_
_ ___j~lC _
_
NUMERICAL
I___
SIMULATION OF THE
TERRESTRIAL PLANET
___
FINAL
_I
_I_ ______~~ ______ ___i__X__~_
_I_ _C _
STAGES OF
FORMATION
by
O
S.B.
Physics,
LARRY PAUL COX
Massachusetts
(1969)
Institute of Technology,
SUBMITTED IN PARTIAL FULFILLMENT
OF THE
REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF
at
MASSACHUSETTS
PHILOSOPHY
the
INSTITUTE
OF TECHNOLOGY
(May, 1978)
Signature of Author
Department of Earth
. . . . . . . . . . . .
and Planetary Sciences,
Certified
.
by
.
.
.
Accepted by
Chairman, Departmen
.
.
.
.
.
.
.
. . . . . . . . .
Thesis Supervisor
-1
Comm
eAaduate
MASSACHUS
TUTE
1978
LIBRARIES
. . . . .
May, 1978
IS
- ..
.
Students
In Anfwer to the Epicurean Syftem,
[he argues] How often might a Man, after
he had jumbled a Set of Letters in a
Bag, fling them out upon the Ground
before they would fall into an exaA
Poem, yea or fo much as make a good
Difcourfe in Profe? And may not a
little Book be as eafily made by
Chance, as this great Volume of the
World? How long might a Man be fprinkling
Colours upon a Canvas with a carelefs
Hand, before they could happen to make
the exaA Picture of a Man? And is a Man
eafier made by Chance than his Picture?
How long might twenty thoufand blind
Men, which fhould be fent out from the
feveral remote Parts of England, wander
up and down before they would all meet
upon Salisbury-Plains, and fall into
Rank and File in the exaa Order of an
Army? And yet this is much more eafy to
be imagin'd, than how the innumerable
blind Parts of Matter fhould rendezvouze
themfelves into a World.
John Tillotson
Archbishop of Canterbury
in Maxims and Discourses Moral and Devine
(London, 1719)
--~^-)
-I~----LLI.I
---
NUMERICAL SIMULATION OF THE FINAL STAGES OF
TERRESTRIAL PLANET FORMATION
by
LARRY PAUL COX
Submitted to the Department of Earth and Planetary
Sciences in May, 1978 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
Abstract
Four representative numerical simulations of the
growth of the terrestrial planets by accretion of large
protoplanets are presented.
The mass and relative
velocity distributions of the bodies in these simulations are free to evolve simultaneously in response to
close gravitational encounters and occasional collisions
between bodies.
The collisions between bodies, therefore,
arise in a natural way and the assumption of expressions
for the relative velocity distribution and the gravitational collision cross-section is unnecessary.
The relation of the present work to scenarios given
by Safronov (1969), Goldreich and Ward (1973), and
Greenberg et al. (1977) for the early stages of solar
system evolution is discussed.
A comparison of predictions of the commonly employed two-body gravitational
model with integrations of the equations of motion which
include the influence of the Sun indicate that the twobody model is inadequate for the treatment of encounters
between large protoplanets, particularly when they move
along low eccentricity orbits resulting in encounters
with low relative velocities.
A new model is, therefore, proposed for single close
encounters between two small masses, m I and m 2 , which orbit
a much larger mass, M.
Comparisons of predictions of the
model with integrations of the three-body equations of
motion indicate that the model is an adequate approximation
nr-~-----II-CI-YYII--~
< --5
10 5 M and E > 4,
he hyperbolic orbit of
ml about m 2 . The current simulations represent to my
knowledge the only model of planetary accretion taking
into account the presence of the Sun in the gravitational
interactions between bodies.
m
for those encounters with m 1
of
where E is the eccentricity
'
The protoplanetary bodies in these simulations are
assumed to move along unperturbed co-planar orbits until
they pass within a sphere of influence distance R s
R 0 [ml (m l + m 2 )/M 2 ]1/ 5 , where R O is the heliocentric
When a close approach to another body occurs,
distance.
new orbital elements are computed for each body either
by the model or by integration of the regularized
equations of motion, depending on the parameters of the
Protoplanets are assumed to accrete and form
encounter.
a body with mass equal to the sum of the masses of the
two incoming bodies when a close approach with separation
less than the sum of the radii of the bodies (assuming
lunar density material) occurs during integration.
All simulations begin with an initial system of one
hundred bodies of equal mass having a total mass equal to
The surface density of
that of the terrestrial planets.
the system is assumed to have an R- 3 / 2 dependence on
heliocentric distance between 0.5 and 1.5 AU and to be
The orbital eccentricities are randomly
zero elsewhere.
chosen from a uniform distribution on the interval from
0 to a given maximum eccentricity, emax These simulations indicate that the growth of bodies
with final masses approaching those of Venus and the Earth
is possible, at least for the case of a two-dimensional
Simulations with emax < 0.10 are found to produce
system.
final states containing too many bodies with a narrow mass
distribution, while simulations with emax > 0.20 result
in too many catastrophic collisions between bodies and
rapid accretion of planetary-size bodies is prohibited.
The emax = 0.15 simulation ends with a state surprisingly
similar to that of the present terrestrial planets and,
therefore, provides a rough estimate of the range of
radial sampling to be expected for the terrestrial planets.
Thesis Supervisor: John S. Lewis
Title: Associate Professor of Geochemistry and
Chemistry
- --i^-I 111~11~-~.-
ACKNOWLEDGEMENTS
The preparation of this
thesis was an effort
involving a number of people besides the author.
The topic emerged
as a result of conversations
with Myron Lecar, W.R. Ward, and J.S.
Lewis.
Throughout
the research the guidance of my advisor, Professor John
Lewis, has been of great help.
He has not only made
many helpful suggestions and
criticisms, but has provided
a good balance of enthusiasm
and healthy skepticism.
During
in 1976,
the sabbatical leave of John Lewis
Professor Charles Counselman ably filled
in as my
advisor and frequent discussions with him were very
helpful in constructing many of the algorithms employed
in the simulations.
This research has
from the suggestions, criticisms, and
Stephen Barshay and Bruce Fegley.
full credit for the
interest of
I, of
course, take
shortcomings which undoubtedly remain.
During my career in graduate
by
also benefited
school I was
supported
the Earth and Planetary Sciences Department and by
the Mathematics Department.
The computing
this research was provided by NASA
time for
Grant NGL-22-009-521.
I especially wish to thank my wife, Candace,
her encouragement,
support, and
love, which were
essential to me as well as to this research.
for
6
I also wish to thank Roxanne Regan for her efforts
in
typing
this
manuscript.
NUMERICAL SIMULATION OF THE FINAL STAGES OF
TERRESTRIAL PLANET FORMATION
TABLE OF CONTENTS
page
Frontispiece
2
Abstract
3
Acknowledgements
5
Table of Contents
7
List of Tables
9
I.
INTRODUCTION
A. The Absence of Known Counterparts of
Our Solar System
B. Requirements for a Successful Theory
of Solar System Formation
C. Cosmogonical Hypotheses of the Present
Simulations
D. Three Scenarios for the Early Stages
of Solar System Evolution
E. The Late Stages of Solar System
Evolution
F. The Numerical Simulations
II.
A NEW MODEL FOR CLOSE GRAVITATIONAL
ENCOUNTERS
A.
B.
C.
D.
E.
Introduction
A Description of the Model
Plausibility Arguments for the Model
The Algorithm for Applying the Model
Tests of the Model Against Numerical
Integration
F. Conclusions Drawn from the Numerical
Comparisons
G. Treatment of Encounters Outside the
Range of Validity of the Model
10
10
13
15
17
33
45
48
48
50
52
57
60
68
70
rir~i---ll~--~-
-ii_-xrla
-1L-ii-urmu
-rrrr~-----^-i
ra~---r^--~----ypWpn----rrrr-
page
III.
IV.
V.
THE MODEL FOR COLLISIONS BETWEEN BODIES
74
A. A Description of the Collision Model
B. An Estimate of Fragment Size After A
Collision
C. The Requirement of Total Energy
Conservation
D. Implications of Total Energy
Conservation
E. Do the Clusters of Fragments Shrink
to Solid Density Before Their Next
Collision?
74
THE COMPUTER SIMULATION PROCEDURE
90
76
78
82
84
A. The Basic Assumptions of the
Simulations
B. The Relative Importance of Close
Versus Distant Encounters
C. The Algorithm Used for Finding
Close Encounters
D. A Flowchart of the Simulations
96
103
THE COMPUTER SIMULATIONS
110
90
92
A.
Results of Four Representative
Numerical Simulations
B. An Estimate of the Evolution Times
Expected for Three-Dimensional Systems
C. Heliocentric Distance Sampling of
Material for the Final Bodies of the
Simulations
D. Conclusions Drawn from the Simulations
Regarding the Formation of the
Terrestrial Planets
E. Limitations of the Present Numerical
Simulations
110
127
130
134
136
References
140
Figure Captions
144
Figures
149
Biographical
Note
173
LIST OF TABLES
page
1.
Two-Body Gravitational Cross Sections
with Keplerian Differential Velocities
40
Gravitational Cross-Sections Obtained
by Integration
2.
Collision Statistics
(e
= 0.05)
116
3.
Collision Statistics
(e
= 0.10)
max
117
4.
Collision Statistics
(e
118
5.
Statistics of the Eccentricity Changes
Occurring for Encounters and Collisions
120
6.
Comparisons of Total System Angular
Momentum and Energy
127
max
max
= 0.15)
CHAPTER I.
A.
INTRODUCTION
The Absence of Known Counterparts
The study of
the origin of
of Our Solar System
the solar
system is
hampered considerably by the absence of any known
counterparts of our
evidence
(van de Kamp,
of Barnard's star,
)
<0.01 M
own system.
1969;
Except for the tentative
Gatewood,
no extrasolar
1972)
in the case
substellar mass
has been discovered and confirmed.
(mass
Because of
the many uncertainties
in the data,
difficult to determine
the characteristics of the Barnard
planetary system.
it is extremely
Black and Suffolk
(1973)
(1976) have suggested that if the residuals
and Gatewood
from recti-
linear motion are assumed to be reliable, two bodies
with masses
similar to that of Jupiter having periods
between ten and thirty years, respectively, are
Black and Suffolk further conclude
cannot be
noted
in co-planar orbits.
that the
However,
two planets
it should be
that the elimination of systematic errors
observations
conclusions
The
has been difficult and,
indicated.
in the
therefore, the above
are subject to change.
tentative detection of planet-like companions
of Barnard's star was entirely dependent on the proximity
of
the
star and the
small mass of the primary.
Since
I _
the existence of
1__
~_1111_11_1~
such objects can be presently determined
only by observing the perturbed motion of the primary and
since the motion of the primary relative to the center
of
mass would be highly complicated by the presence of
several massive planets, the observation of even a single
counterpart of
near future.
like
our own solar
Thus,
system seems unlikely in the
the chance of observing a terrestrial-
system of planets
is quite remote.
At the very
least we are, therefore, deprived of the insights that
might result
systems analogous
If
scheme
from a classification
to
those
for
stars
for planetary
and galaxies.
it is difficult to observe true counterparts of
our planetary system in their completely formed state, it
is that much more difficult to observe them in the process
of
planet formation both because of the small fraction of
their
age that is thought to be spent in planet formation
and because of the probable obscuration of such systems
by circumstellar dust grains.
Indeed, there
is a large
body of observational data on nebulae exhibiting strong
"excess" infrared emission which surround newly formed
stars.
These data yield valuable information about the
properties of the circumstellar dust grains surrounding
these stars and in the case of MWC
(1977)
have
349, Thompson et al.
inferred the existence of a preplanetary
~IY~P~I~~~ 11 ___I__?YI_
_I1YC______^
11
(Amalthea),
Titan),
the seven inner satellites of Saturn
(excluding
and the five satellites of Uranus are comparable in
mass, but are about two orders of magnitude less than those
of the main satellites.
These satellites also move in nearly
circular, except for Hyperion, prograde orbits
in the equa-
torial planes of their parent bodies.
However,
often substantial mutual perturbations
in the orbital
motions of these
of possible clues
satellite systems.
there are
Therefore, a number
to the problem of satellite formation
and perhaps planetary formation have been obscured by the
dynamical evolution of these systems subsequent to
their
formation in much the same way as in the planetary system.
If it
is difficult to observe true counterparts of
our planetary system in their completely formed state, it
is that much more difficult to observe them in
the process
of planet formation both because of the small fraction of
their age that is thought to be spent in planet formation
and because of the probable obscuration of such systems
by
circumstellar dust grains.
Indeed, there is a large
body of observational data on nebulae exhibiting strong
"excess" infrared emission which
stars.
sirround newly formed
This data yields valuable information about
the
properties of the circumstellar dust grains surrounding
these stars and in the case of MWC 349, Thompson et al.
(1977) have
inferred the existence of a preplanetary
._.
II-IC~L^
_I~___*I~___Y__LLIL__-s~~ -(^L11-l~-~
_~
_~_~ _I_
1_____
disk surrounding the star.
in obtain-
The real problem is
ing data on the larger bodies and this is observationally
very
difficult.
Assuming
models for nucleosynthesis,
classical
extinct radionuclides I
various meteorites
129
and Pu
244
studied so far)
have given
the
(for the
time intervals ranging
from 90 to 250 million years for the time of isolation of
the gas and grains later
forming the solar system to the
start of xenon retention in meteorites
bodies
like asteroids).
(presumably in small
These formation intervals then
indicate that no more than about 200 million years of
solar-system history, and probably less than that, preceeded the cooling of the parent meteorite bodies.
retention ages
(K
40
40
-A0 and U,
4
Th-He )
Gas-
further indicate
that the parent meteorite planets cooled approximately
4.5 x 10
years ago.
crust of the Earth is
trial rock.
Finally, the minimum age for the
the oldest measured for any terres-
At the moment
this
age
is
about 3.7 x 10 years.
Therefore, no more than a few times 108 years
for
is available
the formation of at least the Earth and Moon.
The processes leading
to the formation of
the solar
system, therefore, occurred approximately 4.5 x 10
years
ago and presumably consisted of a relatively rapid
succession of non-equilibrium early evolutionary phases.
not then surprising that many of
It is
the clues to the
nature of these processes have been obscured.
Consequent-
ly, a simple extrapolation backwards in time cannot lead
to a plausible picture of the origin and early
We then conclude that the chance of
solar system.
of the
evolution
observing a counterpart of our system in the process of
planet formation
B.
is
indeed
small.
Requirements For A Successful Theory Of Solar System
Formation
With incomplete knowledge of only one planetary
system,
solar
the problem of deciding which properties of our
system are "accidental"
or probabilistic
and which
are expected to be general to other planetary systems is
a delicate one
indeed.
data, we endeavor,
Undaunted by the paucity of our
like the ancient Greeks,
essential characteristics of reality.
propose theories of solar
for a
Yet, if we
are to
system formation, requirements
successful theory must be explicitly stated.
authors, Herczeg
among
to intuit the
(1972) , Woolfson
(1968) , McCrea
others, have constructed
characteristics of the solar
lists
planet formation.
(1969)
of fundamental
system to be used in verify-
ing any cosmogonical hypothesis.
proposed which is
Many
Such a
specific to theories of
list will now be
terrestrial
The criteria proposed here for a
-- ii---..~~
satisfactory theory are listed in a necessarily
order
subjective
importance.
of
(1) The existence and
approximate number of the
terrestrial planets must be accounted for.
