Download On the formation of Uranus and Neptune

O n the formation of Uranus and Neptune
Edward W, Thommes
A thesis submitted to the Department of Physics
in conformity with the requirements
for the degree of Doctor of Philosophy
Queen's University
Kingston, Ontario, Canada
October, 2000
@
J Edward W. Thommes, 2000
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ABSTRACT
The outer giant planets, Uranus and Neptune, pose a challenge to theones of planet formation,
They w-st in a region of the Solar 3ystem where long dynamical timescales and a low yrhordiai
density of material would have conspired to make the formation of such large bodies (- 15 and 17
times as massive as the Earth, respectively) very difEcult. In this work, a weil-estabfished model
of planet formation, together with numerical simulations, are used to show that such bodies are
unlikely to have formed far beyond the region of Jupiter and Saturn. A model which addresses
this problem is proposed: instead of forming in the trans-Saturnian region, Uranus and Neptune
underwent most of their growth among prot-Jupiter and -Satuni, and were scattered outward to
their present orbits when Jupiter acquired its massive gas envelope. Numerical simulations show
this model to be very robust, readily reproducing the configuration of the outer Solar System for a
wide range of initial conditions. Simultaneously, the model may also help to account for the present
state of the asteroid b d t and the Kuiper belt (the trans-Neptunian disk of cornets).
Statement of CO-aut
horship
The foundation for the mode1 developed in this thesis was first presented in Thommes, Duncan and
Levison (1999)-
ACKNOWLEDGMENTS
1 would like to thank the following people for help during the course of this thesis project- My
advisor Martin J. Duncan, for invaluable guidance, support and enthusiasm throughout; Harold JLevison, for many useful suggestions, for answering my SyMBA questions, and for letting me ride
in the Maserati; Man Hoi Lee and John Chambers, for stimdating discussions,
1 gratefully admowledge the financial support received during my graduate work ftom NSERC
and Queen's University. 1 want to thank the Queen's University Astronomy Research Group as a
whole, for providing a pleasant and enjoyable working environment. And I'd Iike to express my
appreciation for the mild Kingston weather, which limited the number of civil emergencies during
my time here to oneLast but certainly not Ieast, 1 want to thank my family and PvIonica Cojocaru for the always
generous supply of moral support (and food!).
Statement of originality
The results presented in this thesis are the origind work of the author, a-cept where referenced
within the text, Numerical simuIations were perfomed using the SyMBA simulation package
developed by Duncan, Levison and Lee (1998),with modifications by the author.
Contents
4.6
vii
4.5.6 Set 3: A shallower disk density profle . . . . . . . . . . . . . . . . . . . . . .
4.5.7 Set 4: An even shallower disk density profile . . . . . . . . . . . . . . . . . .
4.5.8 Set 5: The role of Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Scattered protoplanets and the s m d body belts
72
74
74
80
. . . . . . . . . . . . . . . . . . . 83
5.1 The asteroid belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 -1 Belt-crossing scattered protopIanets . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Sweeping secular resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The Kuiper belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
84
90'
93
97
6. Su~lunary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1 Zn-situ formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Planet formation in the Jupiter-Saturn region . . . . . . . . . . . . . . . . . . . . . . 99
6.3 The fate of Jupiter's neighbours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Evidence and predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
.
A Giossary of symbols. abbreviations and terrns . . . . . . . . . . . . . . . . . . . . . 112
B-The SyMBA integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-1 Symplectic integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Vaq-ing the tirne resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 The democratic heliocentric scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
115
117
118
C. Gas drag implementation in Sy2MBA . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
C.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
C.2 Calculating and appIying the drag acceleration . . . . . . . . . . . . . . . . . . . . . 122
LIST O F TABLES
4-1 Total mass of solids between 5 and 10 AU for different surface density profiles . - . 49
4.2 Oligarchie-growth protoplanet masses for c = u,(a/5 AU)-* . . . . . . . . . . . . . . 50
4.3 Successive orbital radii in the Jupiter-Saturn zone for 10 Ma protoplanets spaced by
n r -~ - .~. . - - . - - - - . . - - . - - - . - - - - - - - - - - - - . -51.
4.4 Final orbital elements of Run 1A . . . - . . - . . - - . - . . . . . . . . . . . . . . . . 66
5.1 Median eccentricities and inclinations in the asteroid belt . -
,
. . . . . . - - . . . . 85
.---.--
LIST OF FIGURES
Oligarchie growth b a l masses (top panel) and corresponding growth tirnes (bottom
panel) for the Solar System beyond 2.7AU (the snow Line) for the pIanetesimaJ and
nebuIar gas densities given by Eqs. 3.9 and 3.10- . . . . . . . . . . . . . . . . . . . .
Growth times for 1 Me, 10 Me and 20 Me bodies in the Solar System beyond 5 AU
for the planetesïxnai and nebular gas densities given by Eqs- 3.11 and 3.12 . , . . , .
Protoplanet masses a t different times as computed from Eq- 3.6, using the planetesimal and gas densities given by Eqs- 3.11 and 3-12, and planetesimal masses of
IO-^ Me (10 km bodies for a density of 1.5 g/cm3) . . . . . . . . . . . . . . . . . . .
Protoplanet masses a t different times, cornputed using the planetesimal and gas
densities given by Eqs- 3-13 and 3-14. . . . . . . . . . . . . . . . . . . . . . . . . . .
arnong 1and 10 km planetesimals, using the planThe timescale for collisions, TCOU,
etesimal surface density of Eq. 3-11, . . . . . . . . . . . . . . . . . . . . . . . . . . .
Top panel: ProtopIanet masses versus semimajor axis a t 10 Myrs, computed for
10m (IO-'' M e ) planetesimds. NebuIa 1 has the moderate planetesimd and gas
densities of Eqs. 3.11 and 3-12, while Nebula 2 has the very high densities of Eqs3.13 and 3.14. Bottom panel: semimajor axïs decay timescale of a 10m body, using
7=
and the gas density of Eq, 3.12, . . . . . . . . . . . . . . . . . . . . . . . .
Cnticai mass Mmit, as defined in Eq. 3.18, versus semimajor axïs for a protoplanet
density of 1.5 g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ProtopIa.net masses at different tirnes as computed Erom Eq. 3.6 with Mo = 1 Me,
using the planetesimal and gas densities given by Eqs. 3-11and 3.12, planetesimal
masses of IO-' Me (10 h bodies for a density of 1.5g/cm3) . . . . . . . . . . . . .
The state of Run A at 1 Myr. Ekom top to bottorn, the panels show eccentricity,
inclination and mass versus semimajor axis. The bottom panel plots the protoplanet
m a s distribution predicted by Eq. 3.6 with an initial mass of 0.2 Ma. . . . . . . . .
T h e s t a t e o f R u n A at2Myrs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The state of Run A a t 4 Myrs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The state at 1 x IO? years of Run B, which takes the pIanetesimal surface density
given by Eq. 3.11 as the initial profile of the pIanetesimal disk- The disk initially
extends from 10 to 35 AU. Gas drag on the planetesimals is calculated assuming a
planetesimal radius of 10 km and a gas density as given by Eq- 3.12. The three
panels show eccentricity, inclination and m a s versus semimajor axis. The line in
the bottom panel shows the protop1a.net mass profile predicted by Eq. 3.6 with an
initial mass of 1Me.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The state at 5 x 107 years of Run B . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
25
26
27
29
31
33
35
39
40
41
42
43
List of Figures
x
3.14 The state a t 1 x 107 years of Run Clwhich takes the planetesimal surface density
given by Eq. 3-13 as the initial profile of the planetesimal disk, which extends from
10 to 35 AU. Gas drag on the planetesimals is calculated assuming a planetesimal
radius of 1 km and a gas density as given by Eq. 3-14. The three panels show
eccentricity, inclination and mass versus semimajor axis. The line in the bottom
panel shows the protoplanet mass profile predicted by Eq. 3.6 with an initial mass
o f l M a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.15 The state of Run B after it has been nui for another 1.5 x 108 Myrs without gas drag. 45
PeriheLion distance q as a function of semimajor axis a for three different d u e s
of the Tisserand parameter relative to Jupiter, The value of a corresponding to
q = aJ = 5.2 AU (the upper lirnit of the plot) is the furthest possible scattering
distance. A body with T = 2 f i can be scattered to infiniSr, çince q w p t o t e s to
a~asa+oo..
.......................................
53
A test particle scattered by Jupiter. The top panel shows the evolution of its semimajor axis (thick solid), perihelion (solid) and aphelion (dotted) over time. The
bottom panel shows the evolution of the Tisserand parameter . . . . . . . . . . . . .
Evolution of the eccentricity of a 10&& body (thick line) embedded in a disk of
0.2 Me bodies, and the evolution of the FUIS eccentricity of the surrounding 0.2 Me
bodies (thin iine)- The semimajor axis of the 10 Me body is approximately constant
during this t h e , so that as the eccentricity decays, perihelion and aphelion converge
in an essentially mirror-symmetric mamer, . . . . . . . . . . . . . . . . . . . . . . .
Eccentricities (top) and inclinations (bottom) ui the outer Solar System at the
present epoch, showing the giant planets and those Kuiper belt objectç which have
been observed a t multiple oppositions. The w e in the top panel shows the locus
of orbits with perihelia at the semimajor axïs of Neptune. . . . . . . . . . . . . . . .
The initial state for runs in Set 1,showing eccentricity (top) and inclination (bottom)
versus semimajor axis. The iarger solid circles denote the four 10 Me protoplanets,
and each of the smail empty circles represents a 0.2 Me planetesimal. The planetesimd density in the vicinity of the protoplanets is decreased to keep the density of
protoplanetss plus planetesimals consistent with the surface density given by Eq. 4.13.
Run A6: Evolution of semimajor axis (bold lines), penhelion distance q (thin lines)
and aphelion distance Q (dotted lines) of the four l O M & protoplanets. The p r o t e
planet which grows to Jupiter mass (314 Me)over the first 105 years of simulation
time is shown in black- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Run A6 continued to 50 Myrs. Between 5 Myrs and 50 Myrs,the net migration
for Jupiter and the protoplanets, going fkom inside to outside in semimajor axis, is
-0.2AU, 1.5AU,4AU,and 1.3AU. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Endstates of the eight Subset A runs, after 5 Myrs of simulation time, e x e p t for
Al and A8, which were continued on to 10 Myrs. Eccenctricity is plotted versus
semimajor axïs, PIanetesimal orbits crossing Jupiter or any of the protoplanets are
generdy unstable on timescales short compared to the age of the Solar System, thus
the region among the protoplanets would be essentially cleared of planetesimals long
before the present epoch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
End states of the eight Subset B runs, after 5 Myrs of simulation time, except for
B3, which was continued on to 15 Myrs. Eccenctricity is plotted versus semimajor
55
56
58
61
64
65
67
axis.
.......................................
69
4.10 End states of the eight Subset C r u s , after 10 Myrs of simulation time, except for
Cl, which was continued on to 15 Myrs. . . . . . . . . . . . . . . . . . . . . . . . . . 70
List of Figures
xi
4-11End states of the eight Set 2 runs. The £ k t four are run to 5 x 106 years; the last
four are run to lo7 years- . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.12 End states of the twelve Set 3 nuis. Al1 are run to 5 x 106 years- . . . . . . . . . . . 73
4.13 End states of the eight Set 4 runs. Al1 are nui to 5 x 106 years. . . . . . . . . . . . . 75
4.14 Set 5: Evolution of semimajor axis (bold Iines), perihelion distance q (thin h e s ) and
aphelion distance Q (dotted h e s ) of the four 10 Me protoplanets. The protoplanet
which grows to Jupiter mass (314 Me)over the first IO5 years of simulation time is
shown in black, In each of the three panels, a different protoplanet grows to Saturn's
mass at 2 x.105years: red (top), green (middle) and blue (bottom) . . . . . . . . . . 76
4.15 Set 5 runs in which Saturn grows at 4 x IO5 years- In the top panel the green
protoplanet becomes Saturn; in the bottom panel, it is the blue one- - . . , , . . , , 78
4.16 Set 5 runs in which Saturn grows at 6 x IO5 years- In the top panel the green
protoplanet becomes Saturn; in the bottom panel, it is the blue one- . , . , , . , , - 79
4-17Set 5 runs in which Saturn grows at 8 x 105 years. In the top panel the green
protoplanet becomes Saturn; in the bottom panel, it is the blue one. . . . . . . . . . 80
4-18Endstates of those Set 5 runs in which Saturn commences growing after the protoplanets have largely decoupled from each other. The start L e ofaatum's growth,
ts ,is denoted on each panel- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Eccentricity (top) and inclination (bottom) vesus semimajor axis of bodies in the
asteroid belt larger than 50 km . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The first 106 years of Run IA8, showing the evolution of the semimajor cwcis (thick
solid), perihelion (solid) and aphelion (dotted) of Jupiter (black) and the three protoplanets. At 105 years, a protoph.net (red) bnefiy crosses the asteroid belt, its
periheLion going donm to 3 AU. The crossing lasts less 5 2 x 104 years. . . . . . .
Evolution of semimajor axis (thick solid), perihelion (solid) and aphelion (dotted)Protoplanets are initially 15 Me As usual, the innerrnost protoplanet (bIack) grows
to Jupiter mass over the first 10' years. One scattered protoplanet (red) crosses the
region interior to Jupiter's orbit numerous times within the first 2 x 105 years. - - Eccentricities and inclinations of planetesimals in the asteroid belt region, interior to
Jupiter (the large dot), a t 3 x lo4 years (top panel) and 2 x 105years (bottom panel)These times are, respectively, just before and just after the penod during which a
protoplanet repeatedly crossed the region interior to Jupiter (cf- Fig. 5.3). Jupiter
in this run has moved inward to 4.8 AU, 0.4AU Iess than its present semimajor
&S.
The line marks the Iocus of Jupiter-crossing orbits . . . . . . . . . . . . . . . .
Evolution of a growing Jupiter and three 15 Me protoplanets, with one 1 Me body
(light blue) interior t o it, in the asteroid belt region. A scattered protoplanet (green)
ejects the body from the belt a t 5 x lo4 years. . . . . . . . . . . . . . . . . . . . .
A Iarger view of the lower pane1 of Fig. 4.15. The protoplanet plotted in blue
begins growing to Saturn's mass (on a 105 year timescale) a t t = 4 x IO5 years. Its
semimajor axis a t the time it cornpletes its growth is about 8.4 AU. At t = 5 x 106
years, "Saturn" has a semimajor axis of about 9.4AU. . . . . . . . . . . . . . . . . .
Counterpart to Fig. 4.8, showing inclination versus semimajor axis for the endstates
of the nrns in Subset 1A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Counterpart to Fig- 4-10,showing inclination versus semimajor axis for the endstates
of the runs in Subset IC- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-
-
-
84
86
87
88
89
92
95
96
1.1
Motivation
The planets of our Solar System are readily divided into two categones: there are the terrestrial
planets planets, the largest of which is the Earth itself. Then there are the giant planets, the
= 1Ma)by more than an order of magnitude.
smdest of which exceed the Earth's mass (6 x 102?g,
Among the giant planets, there are two further categories. Jupiter and Saturn are gas giants; though
they are thought to have solid cores
-
10 Me in m a s , their prim;rrv constituents are hydrogen
and helium, in approximately Solar abundance, They have total masses of 318 Me and 94 ?de
respectively, Uranus and Neptune, on the other hand, beloag to the category of ice gïants (though
they are frequently also referred to as gas giants, this is a misnomer). Aside from a gaseous outer
layer containing molecular hydrogen and helium, they consist mostly of a mixture of &O, CH4
and NH3 ices, likely mixed with rock-very
much like the composition of cornets. At 14.5 Ma
and 17.1 Me respectively, Uranus and Neptune are far smaller than their gaseous neighbours. A
detailed oveMew of the present state of knowledge about the interiors of the giant planets is given
by Guillot (1999).
The significant differences between the gas and ice giants belie the fact that both likely had a
similar origin. Jupiter and Saturn are thought to have started out as ice giants themselves, growing
from mergers among the solid components of the protostellar disk. The individual bodies making
up this disk, called planetesimals, were typically perhaps tens of kilometres in size, and in the region
of Jupiter and beyond, would have had the udirty snowbali" composition of present-day cornets
(themselves just leftover planetesimals). In the case of Jupiter and Saturn, ice giant became gas
1- Introduction
2
giant through the accretion of a massive gas envelope. Uranus and Neptune, however, received
Iittle gas; one can view them as "faiied" gas giants.
The picture so far seems straightforward: Four giant planets accrete from planetesimals in the
outer Solar System ("outer" being taken here to mean beyond the asteroid belt, a t a heliocentric
distance greater than 5 AU), and two of them manage to subsequently also accrete large quantities
of gas- However, the rate of planet formation by planetesimal accretion decreases with heliocentric
distance, Numerical simulation shows that the formation of the terrestrial planets is completed on
a 108 year timescale (e.g. Chambers and Wetherili 1998). But in the region of Uranus and Neptune,
simulations have been unsuccessful at producing
-
10 Me bodies in the lifetime of the plantesimal
disk, unless implausibly extrerne parameters are used.
1.2
Aim of the thesis
The difficulty in thus far accounting for the existence of Uranus and Neptune constitutes a troublesome gap in our understanding of the origin of the Solar System. In this thesis, the timescale
problem will e s t of ail be quantified. It wiI1 be shown that in the conventional scenario for the
formation of Uranus and Neptune, in which they originate at approximately their present heliocentnc distances, the timescale problem is severe- A new model for the origin of Uranus and Neptune
will then be developed. It will be shown that rather than forming in-situ, the ice giants could have
undergone the majority of their accretional growth in the same region as Jupiter and Saturn. Using
numericd simulation, it will be demonstrated that the final growth stage of Jupiter will trigger an
initially violent and rapid evolution in the orbits of these boclies (as well as proto-Satu). The
end resuit tends to be a configuration resembling Our outer Solar System. In many cases, the
resemblance is remarkably strong. Finally, it will be shown that, as a side effect, this model may
help account for the dynamical structure of the present-day Kuiper belt (the tram-Neptunian disk
of cornets), as well as the asteroid belt.
1-
1.3
Introduction
3
Outline of the thesis
Chapter 2 develops a basic theoretical framework for the study of planet formation. ft introduces
the three major influences on plantesima1 dynamics which are relevant to this work: Gravitationai
viscous stirruig, energy equipartition among planetesimals, and gas drag. The concepts of runaway and oligarchic growth are also presented. FinaIly, bnef o v e ~ e w of
s planet formation in the
terrestrial and gas giant regions, as these processes are presently understood, are given-
In Chapter 3, planet formation timescales are investigated. Tirnescde estirnates are made based
on an approxhate analytic description of oligarchic growth. it is shown that the in-situ growth
of Uranus and Neptune duruig the Iifetime of the gas component of the protosolar disk requires
an implausibly massive disk. Numerical simulations c o n h this, and also suggest that accretional
growth in the outer Solar System essentidy stalls once the gas is rernoved- Thus if Uranus and
Neptune cannot form while the gas is present, they are unlikely to form a t all.
Chapter 4 develops the new mode1 for the origin of Uranus and Neptune. Plausible initial
conditions for the simulations are derived- The simulation results are presented- The mode1 is
found to be very robust; Solar Systern analogues are readity produced for a broad range of initial
conditions.
Chapter 5 investigates whether protoplanets scattered from the region of Jupiter and Saturn
might account for the anomalousIy high degree of dynamical excitation and mass depletion in the
present-day asteroid belt and Kuiper belt. It is found that such a mechanism may have played
a Iimited or indirect role in the asteroid belt. In the Kuiper belt, however, excitations similar to
those presently seen c m be directly produced by giant protoplanets as they are scattered.
Chapter 6 siimmarizes the findings of t h e thesis and identifies areas for future work.
BACKGROUND
2.1
Early stages
The Solar Systern is believed to have fomed Grom the collapse of a molecular cloud core into a
flattened disk with the Sun at its center (see Lissauer 1993 for a review). The disk is likely to
have been very sirnilar in composition to the Sun itself- Part of the gravitational potential energy
released during collapse went into heating the disk; once the collapse slowed, the disk cooled and
rnicroscopic condensate grains began to fonn. In the first stage of planetesimal formation, these
grains grew by paùwise accretion as they settled toward the disk rnidplane, forming a thin layer of
up to about centimetre-size dusty aggregates (Goldreich and Ward 1973)- For a larninar layer, the
next stage of planetesimal formation could have proceeded via gravitational instabilities. However,
while the central layer of solids rotated a t Keplerian veloaty, V K = J m a a (G = gravitational
constant, Mo = Sun's m a s , a = heliocentric distance), the surrounding gas nebula, being partially
pressure-supported, was in sub-Keplerian rotation. The resultant shear caused turbulence in the
dusty layer, inhibiting gravitationa1 collapse (Weidenschilling 1980). If the particles were sufficiently
large, gravitational collapse codd still have proceeded; otherwise, the particles rnust have continued
to grow by pâirwise mergers instead.
2.2
Gravitational accretion
2.2.1
The two-body approximation
For partides =S I cm, thermal velocity exceeds escape velocity from their surface a t solar nebula
temperatures, on the order of 10 "K and higher. Therefore, the first stage of accretion-if
it is
indeed a pairwise process-relies
on nongravitational forces, predominantly van der Waals attrac-
tion (Weidenschilling 1980)- However, the details of this growth phase are not yet well understood;
a particular problem is that bodies larger than dust but smaller than about a kilometre in size
experience strong gas drag, and t hus spiral inward on short tirnescales. Therefore, planet esïmds
must have passed through this size range rapidly (Lissauer 1993).
