshoe
region and the horseshoe libration period (see [19] for a description of how these terms are
incorporated). Eccentricity and inclination damping are included [33], with reduction factors
accounting for eccentricity/inclination [26]. A similar reduction factor is applied to the Lindblad
torque, and the corotation torque is multiplied by the reduction factor described in Hellary &
Nelson [19]. Note that this is not the same prescription as described in §4 because the result
.........................................................
(a) The gas disc model
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130074
We now present the results of N-body simulations of planet formation. These are initiated
during the oligarchic growth stage when the system consists of embryos and planetesimals.
The basic model is similar to that described in Hellary & Nelson [19] but incorporates
numerous improvements.
6
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We include gas accretion in our simulations through the implementation of a fitting formula to
the detailed envelope accretion calculations presented in Movshovitz et al. [34]. Coagulation and
settling of grains in the atmosphere decrease the opacity and shorten the envelope accretion
time to 2.6 Myr for a planet with a 3 M⊕ core and to 0.6 Myr when a 10 M⊕ core is present,
substantially shorter than earlier estimates obtained without grain evolution [9]. Gas accretion
is mass-conservative, as we remove gas from the disc in an annulus surrounding the planet as it
accretes onto the planet. Gas accretion onto planets is only initiated once the core mass reaches
3 M⊕ .
Once the gaseous planet reaches 30 M⊕ , accretion transitions to a mode where the rate is
controlled by viscous supply to its neighbourhood. Based on the local surface density, we estimate
the gas mass within a putative circumplanetary disc, and the planet accretes on the viscous time
scale of this subdisc. This mode of accretion is effective at dramatically slowing the accretion rate
onto the planet when it forms a gap in the disc. We note that the mass-conservative routine also
limits the accretion rate for planets with mp < 30 M⊕ , so, if deep gap formation occurs for such
planets, their masses are also limited.
(e) Enhanced planetesimal accretion
The presence of a gaseous envelope gives a planet an increased cross section for accreting
planetesimals owing to aerodynamic drag. We adopt the prescription given in Inaba & Ikoma
[35], which incorporates expansion or contraction of the envelope due to increased or decreased
accretion of solids. The accretion rate obtained in the simulations is used to compute the envelope
structure.
(f) Initial conditions
We have performed more than 100 N-body simulations covering time periods up to 5 Myr. For
each parameter set, we used two different random number seeds for defining initial protoplanet
and planetesimal positions. We have considered discs with mass in the range 1–6 MMSN (where
MMSN is the minimum-mass solar nebula model that contains approx. 0.015 M (solar masses)
within 40 AU). We have also considered two metallicity factors (1 and 2, the factor 2 is only used
for discs with masses in the range 1–3 MMSN), two effective planetesimal sizes (1 and 10 km)
and three EUV photon fluxes (f41 = 1, 10, 100). We emphasize that our aim at the present time is to
examine the interplay between accretion, migration and planet–planet interactions during planet
formation. It is not to undertake a study of planetary population synthesis, as exemplified in the
studies by Ida & Lin [36] and Mordasini et al. [37]. It is for this reason that we do not choose
our initial conditions by drawing them from a distribution based on observations or theoretical
considerations.
6. Simulation results
Space constraints prevent us from presenting a full set of results, so instead we present two cases
that illustrate some of the more common modes of evolution observed.
.........................................................
(d) Gas accretion onto planets
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described there was obtained after the simulations described here were performed. Future work
will adopt this new reduction factor. Migration transitions to type II when a planet forms a gap.
It is noteworthy that the disc can become very thin near to the star, H/r < 0.02, such that planets
with masses approximately 10 M⊕ can open gaps there.
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(a)
(b)
1.5 Myr
10
10–1
10–2
1
10–3
planet mass (M )
1
10–4
3.0 Myr
(d)
10–1
5.0 Myr
1
10
10–1
10–2
1
10–3
planet mass (M )
102
10
10–4
10–1
1
10
semi-major axis (AU)
10–1
1
10
semi-major axis (AU)
10–1
Figure 2. The locations and masses of the planets (dots, blue online) in run 1 plotted against the planetesimal (rapidly varying
curve, red online) and gas (smoothly varying curve, green online) surface densities at various times. Plotted gas surface densities
are reduced by a factor of 10 to allow concurrent plotting. (Online version in colour.)