(2) To a high degree of approximation planetary
orbits are co-planar and circular.
always less than 0.1 and
eccliptic
exceptions
the eccentricity of planetary orbits
of Mercury and Pluto,
is
With the
less than 3.5*.
the inclination
to the
On the other hand,
yielding too nearly circular and co-planar
a theory
orbits would
to
Also, a corollary of the coherence of
my mind
be suspect.
orbital
inclinations is that all of the planets are
observed to have a common sense
of orbital motion.
(3) A theory should lead to the
formation of terres-
trial planets of appropriate masses at appropriate heliocentric distances.
approximate one,
This requirement is necessarily an
since
the eventual discovery of even one
extrasolar planetary system with a terrestrial planet
configuration having planetary masses and orbital spacings
identical to our own would
(4) A
to me be extremely remarkable!
successful theory should also account for
axial rotations of
the planets.
With the exception of
Venus and Uranus, planetary rotation is predominantly
direct.
the
(5) The physical structure and chemical composition
of
the planets must be taken into account.
The terres-
trial planets in particular exhibit some characteristic
deviations from the cosmic abundance of chemical elements
and their
isotopes.
The most important deviations are:
(a) an almost complete absence of hydrogen and helium, and
a strong depletion of
some volatile elements, and
(b) an overabundance of deuterium,
lithium, and other
light elements relative to the present solar composition.
(6) As was discussed earlier, no more than a few
times
10
least
the Earth and Moon.
years are available for the formation of at
(7) A well-developed
theory might include the origin
of
the Earth-Moon system and we can hope for explanations
of
the absence of satellites for Mercury and Venus
the presence of Deimos
and
and Phobos.
C. Cosmogonical Hypotheses Of The Present Simulations
Many classifications of theories
the solar
system have been proposed,
thousands
of years old.
Herczeg
(1968) ,
(1963),
Schatzman
McCrea
(1972),
ter Haar
(1965),
and Huang
for the
formation of
some of which are
In this century Russell
(1948) ,
(1935),
ter Haar and Cameron
Williams and Cremin
(1972) among
(1968),
others have presented
__
In order to categorize the cosmogonical
classifications.
hypotheses of the present numerical simulations,
major divisions common to many of
are presented.
these classifications
Theories of cosmogony can be roughly
divided into three classes depending on the
intimacy of
the connection of the origin of the primitive
with the formation of the Sun
(A) The
the
Sun is assumed
solar nebula
itself.
to be a stable main-sequence
star with essentially the same
structure at the time of
formation of the planetary system as
today.
Here,
the
raw material of the planets is assumed to be captured
from an
interstellar cloud either by the intervention of
another
star or simply by accretion of infalling matter.
(B) The
Sun is assumed
to be
in a late phase of its
contraction at the time the planets originated.
dominant central mass
A pre-
is assumed to exist already, enforc-
ing Keplerian motion of the material in the nebula.
it is
inferred that the Sun was
stellar material,
solar nebula
forming
is
it makes for
formed from diffuse intereconomy to assume that
simply made up of material
the
left over in
the Sun.
(C) Here, the proto-Sun is
phase of
Since
assumed to be in an early
its formation, very extended and perhaps much
more massive
than the final,
developed Sun.
There may be
no strong central condensation at this
accretion is assumed to have started
that of the
Sun within the
stage.
simultaneously with
same nebula.
Capture and cataclysmic
hypotheses are
in
included
The classical hypotheses of Laplace and Kant
class A.
are perhaps the earliest representatives of
C,
Planetary
respectively.
classes B and
Many hypotheses rely on a very dense
gaseous medium to facilitate planetary accumulation, by
reducing relative velocities,
cence of solid particles.
thus promoting the coales-
However,
in the
current simula-
tions only a minimum amount of "circumsolar" matter
is
assumed, with a density corresponding to the total mass
of
the terrestrial planets spread over the terrestrial
region.
The current
simulation is based on theories of
type B.
D.
Three Scenarios For The Early Stages
of Solar System
Evolution
Ward
In this
section the scenarios given by Goldreich and
(1973),
Safronov
(1969),
and Greenberg et al.
for early stages of the evolution of the
solar
(1977)
system are
described.
These investigations provide a background for
the present
study of the growth of the largest bodies
the last
stage of terrestrial planet formation.
in
They also
permit an assessment of
the initial conditions used
in
the simulations to be described here.
The Goldreich-Ward
Scenario:
The Goldreich and Ward
accretion of planetesimals
In the first stage
from
scenario for the
is divided into four stages.
small particles are presumed to condense
the cooling primordial solar nebula as the vapor
pressures
of various constituents
pressures.
fall below their partial
After nucleation a particle is
settle through the gas
toward the
nebula while continuing
still
(1973)
equatorial plane of
at a rate given by Hoyle
The limiting radius a particle can reach
descending particle relative
in this way is
Using the velocity of the
to the
local gas determined
by the balance between the vertical component of
the gas drag force,
a limiting radius of R
%
terrestrial region.
The
estimated by augmenting
3 cm and a characteristic descent
surface density of
in the
the nebula was
the total mass of the terrestrial
planets up to solar composition
solar system.
solar
Goldreich and Ward obtain
time of ten years for the condensation of iron
inner
(1946).
estimating the rate at which a particle
descends to the central plane.
gravity and
the
to grow by collection of material
in the vapor phase
calculated by
expected to
and spreading it over the
The half-thickness
and resulting gas
density of the nebula are determined by the balance of the
vertical pressure gradient and the vertical component of
Because of the uncertainty of the number
solar gravity.
sites, the final particle radii may very
of nucleation
well be much smaller than the estimated 3 cm especially
in
light of the fact that no particles approaching this size
are found in
it is expected
chondrites, but
particles will
that the dust
settle into a thin gravitationally unstable
disk in any case.
Goldreich and Ward next make use of the dispersion
relation for local axisymmetric perturbations of a thin
rotating disk to investigate the gravitational stability
of the dust disk that forms
the solar
nebula.
This dispersion relation
1965a, b)
Goldreich and Lynden-Bell,
2
=
in the equatorial plane of
22
k c
where K2 =
2Q[Q + d(ri)/dr],
density, c
is the
+
K
2
-
(Toomre, 1964;
is written as
1-1
2TGOk
G is the unperturbed
surface
sound speed, Q is the angular velocity,
G is the gravitational constant, w is
the disturbance, and k = 2r/X
the frequency of
is its wavenumber.
Using
the known masses of the terrestrial planets to estimate
the
surface density of the protoplanetary dust disk,
Goldreich and Ward calculate a critical wavelength below
which all wavelengths
become unstable to collapse in the
absence of random motions.
The largest
form when the unstable disk breaks up
have masses of order m
%
2 x 10 g,
critical wavelength X
=
2
47 2G
c
p
fragments that
are estimated to
corresponding
2
/02
containing
total masses as large
unimpeded,
since the required rate of
5 x
10
8
cm.
to a
Regions
as this cannot collapse
increase of rotation-
al and random kinetic energies of the fragment cannot be
met by the release of gravitational potential energy.
If
the equilibrium contraction size of a region corresponds
to spatial densities at least as great as that of
the
solid material, the fragment may collapse directly to
form a solid body.
In the second stage of accretion a first generation
of planetesimals having radii up to R O
of the order of m (
direct collapse to
2 x 10 14g is
expected to form by
solid densities.
unstable regions have masses
0.5 km and masses
Since
the largest
104 times greater than the
regions collapsing directly to solid bodies,
that up to "104
it is expected
first-generation objects will be gravita-
tionally bound in
larger associations resembling partially
contracted clusters.
Collisions between these associations will be induced
by the differential
rotation of the disk.
Some associa-
tions will be disrupted by these collisions while some
will probably grow at
the expense of others.
however, gas drag will reduce
Gradually,
the internal rotational and
random kinetic energy of these associations and they will
contract toward solid density.
accretion ends with the
This third stage of
formation of a
second generation
of planetesimals having masses up to m 'R 2 x 10
radii up to r 1
18
g and
5 km.
Goldreich and Ward argue
this third stage
that further growth beyond
is unlikely to occur by means of collec-
tive gravitational instabilities.
Instead, direct particle-
particle collisions resulting from the gravitational
relaxation of the disk of planetesimals is
expected to be
the dominant accretion process following the formation of
the second-generation planetesimals.
Since gas drag produces a slow inward drift of the
planetesimals toward the sun for as long
solar nebula is present, it is
growth time be
essential that the particle
short compared to the orbital
by this drag at each stage of accretion.
Ward demonstrated
their
as the gaseous
lifetime set
Goldreich and
that at each stage of accretion
scenario the survival
time exceeds
by at least two orders of magnitude.
in
the growth time
Before going on to discuss
the work of Safronov and
co-workers, we will briefly consider the obstacles
encountered by attempts to attribute the planetary formation process to gravitational instabilities of the
gaseous nebula.
entire
The dispersion relation used earlier
to
investigate the stability of the dust disk is not rigorsince
ously applicable to the gaseous solar nebula
derived for a thin disk;
however, it does provide a good
estimate of the surface density required for
When the value
instability.
obtained by augmenting the
of a
it is
terres-
g
trial planets up to solar
g/cm
,
composition, a
'
1.5 x 103
g
is used, the dispersion relation implies
gaseous solar nebula
collapse unless
order of or less
that the
is stable against gravitational
the temperature of
than 0.040 K.
the gas is
of the
This unrealistically
low
temperature forces one to assume a nebula mass greatly
exceeding the value needed to account for
the present
combined masses of the planets in order to obtain gravitational instabilities of the gaseous nebula.
This assump-
tion then leads to difficulties
in later accounting
the removal of this excess mass
from the
In addition,
for
solar system.
the fact that the chemical composition of
the planets differs from the assumed solar composition of
the primordial nebula indicates
that the density of the
gaseous component of the nebula was
to gravitational
of
not so high as to lead
instability and the resulting formation
stable gaseous protoplanets.
The Scenario of Safronov:
Safronov credits Edgeworth
Lebedinskii
(1950) with the
simultaneous
the solar nebula with the
instability of the dust layer
its disintegration
tions.
independent and almost
inference of the separation of the dust and
gaseous components of
gravitational
(1949) and Gurevich and
into a large
The effect of turbulence
subsequent
resulting in
number of dust condensain the primordial nebula
on the formation of a thin dust layer was considered by
Safronov and
it was demonstrated
energy released by
the cloud as
that the gravitational
interactions between
turbulent eddies caused them to move closer to the Sun
was
insufficient to maintain turbulence
Safronov's description
of
(Safronov, 1969) of
the planetary formation process
is
of the dust condensations
to be m '
The initial mass
for the terrestrial
5 x 10
between those obtained
the first stage
in many ways
similar to that of Goldreich and Ward.
calculated
in the cloud.
16
g,
a value which
for first and
planetesimals by Goldreich and Ward.
zone are
falls
second generation
In calculating
the
above mass for the
was
initial dust condensations, Safronov
apparently unaware of the calculations of the gravita-
tional stability of thin rotating disks carried
Toomre
(1964) and Goldreich
More surprising
and Lynden-Bell
out by
(1965a, b).
is the fact that Goldreich and Ward were
unaware during their
investigation of
the work done on
instabilities in a thinning dust disk done by Safronov
(1969),
Edgeworth
(1950).
(1949),
and Gurevich and Lebedinskii
Like Goldreich and Ward, Safronov describes a
stage of accretion in which
"primary condensations"
collide sometimes combining
and sometimes disrupting.
This aggregation process presumably leads
to the formation
of "secondary condensations" having masses of the order of
104-106 times
those of the primary condensations.
Next,
the entire system of condensations was assumed to be
converted
into a cluster of bodies within a cosmogonically
short time.
This is roughly the last stage of
evolution
considered in any detail by Goldreich and Ward.
Initially, the bodies formed
in the flat dust disk
are presumed to move in nearly circular orbits and
quently have low relative velocities.
conse-
Later their mass
increases as does their gravitational
interaction with
other bodies resulting in an increase
in relative veloci-
ties and orbital eccentricities.
Since the
relative
velocity of
the bodies determines the rate of planetary
growth and the extent of fragmentation of colliding
there is a close relation between the relative
bodies,
velocities of the bodies and their size distribution.
Safronov assumes, as is commonly done,
encounters between bodies can be
body problem.
that
treated as in the twoof equal mass m,
For approaching bodies
the relative velocity vector V does not change
tude,
but merely turns
where D is the
analysis of
bodies
through the angle
impact parameter and
the balance between the
in encounters and the energy
Y <<
I
in magni-
1 Gm/DV 2
/2.
In an
energy acquired by
lost in collisions,
Safronov concludes that the relative velocities of bodies
may be conveniently expressed by V =
are the mass and radius
accretion zone.
bodies
/Gm/r,
where m and r
of the largest body in a given
For a power law mass distribution of
in the terrestrial zone,
an estimate of
e
% 3-5
is
obtained.
The
two problems of determining
the velocity distribu-
tion in a system of bodies of varying mass and
determining
the mass distribution of
solved simultaneously.
should be
However, the complete problem is
insoluble in analytic form,
take
the bodies
that of
especially
into account collisions resulting
if one attempts to
in fragmentation.
In order to treat the problem analytically it has
split into two parts, one
of bodies
is
to be
in which the mass distribution
assumed known and another
in which the
relative velocity distribution of the bodies is assumed
to be known.
The coagulation theory method used
in studying the
process of coagulation in colloidal chemistry and the
process of rain droplet growth was used to construct
equations which describe
the evolution of the mass distri-
bution function for protoplanetary bodies.
investigations of
A number of
the coagulation equations,
ing fragmentations and some not,
Safronov and by other workers.
some includ-
have been carried out by
The coagulation coeffic-
ient in the most ambitious of these investigations
based on the two-body gravitational
section.
fore,
The relative velocities
assumed to be given by V =
earlier.
were, there-
/Gm/Gr as discussed
the fragmentation of
quantitative treatment is very difficult,
if not impossible.
tions
collision cross
of bodies
When allowance is made for
colliding bodies,
is
Nevertheless, qualitative considera-
(Zvyagina, 1973)
coagulation equation
and numerical
solutions of
(Pechernikova, 1974)
the
indicate that
the mass distribution of a system of bodies with fragmentations may be approximated by a power function n(m) =
cm
-q
,
where the exponent q of this function lies between
1.5 and 2.0
(more likely closer to 2.0) depending on the
exact treatment of the fragmentation process.
There are a number of reasons why the coagulation
theory method fails to be of any use in the description of
the final stage of accretion during which planet-sized
objects form.
First,
a distribution function description
is only useful when the number of bodies within a given
mass interval is large enough to permit the use of
statistics.
In the case of the largest individual bodies
Second,
statistics cannot be applied.
strated later
in Chapters II and III,
gravitational collision cross
as will be demonthe two-body
section does not adequately
describe the gravitational interactions of bodies in
relatively low eccentricity orbits.
Finally, a simultan-
eous treatment of the evolution of the mass and relative
velocity distributions of the protoplanetary bodies is
much more desirable than a divided
approach.
Numerical
experiments including the coupled evolution of the mass
and velocity distributions have recently been carried
out by Greenberg et al.
(1977).