Once planetesimais reach sizes on the order of a kiIometre, their long-range interactions are
dorninated by gravitation. These interactions impart random velocities relative to circular motion
on the planetesimals. It has been found through direct N-body sùnulations (Ida and Makino 1992a)
that the velocity distribution for a swarm of coUisionless, equal-mass bodies interacting by mutual
gravitational s c a t t e ~ g(also called viscous stirrïng; see Section 2.2-2) is weli approximated by a
Gaussian distribution in eccentricities and inclinations:
where (e2), (i2) are the RMS eccentricities and inclinations, and P(e)de (P(i)di) gives the probability that e (i)falls in the range [ele + de] (Li,i
+ di]). Also, unless eccentricities and inclinations
are low enough to be in the shear-dorninated regime (see Eq-2.6), they are related by
in equilibrium. At the same time, uniess the planetesimals are subject to the gravitational influence of much larger bodies (Le- protoplanets embedded in the swarm), their eccentricities and
inclinations are srnail enough that the viscous stirring timescaie is shorter then the physical collision timescale (Sections 2.2.2 and 3.4). Thus physical collisions do not sigaificantly infiuence the
velocity distribution (Ida and Makino 1992a, 1992b).
2- Background
6
In a swarrn of equal-mass planetesimals rn, the velocity dispersion u, is given by
and the relative velocity is Ur,[
= 6,
For
a body of mass m in orbit around a m a s M, »rn at
a distance a, one can d e h e a radius around m within which its gravity is stronger than the tidal
force of M,. This is analogous to Roche's limit, and is known as the Hill radius
rH,:
The reduced (nondimensional) Hill radius is defined as
The reIative velocities among members of the swarm are considered to be dispersion-dominated,
as opposed to shear-dominated, when they exceed the Keplerian shear across the Hill sphere of a
planetesimal:
where 32 = u K / a =
Jw
is the Keplerian orbital fi-equency.
which a body of mass M
>> m
In this regime, the rate at
within the swarm increases its mass by accreting pIanetesimals is
well described by the two-body approximation (Safkonov 1969, Wetherill 1980, Ida and Nakazawa
1989), also called the particle-in-a-box approximation because of its origin in kinetic gas theory. If
one assumes perfect accretion-that
rnerger-the
is, every physical encounter between M and an rn results in a
mass growth rate is
where n is the number density of the swarm, and A is the collision cross-section for a body m with
the body M. The mass density of the s w m is p,,,
= nm, and this can be written in terrns of
2. Background
7
the surface density a and the swarm's scale height h: p,,,
= a/h- The collision cross-section,
neglecting the radii of the masses rn, is
(2-8)
where RM is the radius of M and v,,,
=
d-
is the escape velocity fkom the surface of
M. The factor in brackets is the enhancement of the physical cross-section due to gravitational
focusing; it accounts for those encounters where a body rn would miss M if it were not for the
gravitational force between them - The gravitational enhancement factor is often written as
F, = 1 + 28,
(2-9)
( V ~ ~ , / V ~ , is
, ) /called
~
the Safionov number. The mass growth rate can therefore be
where 0
wrîtten as
Using h
J
-
= a ( i 2 )I l 2 , ( i 2 )'12 = 1/2(e2)
RM = ( 3 ~ / 4 ~ I3~ ( ~p M) =' protoplanet densiw), v,,, =
= J ~ G ( ~ ~ P M / ~ ) 'v /~ e~ r M( e~2 )/1 ~/ 2 ,v K ,and assuming that gravitationai focusing
is effective so that 8
>> 1, this can be rewritten as
where C = 47r2j3 ( 3 / 4 p M ) 1 / 3(G/Mo)'12.
The timescale to reach a final mass M is given by
2.2.2
Evolution of the eccentricities and inclinations
The RMS eccentncities and inclinations of Eqs. 2.1 evolve over time. Gravitationaliy, two effectsviscous stirring and dynamical friction-drive
this evolution. The former is the increase in randorn
2- Background
8
motions as bodies undergo gravitational encounters- The latter is the tendency of a system of
gravitationally interacting bodies to equaiize the kinetic energies of random motim between bodies
of differing masses- The expressions given below for eccentricity and inclination evolution are
based on ones developed by Stewart and Wetheriil (1988), using the Fokker-Planck equation for
the phase space evolution of a population of bodies undergoing gravitationai encounters with each
other. Non-gravitatîonally, random motions are a£Fected by gas drag, if gas is present, and by
physica.1 collisions- The associated timescak-the
gravitational relaxation time-was
fbst derived
by Chandrasekhar (1949).
Viscous stirring
The eccentricity and inclination rates of change due to viscous stirrïng can be expressed as (Weid e n s c w n g et al 1997)
and
where subscript 1 refers to the bodies being stirred, and subscnpt 2 to the population of stirruig
bodies- The coefficient C is given by
where G is the gravitational constant, V K the local Keplerian velocity, pSwl the spatial mass density
of the swarm, and A is approximately the ratio of the maximum encounter distance between a body
ml and a member of the population of ma,to the maximum separation that results in a physicai
encounter. It is defined as (Stewart and Wetheriii 1988)
2. Background
b,,,
9
is the maximum approach offset, taken as the scale height of the systern. bo is the distance at
which the relative velocify between a body mi and members of the population of m;!is the same
as the Keplerian velocity of a body orbiting ml,
R, is the physicd radius of ml enhanced by gravitationai focusing,
(cf. Eq. 2.9), in other words, the maximum impact parameter of an ml relative to an rn:! which
resuits in a physical collision. Je and Ji are definite integrais depending on the parameter P (Stewart
and Wetherill 1988),where
-
At the equilibrium value (i2)1/2/(e2)1/2
a d Je = J ;
-
1/2 for both the stirred bodies and the stirrers, 8
-
1/2
1-
When planetesimals of different masses coexist in a swarm, gravitational interactions among them
act to make the kinetic energy of their random motions more similar (Stewart and Kaula 1980,
Ida and Makino 1992b)- This tendency toward energy equipartition is c d e d dynamical friction.
For a two-component population of planetesimals, with masses M and m, M
> m, equilibrium is
att ained when
-
(e&)~
(em)m,
( i k )-.~(i2)rn.
(2.20)
Thus the velocity dispersion is higher for smaller bodies when dynamitai fnction is effective:
The eccentricity and inchation rates of change due to dynarnical friction on a body of m a s M
in a swarm of mass m bodies can be expressed as (Weidenscihilling et al 1997)
and
where the coefficient C given by Eq. 2.15- K,(/?)
and Ki@) are definite uitegrals also defined in
Stewart and Wetherill (1988))and /3 is as defined in Eq-2.19, substituting the subscripts M and
m for 1 and 2. When
B
-
1/2, Kt= Ki
-
8. In Section 4.4, these analytic estimates are shown to
agree well with numerical simulation results.
Gas drag
In the presence of gas, the evolution of a body's eccentricity and inclination, as well as its semimajor
axk, is modified. Acceleration of a body subject to gas drag is of the general form
where
vrelis
the velocity of the body relative to the gas. For Stokes drag n = 0, and n = I for
Epstein drag, whicb is applicable at high Reynolds number. The latter state is generally accepted
to have existed in the solar nebula (e.g.
Weidenschilling and Davis 1985). In this case the drag
parameter, K, is given by
where POas is the gas density, Cda dimensionless drag coefficient (E 0.4 for spheres), p,l is the m a s
density of the body, and r,l the body's radius. K has dimension of length-', and is approximately
equal to the distance a body must travel relative to the gas to encounter its own mass in gas.
The evoIution of a body on a circular orbit due to gas drag is easiiy derived. In this case v,,r
is simply the clifference between Keplerian velocity and the circular velocity of the gas. Setting the
2- Background
11
energy rate of change per unit mass due to gas drag,
equal to the energy rate of change per unit mass of a circular orbit due to variation in semimajor
one h d s the rate of change of the body's semimajor a i s ,
and, integrating, the semimajor axis as a function of time,
The evolution of semimajor a ï s , eccentriciw and inclination for the general case of nonzero e
and i are derived in Adachi (1976). The results (including a correction from Kary, Lissauer and
Greenmeig 1993) are reproduced below:
and
where
a!
gives the exponentiai a-dependence of the gas density (pgas(a)=
being the Merence between the gas velocity and
UK.
and 7
= v g / v K with ug
2. Background
12
Pbysical collisions
In the two-body approximation (Section 2.2.1)' the collision tirnescale is
where n is the number density of bodies, A is the collision cross-section, and Ur,[ the bodies' relative
velocity- Thus far, collisions were assurned to be accretiona1, as is to be expected when a small body
impacts a large body a t a velocity much less than the large body's v,,,,
However, for collisions
occurring a t velocities larger than the escape velocity from either body involved, fragmentation will
tend to occur. The role of fragmentation in planetesimal dynamics is investigated, for example, by
Wetherili and Stewart (1993).
2.2.3
Runaway growth
In the regime where relative velocities are dispersion-dominated b u t srnall enough that gravitational
dynamitai fnction results in a "runaway" accretion mode. In other words, larger bodies grow more
rapidly, so that the Iargest body in a given region eventually detaches from the upper end of the
size distribution. This can be shown with a simple argument ( e g - Kokubo and Ida 1996). The
relative velocity between the large body and the swarrn is vre[=
-d
E vm
under condition
2.21. Usïng 2.3 and 2.2,
The scale height is h
= 2 4 9 , and R
a M
~ Also,
/ ~ effective gravitational focusing means that
F, zz 28. Thus, from 2.10,
dM
M ~ / ~
OCdt
(e2,) -
2. Background
13
If M is small enough that the velocity distribution of the planeteshaIs iç determined principally by
their mutual interaction and not gravitational stirring by M , then (em) is not strongly dependent
on M , so that
For two bodies of masses M l , M2y Ml
> M2 in a swarm of smaller planetesimals,
the rate of
change of their mass ratio is
and so runaway accretion takes place-
The condition that the velocity dispersion be dominated by the mutual interactions of the planetesimal swarm breaks down when protoplanets gronring within the swarm becorne sufficiently Iarge.
Ida and Makino (1993) derive the condition for regdation of a swann of mass-m planetesimals' (e$)
and (im) by embedded protoplanets of m a s M rather than interactions among the planetesimds
thernselves as
where
is the surface mass densiQ of protoplanets, and cr,
is the surface mass density of plan-
etesimals- The surface density of protoplanets is defined by Ida and Makino as CM = M/2?raAay
where Aa is the width of the zone dynamically heated by a protoplanet, and Aü
%
Aa/h,a
is
given by
where the approximate expression cornes fiom using hM « 1 and assuming ,e
-
2i,.
Eq. 3.4
in Chapter 3 for the equilibrium between viscous stirring by a protoplanet and gas drag can be
2- Background
14
rewritten in terms of the protoplanet's reduced Hill radius h ~ :
Using p,,,
g at 1AU and m =
= 2 x 10-~(a/lAU)g/m2,
m =
Makino obtain M / m
-
g at 5 AU, Ida and
50 and 100 respectively.
However, these planetesimal masses are somewhat large. Planetesimal size was likely 1 to 100
km (Lissauer 1993), corresponding to a mass of
a planetesimai mass of
-
10-l2 - 1oA6Me (6 x 1015- 102' g), For example,
Me (- 10 km planetesimals) makes M / m
-
3000 at 5, and 15000 at
10 AU- Protoplanets would thus corne to dominate the planetesimal disk dynamics when they are
only
-
10-~
-
Ma. Therefore, runaway growth stops when the Iargest bodies are still very
smalI, and the dominant timescale for growing protopIrnets is that of post-runaway growth.
When the velocity dispersion of the planetesimals becomes dominated by interactions with the
protoplanets, v,,, is approximately proportional to the escape velocity fiom a protoplanet's surface,
J--
Since RM cc M ~ / this
~ , makes
Ida and Makino (1993) codkm this dependence using N-body simulation. For a constant a, Eq2.12 gives Tgrow
a M ~ and
/ ~so
Therefore, in the post-runaway regime, larger protoplanets accrete planetesimals more slowly than
smaller ones for a given semimajor axis, so that the mass ratio among protoplanets localiy evolves
toward unity- This growth mode is cal1ed orderly growth.
Orderly growth, however, only applies among the largest bodies within a swarm. Large bodies
still g o w more quickly than the smdest bodies, as long as the planetesimal density is high enough
that dynarnical fiction is effective and one can make the approximation that J
v
R
EZ
um-
2. Background
15
Then the mass ratio (M/m) grows since, comparing with 2.37,
The situation of accretion in a planetesimal swarm being dominated by its most massive members,
simiIar in size to each other, has been termed "oligarchic growth" (Kokubo and Ida 1998, 2000).
Through a coupling of mutual scattering and dynamitai fiction, the protoplanets evolve toward
circular well-spaced orbits.
When two protopIanets approach closely, they scatter each other,
increasïng their eccentricities and orbital separation. SubsequentIy, dynamical friction reduces
their eccentricities with little effect on their separation. This repdsion mechanisrn keeps adjacent
orbits spaced by around 5 - 10 mutual Hiii radii. The mutual HilI radius for a pair of bodies of
masses Ml and
M2is defined as
Assuming the protoplanets remain well-spaced while growing, their final masses are given by
where a is the protoplanet semimajor axis, b is the spacing between adjacent protoplanets, and p
is the fraction of the total mass in the zone [a - b / 2 , a
Orbital repulsion makes b = n r ~,,n
-
+ b / 2 ] ïncorporated
into the protoplanet.
5 - 10- The protoplanet mass then becomes
Eq. 2.12 gives the growth thescale to a h a 1 mass M:
2.3.1
The terrestrial zone
The process of planet formation in the terrestrial region of the Solar Systern, intenor to about 2 AU,
is relatively well understood, at least as far as formation timescaies are concerned. The occurrence
of runaway growth is a robust result (Wethedl and Stewart 1989, Kokubo and Ida 1996). The
transition from runaway to orderly/oligarchic growth has also been extensively studied (Ida and
Makino 1993, Weidenschiilïng et al 1997, Kokubo and Ida 1998, 2000). Lunas- to Mars-sized
protoplanets are found to form by this time.
In the final stage of terrestrial planet formation, these protoplanets perturb each other ont0
crossing orbits, and accretion proceeds primarily by giant impacts. Simulations of this stage (eg.
Chambers and WethenU1998, Agnor, Camp and Levison 1999) do produce final bodies in numbers
and masses comparable to the present terrestriai planets, but some discrepancies rernain. Most
significantly, h a 1 eccentricities and inclinations of bodies tend to be too high; in simulations, the
majority of planetesimals have been incorporated into protoplanets by the time of the giant impact
phase, so dynamical friction is ineffective. It seems that as-yet unmodeled physical process must
play a role. It is unclear, for instance, how effectively fiagmentationary collisions replenish the
population of small bodies in the late formation stage.
2.3.2
T h e gasgiants
Understanding the formation of the gas giants, Jupiter and Saturn, presents a greater challenge.
Two different mechanisrns have been proposed; either direct formation by gravitational gas instability (eg. Cameron 1978),or formation of a solid core protoplanet from plmetesimals, which then
accretes a massive gas envelope (eg. Safionov 1969, Mizuno 1978, Bodenheimer and Pollack 1986).
The former mode1 has largely fallen out of favour because of some problems which are difficult to
2. Background
17
overcome; it seems to require a gas disk more massive than is supported by obsenations of young
stars, and it tends to produce giant planets Iarger than Jupiter. The latter mode1 is the most widely
supported one. It is able to account for the enhancement of Jupiter and Saturn in high-Z materiais
relative to Solar abundance, and therefore dso for the similarity in the core masses of Jupiter and
Saturn. Detailed simulations usùlg this model have been very successful in reproducing Jupiter
and Saturn (PoUack et al 1996). The diversity of the discovered extrasola. planetary systems has
Ied to a reevalution of the disk instability mode1 (Boss 1998), and indeed, such observations may
eventually serve to determine which is the correct (or dominant) mechanism- However, in our Solar
System at least, the nucleated instability mode1 is better able to account for the characteristics of
the gas giants.
The first requirement, then, is the formation of a solid core large enough to accrete a massive
enveIope. The required mass has been found through hydrodynamical modeling to be around
10 Me (Mizuno 1980), and as high as
-
30 Me (Bodenheimer and Pollack 1986)- Also, hydrostatic
models of the present-day interiors of Jupiter and Saturn, constrained by spacecraft measurements
of the gas giants' gravitational moments, indicate that they have core masses of up to 10 and 17
Me,respectively. At the critical mass, the core's gas envelope can no longer maintain hydrostatic
equilibrium, and it collapses onto the core, while the surrounding gas streams in to fiiI the ~ c a t e d
volume, The hydrodynamic coliapse proceeds rapidly, with Jupiter and Saturn accreting most of
their final mass of gas in on the order of IO5 years. An important constraint is thzt the formation of
the gas giants must occur before the gas is removed fkom the primitive Solar System- Observations
of young stars suggest that this happens in 10 Myrs or Iess, when a star is in its T Tauri phase
(Strom, Edwards and Skrutskie 1990)- This phase is characterized by strong stellar winds, which
are thought to be responsible for the clearing of the gas.
The model of Pollack et a1 (1997), which simulates sirnultaneous accretion of solids and gas,
exhibits three distinct phases. In the first, pIanetesimais accrete almost exclusively
. This phase
2- Background
18
ends when the pIanetesimals are depleted in the feeding zone of the growing protoplanet, and its
solid accretion rate drops. Runaway p w t h is assumed to act, and thus growth is rapid; the fength
of this phase is only a few 105years-
In the second phase, the protoplanet begins to accrete a gas atmosphere- Gas and solid accretion
rates remain nearly constant, and low compared to that of the first phase, for a t h e approaching
10 Myrs.
The third and final phase commences when the soIid core and its gas envelope are of comparable
mas. The gas accretion rates increases sharply until hydrostatic equilibrium breaks down, and
runaway gas accretion sets in.
It is worth noting this model's assumption that the planetesimal accretion proceeds by runaway
growth, in other words, that planetesimal random velocities are regulated by mutual viscous stirring,
is probably unrealistic, In fact, once the protoplanet's mass reaches
-
103 times that of the
planetesimals, its effect on the planetesimals will dominate (see Section 2.2.4). Therefore, Pollack
et al's estimate for the timescale is Iikely to be too short.
Results of a more detailed simulation of the k t phase are reportecl by Weidenschïlling (1998,
conference abstract). A planetesimal surface density of 10g/cm2 is found to allow the formation of
a 10 Me body in
-
107 years. This may exceed the lifetime of the solar nebula, and the growth of
Saturn's core will of course be even more problematic. A moderately higher surface density is said
to alleviate this diaculty.
Apart fiom the above, the author is not aware of any simulations which have successfulIy
produced bodies of
-
10 Me or Iarger at a semimajor axis of 5 AU or greater. It is a challenging
problem to sirdate directly, with a pure N-body code, since a large amount of m a s is involved.
Breaking this mass up into planetesimals that are anywhere near a realistic size results in an
unmanageably large number of bodies. Indeed, the simulation of Weidenschilling was performed
using a hybrid code, which handles Iarger bodies individually but does not directly integrate their
2. Background
orbits, and models bodies below a cutoff mass as a multi-zoned continuum.
19
3. ACCRETION I
N THE TRANS-SATURNIAN REGION
The origin of the outer giant planets, Uranus and Neptune, presents a fundamental problem in the
study of our SoIar System- As outüned in Section 2.3.2, the formation of the solid cores of Jupiter
and Saturn is less well understood than the formation of the terrestrial planets, and a full numerical
simulation of this process has so far not been published in a refereed journal. The formation of
Uranus and Neptune is, not surprisingly, even more poorly understood, as well as more difficult
to simulate- The "ice giants", after dl, attained approximately the sarne final mass as the gas
giant cores- However, in the standard model of in-situ formation, they grew t o this m a s in an
environment significantly less hospitable to accretional growth than the neighbourhood of the gas
giants.
3.2
Previous work
Late-stage accretion in the tram-Saturnian region was investigated by Femandez and Ip (1981,
1984, 1996). They do not use full N-body simulations; rather, they resort to a statistical approach.
Bodies are assurned t o move on unperturbed Keplerian orbits until a pair is selected to have a close
encounter based on an orbit-averaged encounter p r o b a b i l i ~distribution (Opik 1976). The major
drawback of this technique is that it does not model interactions of bodies on noncrossing orbits; in
other words, distant perturbations are neglected, which means that gravitational stirring rates are
underestimated, The most recent simulation (Fmandez and Ip 1996) starts with 750 Mars-mass
(0.1 Me)bodies bettveen about 15 and 35 AU. They 6nd that Uranus- and Neptune-mass objects
3. Accretion in the trans-Satumian region
21
form in a few 108 years. The process is inefficient, requinng at least twice (and preferably three o r
four times) the combined mass of Uranus and Neptune to be present in the disk initidy.
Long-range perturbations in reality have a significant effect on the dynamics, and it has not
been possibIe to reproduce the results of Femandez and Ip using fdN-body simulations (Levison
and Stewart, private communication). Brunini and Fernandez (1999) do obtain simiIar results using
an N-body code which negIects interactions among bodies smalIer than 1 Me. However, in later
simulations which include all interactions, accretion is found t o be far less efficient (Bruini 2000);
t h e previous accretion rates cannot be attained unless the radii of the bodies are increased by about
an order of magnitude, relative to bodies of density
3.3
-
2 g/crn2.
Accretion with gas drag
T h e mass accretion rate of a protoplanet in a swarm of planetesimals is given by Eq. 2.11:
where C is a constant ( d e k e d after Eq. 2.11)- From Section 2-2.4, protopIanets can only reach a
very small mass before they dominate the dynamics of the planetesimals, so that runaway growth
transitions t o oligarchic growth. Thus the oligarchic growth timescale is the dominant one- Ida and
Makino (1993) derive an expression for the equiiïbrïum RMS planetesima1 eccentricity in this regime
by equating the timescale for eccentricity pumping of the planetesimals m through viscous stirring
by pmtoplanets M,
T Z with
~ ,the thescale for t h e dissipation of the protoplanets' eccentricity
through gas drag with the nebula,
Te.The stirring tintescale is approximately given by
where Aü is given by Eq. 2.39. The timescale for eccentricity damping by gas drag is
2 1/2 =
Te = Tcp~/(e,)
3.2 x ~ O - ~ ( r n / l ~ )
years.