(a) Run 1
Run 1 has an initial disc mass equal to twice the MMSN, metallicity factor unity, f41 = 1 and
planetesimal radius of 10 km. Snapshots of the simulation results showing protoplanet masses,
and planetesimal and gas disc surface densities are shown in figure 2. Mass growth as a result
of planetesimal accretion and planet–planet collisions begins slowly, as the solid disc mass is
comparatively low in this run. During the first 0.5 Myr, no planets greater than 1 M⊕ have formed,
and the gas-driven migration has not been effective. By 1.5 Myr, the gas disc mass has halved.
Some Earth-mass planets have formed and migrated to regions where they experience zero net
torque from the gas disc. The contours in figure 3 indicate the direction of migration for planets
on circular orbits as a function of planet mass and semi-major axis at different times during
the evolution. A value of +2.5 indicates strong outward migration and −2.5 indicates inward
migration. For a given planet mass, there may exist disc regions where the thermal and viscous
time scales are approximately equal to the horseshoe libration time, giving a maximally positive
value to the corotation torque that drives outward migration. As the planet migrates outwards,
however, the disc surface density and opacity decrease, and the thermal time scale becomes short,
reducing the corotation torque. Eventually, the corotation and Lindblad torques cancel, creating a
zero-torque zone for the planet. Planets outside of this zone will tend to migrate inwards towards
it, and planets inside will tend to migrate outwards to it, creating a convergence point. In figure 3,
this region corresponds to the terminus between dark grey (blue online) and mid-grey (red online)
areas, where the dark grey (blue online) area is inward of a mid-grey (red online) area. A number
of bodies with mp = 1–2 M⊕ are seen to straddle this region at 1.5 Myr. These planets do not grow
substantially over the following 1.5 Myr but remain in regions of zero torque and move inwards,
as the gas disc accretes onto the star (the zero-torque zone moves inwards, as the disc surface
.........................................................
102
10
(c) 102
surface density (g cm–2)
0.5 Myr
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surface density (g cm–2)
102
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2.5
0.5 Myr
1.5
10
planet mass (M )
0.5
10–1
3.0 Myr
102
5.0 Myr
0
migration factor
1.0
1
–0.5
10
–1.0
1
–1.5
10–1
10–1
–2.0
1
10
10–1
semi-major axis (AU)
1
10
–2.5
Figure 3. Contour plots showing regions of outward and inward migration for run 1 at various times. Regions of outward
migration are shaded dark grey (blue online) while regions of inward migration are shaded mid-grey (red online). Planets are
illustrated as black circles. (Online version in colour.)
density and opacity decrease). By 3 Myr, the gas disc has almost completely dispersed. Note the
characteristic dip in surface density in figure 2c due to photoevaporation. This dip soon grows into
a wide hole in the disc, after which the remainder of the gas in the disc rapidly photoevaporates.
After the disc is completely dispersed (after 3.25 Myr), evolution is driven entirely by solidbody interactions. Accretion is limited to planet–planet collisions and planetesimal accretion. As
the eccentricities and inclinations of bodies can no longer be damped by the gas disc, a sharp
increase in these quantities is observed from this point onwards in figure 4, which shows the time
evolution of masses, semi-major axes and eccentricities for all planets in this run.
This run failed to produce any gaseous planets, because none exceeded 3 M⊕ (required for gas
accretion) during the disc lifetime. All remaining planets lie between 0.4 and 6.4 AU. units and
undergo a final stage of giant impact growth that can last for tens of million years, longer than
our simulation evolution time.
(b) Run 2
Run 2 has an initial disc mass equal to three times the MMSN, metallicity factor unity, f41 = 100
and planetesimal radius of 1 km. Snapshots of the simulation results showing protoplanet
masses, and planetesimal and gas disc surface densities are shown in figure 5. The more massive
disc and smaller planetesimals in this case lead to swifter planetary accretion than that observed
in run 1, and we see the growth of approximately Earth-mass bodies after 0.1 Myr in figure 5a.