'L'he Numerical Experiments of Greenberg et al.:
R. Greenberg, J.F. Wacker, W.K. Hartmann, and C.R.
Chapman
(1977) have developed
a numerical model for the
evolution of an initial
swarm of kilometer-sized planetes-
imals into a distribution of bodies which includes
planetoids with diameters up
to 1000
km.
several
The low- and
moderate-velocity collisions between solid bodies were
carefully evaluated using
ments combined with
principles.
four
The outcomes of collisions were divided into
categories
(depending on the ratio of the masses and
elastic rebound,
impact for
(ii)
surfaces of both bodies,
body and
low-velocity experi-
interpolation based on physical
the relative velocity at
(i)
results of
the given collision):
rebound with cratering of the
(iii) shattering of the smaller
cratering of the larger one,
of both bodies.
velocities
outcomes is
and
(iv)
shattering
Once the range of mass ratios and relative
at impact appropriate to each of
these collision
established, a variety of other parameters
specifying collision outcomes remain to be determined.
For elastic rebound only the coefficient of restitution,
given as the ratio of the rebound velocity to the
velocity, is required.
more complicated,
ejecta mass
As the collision outcomes become
estimates are required for
the total
as a function of the mass ratio, material, and
impact kinetic energy of the colliding bodies;
tioning of
bodies;
impact
the impact kinetic
the parti-
energy between colliding
the cumulative fraction of ejecta with velocity
greater than V;
dependence of
the mass distribution of ejecta;
impact strength on body size;
distribution of
fragments of
the
the mass
shattered bodies;
as well
as the coefficient of restitution.
Despite the unprecedented detail in this model of
collision outcomes, caution must be exercised when extrapolating from results obtained for
to bodies having masses
tude greater.
The
small
laboratory samples
twenty or more orders of magni-
relationship between material defects
in bodies with diameters of
several kilometers to defects
occurring in small laboratory samples
is not at all clear.
In addition, gravitational binding energy becomes more
important than material binding energy
with radii greater than %40
km
for rocky bodies
(cf.Chapter III).
The
effect of this dominance of gravitational binding energy
for bodies with R
40 km on the mass
distribution of
fragments when such bodies are shattered by collisions
is
not understood.
Collisions are assumed
to result
from random relative
velocities between bodies as measured by
eccentricities and inclinations
ential Keplerian velocities.
is
their orbital
rather than from differ-
The probability of collisions
computed by a "particle-in-a-box" estimate using the
two-body gravitational collision cross-section which
30
neglects the presence of the
Sun.
Besides collisions,
which tend to damp the random velocities, Greenberg et al.
have taken into account the rotation of random velocities
due to gravitational
encounters and the gravitational
stirring arising from randomization of Keplerian differential velocities.
All of these ways of re-distributing
the velocities are based on two-body formulae.
Numerical experiments with a variety of
and velocity distributions
performed by Greenberg
from
their
and material properties were
et al..
simulations are
initial mass
The main conclusions drawn
the following:
(a) Collisions
between bodies of 1 km diameter rapidly produce a substantial number of
500 to 1000 km diameter bodies, while
bulk of the mass remains in the form of
bodies even after
assume values
conclusions
to be true
variety of material properties
butions so long as
than 20 V
e
and not of
is maintained by Safronov
are held
, where V
(1969).
the
These
for experiments with a
and initial velocity distri-
the initial velocity is
e
formed.
of the order of the
escape velocity of the original bodies,
large bodies, as
1 km diameter
the larger planetoids are
(b) Random velocities
the
somewhat less
is the escape velocity of the
original bodies.
It
will
be
shown later in
this chapter
that
the
two-body
gravitational cross-section assuming relative velocities
of
the order of the escape velocity of the bodies account-
ing for the bulk of the mass depends on mass as a
rather
= m2/3
than as
as demonstrated
by integrations
we have carried out for encounters between bodies
in circular orbits.
orbits, e 0
5.0 x 10
Greenberg et al.,
then surprising
initially
Given the very low eccentricity
-4
,
occurring in the simulations of
a dependence
rather than the
m 4/3
m
m4/3
a a m2/ 3
dependence assumed.
that the evolution of
in these simulations
is to be expected
It is
not
the mass distribution
indicates a "runaway" growth of the
largest bodies while most of
the mass of the
system remains
in the original 1 km diameter bodies.
Presumably, simula-
tions with a more refined treatment of
the gravitational
encounters between bodies which included
the
the effect of
Sun would produce different evolutions of
distribution,
What
but we cannot be
then can be
the mass
sure of this conclusion.
concluded from these scenarios and
simulations about the initial conditions to be used in our
simulations?
tational
In view of the
inadequacy of treating gravi-
encounters as isolated two-body
interactions, it
is difficult to know what confidence to assign to the
final mass distributions obtained in the preceeding
tigations.
Safronov's
suggestion that isolated
inves-
zones of
accretion will begin to merge at some stage of the evolution of the protoplanetary disk and that the masses of
the protoplanets at that time will be comparable within an
order of magnitude has some physical plausibility, but
proof of
the occurrence
of such a spike
in the mass distri-
bution for some value of the mass does not now exist.
For the special case of circular motion, W.R. Ward
has estimated that such a spike is to be expected
(1975)
at a value approximately equal to a lunar mass.
This
value was obtained by assuming that a body sweeps up
(2fr) (2Ar)G0,
material having a mass m =
heliocentric distance, Ar
annulus
where r is
the
is the half-width of the
from which material is accreted, and
is the
0
surface density computed by "spreading-out" the mass of
the present terrestrial planets over their portion of the
solar
Further assuming
system.
where R
Ar = R L
1/3
= r(m/M )1/3
is the distance to the straight-line Lagrangian
point, a spike in the mass distribution is found at
1/2
[2 x
(M8
+ M + Mc
+ M
)]
1.1 M
m=
(r
= 1 AU).
0
It
is:
not clear
how this
1-2
argument could be modified to
take into account non-zero eccentricity orbits of
accreting bodies.
E.
The Late Stages Of Solar System Evolution
The Safronov Description of the Growth of the Largest
Bodies:
Realizing the inadequacy of distribution functions
for studying the growth of the largest bodies,
(1969, pp.
105-108)
investigation of the
Safronov
takes a different approach in the
late stages of planetary accretion.
This description serves as an example of the assumptions
and physics currently employed
stages of the dynamical
in studies of the
evolution of
and also raises many questions
the solar
late
system
for which it is hoped that
the present numerical simulations will provide
some
answers.
The gravitational collision cross-section along with
the expression for the velocity relative to the circular
velocity, V2
= Gm/er, are used to argue that the
largest
body in a given zone will grow more rapidly than the
second-largest both absolutely and
relatively.
further argued that the ratio of the mass
body, m,
to that of the
to a certain
of the largest
second largest, m',
limit, calculated to be m/m' R
increases only
200 to
Safronov then concludes that a characteristic
the accumulation process was
"embryos"
It is
1000.
feature of
the formation of planet
whose masses were far greater than the masses
of the other bodies
in their zone.
Confirmation of this
conclusion is claimed based on estimates of the masses
the
largest bodies
falling on
the
inclinations of the axes of rotation of the planets
of
the planets as determined by
relative to the ecliptic plane.
The
scenario given by Safronov
largest bodies
is as follows.
for the growth of the
the ratio m/m'
As
increased,
the orbit of m became more nearly circular as m grew due
to randomly oriented impacts and
much smaller bodies.
be an
The
subsequent accretion of
supply zone of m was assumed
a width determined by
isolated annular region with
the mean orbital eccentricity, e,
bodies.
to
of the main mass of
The zone of gravitational
influence of m,
as
measured by the distance to its libration point, R L
L1
to be more than one
(m/3M) 1/3R, was calculated
magnitude smaller than the
supply zone of m as
So long as
accumulation process began.
and eccentricities remained
supply zones were narrow.
the final
the bodies
zone of a planetary "embryo" remained small,
velocities
order of
in the
the relative
small and the
Outside the given
zone were
others in which the relative velocities were assumed to
be determined by the largest bodies
assumed
relation V2
= Gm/8r.
These
inside them via the
embryos,
in turn,
grew relatively more rapidly than other bodies
in their
zones and
the late
their orbits also tended to become circular.
stages of solar
system evolution began,
expected that there would be many planet
it was
"embryos" present.
For as long as they occupied different zones, they were
affected by the law that the
As
largest body of all
not
should
grow relatively more rapidly.
As the bodies grew, the relative velocities also grew
and the
supply zones
broadened.
When adjacent zones began
to overlap, relative velocities equalized and the
embryos began to grow more slowly.
However, the
smaller
smaller
embryos are expected
to continue in nearly circular orbits
for some
the distance between embryo orbits
time, since
for which the smaller embryo is
perturbations
susceptible
of the larger is only three to five times
greater than the distance to R
and
several times
smaller than the width of the supply zone.
between embryos are expected
by
the embryo population due
increased
The gradual reduction in
to the mass
increase of the
embryos is expected to continue until all
surrounding material has been used up and
between embryos have become so large
tions
Collisions
only after m has
1.5 to 2 orders of magnitude.
larger
to strong
are unable to disrupt the
over large
time spans.
This
is
the
the distances
that mutual perturba-
stability of their orbits
expected to be the primary
process determining the law of planetary distances.
This scenario for the late stages
of planetary
accretion is, for the most part, a speculative description
of what might be expected to occur.
It is not clear
whether the assumption that the gravitating system of
bodies can be treated as a set of
isolated zones within
each of which the relative velocity distribution of
smaller bodies is controlled by
the largest body through
the expression V2 = Gm/8r can be justified.
In any case,
the assumed expression for the relative velocities cannot
be expected to be of any use
zones.
in the case of overlapping
Also, as was stated earlier,
the two-body gravita-
tional model is not an adequate description of the gravitational encounters between the largest bodies
when the presence of the sun is taken
Encounter Physics:
(m I M )
into account.
A Comparison of the Two-Body Model with
Integrations Including the Presence of the Sun for Bodies
Initially in Circular Orbits:
In order better to understand the gravitational interactions between bodies when their orbital motion about the
Sun is taken
into account, encounters between bodies
initially in co-planar,
These encounters
circular orbits were studied first.
arq much more tractable than more general
ones, since it is possible to specify these encounters with
Bettis and Szebehely
the regularized equations of motion,
were carried out for a range of orbital
(1971),
for the following four sets of values of
10
=
mI
(i)
-2
M
10
i
=
m
M ,
(Figure
(iv) m
and
-5
2);
(iii)
of influence"
(i) and
(ii).
1/5
m
to m
i
and m
This radius
and where m 2
2
-3
M , m2
= M
e
m
=
= m
(Figure 3);
integrations were initiated
five hundred times the
larger body for cases
is given by Rs = R2[ (m2(ml
+ m2
where R 2 denotes the distance from
is taken to be
are unequal.
(ii)
The separation between
radius of the
(Battin, 1964),
2
10
(Figure 4).
e2
M
=
spacings
the two masses:
(Figure 1);
taken to be
"sphere
]
M
m
e
= m2 = M
and terminated was
2
-2
at which the encounter
bodies
M
10
and
Integrations of
spacing of their circular orbits.
the
two bodies
the masses of the
only three parameters:
When
the larger mass when
the separation value 500 R
resulted in orbital angular separations greater than 7/2,
as occurred for m
ing
= M(,
an initial separation correspond-
to an orbital angular
radius
1 AU.
of the orbit of m
2
separation of
was
2
were
I
integrated, since
inside the orbit of m
2
were expected
symmetric with those outside.
7/2 was used.
The
in each case taken equal to
Only those encounters with m
the orbit of m
mI
2
initially outside
the encounters with
to be approximately
These integrations
reveal
many interesting differences from the behavior predicted
on the basis of the two-body model of gravitational
encounters where the presence of
the Sun is
not taken
into account.
Before these differences are described,
an expression
will be derived for the two-body gravitational crosssection to be
tions.
edges
employed for the comparison with integra-
The limiting case of grazing impacts defines the
of the range of the
collisions result.
m
2
For
and radii R 1 and R2,
impact parameter, D,
two bodies
for which
having masses of m 1
respectively,
and
the conservation of
energy requires
1
mlm2
2 mI + m
where V
2
V
1
mlm2
2 m
+ m2
o
V
2
Gmlm2
R1 + R
1-3
1-3
is the relative velocity of the bodies
0
separation and V.
1
of impact.
is
at infinite
their relative velocity at the instant
For a grazing impact the conservation of
angular momentum requires
m2 Vi(R1 +
R2) = m2 V D .
1-4
Combining the two conservation laws the
D
2
=
(R
+ R )
1
2
is obtained for the
grazing
collisions.
m
2
[1 + 2G
1
1
R1
+m
2 1
+ R 2V 2
impact parameter
If m 1
and m
expression
2
1-5
just resulting in
are assumed
to be
in
circular orbits about the
Sun with
radii R
and R
0
+ D,
o
then the relative velocity at infinite separation may be
taken to be the Keplerian differential velocity
1/2
Vo =
9
[ (i_(
R
1
1/2
)
](
+ D
R
(D <
R
GM
R
D
D
2R
1-6
).
The collision regions with their widths and impact
velocities
(given as a fraction of the
circular orbital
velocity at 1 AU) are obtained by means of
the two-body
gravitational cross-section with Keplerian differential
velocity and tabulated in the upper part of Table
the four sets of values
1 to 4.
density
Also
of
and m 2
I
presented
in Figures
The bodies were assumed to be of constant lunar
(P
=
3.34 g/cm
shown in Table
widths,
of m
1 for
and
)
and R 0 was taken as 1 AU.
1 are the collision regions,
impact velocities as obtained by
their
integration
the regularized equations of motion.
After
these integrations were performed it was
discovered that Dole
(1962) and Giuli
(1968)
in attempting
to account for the rotation of the Earth by the gravitational accretion of small bodies had
integrations, where those
m
carried out similar
encounters with
initially inside that of m
2
their investigations the body m
the orbit of
were also included.
I
was
In
assumed to be mass-
Table 1
Two-Body Gravitational Cross-Sections
With Keplerian Differential Velocities
a(A.U.)
outer
inner
boundary
boundary
Aa(10
-5A
V./V
1 cir
A.LU.)
0.99809
1.00191
382
0.152
ml=m2=10-2M
0.99737
1.00263
526
0.160
ml=10-3M
0.99115
1.00885
1770
0.707
0.98778
1.01222
2444
0.741
m2=10 -2M
ml=10-5 M
,
m =M
ml=m2=M
Gravitational Cross-Sections Obtained by Integration
a(A.U.)
i
Band
m =10
-5
-2
M , m 2=10 M
ner
1 +I-
r
-5
V./V
A.U.) 1 cir
boundary
boundary
1.00446
1.00457
0.071
1.00512
1.00520
0.071
Aa(10
= 2[Aa(1 0 ) + Aa(20)]
-2
ml=m2=10-2 M
1.00551
1.00578
0.074
1.00643
1.00655
0.074
a = 2[Aa(l 0 ) + Aa(2)]
m =10
1
-3
M,
m2=M(
2
1.02040
1.02110
0.329
1.02369
1.02407
0.329
a = 2[Aa(10) + Aa(20)]
m -m 2=M
1 2 @
= 216
1.02560
1.02670
110
0.347
1.02985
1.03042
57
0.347
a = 2[Aa(10) + Aa(20)] = 334
..
~li
dlLll___bLaLIIIIIIl-L. ~IC----l_
therefore, the mathematical structure of the
less and,
restricted three-body problem was used.
Dole found
case
For this
seven distinct inner bands of heliocentric
impacting
orbits which yielded
to the two outer bands found
trajectories in addition
in the present integrations.
Of the seven inner bands found by Dole,
two consist of
directly approaching trajectories which impact m 2 without
looping,
and the remaining five consist of
loops before
which make one or more
major bands together account for
the inner impacting
impacting m
2
The two
.
approximately 85%
of all
As we had expected,
trajectories.
the two directly approaching
trajectories
inner bands are
closely
symmetrical with the two outer bands at least for the
case when m 1
is assumed to be massless.