(Cd/2)drm/1m ) 2 ( p g a s / i g ~ m - (VK
~ ) /1 c m S-1) (e$.,)1/2
(3.2)
3- Accretion in the transSatum-an region
22
Cd is a dimensionless drag coefficient (= 0.4 for spheres), r,,, is the radius of the planetesimals,
where p,
is the mass density of a planetesimal, taken to be 1 . 5 g / m 3 (roughly the density of
cornets), and p,,,
is the mass density of the nebular gas-
Setting T&-~ = Te and solving for
(e2)'j2,
one obtains
( e g) '1'
Substituting Eq, 3.4 into Eq. 2.11, one obtains the protoplanet mass growth rate as a function of
protoplanet m a s , semimajor a.xis, planetesimal surface mass density and gas volume mass density:
Integrating this equation gives protoplanet mass as a function of time:
~ 4 . x6 1 0 - l ~
[(l g / m 2 ) (I g / m 3 )
Orn
han
li3
(A)
-1/9
1 Me
( year)
( ~ 1 - l ' ~
1 AU
1
Growth t h e as a function of mass is thus
-1.8 x
log (-Mo)
1/3
1Me3
years.
(3.7)
Eq. 2.46 with p = 1gives the final mass of an oligarchically-grown protoplanet, attained after
it accretes al1 the planetesimals in an annulus of width n Hill radii centered on it:
3- Accretion in the trans-Saturnian region
23
Using Eqs. 3.7 and 3.8 together with estimates of the gas and planetesimal densities in the prirnor-
dial Solar System, one can make estimates of the time taken for oligarchic growth to complete, and
the final masses of oligarchicdy-grown protoplanets, throughout the Solar System.
The minimum surface density of soiids beyond the "snow line" at a
-
2.7 AU, obtained by
smoothly spreading out the high-Z mass contained in the pIanets, is
(Hayashi 1981). The corresponding gas nebula surface density is obtained by enhancing these solids
with hydrogen and helium to Solar abundance, and thus has the same density profile. The halfthickness of the gas disk is a a 5 / 4 thus
,
the midplane gas volume density is a a-3/2-5/4= a-l1I4.
The gas nebula of Hayasbi (1981) has a midplane density of
(a) = 1.4 x 10-~(a/l AU)-^-^' 9 / n 3 -
(3.10)
Assurning a protoplanet spacing of n = 10 Hill radii and a characteristic planetesimal mass of
I O - ~ M(corresponding
~
to a size of
-
10 km),estimated final masses and corresponding growth
times for this solar nebula are plotted in Fig. 3-1. Looking a t the upper panel alone, it appears
that oligarchic growth can produce bodies of roughly the size of Uranus and Neptune (within a
factor of 2) at their present locations. However, the tirne for oligarchic growth cornpletion already
reaches 108 years at 10 AU. The gas nebula lifetime is only on the order of 107 years; see Section
2.3.2. Furthemore, the final rnass reached between 5 and 10 AU is only 3 to 5 Me. This is unlikely
to have been enough for the nucleated gas instabiIity necessary to accrete Jupiter and Satum's
envelopes. Mergers among some of the endproduct protoplanets subsequent t o the termination of
oligarchic growth would likely have been necessary.
The actual protostellar nebula is likely to have been more massive than the minimum-mas
estimate. The surface density of solids, after dl, is calculated under the assumption that planet
formation proceeded with perfect efficiency, with all available planetesimals being incorporated
3. Accretion in the trans-Sat-an
region
24
Figure 3.1: Oligarchie growth final masses (top panel) and corresponding gronrth times (bottom panel) for
the SoIar System beyond 2.7AU (the snow Iine) for the planetesimai and nebular gas densities
given by Eqs. 3.9 and 3.10.
into the planets- This assumption is aimost certainly false (see Section 3.5). Many estimates
for the protoplametary disk are considerably higher; for instance Lissauer (1987) cites a value of
15 - 3 0 g / n 2 or more at the location of Jupiter as necessary to accrete the Jovian core before
dispersal of t h e nebular gas, while Pollack e t al (1996), modeling concurrent accretion of gas and
soIids, find t h a t a density of
in the outer Solar System works best for producing Jupiter and Saturn. This is 3.7 times the
minimum-mass surface density at 5 AU, The corresponding gas nebula should therefore have a
3. Accretion in the tran&aturnjan
1O1O
t
&
6
5
,
a
10
,
s
, ,
,
,
, ,
,
,
a
20
Semimajor axis (AU)
15
,
25
region
a
,
,
25
, , , ,
I
30
Figure 3.2: Growth times for 1 Me,10 Me and 20 Me bodies in the SoIar System beyond 5 AU for the
planetesimai and nebular gas densities given by Eqs- 3.11 and 3.12
midplane volume density of 3.7 times its value at 5 AU: and a density profile oc
= a-I3j4:
From Eq. 3.8, t h e final m a s of protoplanets is then 20 Me,independent of heiiocentric distance.
Due to irnperfect accretion efficiency, final masses were likely srnaller than this- T h e time to reach
20, 10 and 1 Me as a function of heliocentric distance is plotted in Fig. 3.2- One can see that for
this nebula, bodies of mass
-
10 Me can f o r n in 5 Myrs at 5AU,and in 40-50 Myrs a t I O AU.
Thus the solid core of at least Jupiter may f o r a directly through oligarchie growth, wïthout the
need for subsequent mergers. Growth times will also be further shortened by the enhancement of
3. Accretion in the trans-Satumian region
10
15
20
Semimajor axis (AU)
25
26
30
Figure 3.3: Protoplanet masses at different tirnes as computed £rom Eq. 3.6, using the planetesimal and
gas densities given by Eqs. 3.11 and 3.12, and phetesimal masses of IO-' Me (10 k m bodies
for a dençity of 1.5 g / c m 3 )
the protoplanets' physical cross-section by (pre-gas runaway) accretion of a gas envelope once the
bodies approach a mass of 10 Me.
The relatively short growth times obtained here are possible because the drag force exerted
by the gas nebula on the planetesimals keeps the planetesimal disk dynamically cold even in the
presence of large protoplanets. Once the gas disperses, accretion rates will drop significantly (see
Section 3.5 below). Thus, it is of central importance to determine how far accretion has progressed
by that tirne. Fig. 3.3 plots the protoplanet masses as a function of semimajor axis for the above
nebula (Eqs. 3.11 and 3-12)' as @en by Eq- 3.6. Giant planet core-sized bodies accrete out to
3- Accretion in the trans-Satumian region
27
Figure 3.4: Protoplanet masses at different tirnes, computed using the planetesimal and gas densities given
by Eqs- 3.13 and 3 . 1 4 -
-
10 AU in a few 107 years. However, even after 108 years-weli
to have dispersed-10
Me bodies have formed only out to
-
after the nebular gas is likely
13AU, and a t 20 AU, protoplanets
have only grown to a fraction mfan Earth mass. Once again, therefore, Uranus- and Neptune-sized
bodies cannot form a s oligarchïc growth endproducts before the nebular gas disperses. If they are
to have formed in-situ, it seems that mergers among protoplanets are required.
Does this result change if a n e considers a more extreme case? Adopting the upper Iimit of
Lissauer's (1987) estimate for tfie primordial planetesimal surface density at the location of Jupiter
3- Accretion in the trans-Satumian region
and a profile proportional to
28
one has
This constitutes a very massive disk, containing over 500 Me between 5 and 30 AU. Increasing the
gas midplane density above t h e minimum value by the same factor of 30/2.7 = 11.1 yields
Protopia.net masses at different times are computed for these densities, and are plotted in Fig- 3.4.
Even with this rather implausibly massive nebula, it takes between 2 and 5 x 10' years to
oligarchically accrete Uranus at its present location.
At Neptune's location, 108 years is just
sufficient to form a 10 Me body. If, however, the planetesimal characteristic mass is reduced fiom
10 k m to 1 km (10-l2 Me),
it becornes possible to reach between 10 and 20 Me at 30 AU in 5 x IO7
years.
Collisions among planetesimals
3.4
In the oligarchic regime gravitational interactions among planetesimals can, by definition, be neglected. But what about physical collisions between planetesimals? Ida and Makino (1993) and
Kokubo and Ida (2000) neglect these too, but it is worthwhile estimating the effect they might
have. The collision timescale is given by Eq. 2.32:
where n is the nurnber density of bodies, A is the collision cross-section, and v,r the bodies' relative
velocity. Since in the oligarchic regime planetesimal velocities are large compared to their surface
escape velocities, there is little gravitational enhancement of the interaction cross-section, and so
A
= ~ ( 2 r ) lUsing,
.
&O,
written as
v,.r
zz
&hm= *euK, and n = o/mh = c l m ( 2 a i )
u/me, this can be
3. Accretion in the tram-Satm-an region
29
Figure 3.5: The timescale for collisions, T c o ~ i among
,
1 and 20 k m planetesimals, using the planetesimal
surface density of Eq. 3.11.
Thus the collision timescale is independent of the random velocities. A plot of Tc,lrversus semimajor
axis for planetesimal radii r = 1and 10 km, a planetesimd density of 1.5 g/cm2 and the planetesimal
surface density of Eq. 3.11, is given in Fig. 3.5- It shows that, depending on the heliocentnc distance
and the planetesimal size, planetesimals may undergo a large number of collisions during the
-
107
year lifetïme of the gas nebula. Since relative velocities are well in excess of the planetesimals'
surface escape velocities, these collisions will predominantly result in fragmentation of the bodies
involved.
In the presence of gas, smaller pIanetesimals means greater random velocity damping and hence
increased rates of accretion ont0 the protoplanets. With planetesimals receiving more collisional
3. Accretion in the tranç-Satumian region
30
processing at smalIer heliocentric distances, this steepens the nse of protoplanet growth rate with decreasing distance even more. But since the growth rate eueryulhere-including
region-is
the tram-Saturnian
potentidy increased, could this effect d o w the formation of Uranus and Neptune in-situ
after au?
Without resorting to a detailed fragmentation model, which is beyond the scope of this thesis, one c m nevertheless obtain some constraints. What if colliçional fkagrnentation were able to
produce bodies in the trans-Saturnian region that are, say, 10 m (10-l8Me)in size? At relative
velocities high compared to their surface escape vefocities, collisions arnong pIanetesirna.1~wiii likely
be an ineffective mechanism for damping random velocities- Wethedl and Stewart (1993) perfonn
statistical simulations, calculating the time evoiution of the m a s spectrum of iaitially equal-size
10 km objects. Their work takes fiagmentation into account, and they find gas drag to be the
dominant velocity damping mechanism for collision fragments. However, since their calculations
are perfonned a t 1AU, and terminate when the largest bodies are only 0.02 Me, caution should be
used in extending their results to the growth of bodies of several Ma in the outer Solar System.
An understanding of the role of planetesimal coIlisions in this regime wiU require a new model.
Proceeding under the assumption that gas drag continues to be the dominant source of eccentncity and inclination damping, one can use the same approach as in Section 3.3, this time with IO
m planetesimals. Protoplanet mass versus semimajor axis at 10 Myrs for the two different nebula
models, cdculated using this planetesimal size, are plotted in the top panel of Fig. 3.6. In the
moderate nebula, oligarchie growth produces 10Me protoplanets out to only about 13AU. The
massive nebula forms them a t up to 30AU. -
However, if the planetesimals r e d y were coIlisionaUy ground down to 10 m-sized bodies, this
would raise a serious problem. Since the collision timescale decreases with heliocentric distance,
planetesimals at smaller distances would be ground down to even smaller sizes. The bottom panel
of Fig. 3.6 shows the semimajor axis decay timescale of a 10 m body on a circular orbit due to
3- Accretion in the trans-Satumian region
31
ProtopIanet masses versus semimajor axis at 10 Myrs, computed for 10m
gas densities of Eqs.
3.11 and 3.12, while Nebuia 2 has the very high densities of Eqs. 3.13 and 3.14. Bottom panel:
semimajor auis decay timescaie of a 10 m body, using q =
and the gas density of Eq. 3.12.
Figure 3.6: Top panel:
(IO-la
Me)planetesimals. Nebuia 1 has the moderate planetesimai and
the sub-Keplerian rotation of the gas disk, cdculated using Eq. 2.28- The parameter 7 is taken to
be 1 0 - ~
(Adachi, Hayashi and Nakazawa 1976). The timescale is less than 10 Myrs for a < 8 AU.
Thus 10 m planetesimals would get Iargely cIeared from the Jupiter-Saturn region (and everywhere
interior) during the gas Metirne. Since the migration timescale increases with semimajor axis, and
the surface density decreases, the depleted planetesimals wouId not be effectively replenished from
further out. Thus it is likely that whatever collisions took place, the characteristic planetesimal
size in the tram-Saturnian region remained well above 10 m.
3. Accretion in the trans-Satwân region
3.5
32
Post-gas dispersal accretion
Once the nebular gas is gone, planetesimal eccentricities and inclinations wiJl increase. Under
the assumption that physical coUisions among planetesimals are an ineffective velocity damping
mechanism in the oligarchie regime, the upper limit will be the escape velocity from the surface of
the locally dominant protoplanet. It is usefd to compare the escape velocity from the Sun and the
escape velocity from the s d a c e of a body as a function of heliocentnc distance, From a protoplanet
of mass M , density p and radius
R,the escape velocity is
Setting this equal to the escape velocity at a distance a fiom the Sun, ,.v
=
dm,
one
obtains
MW' ( 4 ~ ~ / 3 =
) ~&/a.
/ ~
One can therefore define, as a function of heliocentnc distance, a critical protoplanet mass Mmit:
For protoplanets exceeding this limit, one expects planetesimal random velocities to be so high that
is plotted versus semimajor m - s in Fig.
m a s loss through planetesimal ejection takes place. Mcrit
3-7, using the usual density of 1-59.At 10 AU,
is 4 hle, and at 20 AU, it is only 1.4Me.In
the absence of nebular gas, one can therefore predict that well before Uranus- and Neptune-mass
bodies-;5
10 Ma-form,
the trans-Satumian planetesimal disk will have high eccentricities and
inclinations, and perhaps a significant mass loss rate through ejections, all of which will act to
throttle the growth rates of the existing protoplanets.
3.6
Simulations
As mentioned in Section 2.3-2, modehg the process of accretion in the region of Jupiter and Saturn
in detail is difficult; in the trans-Saturnian region it is more difFicult still. However, judicious use
3. Accretion in the trans-Satumian region
Figure 3.7: Cnticd mass hf-it, as defined in Eq. 3.18, versus semimajor
1.5 g
33
for a prot
of simplifications can make such simulations tractable, while a t the same time preserving enough
of the physics of the process to make the results rneaningfui. Most of the simplifications which
will be made here will stack the odds in favour of producing large protoplanets, relative to more
realistic conditions. Therefore, if these simulations still fail to produce Uranus- and Neptune-mass
objects, it will be a strong indication that the conventional mode1 of in-situ formation of the ice
giants needs to be revised.
3.6.1
Initial conditions
Three simulations are performed, two using the moderate planetesimal and gas densities of Eqs.
3.11 and 3.12 (Runs A and B), the other using the extreme values of Eqs- 3-13 and 3.14 (Run
3- Accretion in the trans-Satumian region
34
C). Runs B and C simulate t h e trans-Saturnian region, while Run A simulates the Jupiter-Saturn
region. Run A is designed as a test of the analytic growth timescale estimates made above- During
the probable lifetime of the g a s nebula (- IO7 years) the trans-Saturnian region remains largely
in the "tail" of the protoplanet m a s cuve for a moderate planetesimal and gas disk (Fig. 3.3).
Thus, a simulation in this region wil1 not be ideal for testing the semimajor a~Üsdependence of
the protoplanet masses unless one extends the simulation to longer than the gas nebula lifetimeHowever, significant accretion is predicted to take place in the region around 5 AU in only a few
Myrs, and so Run A should previde a good assessrnent of how weli the timescde estimate works.
The first simplification is in the planetesimals used. To prevent the computational expense fiom
being prohibitive, the planetesimal disks must be made up of bodies much Iarger than a realistic
characteristic planetesimals s i z e (- 1 to 100 km). Planetesinial masses of 0.02 Mg,a r e adopted
for Run A, and 0.1 Me for R u n s B and C- The planetesimals are given a density of 1Sg/cm3,
and thus have radii of 2700 and 4600 km. The planetesimals are treated as a non-self-interacting,
c'second-class" population by t 3 e integrator (see Appendix B). This prevents viscous self-stirring
of the planetesimal population; given their large masses, this would result in unrealistically high
eccentncity and inclination growth rates (see Section 2.2.2). Of course, this way there is n o self stir-
ring a t d l , but since t h e oligarchic growth r e g h e is being modeled, where the effect of protoplanets
dominates the random velocity evolution, this is a reasonable approximation. An important added
benefit of a non-self-interacting planetesimal population is that computation time scdes linearly
instead of quadratically with planetesimal number.
Severd protoplanets are plaxed within the pIanetesimal disk. The protoplanets are 'Yirst-classn
bodies, fully interacting with each other and the planetesimals- Also, they are the only bodies able
to merge with other bodies (either planetesimals or each other). Thus the simulation only permits
an oligarchic growth mode; o d y the protoplanets can gain mass. A uniform initial protoplanet m a s
of 0.2 M* is used in Run A, a n d 1Me in Runs B and C . This is not a realistic initial condition- For
3- Accretion in the tranç-Satumian region
5
10
15
20
Semimajor axis (AU)
25
35
30
Figure 3.8: Protoplanet masses at different times as computed fiom Eq- 3.6 with MO = 1 Me,usïng the
planetesimal and gas densities given by Eqs. 3.11 and 3-12, planetesimd masses of IO-' Me
(10 km bodies for a density of 1.5 g/cm3)
instance, fiom Fig. 3.3, it takes -- 10 Myrs to Grst reach 0.2 Me a t 10 AU. However, a protoplanet
mass significantly larger than that of the planetesimals is necessary in order for dynamical friction
to be effective on the embryos. Numencal experirnents show that an order of magnitude mass
difference between the two populations is necessazy in order that the protoplanet eccentrïcities
and indinations are kept reasonably low. In Run A, the protoplanets are distributed between
6 and 15 AU initially, while in Runs B and C, they are placed from 10 to 35 AU. In aJi runs,
successive protoplanets are spaced approximately ten H
ill radii apart. How does one expect the
growth timescales of the protoplanets to be affected by their larger initial masses? Fig. 3.8 plots
3. Accretion in the trans-Satumian region
36
protoplanet masses over time using the same parameters as used for Fig. 3.3 but with an initial
mass of 1 Me instead of zero. As one would expect, the steep decrease of protoplanet mass with
semimajor axis remains, with the mass now asymptotically approaching I Me instead of zero- The
times to reach masses above 1Me at a given semimajor axïs are decreased, but even so, in 108
years 20 Me objects are stiiI only expected to have formed out to
-
16 AU.
The large mass of the pIanetesimals relative to the protoplanets will also have an effect on the
merger rates. The 1 Me protoplanets have radii of 9800 km, whïch is sufEcientIy large relative to
10 km planetesimais that the radu of the latter can be neglected. However, the 0.1 Me "superplanetesimals" have radu of 4600 km, almost half as large as the protoplanets. In that case Eq2-10 must be rnodified by changing RL to
(RM+ &)2.
Therefore the mass growth rate d M / d t
is increased by a factor of about two for 1 Me protoplanets, and less than this for larger ones. So
this approximation also favours accretion.
One important difference between the analytic results and the simulations is in the surface
density of planetesimals- It is assumed to remain constant at its initial value in the above calculations, but in the simulations severaI factors cause it to decrease over time. First, planetesimals are
depleted by accretion onto protoplanets- This mass of course remains in the disk so, in an average sense, the density remains the same. Therefore depletion of planetesimals by accretion simply
increases the importance of the protoplanet-protopIrnet accretion mode relative to the protoplanetplanetesimal mode. Indeed, protoplanet-protoplanet mergers do occur in the simulations, but this
growth mode is dearly slower than planetesimal accretion; the runs below show protoplanet growth
rates dropping relative to the predicted values as planetesimals become significantly depleted in
their vicinity. This is to be expected, since the orbital repulsion mechanism mentioned in Section
2.2.4 wiII tend to keep protoplanet orbits £rom crossing stronglyViscous stirring by the protoplanets causes the planetesimai disk to spread over time, and thus
d s o to decrease in density. Ln Fig. 3.13 below, it can be seen that the disk's outer boundary
3. Accretion in the trans-Sottunianregion
has moved from 35 to
-
37
45 AU over 5 x 107 years, The change in location of the inner disk edge
is diacult to assess because of the inner simulation boundary at 7AU. A sigdicant fraction of
the protoplanets are eliminated in the course of Runs B and C when their perihelia &op below
this boundary. It can be argued, however, that this mimics the effect of a growing Jupiter and
Saturn interior to 10 AU, which wodd be accreting or ejecting most of the material entering their
region. In the case of Run A, the inner boundary is far enough in (3 AU) that no planetesimals are
lost, but an accumulation of planetesimals at the inner bound;trv due to gas drag orbital decay is
clearly visible. In reality, one would expect bodies to have formed even earlier further in- These
bodies would interact with inward-migrating planetesimals, scattering some of them further out. Of
course, making this assumption in the Jupiter-Satu region amounts to asserting that planet-sized
bodies exïsted in the asteroid belt a t this time. Asteroid belt phnets, in tuni, rnay help explain
this region's m a s depletion and dynamical excitation, assiiming such bodies c m Iater be removed
(Section 5.1).
There is no artificial resupply of planetesimals in the simulation. There is, however, a net decay
of planetesimds due to the gas drag, as given by Eq. 2.28, which partially replaces planetesimals at
smaller heliocentric distances. The parameter q, the fractional merence between the disk rotation
speed and Keplerian velocity for a given heliocentric distance, is of order 1 0 - ~ (Adachi, Hayashi
and Nakazawa 1976). RMS eccentricities are
planetesimals- Thus in the former case, 7
2 0.1 for 10 k m planesimals, and 2 0.05 for
1 km
< e, e2 and so planetesimal orbits decay primarily due to
energy loss from damping of the random motions, rather than through headwind from the gas disk.
In the latter case, the effects are comparable. In the simulations, the disk is modeled z s rotating at
exactly Keplerian speed. This simplification should have little &ect on the planetesimal migration
rate of 10 km planetesimals, but wiU reduce the migration rate of I km platesimais by a factor of
-
2/3.