Figure 5b shows that numerous bodies with masses in the range 8–20 M⊕ have grown after
1.5 Myr. The contour plots in figure 6 show the regions of outward and inward migration in
the planetary mass–semi-major axis plane. At 1.5 Myr, the more massive bodies cannot migrate
outwards through corotation torques because their associated horseshoe libration times are short
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2.0
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102
9
1.5 Myr
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10
10–1
a (AU)
10
1
0.4
e
0.3
0.2
0.1
0
0.5
1.0
1.5
2.0
2.5
3.0
time (×106 years)
3.5
4.0
4.5
5.0
Figure 4. The evolution of masses, semi-major axes and eccentricities for all protoplanets in run 1. (Online version in colour.)
planet mass (M )
0.1 Myr
1.5 Myr
10
10–1
10–2
10–3
10–4
10–5
1
10–1
planet mass (M )
(c)
1.6 Myr
(d)
2.5 Myr
10
102
10
1
10–1
10–2
10–3
10–4
10–5
1
10–1
10–1
102
10
1
1
10
semi-major axis (AU)
10–1
surface density (g cm–2)
(b)
surface density (g cm–2)
(a)
1
10
semi-major axis (AU)
Figure 5. The locations and masses of the planets (dots, blue online) in run 2 plotted against the planetesimal (rapidly varying
curve, red online) and gas (smoothly varying curve, green online) surface densities at various times. Plotted gas surface densities
are reduced by a factor of 10 to allow concurrent plotting. (Online version in colour.)
relative to the local thermal and viscous time scales. These protoplanets migrate inwards because
of Lindblad torques. Lower-mass planets located at approximately 2 AU are located in zero-torque
regions, and so migrate inwards slowly, as the gas disc accretes onto the central star.
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mp (M )
10
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1.5 Myr
0
1
–1
–2
10–1
10–1
1
10
10–1
semi-major axis (AU)
1
10
Figure 6. Contour plots showing regions of outward and inward migration for run 2 at various times. Regions of outward
migration are shaded dark grey (blue online) while regions of inward migration are shaded mid-grey (red online). Planets are
illustrated as black circles. (Online version in colour.)
The temperature in the disc inner regions leads to the disc scale height to radius ratio
H/r 0.02, allowing relatively low-mass planets with mp 10 M⊕ to form gaps and transition
to slower type II migration, as illustrated in figure 5c. Nonetheless, the large quantities of gas
lying exterior to these five gap-forming planets drives them into the central star. After 2.5 Myr,
we are left with only two sub-Earth-mass planets in the disc, and these remain until after the gas
disc fully disperses.
7. Discussion and conclusion
We have shown the results from two simulations drawn from a suite of 108 runs in total. These
two simulations are quite representative of the outcomes of all simulations because they result
in populations of low- and intermediate-mass planets, but no surviving gas giants because they
or their precursors always migrate into the central star before gas disc dispersal. Examples of
the following types of planets are formed in our simulations: gas giant planets with substantial
solid cores (i.e. mcore ≥ 10 M⊕ ); gaseous planets with modest core masses (i.e. mcore ∼ 3–5 M⊕ );
Neptune-like and super-Earth-mass planets with significant core masses and moderate envelopes
that account for approximately 10% of the total planet mass; and terrestrial planets whose masses
remain less than 3 M⊕ and so did not accrete significant gaseous envelopes. All gas giant planets
that emerged from the simulations (with high- or low-mass cores) formed gaps in the disc and
experienced type II migration into the central star, leaving no long-term survivors. Significant
numbers of the other classes of planet described above formed and survived long term in the
runs. The maximum mass of any surviving planet was approximately 15 M⊕ .
The formation of gas giants was favoured in runs either where the initial total disc mass was
5–6 × MMSN, or in discs with a total gas mass 3 × MMSN and a metallicity enhancement factor
of 2, and in models where the planetesimal radii were equal to 1 km, as this enhanced the rapid
growth of cores. The rapid formation of cores followed by runaway gas accretion typically formed
Saturn-mass bodies within 1 Myr whose further mass growth was retarded by gap formation. The
existence of a substantial gas disc lying outside the planet orbit essentially guarantees migration
into the central star in our models because the photoevaporative wind is unable to remove
such a large amount of mass within the type II migration time scale. No giants formed late in
the disc lifetime such that photoevaporative loss of the disc could prevent migration into the
central star. Adoption of an inner magnetospheric cavity close to the star would lead to the
formation of hot Jupiters, but this cannot account for the long-term survival of the numerous
extrasolar gas giants that are seen to orbit at large or intermediate distances from their stars,
and so we have not implemented this in our models. Improved statistics in terms of a larger
sample of simulations are required to determine whether or not the basic model adopted here is
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