The integrations of Giuli can most readily be
to our
m
= M
own integrations
.
the two outer bands when the 10%
+ R 2 due to the radius R
for Giuli's integrations)
the
=
10
-1
final
10
-3
M
and
increase
in
=
0
is taken into account along with
integrations
the results to the choice of
separations for
=
(compared to R
M
sensitivity, exhibited in both our
Giuli's,of
I
is found for the location
Very good agreement
and width of
R
in the case where m
compared
the integrations.
the initial
and
and
In computing the
total gravitational cross-sections obtained by integration,
_~
2[Aa(l
o
) + Aa(2
symmetry of
o
)],
in Table
The major differences between
the behavior exhibited for
<
M )
encounters between two small
bodies orbiting the
which neglects the presence
Sun when the two-body model,
of the Sun,
is used and
the behavior observed when these encounters
will
the
inner and outer directly approaching bands
found by Dole and Giuli.
(m
1, we have made use of
be summarized
are integrated
next.
First, the two-body model predicts collisions of m
and m 2
for those orbits of m
distances
having heliocentric
I
in a relatively broad range centered on and
symmetric about the orbital radius of m 2 .
that range is given for four sets of
values
in the first part of Table 1.
ized equations of motion of m
hand, demonstrate
I
of m
I
and m
2
about M,on the other
and m 2 occur
distance.
earlier, Giuli and Dole have shown that five of
that these
stated
these
of all impacting
five bands contribute only
trajectories.
major bands are approximately symmetric
of m 2 .
As
in
trajectories which make one or more loops
before collision and
7 to 8%
and m 2
having orbital radii falling
nine distinct bands of heliocentric
bands consist of
I
Integrations of the regular-
that collisions of mi
instead for orbits of m
The extent of
The remaining
four
about the orbit
The two inner bands are observed much closer and
_
the two outer bands much further from the sun
range of trajectories predicted
Consequently, it is
that m 2 will experience collisions with bodies
expected
further from its own than is predicted by the
in orbits
two-body model.
taken as
Second,
2
collision cross-section,
the total
four major band widths,
the sum of the
obtained
approximately one order of magni-
from the integrations is
m
than the
on the basis of the two-
body model to result in collisions.
tude
*___
F~_~II
-.I~(_I-_IIXIII1LI
smaller for each of the four sets of values of m 1
than is predicted by the
two-body model.
relative velocity upon collision determined
Third,
and
the
from the
integrations was approximately half of the value expected
on the
This result clearly
the two-body model.
basis of
has profound significance for accretion models of
formation of planets.
the
Finally, the dependence of the
gravitational collision cross-section on the mass m
found from both the integration ratios G(ml = m 2
U(m
1
= m
(ml =
by
0
a
10
m
= 10
2
-5
1/3
2
M
.
-2
,
G(m
M ) and
m2
=10
-2
1
M )
= 10
-3
M
@
,
2
is
= M )/
m2 = M )/
2
e
to be given approximately
These same ratios computed for the
two-
body model with Keplerian differential velocities show
the
Sm
same approximate dependence of
1/33
2
.
a on m 2 , namely
When encounters between bodies
in circular
but inclined orbits are considered, the mass dependence
of the cross-section is
expected to be given by a
=
m2
2/3
This dependence of a on mass
is
in sharp contrast
to the dependence obtained by Safronov
in considering
the growth features
(1969, pp.
105-108)
of the largest bodies
The gravitational two-body
in a given isolated zone.
collision cross-section was used to obtain this dependence
assuming the mean velocity of the smaller
relative to the circular velocity of
the larger body to
and r are the mass
be given by V = /Gm/@r, where m
radius, respectively, of the
incident bodies
largest body
in the
and
isolated
zone and 8 is a parameter computed to be in the range 3 to
5.
The dependence of a on mass obtained by Safronov in
this way is given by a " m
with the dependence a
4/3
.
This is to be compared
inferred for
the three-
dimensional case from the integrations of
encounters
between bodies
m 2/3
initially in circular orbits.
ficance of this difference
of
is that for a
m4/3
the mass
the largest body increases by accretion faster than
the masses of the
running
smaller bodies
away to large mass.
in a given zone, thus
For G
a
a given zone will tend towards
m
the bodies in
similar mass values, since
the smaller bodies will increase their mass
the
The signi-
faster than
largest ones.
The
importance of the difference in the present
investigation is
in the choice of the initial mass
Ya B-~~L~C~I~
distribution of bodies to be used in the numerical
simulations.
The choice of equal mass bodies for this
initial distribution appears
case, since
isolated
if
a
m4/3
to be defensible in either
the largest bodies
in each
zone are expected to have similar masses once
most of the material in each zone has been
the largest body in a zone.
bodies are
swept up by
For the case a
m2/3
the
expected to attain similar masses at an
stage of evolution.
Since only one or
earlier
two hundred bodies
can presently be accommodated by the numerical simulation
programs, the choice of equal mass bodies
distribution also seems
F.
for the initial
to be the most convenient.
The Numerical Simulations
The numerical
simulations carried out here have a
number of advantages over treatments
cribed in some detail
such as those des-
earlier in this chapter.
the mass and relative velocity distributions of
planetary bodies are free
the initial conditions are
handling encounters
is
First,
the proto-
to evolve simultaneously.
chosen and the algorithm
established,
Once
for
the distributions
of
the orbital elements of the bodies evolve in response to
the gravitational close encounters and occasional
colli-
sions occurring between bodies.
larger
Events
involving
YY1_____Ul____s___LLI1__XI~
LI~~
XII_.
bodies and moderate mass ratios are included and are
treated
in the same way as events between pairs of bodies
with large mass ratios.
Collisions between bodies arise
in a natural way in these simulations and assumed expressions for the relative velocity distribution and the
gravitational collision cross-section are not needed.
This
is a fortunate feature,
integrations of close
since as we saw earlier,
encounters between small bodies
initially in circular orbits show characteristics that
are not at all explicable by two-body gravitational models.
Integrations of close encounters between bodies initially
in
small eccentricity orbits, e
'
0.02 say, also indicate
more complicated behavior than can be accounted for by
two-body models.
or other
Finally, no reliance on
such artificial constructions
isolated zones
is necessary.
Next, a brief outline for the remainder of this thesis
will be given.
It is the primary purpose of this dissertation to
determine whether
terrestrial-like planetary systems can
be produced as a result of the gravitationally-induced
collisions and accretions occurring in a swarm of approximately lunar-mass
protoplanets, at least when the
bodies form a co-planar system.
gravitational encounters
A new model for close
is described
in Chapter II
and
plausibility arguments are given for
chapter, predictions of
results of
motion
the model are
it.
In the
same
compared with the
integrations of the regularized equations of
and a brief discussion of the treatment of
slow encounters
is also given.
the
In the next chapter
(III)
a scenario is developed for collisions between protoplanets.
Chapter IV describes
and in Chapter V the results of
tions are presented.
the numerical
simulations
several computer
simula-
A discussion of conclusions drawn
from the simulations about the
terrestrial planet forma-
tion process and a short consideration of possible future
investigations are also given in this
final chapter.
A NEW MODEL FOR
CHAPTER II.
CLOSE GRAVITATIONAL ENCOUNTERS
A.
Introduction
A model
is described for single close encounters
between two small masses, m
mass, M(ml,m2 <<
M).
I
and m2 ,
orbiting a large
The special case where m
and m 2
I
comparable will receive particular consideration,
are
shown to be equally valid when m
but the model will be
and m 2
A close encounter is
are not comparable.
to be an approach of mi
taken
and m 2 for which their minimum
separation, as they move along unperturbed orbits about
M,
is
less than the
"sphere
larger of the small bodies.
s
The
(i)
and m 2 are
and m
2
about M into
two parts:
and m
the unperturbed two-body motion of m
the unperturbed
the separation of m 1
The motion of m
analyzed
is
i
serves in this model to divide the
s
and m
when the separation of mi
(ii)
where m
(Battin, 1964),
radius R 1 denotes the distance from M to mi.
The radius R
motion of mi
given by
is
This radius
the larger mass when mi
now taken to be
unequal.
2 1/5
1 + m2
1 [(m
of influence" radius of the
1
into two
2
and m
parts.
2
2
is
is
about m
2
assumed
in
I
and
when
.
less than R
inside the radius
It
about M
is greater than Rs,
two-body motion of m
and m
2
R
is
further
s
this
approxima-
tion that the center of mass of m 1
orbit in the gravitational
and m
describes an
2
field of M and that m
and m
I
2
have residual motions with respect to their center of
The reference frame to which these residual
mass.
are referred
motions
contrast to the usual
Chandrasekhar
is clearly an accelerated one
in
inertial frame employed by
(1941) and Williamson and Chandrasekhar
(1941) in their classic treatment of stellar
encounters.
Indeed, the assumption of conic motion of the m , m
i
center of mass accounts for much of the
2
improvement of
the present model over that of Williamson and Chandresekhar
in which
is
assumed.
The variation of
2
influence of the
m2
2
will give rise
<<
M,
tidal field,
encounters completed within a time
relative to the time required for
small
m
and m
small, by requiring m i ,
only negligible
mil
I
However, we will require that Rs be
to a tidal field.
sufficiently
of mass
the gravitational field of
M over the region occupied by m
for all
m 2 center
straight-line motion of the mi,
to assure
at
least
sufficiently
the motion of the
center of mass about M.
The special
case of encounters between two bodies
of comparable mass
orbiting a much larger mass has been
considered recently only by Horedt
and Zalkalne
(1970).
(1972)
and Steins
Horedt considers only the special
initially moves
case where m 1
the large mass, M,
and m
bolic orbit around M.
studies
seems to be
2
in a circular orbit around
moves in an unperturbed hyper-
The primary interest in these
in encounters
satellites or asteroids and
of comets with
indeed Horedt's model seems
type,
applicable only to very rapid encounters of this
since
no allowance is made for changes in the velocities
of m
i
and m
the
encounter.
2
arising from their motion about M during
On the other hand, our model will be
shown to be an adequate approximation for many purposes
in the more general case where the two
small bodies move
in initially unperturbed elliptic orbits about M,
the eccentricity of the hyperbolic motion
of m
2
provided
about m
1
satisfies the condition E > 4.
Kaula
(1975) has
evolution of a
stated that the
system of planet embryos
accreting much smaller planetesi-
mals may be "significantly affected by the small
of events involving
ratios."
fraction
larger bodies and moderate mass
The present model is uniquely suitable for the
numerical simulation of such a system.
B. A Description of the Model
The
When m
1
essential features of the model are as follows.
and m
2
and velocity of
reach a separation Ar
= R
the center of mass of mI,
s
,
the position
m 2 are used to
determine a conic
section orbit about M.
always an ellipse
in this
be by the model.
The center of mass is
along
study, but
I
and m
is
is not required to
this ellipse for the duration of
Meanwhile, m
This orbit
assumed to move
the encounter.
execute two-body motion, always
2
along hyperbolic orbits here, with respect to their
less
center of mass for as long as their separation is
than R
m
s
.
The frame of reference chosen for the motion of
and m2 about their center
their center of mass and
an initially specified
then accelerated
of mass has
its origin at
its axes parallel to the axes of
inertial frame.
This
frame
in accordance with motion along
is
a conic
The duration of the encounter
but does not rotate.
the extent of the simultaneous motion
mass are determined by the
and
of the center of
time required for m
I
and m2
to execute their hyperbolic motion about their center of
mass from a
separation Ar = Rs
back again to Ar = R
s
.
to closest approach and
The algorithm for treating
encounters using the model will be described
detail in the last part of
close
in more
this section, but we will
next argue for the plausibility of the above features of
the model.
C.
Plausibility Arguments for the Model
First, we examine the assumption that the motion of
the mi,
m
than the
and m2 with M as origin
motion of m
+
R
R
1
+ G(m
+ G(m
are given by
R
R
-R
+ M) -= Gm2(
3
2
3
1
R
R
1
12
2
R
2
+ M) -R
2
Gm
R
R
R
1
3
R2
2
R
R1
- R
13
R
-
R2
12
tional
R2
The
constant.
=
R2
-
R1
2-1
),
2-2
and m
I
2
the gravita-
, and G is
exact equation
)
1
where R 1 and R 2 are the position vectors of m
with respect to M,
less
2
equations of
The
influence radius.
sphere of
and m
for separations of mi
section with focus at M
a conic
is along
center of mass relative to M
2
of motion of the
center of mass of mi, m 2 relative to M
then
is
2-3
+
+
ml1 + m2
m1R1
m22
where R 0
=+
+
( 1 + m)R
2
0 = -G(M
(in
mR
+ m
1
+ m 2 )(
is the position vector of the m
I,
m
R
2
1
mR
2
1
center of
mass.
Denoting the position vectors of m
respect to their
express R
and m
2
with
center of mass by rl and r 2 , we may
-3
-3
and R 2
2
1
in Eq. 2-3
as
2
3
R
2
)
_i~sl~I__1_IX(__nUr~__ ^___~~~
R
-3
i
(R
2
+
0
2
-
2R 0ri
+ r)
0
i
1
-3/2
= R
(1 +
0
2 -3/2
r
-_
2
R
+
+
2R 0 r
2
2
R
i = 1,2
2-4
Taking into account the assumption that m i , m
r
1
,r
rl,r
2-
< R s'
s
2
<<
M
in
1/5
R2
R
R [(m (m + m ))/M
yields the result
0
1
1
2
-3
Expanding Ri
and keeping terms to second
.
R
<<
2
order in r./R
we find
0
R
-3
r
1
3R
-3
R
[i
0
Substituting
+
0
R
*r.
3
2
2
0
r
2
i
2
R
0
(R
15
-2
this result into Eq. 2-3
of order higher
than the second
-+
0
*r )
r i
4
R
0
2
.
2-5
and dropping terms
in r./R
we
1
0
following approximate equation of motion
obtain the
of the mi,
m2
center of mass,
G(M + m
.
R
0
'
+ m2
2
1
R2
-
^
S
[3(R
r)r
0
where r = r 2 -
2
r
R0
(R-C-)
3
^
+ 2
(1-5(R
^
mm
1 2
+ m2
(m
2
r)
0
term
term to the central force
in Eq.
be of order
R
0
] }
0
2
R
R
s
0
2
m
M
4/5
x
2-6
,
The ratio of the non-central
r .
r
)R
2
force
2-6 can be seen to
so the assumption of conic motion of
mass
is a very good approximation.
the m
l
, m
2
center of
Next, we consider
the justification for assuming conic motion
of m 1
and m2
about their center of mass.
The exact equation of motion of m2 relative to m
is obtained by subtracting Eq. 2-1 from Eq. 2-2,
R
R
2
-
R
= r = -G(m
1
1
+ m
2)
3
r
+
GM(
R
R
3
-
2-7
-.
R
3
From this equation of motion we obtain the equations of
motion of m
and m
m.
4
r
+ G
relative to their center of mass
2
-+
r.
3
4+
R.
GM m.
(m.. + m.) 2 r. 3
m
i
=
i + mjj
i
1,2
(
R3
R.
3
4+
R.
- R
R.3
2-8
2
-
j
Again, making use of the condition rl ,r 2 < R
<<
R
in
-3
expanding R. ,
but this time keeping terms only to first
order
we have
1
in ri/R
1
0
0
-3
R.