Finally, the simulations do not inchde the effect of fragmentation. Without a detailed model,
3- Accretion in the transSatumian region
38
it is not clear how this wiIl cause the characteristic pIanetesimal size to evolve. Not including
fragmentation d l overestïmate planetesimal sizes and hence underestimate accretion rates to some
degree. However, as outIined in Section 3-4, the characteristic planetesimal size is uniikely to have
decreased to the point where it would play a deciding role in determining whether or not 10 Me
objects can form a t the location of Uranus and Neptune.
3.6.2
The Jupiter-Saturn region
The simulation in the Jupiter-Saturn region is only nin to 4 Myrs; the srnail timestep and large
number of bodies used make it computationally expensive. The state of the simulation a t 1 , 2 and 4
Myrs is shown in Figs. 3.9,3.10 and 3-11.As expected, accretion proceeds from inside to outside in
semimajor axis. The transition between the accretionaily active and accretionaily largely inactive
region moves outward at close to the rate predicted analyticatly. The growth of protoplanets at the
inner edge of the simulated zone is not as rapid as predicted; the largest protoplanet is only 1.2 Me
at 4 Myrs, and two other bodies of 1Me or larger have formed by this tirne- At the same time, it
is apparent that a lot of pianetesimals are accumulating in the asteroid belt region, out of reach of
any accreting protoplanet. This is an edge effect of the simulation; in the reaI system, protoplanets
in the asteroid belt region would have either accreted these pIanetesima.1~or eventually scattered
them back out, making them a d a b l e in the Jupiter-Saturn region again.
Interactions among protoplanets serve to somewhat blur the predicted sharp falloff in protoplanet mass with semimajor e s - Orbital repulsion will tend to make the larger protoplanets
push other protoplanets away. This appears to be taking place in Run A; between 2 and 4 Myrs,
the largest and the second-largest protoplanet increase their separation, with the latter ending up
several AU outside its original semimajor axis-
3- Accretion in the tram-Satumian region
39
Figxre 3.9: The state of Run A at 1 Myr. From top to bottom, the panels show eccentncity, inclination
and maçs v e m semimajor h.The bottom panel plots the protoplanet mas distribution
predicted by Eq. 3.6 with an initial mass of 0-2 Me.
3. Accretion in the tram-Satm-an region
Figure 3.10: The d a t e of Run A at 2 Myrs.
40
3- Accretion in the tram-Saturnian region
r-
1
I
L
I
I
,
b
1
I
r
,
41
l
,
l
-
-
Figure 3-11: The state of Run A at 4 Myrs.
3. Accretion in the trans-Satumian region
42
Figure 3.12: The state a t 1 x 10' years of Run B, which takes the planetesimal surface density given by
Eq. 3.11 as the initial profile of the planetesimal disk- The disk inïtially extends from 10 to
35 AU. Gas drag on the planetesimals is calculated assuming a planetesimal radius of 10 km
and a gas density a s given by Eq. 3.12. The three panels show eccentncity, inclination and
m a s versus semimajor a i s . The line in the bottom panel shows the protoplanet mass profile
predicted by Eq. 3.6 with an initial m a s of 1 Me-
3.6.3
The trans-Sa t urnian region
The state a t 107years of Run B, using the moderate planetesimal and gas densities of Eqs. 3.11 and
3.12, is shown in Fig. 3.12. The state at the end of the run, at 5 x 107 years, is shown in Fig- 3-13.
Very little accretion has taken place anywhere a t 107years. By 5 x 107 years, as predicted, there still
has been only minimal accretion beyond
-
25 AU. The largest body, a t 22AU, has a mass of only
2.8 Me.It is at a somewhat larger heliocentric distance than predicted, but the stochastic nature
of the accretion process makes such variation inevitable. Furthermore, as noted above, interactions
3. Accretion in the trançSaturnïan region
43
Figure 3.13: The state at 5 x 107 years of Run B
among the protoplanets act to displace them from their original location, further washing out the
predicted mass profile.
AIso of note is the absence of the large bodies predicted near 10 AU. The above-rnentioned
depletion of pIanetesirna.1~a t the inner boundary contributes to this, as does the lack of resupply
t o replace accreted planetesimals.
Since Run C contains more than six times as many planetesimals as Run B, it runs more
slowly by about the same factor, plus additiond tirne due to the larger number of (computationally
intensive) close encounters. Run C is therefore only taken to 1 x 107 years- Its state a t this time
is plotted in Fig. 3.14, together with the analytic estirnate of the protopla.net mass profile at this
tirne. The largest protoplanet which has formed by this tirne is 7.8 Me in m a s ; the second-largest
3. Accretion in the trans-Satumian region
44
Figure 3.14: The state at 1 x 107 years of Run C, which takes the planetesimal surface density given by Eq.
3.13 as the initial profile of the planetesimai disk, which extends from 10 to 35AU. Gas drag
on the planetesimals is calculated assuming a planetesimal radius of 1 km and a gas density as
given by Eq. 3.14. The three panels show eccentricity, inclination and m a s versus semimajor
axis. The Iine in the bottom pane1 shows the protoplanet mass profile predicted by Eq. 3.6
with an initiai m a s of 1 Ma.
is 6 Me. The protoplanet masses are quite close (though a bit higher on average) to what is
anaiytically predicted beyond 20 AU, except for the outermost one. Interior to 20 AU, the masses
are smaller than predicted. This appears, again, to be a consequence of the disk's erosion at the
inner simulation boundary. Since the protoplanets have grown to larger masses than in Runs A and
B, orbital repuision is stronger overail, and the protoplanet mass profile is even more washed-out
relative to the analytic prediction. For instance the o u t m o s t protopIanet, at 43 AU, which exceeds
the predicted mass at that heliocentric distance by a factor of 2.5, has migrated to its location after
3. Accretion in the trans-Satumian region
Figure 3.15: The state of Run B &er
45
i t has been run for another 1.5 x 10' Myrç without gas drag.
forming further in-
3.6-4
Simulating post-gas dispersai accretion
Without eccentricity and inclination damping by gas drag, one expects the accretion rate to s h q l y
decrease (Section 3.5). To test this assumption, Run B is resumed from its endpoint at 50 Myrs, to
2 x 108 years, but without gas drag. The endstate is shown in Fig. 3.15. As expected, eccentridties
and inclinations increase; both are up by a factor of h o by
log years.
Only two mergers (both of
a planetesimal with a protopIrnet) take place, as compared to 23 over the &st 50 Myrs, and none
occur after 75 Myrs. Therefore, under that assumption that the planetesimal velocity evolution due
3- Accretion in the trans-Satumian region
46
to inter-planetesimal collisions is negligible in the oligarchic regime, it appears that no significant
accretion can take place after the dispersal of the nebular gas-
3.7 Discussion
The simulations produce protoplanet growth rates that are reasonably consistent with the estimates
derived from the approximate analytic description of oligarchic growth summarized here. They
reproduce the inside-out growth which the theory predicts, with an outward-sweeping '%aven of
accretion which moves at roughly the predicted rate. At the sanie t h e , the simulations reveal
oversimplifications in the way the estimates are calculated. First, the theory assumes a constant
surface density, whereas in the simulations it decreases with time; in the real Solar System, it
very likely also decreased, though planetesimal resupply £iom other regions may have been higher.
The resultant overestimation of planetesimal-protoplanet rnergers is somewhat mitigated by the
fact that the theory does not take into account protoplanet-protoplanet rnergers or fragmentationNevertheless, the effect is to decrease the growth rate of larger protoplanets, which have consumed
a significant fraction of the planetesimals in their vicinity, relative to what the theory predictsSecondly, the analytic estirnates do not take into account radial migration; they are made under
the assumption that protoplanets are h e d in semimajor axis as they grow. The simulations show
that in fact, protoplanets can undergo signiiicant changes in semimajor axis. Orbital repulsion
tends to make protoplanets move apart in semimajor axis, and the effect is stronger for larger
protoplanets. The result is that protoplanets tend to diffuse from srnalier heliocentric distances,
where many of them are growing rapidly in close proxkity, to larger heliocentnc distances- This
suggests a possibility for producing Uranus and Neptune; if one relaxes the assumption of in-situ
formation, the tirnescale problem can be made l e s severe- Indeed, Run C has a 8 M B body between
20 and 30 AU a t 10 Myrs. However, the simulation produces only once such body, and this is using
a protoplanetmy disk mass that is too high to be very credible- It appears, therefore, that the
3. Accretion in the trançSat-an
region
47
tram-Saturnian region is very unlikely to produce even a single body approaching the mass of
Uranus and Neptune during the probable lifetime of the gas nebula- And once the gas disperses, it
is likely that the lack of random motion damping makes subsequent accretion far too slow t o finish
the job even in the lifetime of the Solar System,
In sitmmary, then, the simulations produce protoplanet masses a t a given semimajor axis and
a given time which exceed the andytic estimate by a t most a factor of 2.5, and almost always
less than that, This is with some approximations ("super-planetesimals" with large cross-sections,
unreaLisitica.Uy large initial masses) which favor the formation of large protoplanets. One may
therefore conclude that the anaiytic estimates provide a plausible constraint, within a factor of
roughly 2, on the m a s of protoplanets growing in the oligarchie mode. The formation of Uranus and
Neptune in the transSaturnian region thus becomes a very doubtful proposition, and alternative
models need t o be considered.
4. FORhlING URANUS AND NEPTUNE AMONG
JUPITER AND SATUR,N
Available materiai
4.1
The minimum surface density of solids, an empincal result obtained by smoothly spreading out the
high-Z m a s contained in the planets, is given by Eq- 3.9:
hi,.,
(a)= 2.7(a/5
AU)-^/' g / n 2 .
(4-1)
However, a surface density of 2.7g/cm2 was likely too low to form the solid core of Jupiter in a
time short enough (< 10 Myrs) th& it could have reached the size necessary to initiate runaway gas
accretion (- 10 Me) before the gas was removed by the Sun's T Tauri phase winds. As outlined
in the previous chapter, Lissauer (1987) fbds that a surface density of 15 - 30 g/cm2 is needed to
allow formation of Jupiter's core on a timescale of 5 x NI5- IO6 years, whiie the mode1 of Pollack et
a1 (1996), which inchdes concurrent accretion of solids and gas, produces Jupiter in less than 107
years with 10g/cm2- Outward dinusion and subsequent fkeezing of water vapor from the inner Solar
System may have resulted in a large IocaI density enhancement around 5 AU, yielduig a surface
density even higher than 30 g / n 2 (Stevenson and Lunine 1988)- A power law surface density
with 10g/cm2 <
-
00
5 3 0 g / m 2 and
1
5
cr
5
2 gives a total mass in the Jupiter-Saturn (J-S)
region in excess of 40Me,and as high as 176Me (Table 4.1). Therefore, it is likely that the region
originally contained significantly more solids than ended up in Jupiter and Saturn's cores.
4- Fodnn
Uranus and Neptune amona Jupiter and Saturn
49
Table 4-1: Total mass of soIids between 5 and 10 AU for different surface density profiles
4.2
Impfications of oligarchie growth
Oligarchic growth has previously only been demonstrated to take place interior to about 3AU
(Weidenschilling and Davis 2000). Though Kokubo and Ida make estimates of protoplanet mass
and growth tirnescales in the giant planet region, they point out that their simulations are restricted
to annuli which are narrow compared to their radii, and thus cannot make strong predictions
about how oligarchic growth works over a wide range in semimajor axis (Kokubo and Ida 2000).
However, their most recent simulations (Ida and Kokubo 2000) span a range of 0.5 to 1.5 AU, and
show oligarchic growth proceeding in an outward-expanding '%aven over tirne, consistent with the
analytic estirnates gîven in Chapter 3, The simulations in that chapter aiso show this behaviour, in
the outer Solar System and over a much Ionger range in a. Of course, those simulations are biased in
favour of oligarchic growth because they only d o w accretion ont0 protoplanets, and because they
begin with a set of fairly large protoplanets. However, they do show that once a set of protoplanets
has detached itself fiom the mass spectrurn of the planetesimals, they can continue to grow in an
oligarchic rnanner in the outer Solar SystemOligarchic growth therefore constitutes a plausible scenario for how accretion proceeds in the
outer Solar System, as weU- At the same time, i t gives analytic estimates for the growth timescales
of the two outer giant planets which are weli in excess of the expected lifetime of the gas nebula,
udess one assumes an extremely massive protoplanetary disk, And with the dispersal of the gas,
growing random velocities Iikely make the planetesixna3 disk in the outer Solar System so diffuse
that accretion is essentially halted. It is then difficult to understand the formation of Uranus and
4- Forming Uranus and Ne~tuneamona Ju ~ i t e and
r
Saturn
50
Table 4.2: OIigarchic-growth protoplanet masses for a = a,(a/5 AU)-^
Neptune on any tirnescaleModels of giant planet formation by concurrent planetesimal and gas accretion suggest a surface
density prome a a a-= (Pollack et et 1996). Substituting this into Eq. 2.46 for the final mass of an
oligarchically-grown protoplanet, one obtains a protoplanet mass independent of semimajor
&S.
This is consistent with the similarity of the high-Z masses of Jupiter and Saturn. Though Uranus
and Neptune's masses are similar to these, the above-mentioned timescale problem suggests that
simple oligarchie growth cannot account for their formation, at least not in-situ.
Table 4.2 shows protoplanet masses for different choices of surface density cro a t 5 AU , spacing
in HU radii n, and fraction of material accreted p. The threshold mass for runaway accretion of gas
from the soIar nebda is uncertain, but 10 Me is a plausible value (eg. Mizuno et al 1978, Pollack
et al 1996). Adopting
= 10g/cm2 and a spacing of 7.5 Hi radii yields a protoplanet mass of
10 Me when the accreted material fkaction, p, is 0.83-
The radii al and a2, ai
< (22, of two consecutive equal-mass (= M) protoplanets' orbits spaced
by n tirnes their mutual H
i
l
l radius as calculated midway between thern, are related by
where
4. Forming Uranus and Neptune among Jupiter and Saturn
51
TabIe 4.3: Successive orbital radii in the Jupiter-Saturn zone for 10 Me protoplanets spaced by nru,
is the reduced mutual
Hill radius (see Eq.
2.44).
Using t his prescription, one c m generate series of consecutive protoplanet orbit al radu- Table
4.3 shows some examples. As can be seen, three or more 10 hie oligarchicaliy-grown protoplanets
could have coexisted in the Jupiter-Satum region. It is likely, therefore, that this region originally
contained more than just the future solid cores of Jupiter and Satum.
4.3
Scattering of the protoplanets
After a period during which gas and planetesimal accretion rates are simiIar for a protoplanet,
runaway gas accretion sets in and proceeds on a timescaie of
-
10' years (Pollack et al 1996).
One of the protoplanets in the Jupiter-Saturn region must have been the k s t to reach this point.
Shorter formation timescales at smaller heliocentric radii argue for Jupiter having formed before
Saturn.
A body increasing its mass fiom
- 10Me
to Jupiter's m a s , M j = 318Me, expands its HiIl
radius (Eq. 2.4) by a factor of about three. The orbital stability of the adjacent protoplanets
will therefore be aEec-ted- As a result, their orbital elements may be significantly changed. This
would have important implications for planet formation in other parts of the Solar System, since it
constitutes a mechanism for delivering large 'teady-made" bodies away from their place of origin.
Given the difficulties in understanding the in-situ formation of Uranus and Neptune, as outlined
in Chapter 3, it is of particular interest to explore the role which protoplanets scattered from the
4. Forming Uranus and Ne~tunea m o n ~
J u ~ i t e and
r
Satuni
52
Jupiter-Saturn region codd have played here.
Onto what sort of orbit can Jupiter scatter a protoplanet? For simplicity one may consider only
Jupiter and a much srnailer protoplanet which does not affect the motion of Jupiter, both orbiting
the Sun, and approximate Jupiter's orbit as circular, with radius a,. The situation then reduces to
the so-called circula restricted three-body problem (eg. IvLurray and Dermott 2000). This problem
has one integral of motion, t h e Jacobi constant:
94 is the effective potential in the frame corotating with the body on the circular orbit, a t angular
veIouty R,:
where 9 is the potentid in t h e nonrotating frame- The second tenn is the potential arising from
the centrif'ugal force in the rotating kame. v is the velocity of the thud, massless body in the
corotating frame.
Using the fact that the Sun is much more massive than both of the other bodies, one can obtain
an approximation to the Jacobi constant, caLIed the Tisserand parameter:
Here a is the semimajor axis of the small body, e is its eccentricity, and i is its inclination. In a
scattering of the small body by the Iarge body on the circular orbit, the Tisserand parameter is
approximately conserved- After the body is scattered, its orbit rnust stiI1 cross that of the scatterer-
If the small body is scattered outward, this means that its perihelion must occur a t or interior t o
the orbit of the scatterer: q
< a,, where
4- for min^ Uranus and Nev tune amona Jupiter and Saturn
1O
20
30
40
50
53
60
Semimajor axïs (AU)
Figure 4.1: Perihelion distance q as a function of semimajor a x h a for three difFerent values of the Tisserand
parameter relative to Jupiter. The value of a corresponding to q = aJ = 5.2 AU (the upper
limit of the plot) is the furthest possible scattering distance. A body with T = 2& can be
scattered to infinity, since q asymptotes to a3 as a -+ oo.
Likewise, if it is scattered inward, its aphelion must occur at or exterior to the orbit of the scatterer:
Q 2 ac,where
Q = a ( I +e).
(4-9)
For planar (i = O) orbits just crossing the scatterer, where the above inequalities are replaced by
equalities, the Tisserand parameter in both cases reduces to
in the planar case. T thus takes on a maximum value of 3 (when a = a,) for any planet-crossing
body. The maximum semimajor axis to which Jupiter can scatter a body in this approximation,
then, depends on the Tisserand parameter of the body with respect to Jupiter before it is scattered.
4. Forming Uranus and Neptune among Jupiter and Saturn
54
This is iliustrated in Fig, 4.1. Perihelion distances are plotted as a function of semimajor aJas for
several different values of the Tisserand parameter. Bodies scattered outward by Jupiter must
end up on an orbit with q
< aJ. Thus the intersection of a curve with Jupiter's
orbit a t 5.2 AU
denotes the maximum semimajor axis to which a body can be scattered by Jupiter for a given
d u e of T. As can be seen, a smaller Tisserand parameter means a higher eccentricity, and hence
a smaller perihelion distance, for a given value of the semimajor f i s . A body with q = 1 and
a = w has T = 2&,
meaning that this d u e of T aliows a body to be scattered to infhity. A
body with T z 2.87 can be scattered to about the semimajor axis of Neptune. Assuming Jupiter
initiaIly has a mass of 10 Me and the next protoplanet's orbit is 10rH further out, at 6.3 AU, then
as Jupiter becomes massive, the protoplanet must acquire an eccentricity of 0.37 (q = 4.2 AU) to
reach T = 2.87.
Thus, a moderate eccentricity shouId be suficient to allow protoplanets to be
scattered to a semimajor axis comparable to that of Uranus and Neptune.
Fig. 4.2 shows an example of a SyMBA run containïng Jupiter and a test particle i n i t i d y
crossing the orbit of Jupiter, with a semimajor axis of 6.3AU. Scattering by Jupiter increases the
semimajor axis of the body to as high as about 17AU within 104 years. The Tisserand parameter,
initially 2.93, is not exâctly conserved by SyMBA, but it is bounded; there is no systematic growth
in error.
Of course, treating the scattering of protoplanets by Jupiter as a circular restricted three-body
problem is a major simplification.
As protoplanet eccentricities becorne high enough to allow
Jupiter-crossing, the protoplanets will also cross each other's orbits, so that energy and anguiar
momentum exchange among them will play an important role- Numerical simulation is the most
promising way to address a problem of this complexity.
4. Fom-ng Uranus and Neptune among Jupiter and Saturn
O
2000
4000
6000
55
8000
IO*
a000
104
Tirne (Years)
L
al
J
O
5 2.94
L
9.
-u
a
2.92
VI
i=
2-9
O
2000
4000
6000
Time (Years)
Figure 4.2: A test particle scattered by Jupiter. The top panel shows the evolution of its semimajor axis
(thick solid), perihelion (solid) and aphelion (dotted) over tirne- The bottom panel shows the
evolution of the Tisserand parameter
4.4
Circularization of a scattered protoplanet's orbit
If no other dynamical processes act, a scattered body continues to have close encounters with its
scatterer and so remains coupled to it, unless i t is scattered ont0 an escaping orbit and leaves the
system altogether. Its eccentricity and inclination will undergo changes on the timescale of the two
bodies' synodic period. Beyond the Jupiter-Saturn region, however, a scattered protoplanet will
encounter the planetesimal disk, consisting of bodies much m d e r than itself. As a result it will
expenence dynamical friction. This will tend to reduce the eccentricity of the protoplanet's orbit.
If the eccentriciîy decays enough, the protoplanet's periheLioo will be lifted away from Jupiter,
4. Forming Uranus and Neptune among Jupiter and Satum
56
time (years)
Figure 4.3: Evolution of the eccentricity of a 10 MB body (thick line) embedded in a disk of 0.2 ilfa bodies,
and the evolution of the RMS eccentricity of the surrounding 0.2 Ma bodies (thin line). The
semimajor axis of the 10 M*.body is approximately constant during this time, so that as
the eccentricity decays, perihelion and apheiion converge in an essentially rnirror-symmetric
manner.
thus decoupling i t £rom its scatterer. In the absence of other effects, such a s encounters with other
larger bodies, its eccentricity will then rnonotonically decrease until it reaches equilibrium witk the
planetesimals (Eq.2.20).
Fig. 4.3 shows an example of dynamical fi-iction in z t i o n in a simulation performed with
SyMBA. Initial conditions consist of a planetesimal disk with surface m a s density a
-
10(a/5AU) g/cm2
from 5 to 60 AU, made up of m = 0.2 Me planetesimals. A M = 10 Me body is placed on an orbit
with a~ = 15AU, e M o = 0.65, making q = 5.25- No other bodies are included in the run. Fig, 4.3
4- Forming Uranus and Neptune among Jupiter and Saturn
57
shows the eccentricity of the large body over 1.5 x 106 years, as weli as the RMS eccentricity of the
surroundhg part of the planetesimai disk. The eccentricity e~ drops to half its initial vdue on a
timescale of 1- 2 x 105 years.