1
R
-3
(1
-
r
3R 0
)
2-9
0
-3
Substituting this expression for R.
dropping
terms on the right-hand
into Eq. 2-8
and
side of higher order than
in ri/R
the first
the following approximate
gives
0
equations of motion of m
and m
I
relative to their center
2
of mass
+
S
Gm.
-
3
1
I
2
+ m.)
(m
3
2 {r
r.
+ m.
1
+ m. (-)
R
i = 1,2
i
0
0
)
^
[3(R 0ri)R
-
2-10
j
#
r.]} •
0
i
The ratio of the non-central force term to the central
2-10 is of order
in Eq.
force term
M
m.
r3
R
M
m.
-
RM 3 M m
1/5
s3 I
1
R
M
so the assumption of conic motion of m
to their center of mass is
m2<
10
-5
M.
It is
not as good an assumption, however,
m
Finally, a brief explanation of
given.
The
and m 2 relative
a valid assumption for mly
as that of conic motion of the mi,
expression for the
I
2
center of mass.
the origin of the
sphere of influence radius, R
s
,
equation of motion of m 2 relative to m
is
is
from Eq. 2-7
r
+ G(
1
+m
2
)
+
r
3
r
The right-hand side of
R
GM(
R
1
3
R1
2
)
3
2-11
R2
this equation is,
of course,
the
Recall from Eq. 2-2
disturbing force due to M.
that
the motion of m 2 relative to M is given by
.
R
+ G(m
R2
+ M)
2
R
+
+
2
-Gml
r
r
2
R
1
+ -- ),
R
1
2-12
where the right-hand side is the disturbing force due to
mI.
According to Laplace,
the advantage of either equation
of motion depends on the ratio of the disturbing force to
the corresponding central attraction.
The preferred
equation of motion is the one providing the smaller ratio.
It can be demonstrated
(Battin, 1964)
that the surface
boundary over which these two ratios are equal
is almost
spherical with radius
1/5
R R[ m 1 (m 1 + m 2 )
s
if
r <<
R0 ,
0
where m
While these
predictions of
integration of
for
tion of
a
to be
lend plausibility to
large variety
will be
presented
for applying
algorithm will
close
of
also
the present
encounters
of numerical
of motion are
desirable.
in Section E,
the model
in a little more detail.
the
larger mass.
the
the model with results
the algorithm
described
taken
the equations
These comparisons
first
is
arguments
model, comparisons
of
I
2
M
but
must be
The following descrip-
serve
to define the
_im~~_
parameter space
D.
to be used in the comparisons.
The Algorithm for Applying
When m
frame
the Model
and m 2 reach a separation equal to the
I
of influence radius of mi,
positions
_~_L~__CX_____~IIL11_1_1~
sphere
it is assumed that their
and velocities with respect to a reference
(X,Y) with origin at M are known.
At this
instant
the positions and velocities of the center of mass of mi
and m
and of m
2
I
and m 2
relative to their center of mass
Next, the eccentricity vector, £1,
are computed.
orbit of mi
with respect to the m i ,
calculated
in the standard way from
and velocity of m
mass.
and m
i
center of mass
is
the initial position
with respect to the m i , m 2 center of
move with respect to their center of mass
2
(Figure 5).
Now, define
(x,y),
second,
(x,y) ,
two sets of orthogonal coordinates
at the mi ,
m
2
center of mass.
has axes parallel to those of
is
rotated relative to
inates and velocities of m 1
and m2
(x,y) are given by
(X,Y)
The
and the
(x,y) through the
P, so that its x axis is parallel to E
system
to
P, in the X,Y coordinate system
which have their origins
first,
is then
by the direction of £1 which corresponds
an orientation angle,
ate
2
the
The orientation of the hyperbolae along which m
1
specified
angle
m
of
.
The coord-
in the rotated coordin-
the transformations
JIII^__YILYI/__I__Ijl~
_~I
X
sino
x
-sinO
cos
yl
S1
cosO
x
cosO
sin
-sin
cos
yl
2-13
x2
2
mx
m1
1
m2
1
m
x
x2
2
1
m2
y1
Making use of the symmetry properties of
i
l
i
i =
i
encounter.
1
=Yi
where the primes
the hyperbola,
i
1,
2-14
2
'
indicate values at the end of the
Reversing the
the final coordinates
transformations in
and velocities
in
(2-13),
the
(x,y)
coso
-sin
sino
cosP
coordinate system become
S'
sinP
"'
xyl
cos4
x
1
2-15
x'
2
m1
x'
1
Y2
m2
Y1
x
2
2
- m
1
1
m2
1
The coordinates and velocities of the mi,
of mass
after
m 2 center
the encounter must be determined next, but
these quantities depend on the duration of
encounter.
For
a partial hyperbolic
the relative
encounter this
duration is given by
-3
a
At = 2/
(E
a
with
=
H
cosh
+
a and E are
respectively,
eccentric
of
-
s),
the semi-major axis
the orbit
of
',
now be determined by
of m2
(eo),
and eccentricity,
about m
the mi,
m
.
The
final
center of mass
2
solving Kepler's
of the duration, At, along with the
(XO 'Y
m
')
2
semi-major axis
center of mass.
and velocities
(X
(E
',Y ')
of
a0
standard formulae
6.4.15 of
e0 , E
,
the
'
by
Eqns.
6.4.3 and
The
final
coordinates and velocities
and m 2 relative to M are
= X '
0
O)
(e.g.,
Fitzpatrick, 1970).
X '
i
(a)
The final
center of mass may then be obtained from
of mi
can
equation making use
and initial eccentric anomaly
computed for the mi ,
coordinates
2-16
E
anomaly, E
eccentricity
IHI)
R
(
a
where
IHI
sinh
+ x
i
'
then
X '
i
= X '
0
+
1
'
i = 1,2
2-17
Yi'
i
= Y0'
O
+
yi'
1
Y.'
1
= Y
0i
' +
y
'
If desired,
the initial and final coordinates and
velocities of m
and m
I
2
relative to M may be expressed
instead as orbital elements of two
ellipses with foci
at M.
E.
Tests of the Model Against Numerical Integration
Changes in the eccentricities and
of the orbits of mi
and m 2
about M are calculated
many close encounters using the model.
are
semi-major axes
for
These changes
then compared to the corresponding changes obtained
by numerical integration of the three-body equations of
motion.
Before presenting these comparisons
necessary to describe the parameter
specifying
are the values of the masses mi
remaining parameters
sons can be divided into
the motion of m
mass
mi,
(a,
m
2
E,
space to be used in
the encounters.
First, there
The
it is
and m
I
,, ±)
and
center of mass
eccentricity of
2
adopted for the present comparitwo groups:
those describing
with respect to their center of
those describing
(a
and m2.
0 ,
e0,
8 0) .
the hyperbolic orbits
the motion of
the
From Section D the
along which mi
and
m 2 move with respect to their center of mass is given by
E.
m
2
The parameter a
specifies
the separation of m
and
at their point of closest approach as a fraction of
the
sphere of influence radius.
hyperbolic orbits,
The orientation of the
specified by
, is defined
to be the
angle between the eccentricity vector of the m
about the mi,
m2
center of mass
and the X-axis of
reference frame with origin at M.
radius, R
s
,
along with a
the encounter.
orbit
I
The sphere of
the X,Y
influence
and £ determine the duration of
The sign ±
specifies the direction of
motion of m
I
and m
2
along
Motion of m
I
and m
2
about their center of mass in the
same
their hyperbolic
orbits.
sense of rotation as the motion of their center of
mass about M
is denoted by +,
about their center of mass
denoted by -.
of mass is
in the opposite
Finally, the motion of
specified by the
eccentricity
(e ) of
mean anomaly
(80).
of mass
and motion of m
orbit relative
be arbitrarily
semi-major axis
The orientation of
chosen.
2
sense is
the mi,
its orbit about M
and m
I
m
center
2
(a ) and
along with its
the mi,
m
center
2
to the reference frame, X,Y may
It will be assumed in all of
comparisons to follow that the orbits of mi
M are co-planar, and that m
and m
2
the
about
and m 2 orbit M in the same
I
direction, but these conditions are not required by the
model.
When all of the above parameters are
the elements of the m
I
and m
2
orbits
specified,
about M prior to
The
the encounter are determined.
integrations
are
started with initial positions and velocities for m
m2
2
is
their
I
and m
2
For certain
and m 2 prior to close
approach reaches a local maximum which is
When this occurs,
less than R0/10.
the initial positions and velocities of
and m2 used to start the integration are
this
and
are terminated when
and
separation again reaches R0/10.
encounters the separation of mi
I
integrations then proceed
The
approximately R0/10.
to closest approach of m
m
to
derived from the above elements, at a time prior
,
close approach for which the separation between m
m
and
I
chosen at
local maximum provided the separation there is of
the order of R /20
or greater.
The numerical
integration,
therefore, includes virtually all of the effect of the
interactions of mi
since the
and m 2 on
their orbital elements,
above choice of initial separation
two orders of magnitude greater than the
influence
radius.
The integrator used
is typically
sphere of
is one developed
by Myron Lecar, Rudolf Loesser, and Jerome R. Cherniak
series expansions of the positions
(1974) using Taylor
and velocities
in time.
appropriate, since
it has been shown
efficient for n-body
one where n
is
The use of
this integrator
to be particularly
integrations similar
small.
seems
to the present
A
of
complete test of
the model for
the nine parameters described
encounters is
a grid spanning
earlier for
clearly not possible.
all
specifying
In the remainder of
this section results of certain parameter
searches are
described where all but one or two parameters are held
fixed while the remaining ones are allowed to vary.
An
attempt has been made to select these searches carefully
in such a way as to gain insight into
various parameters
on the outcome of a close encounter.
First, the dependence on a and
eccentricities
(ae
of the orbits of m
,
I
ae2)
and m
(Figure 7).
The mi,
C of changes
and semi-major
2
axes
in the
(Aal,
about M was investigated
=
two different orientations,
7/10
the effect of
m
2
/
/10
the encounters in both these figures was assumed
sense of motion of m 1
was
and m 2
taken to be opposite
(-)
their center of mass about M.
1 AU.
)
for
(Figure 6) and $
center of mass motion
along a circular orbit with aO = RO =
Aa
=
for
to be
Also, the
about their center of mass
to the sense of rotation of
The masses were chosen
such that m 1 = m 2 = m 0 with the ratio m /M = .01M /M
=
1
2
O
0
-8
denote the mass of the Earth
, where M
and M
3.0 x 10
and of the Sun, respectively.
Values of Ae
1
, Ae
2
, Aa 1 ,
aa2 were obtained by application of the model and by
numerical integration for all of the
encounters represen-
ted by the grid intersection points
The values of Ael, Ae
2
, Aal,
Aa
2
in Figures 6 and 7.
given by the model were
used to produce the contour plots by interpolation.
of E the values of Ae and
For a particular value
decrease as the close approach distance, given by a
fraction of R
s
,
increases as one would
(E
critical value of
decrease as E
decrease
'v 2),
expect.
the values
increases for
of Ae
aa
as a
Above a
and Aa
a given value of a.
This
in Aa and Ae is due to the increasing relative
velocity of m
I
and m
at their point of closest approach
2
as 6 increases.
A measure of the agreement between the model and the
numerical integration is
E
a
afforded by the values
of Ee and
, where
E
E
e
a
=
eael1 -Ae
- Ae '
+ JAe 2 - Ae 'j
2
=
SAa
1
-
Aa 1 '
IAa 1 '1
+ IAa
+ IAa
2
2
-
a2 '
model.
and
the unprimed ones
Contours of E
e
obtained by
to values obtained from
and E
a
2-19
'
The primed quantities refer to values
tion and
2-18
IAe1 ' + IAe2
integrathe
are plotted in Figures
5
6 for the same grid as was used for the contours of
eccentricity and semi-major axis changes.
All of
the contour plots
For instance, the E
is
and E
e
contours show that the model
a
in agreement with the integrations
e with
P =
r/10 than when
understand the effect of
run with
4
assumed that mi
motion was along
= 7r/10.
c
motion
(+) of mi
For mi
= m2
for
= m
2
r.
4
and
0.16,
0.32, 0.64.
Again,
=
and
center of mass
o0 ,
that the
a circular orbit with a
and m
2
0
= R
it was
= 1 A.U.
0
run for both senses of
about their center of mass.
it is not necessary to study encounters with
r, since an interchange of mi
r <
In order to better
4, a new set of encounters was
This sequence of encounters was
>
for lower values of
spaced by intervals of 7/20, with 6 = 4,
with a = 0.04, 0.08,
4
0.
show a dependence on
and m
< 2r which are identical with
The values of Ael',
1"'
Ae
2
',
'
numerical integration for the
Aa
1
Aa
',
2
2
gives results
2
those for
'
0 <
4 <
obtained by
encounters with a
= 0.16 and
E = 4 are plotted in Figure 8 for both senses of motion.
The percentage errors are again given by Eqs.
2-19 and
2-18 and
the absolute errors are given by
E
=
Ea =
(JAel
-
Ae 1 '
+
(I Aa
-
Aa 1 '
+
iAe
2
-
Ae
2
'i)/2
2-20
aa2
-
Aa
2
')/2.
2-21
The near identity of Ael
taa
m2
2
1 is due
and Ae
to the symmetry of these
and with circular motion of
the m
The percentage of error Ee exceeds
= 0 and
vicinity of
for m I = m 2
= 0.
)
,
and Ea
= Tr/2
exceeds 50%
4,
expected for these values of
=
=
(
Large percentage errors
zero near
= 0 and Ae passes
/2.
and
2
iT
IAall
and
encounters with m
I,
50%
of
m2
I
=
center of mass.
only in the
is equivalent to $
= 0
only in the vicinity of
for Ae and Aa are to be
since Aa passes
through zero near
through
= 0 and
At such zero crossings, even relatively small
absolute errors give rise to very
large percentage errors.
The sequence of encounters in Figure 8 then serves to
establish a boundary at c I 4 such that for encounters
with E
Ae
2
>
4 and a
, Aal, and Aa
<
2
.80 the model produces values of Ael
with percentage errors
generally much less,
n/2.
provided
For most purposes it is
errors larger than 50%
tolerable, since
the mi,
is not too close to 0 or
expected that percentage
= 0 and
= 7/2 will be
the absolute errors are quite small
these orientations.
orbits of
at
c
less than 50%,
m2
for
So far, only encounters with circular
center of mass have been considered.
Now, the effects of variations of the center of mass
motion, keeping the remaining parameters fixed,
described.
will be
The dependence of E
(a,
parameters
ferent
e
e
sets of
was determined for
80)
= m0 .
2
eters,
and E
e
a
a
(X,
£.
represents the
were quite independent of
changes in the
e
in Figure 9 then
The boundary case, c = 4,
largest dependence of E
among the sets of fixed parameters
and E
e
on e
a
is relatively
decrease
independent of
slightly as R 0
Even
is not large
e 0 , but both E
e
2
on e 0 *
and E
a
Finally, we consider
increases.
encounters with values of mi
and
0
investigated.
considering the marked dependence of Ael and Ae
a
±)
The largest variations of E
for this case the dependence of Ee on e0
E
,
£,
were observed for encounters with relatively small
values of
R
several dif-
For most of these sets of fixed param-
center of mass parameters.
and E
on the center of mass
a
the remaining fixed parameters
for ml = m
E
and E
e
and m2
other than ml = m2
m0
Only negligible fluctuations
E
a
were found when m /m
12
within the range
m .
0
was
2
1 < ml/m
Next, values of E
of 6, c values used
e
2
<
and E
in the values of E
32
a
subject to
(ml + m2)/2
were determined
16 m
for
= 9/10 for these encounters,
where m = m
since,
=
for the grid
several values
The orientation of the relative hyperbolae was
be
and
allowed to assume values
in Figures 5 and 6
of m in the range m 0/16 < m <
e
= m2 .
taken to
aside from
values of
in the vicinity of the Ae and Aa zero
'
= 97/10 may be considered to be a worst
crossings, $
case in the sense of giving
and E
E
a
the largest values of E e
As m decreases the E
(cf. Figure 8).
= 50%
e
boundaries occur at larger values of
= 50%
and
However,
E.
this movement of the above boundaries is sufficiently
yields E
e
, Ea
a
a value of
50%,
>
On the
for m = m 0/16.
0
50%
E = 4 still
E slightly in excess of
small that a value of
3 for m =
16 m
0
other hand,
already assures E e,
at least for encounters with
c
Ea
not in the immediate
vicinity of the Ae and Aa zero crossings.
F.
Conclusions Drawn from the Numerical Comparisons
The elements of
theoretically be
expressed in terms of
used in Sections D and E
However, no concise,
to specify close encounters.
since the determination of the
final coordinates and velocities
estimate of
(el,
e2)
the parameters
explicit expression for this trans-
formation is possible,
mass requires
and m2 about M can
the orbits of m
of the m
i
, m
a solution of Kepler's equation.
the eccentricities of
for the case ml = m2 and
i
+ m
2 ,
center of
An
the orbits of m
e
but precise values
and m2
= 0 may, however,
be obtained from the approximate relation e 2
where m = m
2
(E/a) (m/M)3/ 5
of el and e 2 can
change by a factor of 2 due to dependence on other
parameters, such as
the values
.
For m
= m2 = m0
I
and
E = 4
of el and e 2 given by the above relation
range from 0.015 at a = 0.80
to 0.068 at a
= 0.04.
These values are typical of exact values taking all of
the parameters into account, and,
estimate of the lowest values
therefore, provide an
of el and
e 2 for which
the model works.
In summary, the present model yields changes of
orbital elements of m
and m
I
2
the
resulting from a close
encounter, which are in reasonable agreement with the
corresponding changes given by numerical integration,
provided the parameters
specifying the encounter
within the region of validity of
region of parameter space
and m I
cal
+ m 2>
$
E
and E
= nT/2
a
(n = 0,
specified by E
comparisons
integrations give E
of
e
m /8,
1,
e
,
2,
the model.
Ea
a
.
.
t
lie
For the
4,
a <
.80,
of the model with numeri50%
except in the vicinity
).
These
larger values of
near zero-crossings of Aa and Ae do not seem
to be a serious concern, since the absolute errors, E
and E
of
, do not increase
.
+ m
1
significantly for these values
If greater accuracy is desired, the model errors
are bounded by E
m
e
2
2
1
2m
0
0
•
e
,
Ea
a
, 10%
for E
'
8, a
.64,
and
When straight-line motion of the m
mass
is
assumed
comparisons with
l
, m 2 center
(Williamson and Chandrasekhar, 1941),
that E
integrations indicate
e
and E
a
E > 128.
for all but the fastest encounters,
exceed 100%
of
Unfortunately, our model is not an analytic one,
as is
that of Williamson and Chandrasekhar, because of our
assumption of conic motion of the m i ,
m
center of mass.
of systems of
In the context of computer simulations
many
2
small bodies orbiting a much larger body, the
present model is nevertheless very
offers a substantial improvement
simple to use and
in handling most close
The model ceases to be a useful approxima-
encounters.
tion only for the very slow encounters between bodies
in
grazing, nearly circular orbits.
G. Treatment of Encounters Outside the Range of Validity
of the Model
Those close encounters having parameters outside the
range of validity of the model and those for which a
collision is predicted by
tion of
of mi
the model are treated by integra-
the equations of motion.
and m
2
about M,
however, when R
=
close approach of mi
Eqs.
R2
-
The
equations of motion
2-1 and 2-2,
R1
and m 2 .
become
singular,
becomes small during a
This
singularity
in the
equations of motion causes a loss of accuracy and a
considerable
increase in the
integration.
Therefore, the
by transformations
computer time required for
singularity is
eliminated
in a process known as regularization
before the equations of motion are
readable account of
integrated.
this process has been given by
Bettis and Szebehely
(1971).
First, a Jacobian transformation of the
is performed where the vectors Q
+
Q =
>
A clear,
+
(mlR 1 + m2R2)/(m 1 + m 2
and R are defined by
4
4+
4
and R = R 1 - R 2
)
coordinates
.
The equations
of motion in Jacobian coordinates become
Sr
G(m
+ m2)
R
4r
r 2
1
F = GM(-- 3)
r
r
2
1
4.
+ F;
R
2-22
and
G(m
+ m
+ M)
r
=
12
m
1
+ m
3
r
2
r
1
+ m2
2-23
-23
r
2
The singularity resulting from close approaches
m 2 now appears only in the equation for R.
Sundman's
in the
and
Next, using
smoothing transformation, dt = RdT,
independent variable
of m
for the
first equation of motion, one
obtains
R
SR'
-
R
R-
+
G(ml
R
m
)
R + R F,
2-24
where
the prime denotes differentiation with respect to
the new independent variable T.
in the
singularity remains
above equation of motion, but its
been reduced,
of Eq.
A
i.e.
the term R
2-22, while
severity has
appears in the denominator
in Eq. 2-24 there is
only a factor of
R as a divisor.
For motion in two dimensions Levi-Civita introduced
a coordinate transformation in addition to a transformation of the
independent variable.
Levi-Civita's trans-
formation may be written as
4+ul
R =(u)u =u
-u 2
U
u2
u1
u2
u2
2
u2
u
R
where ul and u 2 are the new dependent variables.
2
, 2-25
Levi-
Civita's transformation gives the important relation R =
2
2
ul + u 2 so that the calculation of a square root is not
required to determine
Introducing
the relative distance R
the new dependent variables
equation of motion for R,
G(m 1
u"
+
)
+
-
Eq. 2-24,
into the
"smoothed"
one obtains
2(u'.u')
u =
in u-space.
2
u
(u)F.
2-26
2-26
2(u.u)
The coefficient of u in the last equation can be
be
equal
to one-half of
the negative of
shown to
the two-body bind-
73
ing energy per unit mass, E.
The transformed
"smoothed"
equation of motion for R can now be written as
-
u"
E
- - u =
T
u-u
(u)F,
2-27
where
+
-
-
2(u'-u')
E =
G(m
+ m)
4 u-u
-
+ m2
G(m
2
-
2-28
R
The above expression for 8 contains a singularity and
if
used in the above equation of motion would result of
course
in a singularity in the equation of motion.
However, a regular differential equation for the binding
ene'rgy can be found
6'
=
and is given in u-space by
2(•(u)u
- F).
2-29
The system of regular differential equations
t
= u
-9
u"
-
u
5
=
8' = 2u'
may
then be
u'u .- (u)F
(u)F
(u)F
solved for u and for
-
be obtained from R =
2-30
-*. _
(u)u.
the time,
t.
R may
then
A seventh-order Runge-Kutta
algorithm with automatic step-size control proposed by
Fehlberg
(1969) was
employed for the numerical integration.
CHAPTER III.
THE MODEL FOR COLLISIONS BETWEEN BODIES
A. A Description of the Collision Model
A
scenario for events following the collision of
two bodies
is described and various numerical estimates
relevant to the subsequent behavior of
given.
the system are
In particular, the total energy
of the two
colliding bodies at impact and the total energy of the
system at later
order
stages of the
scenario are
to demonstrate that the model
computed in
satisfies the
minimum requirement of consistency with total energy
conservation.
First, however, the various
stages of
the
scenario are outlined.
In a model in which the immediate formation of a
solid body is
sipating
assumed, the primary mechanisms for dis-
the impact kinetic energy are heating of the
resultant body and escape of the remaining material.
For bodies of equal mass with m
>
M(,
the impact kinetic
energy, if converted entirely into heat, would be
cient to melt all of
the material in both bodies.
heat capacity and melting
respectively.
The
temperature of the material
the bodies were taken to be
0.166-0.182 cal/g and
suffi-
in
those of ordinary chondrites,
1180-13500 C
(Alexeyeva, 1958),
Further, for bodies of
this size
the
75
comminution energy is
binding
energy.
far less than the gravitational
Since heat conduction is quite
cient over times as short as those elapsing
it seems much more
are
likely that the two
shattered into smaller fragments.
ineffi-
in collisions,
impacting bodies
From the computer
simulations it is observed that the fraction of collisions
with sufficiently high impact kinetic energy
disperse the material of both bodies
to completely
to large distances
is
quite small provided the orbital eccentricities are not
too large
(Tables 2-4).
We then conclude
that although a
few high velocity fragments may escape, they carry
considerable kinetic energy off with them and most of
the fragments, having
smaller velocities, will in general
form a gravitationally bound cluster of fragments.
The outside radius of such a cluster will necessarily
not exceed the distance to the straight-line Lagrangian
point corresponding to the total mass of
the cluster,
Since
since more distant fragments escape the cluster.
no data are available
(or, for that matter,
likely to
soon become available) on the size distribution of
fragments for
shattered bodies of this
gravitational binding
size, where
forces dominate over material bind-
ing forces, we make the plausible assumption that the
mass
of a typical fragment is equal to that mass
for which
76
comminution energy equals gravitational binding energy.
Since the fragment clusters are
AU)
(-10-3
and the fragments are on the basis of the above
assumption small
and,
relatively small
(a few tens
of kilometers
in radius)
therefore, numerous, collisions between fragments
will be frequent.
These
sipate the kinetic
energy of the cluster as heat, addi-
collisions will eventually dis-
tional comminution energy, etc.
disrupted in the meantime by
If the cluster is not
collisions or close encoun-
ters with other clusters or large
bodies,
it will shrink,
primarily by flattening, at least at first, as its
kinetic
energy is dissipated,
eventually forming
body of-mass approaching that of
a solid
the cluster.
B. An Estimate of Fragment Size After a Collision
Before the total
energy of the system upon formation
of the cluster can be computed, we must estimate the mass
of the fragment for which material binding energy equals
gravitational binding
energy.
Experiments have been
performed by Greenberg and Hartmann
(1977),
with both
finite and semi-infinite targets over a range of velocities
(3-300 m/sec),
to determine the critical energy per
unit volume for catastrophic
parameter,
failure of the
referred to as the "impact
target.
strength,"
was
This
found
to be I
This impact
s
10 7
3 x
'
ergs/cm 3 for
rocky material.
strength has been confirmed
much higher impact velocities
by Moore and Gault
and these results
dent of velocity.
for basalt at
km/sec)
(1-3
in experiments
(1965) and by Fujiwara et al.
(1977)
indicate that impact strength is indepenThis number most likely represents
an
upper limit for the material binding energy because of
probable
flaws in bodies
larger than a few kilometers
radius.
Since an order of magnitude
in
estimate is sufficient
for our purposes, the impact strength cited by Greenberg
and Hartmann for small bodies
is taken as representative
of the material binding strength of
considered here.
the fragments being
Equating this impact
gravitational binding
strength to the
energy per unit mass,
the radius
of a typical fragment is found to be
I
R
f
= (
47r p
G
)
1/2
1
The corresponding mass of this
mf '
1.2 x
10
-5
m
.
3-1
40 km.
The total
fragment is
energy upon formation of a
self-gravitating fragment cluster
can now be determined
on the basis of the above assumption of fragment size.
78
C. The Requirement of Total Energy Conservation
The gravitational potential energy of the
system at
the instant of impact is
Gm
U
i
S
=
3
5
R
2
Gm
2
1
3
2
1
5
R2
Gm m
1 2
R
1
3-2
3-2
+ R
where
the first two terms
bring
the material comprising the bodies,
of
constant density,
tions,
represent the work done to
into the given spherical distribu-
and the third term is
of the bodies.
T.
where V.
The incoming
1
2
-pV.
2
=
i
assumed to be
=
the mutual potential energy
kinetic energy at
mm
1 2
m
+ m2
3-3
is the relative velocity at impact.
simplicity the bodies are
For
for now assumed to be of equal
mass and density, ml = m 2 = m,
case the total
impact is
energy of the
R
= R
= R.
For this
system at the instant of
impact is then
E
As was
occur
=
m
2
V
4
i
stated earlier,
17 Gm
10
R
3-4
collisions with E.
> 0 very
seldom
in the simulations.
Next,
me
i
let m
c
denote the mass
the total escaping mass.
of the bound cluster and
Total mass
conservation then
requires m 1
+ m
= 2m = m c + m e .
2
The total energy of
the
system, once the cluster has been established and any
escaping fragments are
E'
Here, Uf
of
far away, may be
expressed as
= Uf + U
+ T
+ T
+ Q.
f
c
c
e
3-5
is the sum of the gravitational binding energies
the cluster fragments, U
is the gravitational poten-
c
tial energy of the fragment cluster,
T
c
is the total
internal kinetic energy of the fragment cluster, T
e
is
the kinetic energy carried away by escaping material,
and Q
is a term including energy dissipated
in a variety
of ways such as in comminuting or compacting material,
in phase transitions, etc.,
dissipated as heat.
but primarily includes energy
(Note that the energy
required to
raise the temperature of the rock by a little more than
10 K
is sufficient to crush the rock!).
Since the cluster
satisfies
the virial
is a self-gravitating system,
theorem which states that
2<T > = c
<U >
c
where the brackets indicate
brackets,
it
,
3-6
time averages.
Dropping
the
the total energy of the cluster may be written
as
E
c
= U + T
= U /2
c
c
c
,-
3-7
80
and
the cluster may be said
U /2.
c
If for
to be bound by an energy
simplicity all
fragments in the cluster
are assumed to have the same mass, mf,
is
taken to be equal to that mass
and binding energies are
equal,
and their mass
for which gravitational
then Uc is
given in this
case by
c
i,j
ifj
where Rc is
r.
13
mm.
U
G
r.
13
-Gm
f
2
ij
i#j
1
r.
13
the harmonic mean value of
, over all pairs of fragments.
potential
Gm
The quantity Rc is
2 n(n-l)
2
the total
c
3-9
2R
a characteristic
scale size for
the
to the
extremities of
Since the distance from the cluster center
to the outlying extremities
is
limited by the distance to
the straight-line Lagrangian points,
taken to be a fraction, f,
extremities.
3-8
c
expressed as
as a harmonic mean it clearly refers
the cluster.
R
the separation,
central bulk rather than to the outlying
be
f
Since n >> 1,
energy of the cluster may be
U =
c
cluster;
Gm
the
core radius may
of the distance to the
It may then be written as
81
R
where R 0
and Mg
is
= f R=
f
R
1/3
m
( -)
3-10
is the heliocentric distance of
is
the solar mass.
the cluster
The total energy of the cluster
then given by
E
The
G
f
4R 0 f
c
(3M)
1/3
m
5/3
c
3-11
.
total gravitational binding energy of the cluster
fragments is
U
=
f
3
-3-12
G
5
If the fragments
2
m.
i
R
i=l
i
n
are assumed to have the same mass, mf,
and to have uniform densities equal to the density of the
colliding bodies,
p
,
then
3-13
1/3
2
f
3
mf
3 G
cmf
3 G(
5
Rf
5
Rf
5
2/3
3f
In the absence of predictions of the total
the distributions of mass
fragments, and
and velocity of
c
escaping mass,
the escaping
the energy dissipated as heat for collisions
between bodies of mass m > Mi,
we will choose
conservative assumption and set Tee = Q = 0.
the most
With either
82
calculated
total
or assumed values for each term in E',
energy of the
the
system upon establishment of the
cluster may be written as
4R
E
mc 5/3
4R
f
-
5
G(
3
) m
We may now ask how much of the initial
energy could be absorbed in fragmenting
f
m2/3
c
3-14
impact kinetic
the colliding
bodies, assuming the formation of a cluster of fragments
of equal mass, mf?