One can compare this result to what is predided by the expression for the dynamical friction eccentricity damping rate, Eq- 2.22- The initial RMS inclination of the planetesimai disk is
=
-
(%)1/2 = 0.025. Thus the disk scale height is 2 ~ ( i : ) ' / ~ 1.1 x 1oL3cm-Over the radid extent of
the Iarge body's initial orbit, the average surface density is a
of the planetesimals is therefore p, = i ~ / 2 a ( i % )zz~1.2
/~ x
disk scde height as b,,,,
1.4g/cm2. The spatial m a s density
g / n 3 . Using Eq. 2.16, with the
one obtains A = 560. The Keplerian velocity at 15 AU is 7-7x 105cm/s,
-
With e M = 0.65, e, = 0.05, i~ = O, im = 0.025rad, Eq. 2.19 gives /3 = 0.04, From Fig. B1 of
Weidenschilling et al 1997, Ke(0.04) E 8. Thus, debo/dt
-2.8 x 106yr-'.
The time for the eccentricity to be reduced to half its initial value can be estimated as
.,,
which gives Tdf
cz 1.5 x 105 years. This value agrees well with the simulation result of between
I and 2 x 205 years (Fig. 4.3). It is important to note that the eccentricity decay timescale,
though proportional to the spatial mass density of planetesimais, is essentialiy independent of the
individuai masses of the planetesimais as Iong as
~ e >kme:.
4.5
N- body simulations
Putting the pieces together, one can now envision a scenario in which a) a rapidly growing Jupiter
scatters its smalIer neigbours outward, and b) these 'Yailed coresn are decoupled from Jupiter
and ultimately evolve ont0 circular, low-inclination orbits in the outer Solar System- Can such a
scenario reproduce the configuration of the present-day Solar System (Fig. 4.4)? This question will
be addressed using numerical simulation.
U m u s and Neptune among Jupiter and Saturn
4. Fo&g
O
1O
20
30
Semi-major axis (AU)
40
58
50
Figure 4.4: Eccentricities (top) and inclinations (bottom) in the outer Solar System at the present epoch,
showing the giaot planets and those Kuiper belt objects which have been observed at multiple
oppositions. The c u v e in the top panel shows the locus of orbits with perihelia at the semimajor
axis of Neptune.
4.5.1
Initial conditions: Set 1
For the first set of simulations, a planetesimal disk of surface density
is used. The disk extends from 5 to 60 AU. The truncation at GO AU is to keep the number of
bodies in the simulations tractably small; in reality the planetesimal disk may have extended for
hundreds of AU. The part of the planetesimal disk beyond the present-day orbit of Neptune is
referred to as the Kuiper belt, and stili exists. Beyond
-
50 AU, it has Iikely not evolved far f?om
4- Forming Uranus and Neptune among Jupiter and Saturn
59
its primordial state. (for a review, see for example Weissman 1995).
The simulated disk is made u p of equal-mas "planetesimals", each having a mass of 0.2 Ma.
At tmîce the mass of Mars, these bodies far exceed the actual characteristic mass of planetesimals
in the early outer Solar System- Zn reality they are fikely to have been on the order of 1to 100 km
in size, thus with a mass of
-
10-l2 -
Me.The unrealistically large masses are chosen, again,
to keep the number of bodies manageably low. As mentioned in Section 4.4, when the large bodies
have eccentncities high enough t h a t Eq. 4-12 is satisfied, the eccentricity decay tirnescale will be
effectively independent of the planetesimd masses. Thus the large masses used in the simulation
will not produce unphysical timescales initially. The equilibrium eccentricity condition Eq. 2.20,
however, does depend on the planetesimal rnass, so the equilibrium eccentriciw reached by a large
body among the planetesimals in t h e simulation will be unrealistically high.
The eccentricities and inclinations of the planetesimals have Gaussian distributions, as given by
2.1, with
(e2)'j2
= 0.05,
velocities ranging from
(i2)li2
-
= 0-025 = 1.4O. This yields a moderately excited disk, with random
9 k m / s at 10AU to
-
4 k m / s at 60 AU, Such velocities correspond to a
disk which is gravitationally stirred by bodies of up to
-
1Me-
In the numericd integration, the planetesimals are treated as "second-class" bodies by SyMBA
(see Appendix B). Thus they do not interact with each other. This serves the same two purposes
as in the numerical simulations of Chapter 3: First, it makes the simulations run much faster, since
for N second-class bodies, the computation t h e scales a s N instead of N 2 - Secondy, it prevents
unrealistically strong self-stirring o f the disk, since the viscous stirring rate is also dependent on
the planetesimai mass (Eqs. 2.13, 2.14)-A disk made of interacting usuper-pIanetesimals" would
therefore have artificidy high RMS eccentricities and inclinations. Of course, not modeling interactions means that collective planetesimal effects are not accounted for- Wave phenomena could
have had an important effect on the evolution of the planetesimal disk velocity distribution (eg.
Ward and Hahn 1998), provided it was sufficiently massive and dynamically cold. However, it is
4. Forrmhg Ui-anusand Neptune among Jupiter and Satum
60
unlikely that significant wave phenomena could persist once the initial scattering has taken place
and the planetesimal disk has been stirred by eccentnc 10Me bodiesThe 10Me bodies are assumed to have a densiw p = 0 . 2 5 ~=~ 1.5g/cm3, roughly equal to
that of Uranus, Neptune and cornets- This gives them radii of 2.18 x IO4 km- Four such bodies are
put in the simulation, initially on nearly circuiar and uninclineci orbits. The orbits are spaced by
7.5 mutual Hili radii according to Eq. 4.3, starting fiom an innermost distance of 6 AU to d o w
for inward migration by Jupiter, once it has formed (see Section 4.5.2). Thus the bodies' initiai
semimajor axes are 6-OOAU, 7.37AU, 9.04AU and 11-1OAU. Between 5 and 12 AU, the disk is
depleted in planetesimals so that the surface density is still given by Eq. 4.13. Since the large
bodies are spaced proportionally to their semimajor axes, their distribution is consistent with a
surface density a a-2:
AM = 2.iraoAa,
so for a fixed AM-in
(4.14)
this case, AM = 10 Me-
Integrating fkom 5 t o 12 AU, the total mass is 51.5 M@-With 4 x 10 = 40 Me of this in the large
bodies, this Ieaves 10.5 Me in planetesimals in the Jupiter-Saturn region, in addition to 94.7 Me
beyond 12 AU, In siimmary, then, the initial conditions amount to a state where
-
80% of the
planetesimals between 5 and 12AU have accreted through oligarcbic growth into four bodies of
10 Ma each, while no large bodies have yet formed beyond this region (Fig. 4.5). The adoption
of equal-mass protoplanets is a simplïfkation; from the results of Chapter 3, one can expect some
intemediate-mass bodies, of perhaps up to a few Me, among the oligarchie-growth endproducts.
Such bodies are likely to ultimately be cleared, dong with the planetesimals, from the giant planet
region. However, they may end up playing a role in the dynamics of the tram-Neptunian region;
4. Fo-ng
Uranus and Neptune among Jupiter and Saturn
61
Figure 4.5: The initial date for runs in Set 1, showing eccentriciQ (top) and inclination (bottom) versus
semimajor axïs. The larger soüd circIes denote the four 10 Me protoplanets, and each of the
mall empty circles represents a 0.2 Me planetesimal. The planetesimai density in the viciniv of
the protoplanets is decreased to keep the density of protoplanetss plus pimetesimais consistent
with the surface density gîven by Eq. 4.13.
see Petit, Morbidelii and Valsecchi 1999, and Chapter 5.
The imer simulation radius is chosen as 1A& any body whose orbit penetrates this boundary
is elirninated Tom the systern. The base timestep is chosen as 0.05 years, giving 20 steps per orbital
period for an orbit with its semimajor axis at the inner radius, and over 200 steps per orbit for
Jupiter. Ekperimentation shows that this timestep is srnall enough that the energy of the system is
well-conserved. Runs initiaily go to 5 Myrs; in cases where the system stili appears to be undergoing
rapid evohtion a t this point, the runs are extended by another 5 Myrs.
4. Forming Uranus and Neptune among Jupiter and Saturn
62
The next consideration is to simulate the process of m a w a y gas accretion. One of the bodies
must be the first to reach this stage. Though the four protoplanets are initially of identical mass
in the sùndation, in reality this would of course not have been the case- In addition to stochastic
variations in accretion rate a t a given radius in the planetesimal disk, the oligarchie growth timescale
varies considerably over the extent of the Jupiter-Saturn region. Eq. 3.7, with cr, oc aA2,bas cc
a-13/4
and simïiar final masses M, gives
Thus between 5 and 10 AU, tg,,, increases by a factor of more than 9, and so the inner protoplanets
will reach a mass of
-
10 Me e s t - At this point they have depleted the majority of the pIanetesimal
m a s in their respective feeding zones, and they enter the second phase of giant planet core growth,
as described in Section 2.3-2; the rate of planetesimal accretion becomes low, and gas is accreted
at a comparable rate. The second phase lasts for on the order of 107 years, similar to the probable
Iifetime of the gas nebda. Thus, the protoplanets beyond 5 A U have some time to catch up in
mass, leading to a scenario Lise the one represented by the protoplanet initial conditions chosen
above, The timescale to reach 10 Me a t 5AU with a surface density of 10g/cm2 is estimated Ki
Chapter 3 as
-
5 Myrs, kom
Fig. 3.2. One shodd recall that this estimate does not take into
account concurrent solid and gas accretion, thus the actuai timescale may have been shorter due
to enhancement of the protoplanets' physical radii by a growing gas atmosphere.
4.5.2
Subset 1A
To mode1 gas accretion, the author modified SyMBA to allow a subset of bodies to have artificially
time-varying masses (in addition to any changes in mass resulting fiom the accretion of other
bodies). For the nuis in Subset A, it is assumed that the innennost protoplanet undergoes Nnaway
gas accretion first, and grows into Jupiter-
This is simuiated by increasing the body's mass over the
first IO5 years of simulation time, £tom its original mass of 10 Me to 314 Me, approxirnately the
4- Fo-ng
Uranus and Neptune among Jupiter and Saturn
63
present Jupiter mass- A linear growth in m a s is used; this is deemed appropriate since the actual
time evolution of mass during runaway growth is highly uncertain. Also, as the simulations will
show, 105 years is roughly the response time of the system, a n d the system's subsequent evolution
is therefore unlikely to be affected in a systernatic way by t h e exact form of the tirne evolution of
the runaway-phase mass growth.
Subset A consists of eight aiternate realizations of a run, differing only in the initial phases
of the four protoplanets; for each, the angles $2, w (which define the orientation of the orbit in
space) and M (the mean anomaly, a measure of the location of a body on its orbit) are randomly
generated. As will be seen, the stochasticity of the system ensures that this difference in phases is
sufEcient t o bring about a very different evolution in each of t h e versions of the run.
Fig. 4.6 shows the evolution of one of the eight runs, denoted as Run A6. Semimajor axis
versus time is plotted for each of the four protoplanets; the innennost one has grown into Jupiter
after the e s t IO5 years. By this time, the protopIrnet orbits begin to mutually cross, and strong
scattering occurs. The protopIa.net plotted in red briefly has i t s semimajor axïs increased to greater
than 100 AU. However, dynamicd friction acts to reduce eccentricities universally, decoupling the
protoplanets Erom Jupiter and fiom each other. By about 1.2 Myrs, none of the protoplanets are
on crossing orbits anymore. B e r about 3 Myrs, the bodies no longer undergo any changes in
semimajor axis greater than a few AU on a million-year tirnescale. At this point the orbits are
well spaced and dl eccentricities a r e S0.05, ~ 6 t hno large fiuctuations. Subsequent semimajor
axis evolution proceeds by scattering of planetesimals by protoplanets, rather than scattering of
the protoplanets off each other and Jupiter- As planetesimals are scattered among Jupiter and the
protoplanets, the former experiences a net loss of anguIar rnomentum while the latter experience
a gain. Thus Jupiter's orbit shrinks, while those of the protoplanets e q a n d (Fernandez and Ip
1996, Hahn and Maihotra 1999). This phase takes place over a tirnescale of several te-
of Myrs,
and stops when ail planetesimals have been cleared from among the planets. Because the length
4. Forming Uranus and Neptune among Jupiter and Saturn
64
Figure 4.6: Run A6: Evohtion of semimajor axk (bold lines), periheiion distance q (thin lines) and apheiïon
distance Q (dotted lines) of the four 10 Me protoplanets. The protoplanet which grows to
Jupiter mass (314 Me)over the f h t 10' years of simulation time is shown in black.
of migration during this phase is only a few AU, and to Save tirne, most of the runs are stopped
after 5 Myrs- As an example, Fig. 4.7 shows Run A6 continued to 50 Myrs; the net migration of
the outer two protoplanets subsequent to 5 ~Myrsis o d y
-
2 - 4 AU outward.
The semimajor axes at which the scattered bodies end up are very noteworthy, if one compares
them to the present orbits of the giant planets (Fig. 4.4). At 5 IMyrs, the outer two protoplanets are
at 16 and 26 AU. The innermost one is at II AU, whiie Jupiter is at 5.5 AU. This configuration of
orbits is very similar to that of the present Solar System, where Uranus and Neptune are at 19 and
30 AU respectively, Saturn is at 9.6 AU, and Jupiter is at 5.2 AU. And at 5 x 107 years, after some
4. Forming Uranus and Neptune among Jupiter and S a t w
A ,+,$,
y
*p!$
' &@p&,@'
-.f,!!!#ydj*,
/Yb
f i 'iiql,q '
1!
11
i
9$q!&
!de
i b7bLpk,$
j
q 4A ,
\j{/W*:
A
65
1 ;y:
fw,
;+&
bq
1
~~$.&.-".~
v-L\'
' P d :>j! \
,JlfiJ
Figure 4.7: Run A6 continued to 50 Myrs. Between 5 My-m and 50 Myrs, the net migration for Jupiter and
the protoplanets, going kom inside to outside in semimajor axis, is -0.2 AU, 1.5 AU, 4 AU,
and 1.3 AU.
more net outward migration, the outer two protoplanets' semimajor axes are even closer to those
of Uranus and Neptune (though UpmtoSaturn" , having also moved outward, is further away from
its present orbit). Eccentricities and inclinations are likewise very close to their present values; a
detailed cornparison (to Run A6 as well as the other seven versions) is made in Table 4.4.
The end state of all the runs is summarized in Fig. 4.8. Values for semimajor a x i s , eccentricity
and inclination are given in Table 4.4. Depicted are cLsnapshots"of eccentricity versus semimajor
aKis at 5 Myrs, except for Runs A i and A8, which were continued ta 10 Myrs. These were extended
because one or more of the protoplanets stiu had a high but decreasing eccentricity at 5 Myrs.
4- Fo-g
Uranus and Neptune among Jupiter and Saturn
66
Table 4.4: Final orbital eIements of Run IA
Real System Al
~ J ( A U ) 5.2
5-4
eJ
0-048
0.007
A2
5.6
A3
A4
5.8
5.4
0.006
0.015
0.049
A5
5.4
0.014
A6
5.5
0.008
A7
A8
5.4
5.4
0-007 0.012
Runs Al, A4, A6, A 7 and A8 result in a final ordering of orbits that is a t the very least broadly
consistent with the present Solar System: Jupiter is the innennost body, with the other three bodies
interior to the region of the Kuiper belt, and eccentricities low enough that no protoplanets cross
each other or Jupiter. Out of these five runs, A4 and A6 in particular resemble the present Solar
System, Of course, t o actually reproduce the Solar System, another important event has to take
place: The next protoplanet beyond Jupiter must also undergo a runaway gas accretion phase to
acquire an envelope of mass 2 8 0 Me ônd become Satum. With the innermost core on a stable
orbit in the vicinity of Satuni's present location, though-as
it is in runs A4 and A6-the
time
delay between Jupiter and Saturn's runaway phases is not strongly constrained- A more detailed
investigation of the role of Saturn's growth wiil follow in Section 4.5.8.
The final state of the planetesimal disk differs substantidly among the runs. In A2, A3 and
AS, the planetesimal disk is largely unperturbed over most of its radial extent, while in A6 and
A8, eccentricities have been greatly increased throughout the entire disk. Al, A4 and A7 are
intermediate cases.
The extent of the perturbation simply depends on honr much of the disk
is crossed by the protoplanets over the course of the run- In Run A6, for example, Fig. 4.6
shows that one body's aphelion spends some time beyond 100 AU. On the other band, in A3, no
4. Fonnïnn Uranus and Neptune amonnJupiterand Saturn
67
Figure 4.8: Endstates of the eight Subset A rus, after 5 Myrs of simulation tirne, except for A l and AS,
which were continued on to 10 Myrs- Eccenctricity îs plotted versus semimajor axis. Planetesimal orbits crossing Jupiter or any of the protoplanets are generalIy unstable on timescales
short compared to the age of the Solar System, thus the region among the protoplanets wodd
be esentidly cleared of planetesimals long before the present epoch.
protoplanet's aphelion ever goes further out than about f8AU. In those runs where the Iargest
aphelion of any protoplanet f d s within the disk (Le.
< 60 AU), this lirnit corresponds closely to
where the corresponding pIanetesima1 disk transitions from perturbed to unperturbed- The outer
b i t of the planetesimal disk's excitation by scattered protoplanets has been referred to as the
"fossilized scattered diskn (Thommes, Duncan and Levison 1999). The conternporary scattered
disk, by contrast, consists of objects which were scattered by Neptune after the latter had attained
its curent orbit (Luu et al 1997, Duncan and Levison 1997). Both classes of objects are discussed
4, F o d g Uranus and Neptune among Jupiter and Saturn
68
in more detail in Section 5-2.
Three of the runs produce systems at 5 Myrs that are irreconcilably merent from our own,
In A2, one of the protoplanets has merged with Jupiter- In A3, a protoplanet has been scattered
onto an orbit interior to Jupiter, in the region of the present-day asteroid belt. Even assuming
the protoplanet could be subsequently removed from this region, such an event wouid have cleared
rnuch of the asteroid belt- AIso, it couid have had a significant adverse effect on the stability of the
terrestriai planets (Quintana et d 2000)- The potentid effect of giant planet core-sized protoplanets
on the region of the asteroid belt is explored in detail in Section 5.1. Fmally, in A5, two of the
protoplanets merged, It should be noted that mergers which reduce the number of protoplanets
are not i n t ~ s i c a l l ya problem, since extra ones may have existed. Scenarios with five protoplanets
(Jupiter
+ Satum + 3) will be explored in Section 4.5.4.
4.5.3
Subset 1B:Dependence on initial orderhg
HOWstrong1y does the find configuration of the system depend on the initial ordering of the
protoplanets? The next subset of r u s , 1B, uses the same initial conditions, within a random
variation in the protoplanet phases, as 1A- However, for these runs it is the second-innennost
protoplanet, rather than the innermost one, which undergoes simdated runaway growth- One can
reasonably expect that this will favour an outcome like 1A3 (Fig. 4.8), where a protoplanet is
scattered inward into the region of the asteroid belt.
The end states of the runs are shown in Fig, 4.9. In six of the eight runs, a protoplanet has
indeed ended up interior to Jupiter- However, in two cases (B3 and B4), Jupiter is the innermost
body. Thus it appears that if Jupiter does not grow from the innermost protoplanet, the likelihood
of ending up with a final configuration similar to our Solar System declines, though such an outcome
continues to be quite possible.
4. Forming Uranus and Neptune among Jupiter and Saturn
69
Figure 4.9: End states of the eight Subset B ruus,after 5 Myrs of simulation time, except for B3, which
was continued on to 15 Myrs. Eccenctriâty is plotted versus semimajor cuos.
4.5.4
Subset 1C: Dependence on number of cores
How sensitively does the end state depend on the initial number of core-sized bodies? In the aext
set of simulations, an extra 10 Ma protoplanet is added.
AU
protoplanets are more tightly spaced,
by 6.5 instead of 7.5 mutual Hill radii- Starting, again, fiom 6.0A4U,the outermost protoplanet is
therefore i n i t i d y at 12.2 AU- The surface density of planetesimals in the region of the protoplanets
is reduced t o keep the average surface density unchanged a t
01
= l0(a/5 AU)-2 g/cm2. The
innermost protoplanet is, again, increased in mass to that of Jupiter over the first 105 years.
Fig. 4.10 shows the endstates of the runs. This time ail initially have a length of 107 years,
4. Forming Uraaus and Neptune among Jupiter and Saturn
O
20
40
60
O
20
40
70
60
Semi-Major Axis (AU)
Figure 4.10: End states of the eight Subset C runs, after 10 Myrs of simulation time, except for Cl,which
was continued on to 15 Myrs.
since the larger number of bodies take longer t o decouple from each other. Run Cl was continued
to 1.5 x 107 years, because after IO7 years some of the protoplanets are still on crossing orbits.
Eccentricities are uniformly low. A.ll of the protoplanets remain in six of the eight runs, thus
resultuig in systems with one too many planets relative to the present Solar System. However, in
Runs C2 and C6, one of the protoplanets is ejected fiom the Solar System, leaving the right number
of bodies behind- In both of these runs, a protoplanet ends up with a semimajor axis within 10%
of Uranus and Neptune, though both "Saturns" are too far out. One may conclude that with one
extra initial protoplanet in the Jupiter-Saturn region, scattered protoplanets continue to be readily
circularized, and the resulting systems tend to look like ours with one extra outer planet. However,
4- Fonning Uranus and Neptune among Jupiter and Saturn
Semi-Major
71
Axis (AU)
Figure 4.11: End states of the eight Set 2 m. The first four are run ta 5 x 106 years; the last four are
nin to la7 years.
a system with the right number of giant planets remains very much a possible outcome.
4.5.5
Set 2:
A more massive planetesimal disk
In this set of runs, a number of parameters are dianged. The protoplanets are now 15Me bodies.