D.
Implications of Total Energy Conservation
Earlier in this chapter the
instant of impact was determined
m
2
E. = T. + U. = V
1
1
4
i
17 Gm
1
10
R
total energy at
the
to be
2
3-4'
for the case of colliding bodies of equal mass and
density.
The kinetic
to be a fraction,
impact.
be
energy at impact is further taken
L, of the
total potential energy at
The total energy of the system at impact may then
expressed as
E.
1
=
(1
-
)U.-
1
3-15
where X <
1 for E.
1
< 0.
course for E. > 0.
1
material was
also
zero
terms
c
=
to zero, the
2m.
of A = T./-U.
the total
is to be expected of
Since the kinetic energy of
set equal
and mc
No accretion
An
energy of
escaping mass
expression
and m f/m
escaping
for
f
= R
c
is
/R
may be obtained.by
L
in
equating
the system at impact and upon forma-
tion of the fragment cluster,
(3M )1/3
4R
1
0
f =
2/3"
(
For the
S)1/3
)
[
3
10*2
5/3
(1 -
small value of
considered here,
value of f.
17
A is
With pI
)
-
3 mf
5
(
m
3-16
)
c
the ratio m f/mc being
the primary factor determining the
= 3.34 gm/cm
(lunar density),
mf =
-5
1.2 x 10-5 m
(the mass for which impact
strength equals
gravitational binding energy per unit mass),
RO = 1 AU, and A = 0.95,
f <
1/20.
Thus,
mc = 2m(,
the above expression gives
energy conservation permits the
forma-
tion of a quite compact cluster even in the case of a
collision with relatively high impact kinetic energy.
When, in addition, the
impact kinetic
in heating, comminution,
taken into account, we
energy dissipated
and compaction of material is
conclude that at
least for
84
A < 0.95 practically all of
impacting bodies is
cluster.
the material from the
gravitationally bound
in the fragment
Having hypothesized the formation of gravita-
tionally bound clusters of fragments of the shattered
colliding bodies as a stage
in the process of accretion,
it remains to show that these clusters will shrink, by
dissipating
energy in collisions between fragments, to
solid densities
in a time less
than the characteristic
collision time for the protoplanetary bodies.
E.
Do the Clusters of Fragments Shrink to Solid Density
Before Their Next Collision?
The collision probability per unit time for one
fragment is given by
1
where N is
is
-
-
V N
0,
the number of
3-17
fragments per unit volume, V
the mean relative speed of the fragments, and a
the collision cross-section of the fragments.
is
Suppose
that in each collision between fragments an amount of
kinetic
energy
AT
=
mm
1
1 2
8
2
m1 + m
12
-2
V
3-18
is dissipated as
heat, comminution energy,
etc.,
where
is the fraction of the relative kinetic energy dissipated.
For m
= m
i
2
= mf
this dissipated energy
AT =
f
8 -V
4
per collision.
is
-2
3-19
In the mean collision time,
T,
there will
be n collisions in the cluster dissipating an amount of
kinetic energy
T
where n is
T
mf -2
V
4
= n-
the number of fragments in the cluster.
The total kinetic
T
where v2
c
= -m
2
c
is the mass
energy of the cluster is
v
,
average
facilitate comparison of T
and
v2 is
needed.
2
2
= 2v .
3-21
speed of the fragments.
,
and T
To
a relation between V
Identical fragments have been assumed
in this computation,
case V
3-20
and it is
easy to show that in this
For present purposes the difference
between rms and average values may be neglected,
-2
2
relation V
= 2 v results.
The
and the
total kinetic energy of
the cluster may then be written as
86
T
c
=-
1
-2
m
V
4
c
3-22
It comes as no great surprise that the number of collisions
per body required
to dissipate the kinetic energy of the
cluster is
Tc
1
•
3-23
T
Assuming that T remains constant as the cluster
the time required for
evolves,
the cluster to dissipate enough
kinetic energy to shrink to
solid density may be approxi-
mated by
t
=-
T
.
3-24
The behavior of a cluster of fragments
kinetic
energy and
as it dissipates
subsequently shrinks is
certainly
much more complicated than the simple model described
here indicates, but an order of magnitude estimate of
settling time will suffice.
It remains then to evaluate
T.
Invoking the virial theorem once again and recalling
the total potential
energy of
-2
1
2<T > =m
V
c
2
c
the cluster
Gm
= -U
c
=
2R
2
c
c
,
3-25
the mean relative speed of a cluster fragment is determined
to be
Gm
V = /
R
3-26
c
c
Given the mean relative speed of
the fragments and their
mass, the effect of gravitational
focusing in encounters
between fragments can be readily estimated and is found
Therefore,
to be negligible.
for the fragments
is taken to be their effective
geometric cross-section, o =
mf =
the collision cross-section
4W
3
and R
- P(Rf
R
= f
2
(2R f)
(m /3M
)
With the relations
.
1/3
,
the mean
collision time may be written as
1/3
S=
f 7 /2(f)
3-27
m
c
where the proportionality constant C is
2/3
3 7/6
C =
-
= 1210 years
(-
S3Mo
given by
3
at R =1 AU.
0
The largest mean collision times occur for
3-28
the least
massive fragment clusters and
for those clusters with the
largest values of f = Rc/R
The mean collision time
L .
between fragments for the least massive clusters
simulations, m c '
0.04 M
®
,
assuming
f
'O
1/20
is
in our
of the
order of 5 hours.
and
= 1/10,
extreme values of f =
Even for the
1/4
the mean collision time between fragments
is expected on the basis of these estimates to be of the
order of
50 days with a corresponding
settling time for
1.5 years.
On the other hand,
the mean collision time between bodies
in the early stages
the cluster of the order of
(t<
1000 years) of the planar simulations
carried out here
are of the order of 40 years.
Considerably greater mean
collision times between bodies
are expected for a more
A more detailed treatment of
realistic non-planar system.
the
collisions between protoplanets and
evolution of their
the subsequent
collision remnants would of course be
highly desirable, but is certainly too
ambitious for the
present computer simulations.
Collisions
between bodies are
simulations as follows.
tion
then treated
When a close approach with
less than the sum of the two radii is
integration,
in the
found by
the two colliding protoplanets are assumed
to accrete and form a body with mass equal to the
the masses of the two incoming bodies.
elements of
separa-
sum of
The orbital
the newly formed body are determined from
the position and velocity of the
two colliding bodies at the
center of mass of the
instant of
impact.
The
89
relative velocity of
the bodies upon impact
and used to calculate a value for X
is recorded
(Eq. 3-15) which in
turn is used to decide whether the assumption of accretion
was
a valid one.
Relative
ding values of A are
impact velocities and correspon-
tabulated for the collisions in
three simulations in Chapter V.
90
CHAPTER IV.
A.
THE COMPUTER SIMULATION PROCEDURE
The Basic Assumptions of the Simulations
In the present simulations of the growth of the
terrestrial planets by accretion of large protoplanets
(mi
M /50),
the protoplanetary bodies are assumed to
move along unperturbed co-planar orbits until they pass
within a sphere of influence distance,
body.
R
s
, of another
If a close approach to another body occurs,
orbital elements are
computed for
each body.
new
When the
parameters of the encounter fall within the region of
in Chapter
II
for close
validity of
the model described
encounters,
the new orbital elements are computed using
the model.
Those encounters having parameters outside
the range of validity of the model and those predicted
by the model to have separations at closest approach less
than the sum of the radii of the two protoplanets are
treated by
motion
integration of the regularized equations of
(cf. Section G of Chapter
bodies during these
starting
II).
The motion of the
latter encounters is
at an initial
integrated
separation of 50 R
, and continuing
through close approach to a final separation of
where
new orbital
elements
are computed
50 R
s
for each body.
,
As stated
in the previous chapter,
two protoplanets are
assumed to accrete and form a body with mass equal
the
to
sum of the masses of the two incoming bodies when a
close approach with separation
radii of the bodies
(assuming lunar density material)
occurs during integration.
newly formed
less than the sum of the
The orbital
elements of the
body are of course determined by the position
and velocity of the center of mass of the colliding
bodies at the instant of impact.
Several considerations
influence the choice of
50 R
s
as the separation distance at which integrations are
initiated and terminated.
Clearly, the separation must
be large enough to include in the integration as much as
possible of the effect of the encounter on the orbital
elements of the two protoplanets.
At the same time the
separation should be sufficiently smaller
than the
typical spacing between protoplanets that the effect of
the nearest neighboring protoplanets on the encounter
between the two under consideration can generally be
neglected.
For a system of one hundred protoplanets hav-
ing a total mass equal to the sum of the masses of the
terrestrial planets, with uniform spacing of the protoI)lanets in
the ecliptic plane between the heliocentric
distances 0.5 and 1.5 AU,
the typical
spacing is
L170 R s
The separation 50
Rs was chosen as a compromise between
the above two requirements.
It should be noted that
only the effects of those encounters
expected on the
basis of unperturbed orbits to result in separations
less
than the sphere of influence are included in this
simulation.
B. The Relative Importance of Close Versus Distant
Encounters
One can ask what fraction of the changes
orbital elements of the protoplanets might be
result
from close encounters,
more distant encounters,
< R ,
(Ar)m
min
s
(Ar)m
min
> R
s
of
the importance of close encounters
of
the protoplanetary
.
in the
expected to
as opposed to
A rough estimate
in the relaxation
system will now be given.
The
argument here is very similar to one made for twodimensional disk systems of
(1972).
stars by Rybicki
In estimating the relaxation time due to encounters,
we assume that encounters between bodies can be
as two-body interactions neglecting
Sun.
the presence
Including the influence of the Sun on
where
relaxation time
of the
the encounters
by means of the model described in Chapter II
the estimation of
treated
would make
impractically difficult,
an order of magnitude estimate is
sufficient and
much simpler.
It is not difficult to show that in the
center of mass system
the orbits of both bodies are
hyperbolae, the asymptotes of which make an angle 7-6
with the deflection angle
6 given by
2 4
SD
sin 6 =
2
where V is
[1 +
G
2
2
-1/2
V
D V
(m 1 +m2
4-1
2
the relative velocity of the bodies and D is
For Do = G(ml + m2)/V
the impact parameter.
the
deflection angle in the center of mass frame is
The relaxation time,
TR
6 = 7/2.
can then be estimated as the
time for a typical body to experience an encounter with
D < D
.
This will be an overestimate of the true relaxa-
tion time,
since the cumulative effect of long-range
The number of encounters
encounters has been neglected.
with D < D
bodies,
during T R
is equal to the surface density of
a, times the area
(VT
R )
-(2D
) within which
another body must pass to satisfy D < D O .
number of encounters with D < D O
T
R
=
V
2G(m 1 + m2)
2
Setting
equal to one, we
this
find
4-2
94
A more
refined estimate including the cumulative
effects
of long-range encounters, D > D O , will now be
given.
The relaxation time
is defined for this case as
the time in which the root mean square velocity change
the same order of magnitude as the
due to encounters is
typical relative velocity between bodies.
An expression
for the deflection angle 6 equivalent to the one given
earlier
is
tan
6 G(ml + m2
=
2
2
DV
4-3
-
For long-range encounters the deflection angle
is
small
and may be approximated by
2G(m
S
1
DV
+ m2
The velocity perturbation
G(m
AV
1
4-4
for distant encounters
+ m2
DV
is then
4-5
The condition giving T R then becomes
2
V
2
E (AV)
2
G(ml + m2
)
V
(
(VTR)a
D
0
2dD
2
D
4-6
with the lower
integral taken to be DO
the upper limit
formula for AV breaks down, and
where the
taken as o
D.
limit of the
since the
integral converges rapidly for
The relaxation time for
large
long-range encounters is then
V
TR
=
2G(ml
4-7
+ m 2)C
a result identical to that for
encounters with D -0
< D .
Therefore, the relaxation is substantially due to
and the cumulative effect of
encounters with D < D0,
long-range encounters is of no more than the
same order
of magnitude.
It remains to establish the relationship between
D
and the sphere of
define
influence radius, R
close encounters
For this
,
used
to
in the current simulations.
the typical relative velocity between bodies
is expressed as a fraction of the circular orbital
velocity at their mutual heliocentric distance, V = XV c ,
the relative velocities
observed for bodies
in the
current simulations generally fall within the range
.01 < X <.06 with X =
D /Rs
assuming m
.025 being typical.
= m2
= m,
The ratio
may then be written as
96
2 Gm
D
2Vc 24/5
c
R
2
2
1/5
2)
2
C
3/5
m
-
(-)
M
2
,~
4-8
R
M
where R
is
than Rs
for
in our
the
A
heliocentric
>
.0078, which is
simulations.
When
relative velocities are
than for
those
with D
of
argued
that encounters
contribution
bodies
to the
algorithm used
the
basis of
less
C.
than
for
i.
The
The
is
one
less
if
less
the
case
that
long-range
.
of
it
<
min
the
R
s
make
The
expected
close
be
the major
system.
to give
is
Keplerian
can reasonably
encounters
influence
the
encounters
to differential
those
orbits
then
on
approaches
now described.
Finding Close Encounters
Part
algorithm
possible only
for
(Ar)
relaxation
is
also noted
system,
with
finding
particular pair of
is
the
Used for
Spatial
separate parts,
orbits
in
sphere of
The Algorithm
is
due
D
almost always
greater
unperturbed
the
it
< DO
motion
the
distance.
naturally
divided
spatial
bodies,
and
a close
the minimum
than
the
one
into
two
temporal.
encounter
is
distance between
sphere
of
influence
For
a
spatially
the
two
radius,
R
s
,
of the
is
the two bodies.
larger of
specified by its semi-major axis,
angle of perihelion; m.(a.,e.,w.)
1
1 1
and m.(a.,e.,w.).
J I
3
Two
I
the
bodies which must be considered.
number of pairs of
encounter is possible if
Clearly, no close
-
e.)
1
-
a.(l
1
1
eccentricity, and
immediately available to limit
simple tests are
or
The orbit of each body
R
s
> a.(l + e.)
3
3
+ R
s
4-9
if
e.)
3
a.(l 3
-
R
s
> a.(l + e.)
1
1
+ R
survived these
Assuming that the given pair of bodies has
tests,
intersection between the
the orbit of m*
1
there is an
to be determined whether
it remains
s
of
locus of points within R s
and the locus
of points within R
s
the
of
orbit of m..
3
The
locus of points within R
semi-major axis, a,
and eccentricity,
be bounded by two ellipses having
the original ellipse;
and
e'
=
(a/(a -
and
e"
=
(a/(a + R ))e.
s
R
s
of an ellipse with
an
the same orientation as
inner ellipse with a'
))e and an outer ellipse with
The
along
its
semi-major
axes; however,
= a -
Rs
a" = a + R s
ellipses
inner and outer
of the ellipse
exactly bound the locus of points within R
(a,e)
shown to
e, can be
along
the
semi-
the outer ellipse "bulges"
minor axes
inner ellipse
slightly and
include an area slightly
and outer bounding ellipses
greater than the locus of points within R
the
For Rs << a the error along
almost always
s
of
less than 1%
and in any case no close
because of this error
of R
(a,e).
semi-minor axes is
error is
current simulations this
In the
'\R e /2.
s
the inner
Therefore,
is slightly "flat."
the
, usually much less,
encounters will be missed
the bounding ellipses
since
enclose
slightly more area than necessary.