The innermost is initially at 5.3AU, and successive protoplanets are spaced by only 5.8 mutual
Hill radü. Thus the outerrnost protoplanet is initially a t 9.0 AU. The innermost one, again, has its
mass increased t o 314 Me over the first 105 simulation years. The planetesimal disk surface density
profile is still a a-*, but it is scaled up to be 15g/crn2 a t 5 AU; that is,
4. Formian Uranus and Ne~tuneamona .hpitt?r and Saturn
72
The individual planetesimals have a mass of 0.24 Me;this slightly larger mass is a Iegacy of chronologically earlier runs (Sets 3 and 4) with more massive planetesimal disks, Again, initial con ditions
for each member of the set differ only by the phases of the protoplanets,
The end states of the runs are shown in Fig. 4.11. Al1 except C yield the correct number and
ordering of bodies, and eccentricities are unifonnly low- This "success raten is higàer than rthat of
Subset iA, in which only five out of eight runs yield qualitatively the correct orbital configuration.
This is accounted for by the more massive planetesimal disk; it provides stronger dynamical friction,
so that scatterings of protoplanets tend to be less violent, and subsequently, orbits tend to be
circularized more quickly- Runs A, D and R end up with protoplanet orbits that are particulary
close to those of Saturn, Uranus and Neptune- Jupiter systematically ends up at too smaU a
heliocentric distance, indicating that the initial distance of 5 . 3 A U is too s m d - Also, the Iarger
disk mass gives the protoplanets and Jupiter more planetesimals to scatter, and thus increases the
distance they travel due to angular momentum exchange (Section 4.5.2)-
4.5.6
Set 3: A shallower disk density profile
In this set of runs, a shallower planetesimal disk surface density, a
is used. The disk now
begins a t 10 AU, with the interior region being i n i t i d y occupied solely by the protoplanets- In
other words, it is assumed that in the epoch from which the runs start, aii but a neglible mass of
the planetesimals among the protoplanets has been swept up or scattered hom the region. The
surface density profile is given by
and thus a total mass of 123Me is contained in the disk between 10 and 60 AU. Extendiag this
planetesimal disk inward to 5.4U would yield a surface density there of only 5 g/cm2,and a total
m a s between 5 and 10 AU of only 25 Me.However, the four pIanetesimalç in this region a r e still
each 15 Me bodies; for this case it is assumed that the original planetesimal surface density profile
4. Forming Uranus and Neptune among Jupiter and
Semi-Major
Saturn
73
Axis (AU)
Figure 4.12: End states of the twelve Set 3 runs. Ail are run to 5 x 106 years.
in this region was steeper, perhaps due in part to redistribution of water vapor from other parts of
the disk to the vicinity of the "snow line" (Stevenson and Lunine 1988).
This set consists of twelve runs, each to 5 Myrs. As before, the inner body's mass is increased
to 314 Me over the first IO5 years of simulation time. The endstates are shown in Fig. 4.12- All
except E, G,H and K possess the right number and ordering of bodies. Eccentricities are S 0.1 in
A, C, D, F, 1and LVarious degrees of disruption of the planetesimal disk can again be seen- Most show a sharp
transition between a disrupted region crossed by the scattered planetesimals, and a largely undisturbed outer region- In Run A, for example, this transition occurs at slightly below 40 AU, while
4. Forming Uranus and Neptune among Jupiter and Saturn
in Run
74
F,it is located between 45 and 50 AU. Runs B and L show strong disruption throughout
the entire disk, indicating that all of it was crossed by one or more protoplanets. Another state can
be seen in Run G, where all eccentrîcities in the outer part of the belt are u n i f o d y raised. This
occurs when a protoplanet crosses the outer disk with an inclination high enough that it spends
most of its orbit above or below, rather than inside, the disk- The disk planetesimals are then
excited prïmarily by long-range secular effects rather than by short-range scattering encounters
(Thornmes, Duncan and Levison 1999)- Further examples of this eEect c m be seen in Set 1, Run
A7, and in Set 4, Run C below4.5.7
Set 4: An even shallower disk deasity profile
In this set of runs, the planetesimal disk surface density profile is even more shdow:
As a result, the disk is more massive, containhg 212 Me in planetesimals between 10 and 60 AU.
The endstates a t 5 Myrs are shown in Fig. 4-13. In d runs except C, the three protoplanets
end up on low-eccentricity orbits in the outer Solar System- This is in keeping with the trend of a
more massive planetesimal disk providing a higher rate of successfully circularized protoplanets-
4.5.8
Set 5: The role of Saturn
Thus far, the only gas giant in the simulations has been Jupiter. The innermost of the scattered
protoplanets does tend to end up near the present location of Saturn. However, to reproduce the
Solar System, it must at some point accrete
-
80 Me of nebular gas. Investigating the effect of
Saturn growing to its full mass may make it possible to place constraints on the timing of Saturn's
gas accretion phase in this model.
The initial conditions of a Set 1A run which produced a good S o h r System analogue (1A6) are
used. As before, the innermost protoplanet's mass is increased t o Jupiter's over the &st IO5 years.
4- Forming Uranus and Neptune among Jupiter and Saturn
20
O
40
60
O
20
40
75
60
Semi-Major Axis (AU)
Figure 4.13: End states of the eight Set 4 mns- Al1 are run to 5 x 106 years.
The &st objective is to investigate what happens if Saturn f o m s while the protoplanets are still
crossing, which in
Run 1A6 continues until 1 Ivlyr.
In three runs-one
mass startùig a t 2 x
for each of the remaining protoplanets-a
los years, over IO5 years.
protoplanet grows to Saturn7s
In other runs, a protoplanet grows into Saturn at
4 x 105, 6 x 105,and 8 x 105 years. For each of these times there are only two runs; by 4 x
los
years, one of the protoplanets has has been scattered onto a high-eccentricity orbit with a 2 50 -4U.
It is therefore not considered a viable Saturn candidate.
The runs in which "Saturnn grows at 2 x IO5 years are shown in Fig. 4.14. In the top panel, where
the red protoplanet-initially
closest t o Jupiter-becomes
Saturn, both others are ejected from the
4. Forming Uranus and Neptune among Jupiter and Saturn
O
1O'
2 x 1O6
3x 106
4 x IO6
76
5 x 106
Time (years)
Figure 4.14: Set 5: Evolution of semimajor a x i s (bold Iines), perihelion distance q (thin lines) and apheiion
distance Q (dotted lines) of the four 10 Me protoplanets. The protoplanet which grows to
Jupiter mass (314Me)over the
10"ears
of simulation tirne is shown in black. In each
of the three panels, a different protoplanet grows to Saturn's masç at 2 x 10' years: red (top),
green (middle) and blue (bottom)
system, In the other two cases, both protoplanets survive to 5 x 106 ~ e a r s though
,
in the bottom
panel (blue protoplanet becomes Satuni), one of the protoplanets stiil has a high eccentricity and
is crossing the other's orbit ai the end of the m.
Figs. 4-15,4.16, and 4.17 show the runs in which Saturn begins its runaway gas accretion phase
at, respectively, 4, 6 and 8 x105 years. A trend is visible in these runs; namely, the closer to Jupiter
the protoplanet which becomes Saturn is, the more likely the other protoplanets are to get ejected.
This happens because the protoplanets closer to Jupiter receive stronger perturbations and hence
4. Forming Uranus and Neptune among Jupiter and Saturn
77
tend to have higher eccentricities- A protoplanet with a mass approaching Saturn on an eccentric
orbit, crossing the orbits of the other two protoplanets, has a high probability of ejecting one or
both. On the other hând, if the growing Saturn maintains a low eccentricity, the other protoplanets
tend to not be scattered as strongly, and have an increased chance of not being ejected from the
system. It is important to recall, however, that the accretion of Satuni's gas envelope is, as for
Jupiter, modeled only in the form of a simple mass increase. If the eccentncity of a protoplanet is
reduced quickly enough during runaway gas accretion, then Solar System analogues can be readily
produced even if Saturn accretes its gas envelope close to Jupiter's orbit and within a short time
of Jupiter's runaway gas accretion phase, when the scattered protoplanets are still on mutually
crossing orbits,
The effect of Saturn growing at later times, when the protoplanets have decoupled, is also
investigated. This happens after
-
1 Myr in the onguial Run 1A6, with the initially outermost
protoplanet (plotted in blue) now the innermost protoplanet, closest to Jupiter a t
-
10 AU. Another
series of mns grows this protoplanet to Saturn m a s over a 10' year interval, starting at 1, 1.2, 1.4,
1.5 and 1.6 Myrs. The endstates are shown in Fig- 4.18- No protoplanets have been lost fkom the
system by the end of the runs- A protoplanet (red) stiil has a high eccentricity in the 1 Myr case;
this is because this protoplanet's perihelion is still very dose to the innermost protoplanet's orbit at
1 Myr, and is suffers strong perturbations as the latter grows to Saturn's rnass- In the other cases,
however, Saturn's formation does not cause large eccentricities in the protoplanets. This is as one
would expect; at its h a I mass, and at 10 AU, Saturn's HiU radius is 0.45 AU, and by 1.2 x 1o6
years, the closest protoplanet's perihelion is a t
- 14AU'
a h o s t 9 r away,
~
thus out of reach for
strong scatteringOne effect visible in Fig. 4.18 is that the semimajor axis of Saturn a t 5 Myrs tends to be
smaller than that of the innermost scattered protopfanet in those
nuis
where the formation of
Saturn is not modeled (see for example Fig. 4.8). When a protoplanet grows to Satum's mass,
4. Forming Uranus and Neptune among Jupiter and Saturn
78
Time (yeors)
Figure 4.15: Set 5 runs in which Satum grows a t 4 x 10' yearç. In the top panel the green protoplanet
becornes Saturn; in the bottom panel, i t is the blue one.
its subsequent migration speed is much slower, since the rate of migration depends on how much
mass in planetesimals it scatters relative to its own mass. This counteracts the tendency of the
innermost scattered protoplanet to end up at a semimajor axjs larger than that of Saturn in the
other m,where only Jupiter grows.
There is one obvious constraint on the timing of Satuni's growth: It must have occurred within
the e s t
-
10 M ~ T of
S the Solar System's existence, when there was stiil gas available. Another
constraint is that it (probably) occurred f i e r the formation of Jupiter. Jupiter's formation tirnethe time to form its core (Chapter 3) plus the time it spends in the second phase (Section 2.3-2)is probably not much less than the gas lifetime, leaving a window on the order of a few Myrs-
4.
Forming Uranus and Neptune among Jupiter and Saturn
79
Time ( y e o r s )
nuis in which Saturn g o w s at 6 x 105 years. Im the top panel the green protoplanet
becomes Saturn; in the bottom panel, it is the blue one.
Figure 4.16: Set 5
The scattered protoplanet mode1 h h e r suggests that Jupiter and Saturn's runaway phases were
separated by not much less than 1 Myr, the time needed fox the protoplanets to stop crossing
each other and Jupiter strongly, otherwise there is a significan& increase in the likelihood that one
or more protoplanets wiii be ejected £rom the Solar System- A t the same tirne, the requirement
that proto-Saturn not migrate beyond its present semimajor axis implies that the onset of Saturn's
runaway growth occurred not very much more than 1 Myr a f t e r that of Jupiter.
4. Forming Uranus and Neptune among Jupiter and Saturn
o
O
t
'
'
'
~
~
106
'
'
2x10e
~
'3% los
~
'
l
'
'
4x 106
~
80
' 5x1Oa
' '
'
~
'
Time (years)
F i g u e 4.17: Set 5 runs in which Saturn grows at 8 x 10' yeats. In the top panel the green protoplanet
becomes Saturn;in the bottom panel, it is the bIue one.
4.6
Discussion
As shown in Chapter 3, the formation of Uranus- and Neptune-sized objects near their present
locations in the outer Solar System is unlikely to have taken place unless the primitive solar nebula
was considerably more massive than is generaiiy accepted, In a nebula with a plausible gas and
planetesimal density profle, the outward-sweeping "wave" of oligarchie growth wouid only have
reached the region of Jupiter and Saturn by the time the gas is thought to have dispersed, S 107
years. However, in the future Jupiter-Saturn region, this wave of accretion is likely to produce more
t han just two gas giant core-sized bodies. Such bodies will tend to exist on Iow-eccentricity coplanar
'
'
'
~
4. Forming Uranus and Neptune among Jupiter and Saturn
O
20
40
81
60
Semi-Mojor
Axis (AU)
Figure 4.18: Endstates of those Set 5 mns in which Satuni commences growing &ter the protoplanets have
largdy decoupled fiom each other. The &art tirne of Satum's growth, t s , is denoted on each
paneI-
orbits- The numencal simulations performed here show that the formation of Jupiter &om one of
these bodies will resdt in a violent short-term evolution of the remaining bodies, which tends to
scatter them outward. Dynamicd fiction with the planetesimal disk will then act to reduce the
eccentricities and inclinations of the bodies, and they have a strong tendency to ultimately acquire
orbits similar to those of Saturn, Uranus and Neptune.
The simulations show that this result does not depend sensitively on the surface density profile
of the planetesimal disk- Also, Solar System-like configurations can be still be produced if an extra
body is added, or if the ordering of proto-Jupiter among the other bodies is changed. A strong
4- Forming Uranus and Neptune among Jupiter and Saturn
82
coïncidence in the timing of Jupiter and Saturn's ninaway gas accretion phases is not required; in
fact, it works best if Saturn's final growth phase occurs 1- 2 Myrs after Jupiter's- The robustness
of this mode1 stands in sharp contrast to t h e d i E d t y of constructing scenarios for the in-situ
formation of Uranus and Neptune,
5.
SCATTERED PROTOPLANETS AND THE SMALL
BODY BELTS
The simulations presented in Chapter 4 show that scattering of giant planet coresized protoplanets
is a violent event, which leaves a strong dynamical signature on the surrounding planetesimal disk.
Can any trace of such an event be detected in the present-day Solar System? The asteroid belt
and the Kuiper belt constitute remnants of the planetesimai disk. Therefore, they are the naturd
places to look for direct evidence of large scattering events in the Solar System's early history.
5.1
The asteroid belt
It is those members od the asteroid belt larger than about 50 lan in diameter which are of interest
in inferring properties of the early Solar System; smaller bodies cannot be primordial because they
could not have survïved intact for the age of the Solar System (Petit, Morbidelli and Valsecchi
1999). Fig. 5.1 shows the curent eccentricity and inclination of all asteroids above this size cutoff.
They display a degree of dynamical excitation that is not readily explained. The asteroid belt is
customarily divided into an inner, middle and outer belt- The median eccentricity and inclination
in each region is given in Table 5.1.
The other puzzling feature of the asteroid belt is its severe mass depletion, relative to the amount
of mass the region is thought to have originally contained. The total present mass of the asteroid
belt is estimated a t
-
5x
Ma, which is more than 103 times smailer than even the minimum
mass disk of solids (Chapter 3). At the sanie tirne, the necessity of forming the largest asteroids
on a tirnescale consistent with the meteoritic solidification age requires at least 100 times more
5- Scattered protoplanets and the small body beits
84
Figure 5.1 :Eccentricity (top) and inchation (bottom) vesus semimajor axis of bodies in the asteroid belt
Iarger thaa 50 k m
material than the present amount (Wethedl 1989b)- Thus the asteroid belt's low mass must be
the result not of a primordial gap in the planetesimal disk, but of later processes which depleted
its m a s .
5.1-1 Belt-crossing scattered pro toplanets
In the context of the mode1 for the origin of Uranus and Neptune presented in Chapter 4, the initial
violent scattering of protoplanets is perhaps the most obvious candidate to look to for perturbation
of the asteroid belt. Indeed, in numerous runs, protopfanets spend some tirne interior to the orbit
of Jupiter, crossing part of the asteroid belt region. An example of this is shonm in Fig. 5.2, where
5- Scattered protopIaneb and the mal1 body beits
85
Table 5.1: Median eccentricities and inclinations in the asteroid belt
BeIt region
Extent
Median e
Mediani
inner
a < 2.5 AU
0-15
6O
midde
2.5 AU
< a < 3.28 AU
0.14
10.7O
outer
a > 3.28 AU
0.10
12.1°
a 10 Me protoplanet crosses the region of the asteroid belt for a time of less than 2 x IO4 years,
its perihelion penetrating inward to
-
3 AU while its aphelion remains at Jupiter's orbit. One
might expect that such a n occurence would wreak havoc in the the asteroid belt, at least in the
outer and central parts- To assess the effect of belt crossings of this sort, another set of simulations
is
nui,
in which bodies are added intenor to Jupiter's orbit to simulate the asteriod belt part of
the planetesimal disk- The individual bodies have a mass of 0.024 Me, and are distributed with a
surface density of
m,dt
= 8-0(a/l AU)-' g/crn2
(5-1)
between 2.5 and 4.5 AU. This shallow density profile is the same as the one used by Chambers and
Wethenll (1998) in the terrestrial region, which in turn was chosen to be more consistent wïth the
large densities required a t larger heliocentric distances to form Jupiter and Saturn. The protoplanet
masses in this case are 15 MB- Out of a set of six runs (with the initial conditions differing, again,
only in the phases of the Iarge bodies), the one with the longest total time of crossing the region
interior to Jupiter is shown in Fig. 5.3- Between the start of the run and 2 x 105 years, one of the
protoplanets penetrates interior to Jupiter several times, with the duration of individual crossings
being at most 104 years- Fig. 5.4 shows the effect on the planetesimals interior to Jupiter. The
top and bottom panel show before-and-after snapshots of the planetesimal eccentricities and inclinations. As can be seen, eccentricities up to
-
0.4 are caused by the protoph.net (neglecting those
planetesimals which are close to crossing Jupiter). There is a sharp edge to the high eccentricities
a t about 2.4AU; inside this radius, eccentririties are significantly lower. This boundary marks
5. Scattered protoplanets and the smaU body bdts
86
Figure 5.2: The h t 106 yearç o f Run LA8, showing the evolution of the semimajor axis (thick soiid),
perihelion (solid) and apheiion (dotted) of Jupiter ( b l d ) and the three protoplanets. At 10'
years, a protoplanet (red) briefly crosses the asteroid belt, its perihelion going down to 3 AU.
The crossing lasts l e s 5 2 x 104 yearç.
-
the srnailest perihelion of the protoplanet. Inclinations are also excited, though the edge is not as
strong, and few bodies (apart boni those near Jupiter-crossing) attain inclinations above 10"- A
significant number of planetesimals is not ejected during or immediately after the periods when the
protoplanet crosses the belt.
Of the six runs performed, the one shown here received the greatest amount of dynamical
excitation in the asteroid belt region. It appears, therefore, that it is difficult to reproduce the
structure of the asteroid belt with scattered protoplanets done. Eccentricities can get excited to
t heir present d u e s , though only down to the crossing protoplanet's minimum aphelion distance,
5. Scattered protoplanets and the srnaIl body bdts
87
Figure 5.3: Evolution of semimajor a x k (thick solid), perïhelion (solid) and aphelion (dotted). Protoplanets
are initiaiiy 15 Me As usual, the iMerrnost protoplanet (black) grows to Jupiter m a s over the
fkst 105 years. One scattered protoplanet (red) crosses the region interior to Jupiter's orbit
numerous times within the first 2 x 105 years.
which in none of the nins performed reaches the inner edge of the belt, even when Jupiter is at a
smaller semimajor axis than in our Solar System (as in the run depicted in Fig. 5.3). Inchations
fare more poorly; protoplanets seldom raise them much above 10". Aiso, very iittle m a s depletion
takes place whiie the protoplanets are in the bdt region. C e r t d y the raising of planetesimal
eccentricities wili increase the number of planetesimals which undergo encouaters with Jupiter (or
the terrestrial planets, once these fonn) Iater on, but this may not be sufficient to decrease the
total contained in the asteroid belt by the required factor of
-
100.
Scattered protoplanets may have had a more indirect role in the excitation and mass depletion
5- Scattered protoplanets and the small b o d y beits
88
Figure 5.4: Eccentricities and inchations of pIanetesimals in the asteroid belt region, interior to Jupiter
(the large dot), a t 3 x fo4 years (top panel) and 2 x 10' years (bottom panel). These times are,
respectively, just before and just after the period during which a protoplanet repeatedly crossed
the region interior t o Jupiter (cf- Fig. 5.3). Jupiter in this nui has moved inward to 4.8 AU,
0-4 AU less than its present semimajor axis. The line marks the locus of Jupiter-crossing orbits
.Y
5. Scattered protoplanets and the small body belts
89
Figure 5.5: Evolution of a growing Jupiter and three 15 Me protoplanets, with one 1 Me body (Iight bIue)
interior to it, in the açteroid belt region. A scattered protoplanet (green) ejects the body fiom
the belt at 5 x 104 years.
-
of the asteroid belt. The "insideout" nature of planetary accretion (Chapter 3) implies that
the asteroid belt already contained planet-sized bodies when Jupiter and Saturn started to forrn;
the perturbations from the gas giants came too Iate to inhibit planet formation in this region.
The existence of such bodies in the b d t for Myr timescales could have accounted for much of its
excitation and mass depletion. The problem is then how these planets were subsequently removed
from the asteroid belt. This is where protoplanets scattered by a growing Jupiter may corne in. Fig.
5.5 shows the evolution of a simulation iike those described above, but with a 1 Ma body added
in the belt region, initidy on a circular orbit with a semimajor mis of 3.5 AU. At t
-
5 x
IO4
5- Scattered protoplanets and the smaii body belts
90
years, one of the 15 M . protoplanets penetrates the asteroid belt, and its orbit crosses that of the
s r n d e r body, causing it t o be ejected from the belt. A total of eight mns with a body a t 3.5 AU is
perfonned, and in three of these, the body is ejected (though in one case, the protoplanet involved
is deposited in the asteroid belt). It may thus be possible for planet-sized bodies to exist in the
asteroid belt region long enough t o cause substantial dynamical excitation and mass depletion,
oniy t o be removed later by Jupiter-scattered protoplanets, leaving behind the asteroid b d t in its
present form-
5.1 -2
Sweeping secular resonances
h o t h e r rnechanism suggested for explaining the asteroid belt's high degree of excitation is the act ion of secdar resonances involving Jupiter and Saturn. Secular resonances are commensurabilities
between the rates of change of the "slown angles of orbits, namely the longitude of the ascending
node $2, and t h e longitude of pericenter G- In contrast, commensurabilities between the rates of
bodies' motions on their orbits are mean-motion resonances. The two important secular resonances
in the region of the asteroid beIt are the u s , the commensurability with the precession fiequency
of Saturn's G , and the
V16
, the commensurability with the precession frequency of Saturn's 0.The
effect of the vg is t o excite eccentricities, while the
vlg
excites inclinations. A detailed discussion
of resonances can be found in celestial mechanics texts, for exarnple Murray and Dermott (2000).
Ward et al (1976) first suggested that the secdar resonances, their positions displaced by the
changing gravitational potential as the primordial gas nebula dispersed, could have swept across
the asteroid belt and increased the eccentricities and inclinations therein. Ward and Heppenheimer
(1980) found that the v~ could indeed have accounted for the present eccentricities, though not
the mass depletion, in the asteroid belt. Lemaitre and Dubru (1991) looked at the change of the
positions of the secdar resonances resulting from the dispersal of the gas nebula- They found that
although eccentricities throughout the belt could be excited in this way, the range swept by the
5- Scattered protoplanets and the small body belts
vl6
91
resonance depends sensitively on the density profile assumed for the nebula, which is uncertain.
However, Nagasawa, Tanaka and Ida (2000) showed that if the nebda were nonunifonnly depleted,
(as meIl as the
the
~
1
the
5 ~commensurability with Jupiter's R precession) could be made to
sweep the entire asteroid belt.
Lecar and F'ranklin (1996) included the effect of gas drag. They found that bodies strongly
affected by gas drag, S 50 km in size, would lose energy rapidly when their eccentricities were
pumped up (as described by Eq. 2.28) and wodd thus spiral inwards. This provides a way of
clearing much of the mass from the asteroid belt region, while simultaneously exciting eccentricities
in the belt to values consistent with the present ones. Since the model used was planar, it provided
no results regarding inclinations.
Gomes (1997) investigated the effects of resonance sweeping due not to the dispersal of the
gas nebula, but instead to the radial migration of Jupiter and Saturn (see 4.5.2).
He presumed
a migration of Jupiter fiom 5.4 to 5.2AU, and of Saturn from 8.7 to 9.5 AU. He found that this
moves the vo inward kom about 4 to 2 AU, so that it sweeps the entire belt. A migration t h e
of 5 x 105 years is long enough to increase eccentricities from near zero up to 0.3. At the same
tirne, the
to
-
V16
resonance only moves from 2.7 to 2 AU. Over this range inchations are ezited up
15O. But the rest of the middle belt, and the outer belt, is largely unaffected.
Of course, the
246
resonance can be made to sweep the entire belt if the ranges of migration
of Jupiter and Saturn are changed. Moving the initial orbits of Jupiter and Saturn closer together by moving Jupiter outward, Satuni inward, or some combination thereof, moves the
y16
outward. Indeed, a Jupiter and Saturn initially in close proximity is a possible side effect of the
scattered-protoplanet model. The existence of several potential gas giant cores, initially closely
spaced between
-
5 - 10 AU, may make it possible for Saturn to fonn more closely to Jupiter than
is usually assumed.
An exampIe can be seen in the lowest panel of Fig. 4.15, which is shown enlarged in Fig. 5.6.
5. Scattered protoplanets and the s m d body beits
92
Figure 5.6: A larger view of the lower panel of Fig. 4.15. The protoplanet plotted in blue begins growing
to Saturn's mass (on a 105 year timescale) at t = 4 x 10' years. Its semimajor axis at the time
it completes its growth is about 8.4 AU. At t = 5 x 106 years, uSaturnn has a semimajor a x k
of about 9 -4 AU.
In this run, Satuni's growth commences 4 x 105 years after Jupiter's, with the initially outermost
protoplanet (blue) as the core. At the t h e Saturn's growth is complete, Jupiter and Saturn are at
5.7 and 8.4 AU respectively. Thus Saturn's orbit is ?.O Jupiter Hill radii away. In the absence of a
gas nebula, this places the v l ~resonance at
-
3.5 AU. Over the next
inward to 5.4 AU, while Saturn ends up at 9.4 AU. In the process the
-
3 ~Myrs,Jupiter migrates
~ 1 resonance
6
moves inward
from 3.5 AU and hence most of the asteroid belt is swept by it. The fastest migration takes place
within the first IO6 years. Between 5 x
Satuni moves fiom 8.4 to 8.6 AU.
los and 106 yyears, Jupiter moves from 5.7 to 5.6 AU, while
5. Scattered protoplanets and the smaü body beits
93
The final semimajor axes of Jupiter and Saturn are 5.2 and 9.6AU' so at the endpoint of
the above nui, they must migrate 0.2 AU inward and outward, respectively. But since Satum is
less massive, it would migrate further than Jupiter- In reality, therefore, Jupiter and Saturn would
probably both have been a t a slightly smailer semimajor axis a t the t h e Saturn finished its growth.
However, the difference involved does not cause a large change in the location of the v
l resonance.
~
For example, if Jupiter is at 5.5 AU at the tirne Saturn reaches its full size 7 r~,,, away, at 8-1AU,
then this places the v l at~ 3.4 AU-
5.2
The Kuiper belt
In the present Solar System, a new class of Kuiper belt object (KBO) has recently been identified
(Luu et al 1997, Duncan and Levison 1997)- These objects have semimajor =es and eccentricities
such that they lie near the locus of Neptune-encomtering objects shown in Fig, 4.4- They are
thought to be part of a population referred to collectively as the scattered disk-formerly
low-
eccentricity KBOs which have had their orbits changed by close encounters with Neptune. Many
of the simulations in Chapter 4 show an analogous class of planetesimals in their "Kuiper belt"
regions. However, these fall on the locus of orbits crossing not the final semimajor axis of the
outermost protoplanet, but the furthest aphelion distance of any of the protoplanets during their
initial high-eccentricity phase- Since these orbits are no longer being crossed by a protoplanet,
they will be stable over long times. One can refer to these structures as "fossilized" scattered disks
(Thommes, Duncan and Levison 1999), because they preserve part of the dynamical history of the
planetesimal disk.
Observations of our Solar System's Kuiper belt (as surnmarized in Fig. 4-4) do not rule out
the existence of a fossilized scattered disk, in addition to the active scattered disk associated with
Neptune. In fact, the observed part of the Kuiper belt exhibits an anomalously high degree of
excitation, impIying that as-yet uncertain d y n d c a l processes acted on it (eg. Petit, Morbidelli
5. Scattered protoplanets and the smaii body beits
94
and Vdsecchi 1999). The eccentricities and inchations of bodies in mean-motion resonances with
Neptune, particularly the 2:3 resonance at 39.5 AU, can be explained by resonance sweeping during
Neptune's migration, as can the paucity of objects on nonresonant orbits interior to 39 AU (Malhotra 1995). However, the high eccentricities and inclinations found beyond
-
41AU in what is
commonly cailed the "cIassicaln Kuiper belt, cannot be explained in this way. Petit, Morbidelli and
Vdsecchi (1999) propose large (up to 1Ma) Neptune-scattered planetesimals as the mechanism
which stirred and cleared the beIt- However, even when such bodies remain in the belt for 100
Myrs, the inclinations they raise are alrnost always Iess than 20".
The excitation and mass depletion of the Kuiper belt may be accounted for if nre are thus
far only seeing its fossilized scattered disk component- If future observations do indeed reveal a
sharp-edged transition to a more dense, dynamicaily cold disk further out in the Kuiper belt, then
this wilI constitute strong observational support for the scattered protoplanet model for the origin
of Uranus and Neptune. At the same tirne, however, a detectable fossilized scattered disk is not
necessarily a consequence of this model. As mentioned in Section 4.5.2, the protoplanet semimajor
axes evolve quickly during the strong scattering phase, and then migrate outward slowly due to
planetesimai-mediated angdar momentun exchange. If the initial scattering is strong enough to
place the aphelion of a t least one body further out than post-migration semimajor axis of the
ultimately outermost body, then a fossilized scattered disk has the chance to persist. Otherwise,
the fossilized disk fatls interior to the final planetary system, and ail trace of it is cleared.
Can scattered protoplanets crossing the outer Solar System provide sufficient excitation of planetesimais to reproduce the present-day Kuiper belt? The eccentricities and inclinations raised by
a protoplanet can be directly obtained ftom the simulations of Chapter 4. The results of Runs 1A
and 1C will be used for cornparison. Planetesimal eccentricities can be seen in Figs. 4.8 and 4-10,
respectively. Inclinations for 1A are shown in Fig. 5.7, and for lC, in Fig.5.8.
From Petit, Morbidelli and Valsecchi (1999)' the median eccentricity and inclination in the
5. Scattered protoplanets and the small body belts
95
Semi-Major Axis (AU)
Figure 5-7: Counterpart to Fig. 4.8, s h o w h g inclination versus semimajor a.xisfor the endstates of the r u s
in Subset 1A.
classical Kuiper belt (a2 41 AU) are
-
0.07 and 4.0°, respectively. From Fig. 4.4, eccentricities in
this region range as high as about 0.4, and inclinations as high as about 30°. Tn both 1A and 1C,
the eccentricities are readily attained in the part of the planetesimal disk between the outermost
protoplanet and the maximum extent of protoplanet crossing. In fact, the eccentricities in the
nins are even higher on average; the median is
of
.-
-
0.25 or higher, wïth maximum eccentricities
0.8. The la& of such high eccentncities in the present-day Kuiper belt is to be expected,
however; orbits with eccentncities significantly higher than those presently observed cross Neptune
and are eliminated on timescales short compared to the age of the Solar System (Duncan, Levison
and Budd 1995). In any case, scattered protoplanets are e a d y able to excite eccentricities in the
5. Scattered protop1anet.s and the small body belts
96
Figure 5.8: Counterpart to Fig. 4.10, showing indination t-ersuç sernimajor axis for the endstates of the
runs in Subset 1C.
classical Kuiper belt as high as the present ones.
The median inclinations in 1A are
high as
-
-
5" - IO0 in lA, and IO0 or more for 1C. Inclinations get as
30°. The on average higher degree of excitation of the planetesimais in 1C is due to the
fact an extra protoplanet perturbs them. Thus the scattered protoplanets readily reproduce (and
exceed) the median inclination in the classical Kuiper belt, and o d y just reproduce the maximum
inclinations. Bodies on highly inclined orbits, unlike bodies on highly eccentric orbits, are not more
likely to later undergo catastrophic close encounters with Neptune; on the contrary, since they
spend much OF their time above or below the orbital plane of Neptune, they are less likely to have
close encounters than low-inclination bodies.
5. Scattered protoplanets and the smaü body bel&
5.3
97
Discussion
In addition t o explaining the ongin of Uranus and Neptune, the scattered protoplanet model may
also help account for the anomalously high dynamical excitation and m a s depletion found in the
present-day asteroid and Kuiper beIts- Scattered protoplanets are unable to significantly excite
more than the outer asteroid belt. However, bodies of
-
1Me or larger which actuaily grow in t h e
belt wodd have a long enough residence time-several Myrs-to
cause the required excitation and
mass depletion. Such asteroid belt planets might then be destabilized and removed by interactions
with scattered protoplanets, leaving the asteroid belt in its present planet-Iess state.
The sweeping of the gas giants' secular resonances, either from t h e changing gravitational potential as the nebular gas disperses or from radial migration of Jupiter and Saturn themselves,
also provides a means of increasuig planetesimal eccentricities and inclinations in the asteroid belt.
In the latter case, the normally assurned range of migration for Jupiter and Satuni passes the
inclination-purnping v l ~resonance through only the inner and part of the middle belt. However,
if Jupiter and Saturn start out on more closely-spaced orbits-which
protoplanets orïginally existing in the 5 - 10 AU region-the
is not unlikely if numerous
required inclinations can be excited
throughout most of the belt- If gas is still present in the asteroid belt a t this time, bodies a few
10s of kilometres in size or less o n very eccentric/inclined orbits will experience significant orbital
decay, though the targer ones will not be d e c t e d directly.
In the mode1 of Petit, Morbidefi and Valsecchi (1999),large Neptune-scattered planetesimals
(LNSPs) are responsible for exciting the Kuiper belt. In the scattered protoplanet model, however,
it is proto-Neptune and/or -Uranus themselves which stir the belt. Both the eccentricities and
inchations of the Kuiper belt can be raïsed to their present value in this way. The issue of
mass depletion was not investigated in detail, but it appears that on the timescale over which the
protoplanets cross the Kuiper belt,
2 10 Myrs,
a significant fraction of planetesimals is not lost,
5. Scattered protoplanets and the smaii body bel&
98
However, mass Ioss near the inner edge of the belt d l take place over longer timescales, as objects
evolve onto Neptune-crossing orbits under the action of Uranus and Neptune resonances located in
the belt. LNSPs may also be Ïnvolved- According to the analyis in Chapter 3, bodies of
1 Me are
unlikely to have formed in the outer Solar System, but such bodies would have inevitably existed
together with the 10Me protoplanets in the 5 - 10 AU region, and would have also been scattered
out during Jupiter's final growth phase.
Some caveats need to be pointed out, however. Fust, there is a moderate observational bias
against detecting KBOs on hi&-inclination orbits, thus objects with higher inclinations than the
highest currently detected,
-
33", likely ecist- AIso, the observed KBOs are all
-
100 km or
larger in size, and constitute the upper end of the mass spectrum of the bodies in this region.
The tendency toward energy equipartition then implies that the smaller, unseen bodies have even
higher random velocities, However, both of these problems apply equally whether the excitations
orïginate fkom protoplanets or fkom LNSPs-
SUMMARY
6.1
In-situ formation
The conventional picture of Uranus and Neptune's formation, whereby the ice giants accrete near
their present heliocentric distances, has grave problems. Numerical simulations have not been able
to produce
-
10 Me objects in the trans-Saturnian region in the lifetime of the Solar System without
sigaificantly increasing protoplanet radii, neglecting long-range gravitational forces, increasing the
mass of the protostellar disk well above realistic values, or some combination thereof. During the
time the nebular gas is present, gas drag is able to keep the planetesimal disk dynamically cooler
than it wodd otherwise bey by serving as a sink for the energy of random velocities raised by
growing protoplanets. However, for reasonable gas and pIanetesimal densities, this is not enough to
produce anythïng even approaching the m a s of Uranus and Neptune by the time the gas disperses,
-
10 Myrs. ,4fter the gas is gone, growing Uranus and Neptune becomes an even more difficult
proposition; in the simulation performed, less accretion takes place on a Gyr timescale after the gas
disperses, than during the previous 10 Myrs with gas. If the gas giants are to have grown in the
absence of gas, as-yet unmodeled effects (eg. collective phenornena, collisions) would have needed
to be very efficient at keeping the disk dynamically cold-
6.2
Plzt.net formation in the Jupiter-Saturn region
In the andytic approximation of Chapter 3, the sharp falloff of protoplanet size with heliocentric
distance a t a given t h e means that planet formation should proceed in an outward-expanding
wave of oligarchie growth. In a moderate-mass solar nebula, this wave will reach the Jupiter-Satum
region in the lifetime of the gas. When the gas is then removed, it appears the wave will essentidy
halt, at least as far as
-
1Me or larger objects are concerned- Although this means Uranus and
Neptune will not form in the trans-Saturnian region, one expects that the Jupiter-Saturn region
will produce more than just two giant planet core-sized bodies. It shodd be emphasized that
the simulations perfonned here are not intended to directly dernonstrate this; fully sirnulating the
growth of these protopianets is beyond the scope of this thesis- The simulation of the JupiterSaturn region performed in Chapter 3, Run A, does show that the aforementioned accretion wave
reaches this region in the predicted t h e , but it is only run for 4 Myrs. The simulation does suggest
that a gas and planetesimal density somewhat higher than t h e assurned one is required to grow
giant planet core-sized bodies during the gas Iifetime. The formation of a gas envelope on the
protoplanets should also increase the growth rates, as shodd the production of a population of
small collision fragments, whose random velocities will be even more aectively damped by gas
drag. The important point is that bodies of the right order of magnitude in mass can be fonned in
107 years at a 5 10 AU, but not a t a 2 10 AU.
6.3
The fate of Jupiter's neighbours
The focus of this thesis is to simulate the evolution of the outer Solar System starting toward the end
of the gas lifetime, when a number of
-
10Me objects are assurned to have formed at a heliocentric
distance of roughly 5 to 10 AU. It is shown that the accretion of Jupiter's gas enveIope, resulting
in a thirty-fold increase in mass on a timescale of
-
105 years, causes the remaining protoplanets
to become violently unstable- In most cases they are scattered ontu high-eccentricity orbits in the
trans-Saturnian region. With most of its orbit now crossing t h e largely pristine trans-Saturnian
planetesimal disk, a scattered protoplanet experiences dparnical fiction and has its eccentricity
rapidly darnped- As a result, the protoplanets tend to end u p on circular, coplanar and welispaced orbits on a Myr timescale, with semimajor axes comparable to those of Saturn, Uranus and
6- Slimmary
101
Neptune. Thus, Uranus and Neptune could have shared the same birthplace as the gas giants-
A mode1 with some similar features was proposed by Zharkov and Kozenko (1990)- They
suggested a more drawn-out scenario: Jupiter &st ejects a
-
5 Me planetary embryo d e g its
final growth phase. This ernbryo than acuetes more planetesimaIs, as weU as gas, and dtimately
grows into Saturn. Once Saturn's growth is complete, more embryos of a few Me are ejected
outward- These then grow into Uranus and Neptune- Ipatov (1991) perfonned simulations based
on this idea, using a code which considered only close encounters with embryos, and selected
encountering objects probablistically- He found t h a t under the influence of Jupiter and Saturn,
bodies of a few Ma initially slightly outside the orbit of Saturn would be scattered outward, that
interaction with the planetesimal disk would reduce their initially high eccentricities, and that they
could eventually end up at semimajor axes similar to those of Uranus and Neptune- In contrast,
the simulations performed here show that the growth of Jupiter to n e a . its final mass will clear
the region around it of all protoplanets on a short tirnescale, and interactions among the crossing
protoplanets will ensure that their orbits are weU-spaced, so that all except perhaps the innermost
(proto-Saturn) end up a t heliocentric distances where accretiond growth is very slow. Also, a t
this point, the nebular gas will only persist for a few more Myrs. Therefore, it is more likely that
Saturn, Uranus and Neptune's cores have completed most of their growth by this tirne.
The scattered protoplanet mode1 is attractive not only because it sidesteps the t h e s c a l e problem of in-situ formation of Uranus and Neptune, but also because it is remarkably robust. The
simulations show that the final growth phase of Jupiter inevitably triggers scattering of the nearby
protoplanets. Also, the subsequent circularization of the orbits occurs with a high probability, without strong dependence on the density profile of the trans-Satumian planetesimal disk. Furthermore,
a Solar System-like configuration can be achieved even if an extra protoplanet originally existed,
or if Jupiter did not grow from the innermost protoplanet. In fact, if the Jupiter-Saturn region
did indeed produce surplus giant protoplanets, in addition those which became the gas giants, it
becomes a challenge t o explain why such bodies wodd not be present in the SoIar System today.
An aspect not modeled in any of the simulations is the gravitational interaction of the bodies
with a gaseous disk, It has been shown that gas disk tidd forces can cause rapid inward migration
of protoplanets (eg. Ward 1997). In fact, the speed of migration may be peaked for objects of
-
10 Ma, taking pIace on timescales of 10' years or less- This peak corresponds to the transition
between so-called Type 1 migration, where a body's resonant interaction wîth the gas disk gives
rise to a torque imbaiance, to Type 11 migration, where the object opens a gap in the gas disk and
is subequently locked to the disk's viscous evolution. Tidal migration therefore poses a problem
for any model of giant planet formation: how do they avoid spiralling into the central star as they
fonn?
The above results, however, were obtained for a singIe body interacting with a disk. Multiple
bodies in close proxünity, perhaps with overlapping resonances, may weaken the influence of t i d d
torques, Indeed, recent work shows that a t least for Type II migration, adding a second body can
result in fundamentally different behaviour (Kley 2000, Masset and Snellgrove 2000).
The possibility that the gas disk was truncated early on also has a bearing on the issue of tidai
migration (NAMurray, private communication). Since the larger, negative part of the tidd torque
arises fiom the part of the disk outside of a planet's orbit, a planet near the outer edge of a gas disk
wiil migrate outward instead of inward; in the absence of competing effects it d l increase its orbital
radius until i t experiences no more net torque, which happens when i t is beyond the disk edge by
a distance of about twice the disk scale height (Ward 1997). Of course, any scattered bodies which
end up on orbits a t that distance or more will also not be subject to disk torques. Shu, Johnstone
and Hollenbach (1993) show that photoevaporation by the proto-Sun provides a way of removing
the outer part of the gas disk; furthemore, they find that the natural boundary outside which the
gas is lost is near Saturn's orbit. Another mechanism for truncating a gas disk is photoevaporation
by external sources. Structures called proplyds, seen in the Trapezium region of the Orion Nebula,
show this mechanism in action. Proplyds are young stars surrounded by cornet-shaped ionized
envelopes. The envelopes are produced by radiation fiom the brightest Trapezium stars interacting
4 t h the young stars' circumstellar disks- (Johnstone, HoIlenbach and Bally 1998).
6.4
Evidence and predictions
Both the asteroid belt and the Kuiper belt exhibit a degree of dynamical excitation and mass
depletion which is not readily explained, though several different models &st-
The simulations
perforrned in Chapter 4 show that the scattered protoplanets tend to cross both regions, Fvhich
suggests that they may play have played a role in dynamicdy sculptixig the small body beltsThe simulations presented in Chapter 5 show that scattered protoplanets are unlikely to have
directly afFected more than the outer asteroid belt. More indirectly, they may have played a role
in clearhg planet-sized bodies from the asteroid belt- Such bodies, in turn, could account for both
the excitation and mass depletion of the belt pnor to their rernoval, Also, an existing mode1 for
expIaining the asteroid belt's high random velocities-sweeping
by the giant planets-can
by secular resonances produced
be made more effective if Jupiter and Satum are originally in closer
proximity than is usually assumed, A Jupiter-Saturn region crowded with protoplanets is a good
precondition for this to occur.
The scattered protoplanets spend more time crossing the Kuiper belt than the asteroid belt,
and so they can raise Iarger random velocities there. In fact, the simulations of Chapter 4 show
that the protoplanets raise eccentricities and inclinations comparable to those in the present-day
Kuiper belt in the part of the planetesimal disk they cross. The disk will be strongly perturbed out
to the maximum aphelion distance any of the protoplanets acquires- Once the protoplanets' orbits
are circularked, this leaves behind a population of high-eccentricity, high-inclination planetesimals
which are no longer encountering the source of their dynamical excitement. If this 'Yossilizedn
scattered disk extends out to
-
50 AU (the extend of the observed Kuiper belt) or more, this would
account for the Kuiper belt's high random velocities. But then one would also expect to h d ,
at some larger heliocentric distance, a sharp transition from an excited disk to a two-component
disk-a
cold population plus a scattered hi&-eccentncity popdation with perihelia a t the transition
distance. Future observations of the Kuiper belt will reveal whether or not such a transition exists.
A recent h d i n g regarding the deutenum to hydrogen (D/EI) ratios of Uranus, Neptune and the
comets is particdarly interesting in the context of the model proposed here. Using observations
made with the Infrared Space Observatory, Feuchtgruber et ai (1999) find that the D/H ratios
of the ice giants are Iower than average cornet values by a factor of appromately three. This
discrepancy presents a further problem for any scenario in which Uranus and Neptune fonn in
the trans-Saturnian region, since they should then share the chemicd composition of the comets.
However, this is exactly what one would expect if the ice giants origina.iIy formed a t a srnader
heIïocentric distance, where higher temperatures would have made for a lower D/H ratio.
On a more speculative level, it is interesting to consider the scattered protoplanet model for
Uranus and Neptune's origin in the context of the discovered extraso1a.r pIanets (see Marcy and
Butler 1999 for a review)- The majority of extrasolar planets with semimajor axes greater than
0.2 AU, not subject to tidal circularization, have eccentricities greater than 0.2. Some have eccen-
tricities as high as
-
0.7- Such eccentricities are likely t o have arisen fkom scattering events among
giant planets (Weidenschilling and Mamari 1996). This implies the formation of giant pIanets in
close proximity to each other (unless migration subsequent to their formation brought them together; see Kley 2000). Such a scenario is very much consistent with the initial condition proposed
here, of numerous potential gas giant cores originally exïsting in a fairly compact region- One can
easily envision a variation on the model, in which two adjacent protoplanets undergo runaway gas
accretion a t nearly the same time, and scatter each other onto eccentnc orbits. An eccentric body
of
-
102Me would encounter less than its own mass in planetesimals interior to .- 5 AU, and thus
its orbit could not readily be cucularized by dynamical friction (though interaction with the gas
6. Summary
105
disk, a s long as it still persists, wiil likely pIay a role). A phase of violent evolution during giant
planet formation-dbeit
with a different outcome-may
be something our Solar System has in
cornmon with eccentric-planet systems like HD89744.
The scattered protopIanet mode1 impiies that any planetary system containing gas giants is
Iikely to have ice giants a t larger distances fiom its parent star- Conversely, the mode1 aIso predicts
that systems which do not fonn gas giants will not contain ice giants with orbital radii as large 9
those of Uranus and Neptune. In fact, since the reason for not fonning gas giants is likely to be a
less massive initial protoplanetary disk (Ida and Kokubo 2000), systems devoid of gas giants may
not contain any Uranus- and Neptune-sized planets at d.
6.5
fiture work
The model developed here starts the dock, so to speak, a t a time when the planetesimal accretion
phase of the gant planets is Iargely over. The most fundamental improvement t o make to this
model is to start the clock at an earlier time, and to assess how realistic the initial conditions
assumed here for the Jupiter-Saturn region really are. In essence, two problems have to be solved
at the same time: one must simulate the formation of Jupiter and Saturn, and see if Uranus and
Neptune really do form in the bargain.
A new mode1 should also treat the as-yet unmodeIed eEects: gravitational interactions with
a gas disk, and interactions among the planetesimals. Both changes will be nontrivial to impIement numerically, and a hybrid code will be required. In addition to a full N-body treatment
for protoplanet-cIass bodies, such a code must model gas hydrodynamics (a smoothed-particle hydrodynamics approach is the obvious choice), and handle the interactions among a huge number
of planetesirnds in a computationally feasible way (perhaps using tree or g15d rnethods, or treating them statisticdy). The high-velocity impacts among planetesimals which characterize the
oligarchie phase also necessitate the inclusion of a realistic fragmentation model-
Another issue intimately related to the accretion of the giant planets is the formation of the
Oort cloud, a spherical dist~butionof cornets extending fÏom 2000 AU to greater than 15000AU
(eg. Duncan, Quinn and Tremaine 1987, Dones et ai 2000). These aze bodies which were scattered
outward by the giant planets, then decoupled by Galactic tidal forces. Thus in a simulation of the
type described above, the flux of bodies scattered to such large heliocentric distances can give an
indication of whether the formation of the Oort cloud began as the giant planets were formîng, or
was delayed until the planetesimal disk had been substantially depleted, as has been suggested by
Stern and Weissrnan (2000).
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A. GLOSSARY OF SYMBOLS, ABBREVIATIONS AND
TERMS
77 The fractional difference between the local Keplerïan velocity and the gas orbital velocity (dehed
after Eq- 2.31).
VG
T'ne resonance involving comrnensurability of W precession with that of Saturn.
~ 1 6The
resonance involving comrnensurability of fl precession with that of Saturn.
p Volume mass density
p,,,
PM
Volume mass density of gas
Volume mass density of a body of mass M.
Volume mass density averaged over a swarm of bodies, each having mass rn or
resp ectively.
ps,,,ps,,
ml
u The surface mass density of a disk of solid bodies.
um The surface mass density of a disk of bodies each having mass m.
cmi, The minimum primordial surface density of planetesimals, Eq. 3.9.
R Angle between a reference directing and an orbit's line of nodes (the line of intersection between
the orbit's plane and the reference plane)w Angle from an orbit's intersection with the reference plane to its perihelion, measures in the
plane of the orbit.
W The longitude of perihelion of an orbit, S I + W .
flK The Keplerîan frequency, vK/a.
a Semimajor axis of the conic section described by a Keplerian orbit-
Aa In the oligarchie phase, the width of the zone dynamïca3ly heated by a protoplanet (Eq. 2.39)-
A U Astronomicd unit, the length of the Earth's semimajor axïs (1.5 x 1013c m )
Cd Drag coefficient (first used in Eq, 2.25).
CJ The Jacobi constant, the only integral of the circu1a.r restricted three-body problem; Eq. 4.5.
e Eccentricity of an orbit. The paths of unperturbed Keplerian orbits are conic sections; a circle
for e = O, an ellipse for O < e < 1, a parabola for e = 1, and a hyperbola for e > 1.
em Eccentricity of a body of mass m.
A. GIossary of symbols, abbrevicrtions and terms
113
(e$)'i2 Root mean square eccentricity of a popdation of bodies, each having m a s m.
F, The factor by which gravitationai focusing enhances a body's collision cross-section relative to
its physicai cross-section, Eq. 2.9,
h The scaie height of a planetesimai disk
He,, Difference hetween origind and surrogate Hamiltonian; Eq. B -4.
Hint The interaction part in the Hamihonian decomposition of Wisdom and H o h a n (1991) (Eq.
B.7) or that of Duncan, Levison and Lee (1998) (Eq- B.15)HKep Th2 I<eplerian part in the Hamiltonian decomposition of Wisdom and Holman (1991); Eq.
B.6 or that of Duncan, Levison and Lee (1998) (Eq-B.13)-
HsU, The part of the Hamiltonian due to the motion of the Sun in the decomposition of Duncan,
Levison and Lee (1998); Eq. B.14.
i Inclination of the plane of an orbit, usually measured relative to the plane of the net angular
momentum of a system.
,i
Inclination of a body of mass m
(iL)'/2 Room mean square inclination of a population of bodies, each having mass m.
hm reduced (nondimensional) HiIl radius, rH,/a
high-Z material Elements with an atomic number > 3, i-e- heavier than hydrogen and hefium.
K Gas drag parameter; Eq. 2.25.
KBO Kuiper bek object.
M Usually used to denote a m a s , except on p. 4.5-2, where it denotes the mean anomaly- The
mean anomaly of a body is its angu1a.r location on its orbitMeit Mass at which the escape velocity from the surface of the body is equal to the escape velocity
fiom the Sun a t the body's heliocentric distance; Eq-3.18.
Me The mass of the Ecuth, 6 x
Ma The mass of the Sun,2 x
l ~ ~ ~ g .
g-
M, The mass of a general central body, not necessarily the Sun.
oligarchie growth State of accretional growth within a planetesimal disk in which the dominant
growth mode is rnerging of planetesimals with an embedded population of protoplanets.
Also, it is gravitational interaction with the protoplanets, rather than interactions among
the planetesimals, that controls the velocity evolution of the planetesimals. See p. 15.
p The fraction of the m a s in its feeding zone which a protoplanet has accreted (first usage:
2.45).
planetesima2 A solid body
planetary system.
5
Eq.
100 km in size, usually used t o refer to a body in a primordial
A. G1ossa1-yof symbols, abbr-ations
and terms
114
protoplanet A body somewhat arbitrariIy d e h e d as %uch larger than a pkinetesimaln. For the
purposes of this work, a protoplanet is usually taken to be any body large enough to IocaUy
dominate the dynamics of the planetesimai disk.
q Perihelion distance: q = a ( 1 - e)-
Q Aphelion distance: Q = a(a + e).
TH^ The Hill radius of a body of m a s m,
Eq-2.4- f i o written rH, when not referred to a specific
mas
THM~M?
The mutuaf Hill radius of two bodies of masses M land M2; Eq. 2.44,
R, The collision cross-section for two bodies, taking into account gravitational focusing (Eq, 2.18
RM The physical radius of a body of mass M.
T Tisserand parameter (Eq.4.7)
TconTimescale for physical collisions among planetesimals in the oligarchie phase; Eq. 2.32.
Te Timescale for eccentricity decay due to gas drag; Eq-3-1.
TeeIl The timescale for a body undergoing dynamical fnction to have its eccentricity reduced by
a factor of two; Eq. 4-11.
Tg,,Timescale used in calculating gas damping rates; Eq. 2.31.
Tg,,, The growth timescale of a protoplanet (Eq. 2.12)T . - m The timescde for increase of the R M S eccentricity of a swarrn of pIanetesimals of mass m,
stirred by a protoplanet of mass M.
v e , ~The escape velocity fkom the surface of a body of mass M; u . , , ~ =-2J
VK
Keplerian velocity- A body on a circular orbit of radius a about a body of mass M has
VK =
u,
VeIocity dispersion, relative to local Keplerian velocity, of a population of bodies of mass m
(Eq.2.3).
v,,l
d m .
Relative velocity.
B. THE SYMBA INTEGRATOR
The simulations in this work are performed using the SyMBA (Symplectic Massive Body Algorithm)
integrator, either in its m e n t base form (SyMBA5) or with some modifications by the author.
SyMBA is described in detail by its authors in Duncan, Levison and Lee (1998), as weii as in
Lee, Duncan and Levison (1997). This appendix serves only as a brief summary of the principles
underlying the SyMBA algorithm.
B.1 SympIectic integrators
The time evolution of a Hamiltonian system is a canonical or symplectic mapping, which has the
consequence that the mapping preserves volume in phase space (a review is given by Sam-Serna
and Calvo 1994). A symplectic integrator has the same property, withii machine precision- Given
a Hamiltonian H, a symplectic integrator actually solves a surrogate Hamiltonian H which ciiffers
slightly fiom H, i.e.
all-symplectic
H
= H
+ He,.
Another desirable quality possessed by many-but
not
integrators is time-reversibility,
An autonomous Hamiltonian system of N bodies has equations of motion
where H(w)is the Hamiltonian jwithout explicit time dependence), w = (q, p) are the 6N canonical
phase space coordinates and the braces denote the Poisson bracket- The solution of B.1 is
Of interest here is the case where H c m be broken up into two parts: H = Hoand H l , each of
115
B. The S y m A integrator
116
which is seperately integrable. Then the evolution over a timestep r can be approximated by this
second-order operator composition scheme:
It can be shown that the Merence between the original and the surrogate Hamiltonian is a series
in r ; for the second-order scheme, it is
Thus for a suEcient1y s m d r, He,, remains smaIi, and the energy error is bounded.
In the case of the SoIar System, where to lowest order al1 bodies are in KepIerian orbits around a
dominant central mass, the Hamiitonian can be divided into a part describing the Keplerian motion
of the bodies about the Sun, and a part due to gravitational interactions among the bodies:
In the scheme of Wisdom and H o h a n (1991),
and
in^ =
Grni,,)
(-GIJTL~TTZ.
-"i
where
rio
Ti0
-= 2
i=i j=i+i
Gmimj
7
(8-7)
Tij
is the distance from the i t h body to the Sun, and the primed varîables are Jacobi
coordinates- In Jacobi coordinates, the position and momentum of the ith body are taken relative
to the center of mass of the bodies with index j
< i. This method of dividing the the Hamiltonian is
commonly referred to as the mixed-variable symplectic (AWS) scheme- As long as close encounters
arnong bodies or between a body and the Sun do not take place, this method has a large speed
advantage over other planetary system integration schemes; the condition
B. The SyMBA integrator
117
can typicdy be met to a degree sufficent to keep the energy error acceptably low with a timestep
r around 1/20th of the shortest orbital penod-
6.2
Varying the time resolution
In order to obtain both accuracy and speed in a problem where timescdes vary widely, it is desirable
to have an integrator with an adaptive timestep. In this way close encounters and small perihelia
can be resolved when they arise, without the integrator taking a computationdy costly number
of steps when it isn't necessary. However, simply detennining a new timestep based on the phase
space coordiates at the beginning of the timestep is problematic- It destroys the time-reversibility
of the a l g o f i t h , since the new timestep is not determined in a time-symmetric manner. Also,
when r changes, so does He,,,
and thus the surrogate Harniltonian will in generai diverge from the
true Hamiltonian over tirneThe principle used in SyMBA to overcome this problem is to decompose the interaction part of
the Hadtonian. For simpiicity, one can first consider the Kepler problem with Ho = T ( p )= lpj2/ 2
and
Hl= V ( r )= -Gmomi/r,
r = 1q1- Concentric shelis with radii rl
> r2 > ... are placed around
the central body, and the poteatial V, or equivalently the force F = -LW/&-, is decomposed into
a series:
The force Fi is applied with a timestep
Ti.
If the ratio
T ~ / T ~=
+ ~Mit*
is an integer, then a
second-order scheme analogous to that of B.3 can be applied recursively. Wnting D = (, T') and
Ki = (, F) (visuaüsable as a "drift" and a "kick", respectively), one obtains
The base timestep of this algorithm is 70,but it is effectively a multiple-timestep method because
the truncation point of the series in Ki for a body is determined by the location of the body- For
B. T h e SyMBA iitegrator
a body a t radius ri
be
118
> r > ri+i, Ki= O, i > i. In SyMBA, the ratio of successive radii is chosen to
~ k / r k + 1= 32/3= 2.08.
The ratio of the timestep in successive shells is M = 3-
The force decomposition chosen for SyMBA is
with
where fi(x) = Sx3 - 3x2 4- 1 is a partition function which makes
l/r2 at r
2
rk
to O at
T
IFk-r
1
decrease smoothly from
< k f 1. The rationale behind this force decomposition is described in
detaii in Duncan, Levison and Lee (1998).
B.3
The democra tic hefiocentric scheme
The above force decomposition could in principle be used for an MVS integrator, by applying it to
the second term of Eq. B-7. This would retain the speed of the MVS method far from encounters.
However, because of the MVS method's use of Jacobi coordinates, this approach does not provide
a satisfactory resolution of close encounters. There are two reasons for this- First, in the MVS
scheme with a fixed timestep, HKep and Elfit add up the the actual HamiItonian because the
terms Gmimo/ri cancel. However, if EKepand the first term of Hint are applied with different
fiequencies, as happens when the force is decomposed, this canceilation no longer occurs. Secondly,
since in the Jacobi coordinates each body's position and rnomentum are taken relative to all bodies
wïth lower indices, the ordering of bodies in semimajor axis must not undergo large changes, or
energy conservation will be poor- But such teordering is exactly what happens in a system where
bodies undergo close encounters-
In SyMBA, this problem is solved by using a "democratic heliocentric" scheme, whereby al1
bodies orbit a common central mass in the Kepler part of the decomposed Hamiltonian, A new set
B. The SyMBA integrator
119
of conjugate coordinates, Qi and Pi, are used- The former are the heliocentric positions for i # O,
and the position of the center of mass for i = O. The latter are the barycentric momenta for i # 0,
and the total momentum of the system for i = O. The Hamiltonian in terms of the new coordinates
is H(Qi, Pi) = H
+ Nsun + Hint, where
~ e p
and
This decomposition of the Harniltonian is used to construct a second-order symplectic integrator:
where E, denotes the evolution operator -mder Hz.
With respect to Hint, there are two classes of bodies in SyMBA- Tbose in the first class interact
with each of the other bodies. The ones in the second class interact only with those in the &st class,
e
bodies are useful in situations
thus they form a non-self-interacting population. ~ h second-class
where a simulation contains a large number of bodies whose dynamics are dominated by a s m d e r
population of larger bodies, so that the smaller bodies' mutual interactions can be neglected. This
can provide an enormous savings in computational tirne, which for N second-class bodies scales as
N, as opposed t o N2 for the My-interacting first-class bodies.
C . GAS DRAG IMPLER.IENTATION IN SYMBA
The author modified the swiftsymba.5 version of the SyMBA package to add m o d e h g of gas drag;
the modified version is called swiftsymba5dr. The relevant subroutines are avaiIable on request.
As discussed in Appendix B, the fundamental SyMBA integration step consists of a haif-timestep
"kick" (calculation of accelerations due t o other bodies), a full-timestep ''driftn (movement dong
a Keplerian orbit around the central mass) and another hdf-timestep kick- ActuaI movement of
bodies only occurs during drifts, so this is where the drag acceleration is applied.
C.1
Input parameters
Modelling of gas drag requires additional input parameters. First, there are the values needed
to calculate the drag parameter, 3p,,,Cd/8fpr7-. The gas density is specified by RHO-GAS-IAU
and M O - G A S E X P , where rho,,. (a) = R H O - G A S I A U ( ~ ) ~ - ~ * C
~D
. is the drag
coefficient- The user is given the option of either entering the planetesimal density RHOPL and
radius S P L manudy, or using the actual density and radius of each body experiencing drag.
Another input parameter is the fraction of Keplerian velocity a t which the simulated gas disk is
to rotate (KEPBACTOR). This is used t o fùid v,,~. Since the gas is partially pressure-supported,
this value will be Iess than one for a real gas disk.
Finally, the user can determine a t how deep a recursion level the drag will continue to be
appiied (MAXDRAGIREC). In the past, other versions of SyMBA modified to include gas drag
have simply switched off drag for bodies undergoing close encounters. However, swiftsyrnba5dr can
continue to apply the drag at any level- This is important because during close encounters, high
C. Gas drag implementation in SyMlBA
121
velocities arise. Consequently, drag forces are also high, and a body can lose a significant amount
of energy in the course of one encounter-
In choosing a maximum drag recursion level, two considerations should be kept in mind, F i s t ,
the deeper the drag is applied, the higher the computational cost. The energy loss during close
encounters c m pIace bodies in decaying orbits about each other. Thus, two bodies which would normaiIy only require short timesteps during a brief encounter, now must be tracked through numerous
orbits with short timesteps. This causes a major increase in the integration tirne. Therefore, it is
not recornmendable to use too large a maximum drag recursion level; a reasonable d u e depends
on the density of bodies (and thus the frequency of close encounters), and the strength of the drag
force, in a given integration.
Secondly, it is uncertain how a gas nebula would behave in the immediate vicinity of a larger
body. Modelling the drag coefficient a s being constant right down to the body's surface (or the top
of its atmosphere) is aimost certainly an oversimplifkationThe ratio of successive recursion shells, RSHELL, as well as the distance a t which a close encounter commences, RRSCALE, can be adjusted in the include file symba5.inc. Both are in units
of H i radii. The default values are RSHl3LL= 3-2/3,and RHSCALE= 6.5- Thus, successive shell
radii R, have values of
& =6 . 5 ~ ~
Ri = 4.51
R2 = 3.12
R3 = 2.17
& = 1.50
R5 = 1.04
Rs = 0.72
C- Gas drag implementation in SyMBA
C.2
122
Cdculating and applying the drag acceleration
Calculation of the acceleration on a body due to gas drag is carried out in the subroutine SYMBA5DRDRAG
and proceeds in several simple steps. The positions and velocities of the body as well as the cur-
rent tirnestep are input to the subroutine in Cartesian barycentric fonn, and it outputs the new
velocities. First, the heliocenfxic distance of the body is computed, and the Keplerian velocity at
that distance- The velocity of the gas disk at this distance is then computed, by mdtiplying the
Keplerian velocity by KEPJ'ACTOR. Then the velocity of the body relative t o the gas is cornputed. The above information is then substituted into Eq. 2-24, and the x, y and z components of
the acceleration due to gas drag are computed- Using the current timestep, the new velocities are
computed.
This subroutine is called from the drift subroutines, HELIODRIFT and SYMBA5HELIODRIFT.
What was a Keplerian drift for a time dt, D(dt), now becomes bracketed by two "drag kicksn,each
for a time dt/2: &,,,(dt/2)D(dt)Kd,,,(dt/2),
The subroutine is not called for the J t h particle if
BLEV(J), the Jth particle's m e n t recursion level, is greater than MAXBRAGIREC-