Denoting the inner and outer bounding ellipses for
the orbit of m.
1
i' and i",
by
those for m.
denoting
intersection of the
respectively, and similarly
and j",
by j'
the condition of
locus of points within R
of the
s
of the
orbit of m.
with the locus of points within R
orbit of m.
is equivalent to the requirements that i
1
intersect j' or
j" and
j intersect i'
or i".
s
of each orbit, i intersects both j' and
j intersects both
a close
In the
containing points
most common case of overlap of the bands
within R
s
i' and
i".
Two separate regions where
encounter might occur then exist.
exists.
distinct possibilities:
Less frequently
the bands occurs and only one
a "grazing" intersection of
region of overlap
j" and
In this case
(i)
there are
two
i intersects j' but not j"
and
j intersects i" but not i',
lying
(ii)
j"
i intersects
i' but not i".
but not j' and j intersects
Since
the
intersections between two ellipses
finding
problem of
or
in the same plane has a simple explicit solution,
the tabulation of spatially possible close encounters
is
relatively straightforward.
The values of the angle 0 at which two co-planar,
(al,el,
1 t)
confocal ellipses,
(a 2e2 W 2),
and
intersect
can be determined by equating the radii from their
common focus
-
a.(l
r
=i
i
e.
)
1
4-10
1,2.
=
1 + e. cos(O - w.)
1
It can be shown without much difficulty that the values
of 0 for which rl = r 2 are specified by
cos(0
where
=
w =
-
F 1
e 2 sin
tan-l
02 -
l'
(sin w)/(cos w -
=
4-11
/1 +
2
2
(a2(1 - e 2
elF/e2).
the relation
))/(a
(1 -
2
2)),
and
There are in general two
values of 0 which are determined to modulo
of course
e
2r, provided
100
e
This
-1
-<
sin w
2
1.
1
last inequality is
then the condition for
section of the two ellipses.
simplifying,
succinctly
2
e
2
Substituting for
the condition for
written
4-12
2
- 1)
2 2
- 2ele 2
e
F
1
12
<
1.
4-13
cos W
for the intersection of two
ellipses and with the ellipses
the locus of points within R
found earlier to enclose
of an elliptic orbit, one
can quickly determine whether close encounters
spatially possible for
are possible
tabulation of one or two
for the pair,
the explicit
(depending on whether the
of each orbit is
intervals.
If
angles make possible a
the
intersection of bands containing
of angular
are
the given pair of bodies.
solutions for the intersection
s
and
as
With the above condition
within R
i
intersection may be more
(
close encounters
inter-
locus of points
"grazing" or
The points
segments then have the property
along
"general") pairs
each of these arc
that their distance from
the nearest point in the other arc segment is no greater
101
than R
s
.
The temporal part of the algorithm for determin-
ing potential close
encounters consists of determining
that time interval for which the bodies next simultaneously
occupy corresponding orbital
intervals.
ii. The Temporal Part
First, time intervals corresponding to the pairs of
angular intervals are
computed using Kepler's equation.
The time intervals are referenced to the
time since
pericenter passage either at the time the last encounter
involving the given body ended or
of the
1
and t.
3
let 6. and
with m.
For a pair of
denote the mid-point times
time intervals
and P.
and m.,
let
of the intervals,
6. denote the widths of the
finally let P.
time
involving this body
simulation, if no encounters
have yet taken place.
t.
at the starting
intervals, and
be the orbital periods associated
respectively.
The
condition that the
bodies next simultaneously occupy these corresponding
orbital
intervals after n orbits of m.
can be shown to
be
A <
where A is
(6 1 +
,
62)/2
2
the lesser of
4-14
102
A
=
Imod(nP.
+
t.,
P.)
-
t.
4-15
and
A
In
2
= p
1
-
4-16
A .
1
the expression for Al
x -
[x/y]y,
,
mod(x,y) is defined as
where the brackets indicate that the largest
integer whose magnitude does not exceed the magnitude of
x/y is used.
The above condition that m.
1
simultaneously
and m,
j
occupy corresponding orbital intervals for which interaction is
for a close
one.
a necessary condition
spatially possible is
encounter of m.
and m.,
their intervals
The bodies may still pass through
in such a way that their minimum
than R
but not a sufficient
.
To
establish whether the minimum separation
between the bodies
time they are
separation is never less
is
in fact less than Rs during the
simultaneously occupying their corresponding
intervals, a minimum search routine is employed.
routine used, Brent
(1973),
combines golden section
search and successive parabolic
interpolation.
simulations it is possible in most cases
whether the minimum separation
evaluations of the positions
The
is less
In actual
to determine
than Rs with
of the bodies
and their
*--_-r__-r-r;~;rr~-~oEnusxe,~ei*~
103
corresponding
separation at only two distinct times.
This
is so because it has been observed that for relatively
small eccentricities the
separation does not change
very rapidly during the time
simultaneously occupy their
intervals.
Therefore,
separation after
interval in which
the bodies
corresponding orbital
if the
estimate of minimum
two evaluations
is greater
than 4R
it has been found in practice that no close encounter will
occur.
This completes the discussion of the algorithm
for
finding close encounters.
the
simulations follows.
A verbal flowchart for
D. A Flowchart of the Simulations
Step 1.
(Initial Conditions) First, starting values
are chosen for the masses and orbital elements of the
bodies in the system whose evolution is to be simulated.
All simulations to be presented here begin with one
hundred identical bodies with mass equal to 0.02 Earth
masses.
The total mass
equal to the total mass
The
of the
system is
of the present
semi-major axes of the orbits of
then very nearly
terrestrial planets.
these bodies
are
required to fall within the range 0.5 to 1.5 AU, corresponding approximately to the range of semi-major axes
the terrestrial planets.
The surface density in all
of
of
104
these simulations was
assumed to have an R
on heliocentric distance between 0.5 and
zero elsewhere.
-3/2
dependence
1.5 AU and to be
The initial eccentricities of the orbits
were randomly chosen from a uniform distribution on the
interval from
0 to some maximum
eccentricity, ema
angles of perihelion for the orbits are given by
x
.
The
a
sequence of random deviates from a uniform distribution
on the interval
(0,
27) .
Finally, the time since peri-
helion passage for each body is
distribution on the interval
taken from a uniform
(0, P)
where P
is the
orbital period of the given body.
Step 2.
chosen;
An initial simulation
usually this
Step 3.
time step, At,
is
is taken to be five years.
Once the masses and orbital elements
all of the bodies in the system are specified,
for
it can be
determined as described earlier whether a close encounter
is
spatially possible for each pair of bodies.
pairs of bodies
for which a close encounter
angular intervals lying within R
s
of
For those
is possible,
the other orbit
are determined for each orbit.
Step 4.
encounter is
and
For each pair of bodies
for which a close
spatially possible, determine the starting
ending times using Kepler's equation of
intervals during which the bodies
those
time
simultaneously occupy
105
the
sections of their orbits for which spatial interaction
is possible.
Naturally, only those time intervals with
starting times within the current simulation
time step
are considered.
For each time
Step 5.
a minimum
interval obtained in Step 4,
search routine combining golden section search
and successive parabolic interpolation is used to determine
separation of the bodies, (Ar)
whether the minimum
in fact less than Rs
interval.
is
. ,
min
for any time within the given
Those times corresponding
to minimum
separations
comprise a set of potential close encounter
less than R
times.
Step 6.
The
< R
(Ar)
min
s
times associated with
next sorted into ascending numerical order.
of these times determines
The
are
least
the interval within which the
next close encounter will take place along with the
which will experience
pair of bodies, m.1 and m.,
J
that
encounter.
Step 7.
the time during
separation
Again, using
the minimum search routine,
the approach of m.
is equal
to Rs
and m.
is obtained.
for which their
For
this time
the positions and velocities for each of the bodies are
calculated
of
and orbital
elements for the
relative motion
the bodies are then computed from these positions
and
106
The eccentricity, E,
velocities.
a,
of
semi-major axis,
and
the relative motion determine the method to be
used in the next step to compute the new elements of the
and m.
orbital motion of m.
about the Sun following the
encounter.
Step 8.
If 6 > 4 and
(Ar)mi n = a(
mmn
-
1) < 0.8 R
and m.
the new orbital elements for the motion of m.
0.8 R s,
If c < 4 or
in Chapter II.
the new orbital elements
(Ar)mi
For
m.
n
=
a(E -
1) >
are found by integrating
the regularized equations of motion of m.
presence of the Sun.
about
the close encounter model
the Sun are computed using
described
s
in the
and m.
E < 4 encounters between m.1 and
take place relatively slowly and these encounters are,
therefore,
integrated starting with an
prior to close approach of
50 Rs
justification of this choice).
to close approach of m.
separation between m.
1
and m.
(cf.
initial separation
Section A for
The integrations proceed
and terminate when the
and m. again exceeds
J
positions and velocities of
50 R
s
.
The
the bodies upon termination
of the integration are used to compute new orbital
elements
for the motion of m.
Those
and m.
about the Sun.
encounters predicted by the model to have
separations at closest approach less than the sum of the
radii calculated for
the two protoplanets,
assuming
107
homogeneous composition and
tions between bodies used to initiate and
integrations of these encounters were
s
.
terminate the
again chosen to be
When a close approach with separation less
the sum of the two radii is
incoming bodies.
form a body
upon impact is
the masses of the two
elements of the newly formed
from the position and velocity of
the center of mass of
instant of impact.
sum of
The orbital
body are determined
the two colliding bodies
accretion was
a valid
at the
the bodies
The relative velocity of
recorded and used later
the assumption of
than
found by integration, the
two protoplanets are assumed to accrete and
with mass equal to the
separa-
The initial and final
also treated by integration.
50 R
are
lunar density material,
to determine whether
(cf. Tables
one
2
to 4).
With the new orbital elements for m
Step 9.
(only the elements for m.
are required following a
collision),
determine whether a close encounter is
ly possible
for each pair of bodies with i or
members.
and m.
i
spatial-
j as
For those pairs for which a close encounter is
spatially possible, determine
intervals for
as
in Step 3 the angular
each orbit which lie within R
s
of the other
orbit along with their corresponding time intervals.
108
Step 10.
For each pair of bodies
including i or
j
as members and having at least the spatial possibility
of a close encounter, determine
times of
the starting and ending
those intervals during which the bodies
simultaneously occupy the sections of
their orbits
which spatial interaction is possible.
time intervals with
previous step,
Again, only those
starting times within the current
simulation time step are
Step 11.
to be considered.
For each time interval computed in the
the minimum search routine is used to
determine whether the minimum separation of
is
for
the bodies
in fact less than Rs for any time within the given
interval.
Those times
corresponding to minimum
separations
less than Rs form a new set of potential close encounter
times.
Step 12.
If any of
encounter pairs includes
the old set of potential
i or j, delete
the new and old sets of potential
them and
encounter pairs
combine
along
with the times associated with their minimum separation.
If the combined set of potential
empty, all close
encounters
is chosen.
is
in the current simulation
time step have been treated.
tion time step
encounter pairs
In this case a new simula-
If the number of close
109
encounters occurring in the time
step just completed is
greater than one, the new simulation time step
to be one half of the
no close
is
If
length of the preceeding one.
encounters have occurred in the previous
simulation time steps,
taken
the new time
step is taken to be
Otherwise, the
twice the length of the preceeding one.
simulation time step remains
If the starting
the same.
time of the new simulation interval
two
exceeds a termination
time estimated to be greater than that for which the
system ceases to experience further
the simulation is
then ended;
close encounters,
otherwise, branch to
Step 4 and continue, once the new time step has been
chosen.
When the combined set of potential encounter pairs
is not empty, sort the times
separation into ascending
these times
corresponding to minimum
The
numerical order.
least of
is then the time of the next close encounter
and determines the pair of bodies,
the next close encounter.
i'
j',
involved in
Branch to Step 7 and determine
the new orbital elements for
i' and
j'.
This completes the description of
the computer simulations.
and
the algorithm for
In the next chapter the
of four numerical simulations are discussed.
results
110
CHAPTER V.
A.
THE COMPUTER SIMULATIONS
Results of Four Representative Numerical Simulations
1. The Plots
Results are presented
in Figures 10 through 24 of
four representative numerical simulations
of the terrestrial planets by accretion of
planets.
of the growth
large proto-
These simulations primarily differ in the
values chosen for e
max
,
cf. Section D of Chapter IV.
The
time evolutions of the orbital eccentricities for the
bodies
in each simulation are given in Figures
and 19;
the corresponding
presented "in Figures 11,
10,
13,
16,
semi-major axis evolutions are
14,
17,
and
20.
An equal
abscissa increment has been assigned to each gravitational
interaction in the above figures
independent of whether
the interaction results in a collision or not.
associated times are given as
Collisions and
in these
and 21
logl0 of the
time in years.
subsequent accretions of bodies
figures by
small boxes.
the semi-major axes,
the system bodies are
The
are denoted
In Figures 12,
15,
18,
eccentricities, and masses of
shown for nine different stages of
The
each simulation including
the initial and
final ones.
boundaries of
semi-major axis
and eccentricity
the initial
111
distributions
frame.
of
are indicated by the rectangles
The areas of
the small boxes marking
in each
the location
each body are proportional to the masses of the
bodies.
Finally,
the number of
system bodies for each of
the nine stages is indicated at the top of the frame.
Next, several features
apparent in these plots
of the
simulations will be described.
First,
in the
eccentricity evolution plots we note
a redistribution of eccentricities for all of
tions:
the simula-
the number of bodies with very small values of
eccentricity decreases and
a "tail"
extending
higher values of eccentricity develops.
toward
Also, a rapid
initial increase in eccentricity is apparent for
e
max
=
0.025 and 0.05
simulations.
Where the
these figures are not too dense, one
range of typical
measures of
Later
Also note
values.
in the chapter quantitative
(N > 90)
that in the late
each simulation the
perturbed
occurring during
these eccentricity changes are tabulated and
discussed for the early
tion.
lines in
can observe the
eccentricity changes
close encounters.
the
evolution of each simulastages of
smallest bodies
evolution of
are commonly
into orbits having relatively high eccentricity
112
the plots of semi-major axis
In
effects" are
that "edge
evolution one
observes
fairly modest in that bodies
only infrequently scattered out of their
of heliocentric distance,
0.5 to 1.5 AU.
are
initial range
Further, bodies
are more frequently scattered outward beyond 1.5 AU
than inward of 0.5 AU.
One can also make qualitative
estimates from these plots of the range of semi-major
axis changes
typically occurring during close encounters.
A detailed examination of the semi-major
axis plots
reveals that pairs of bodies in orbits having either
nearly
identical or very dissimilar values of their semi-
major axes rarely collide and accrete, even when the orbits
approach each other
place.
closely enough for collisions
to take
Most collisions then occur for pairs of bodies
with only moderately dissimilar values of their semimajor axes.
Many instances exist in the simulations
where bodies
in orbits having nearly identical values
for their
collision
semi-major axes repeatedly interact without
until an event happens to scatter one
other of the bodies
into an orbit with a moderately dis-
similar semi-major axis value.
follows.
or the
Usually, a collision soon
I would also like to draw attention
to the
many instances of repeated close encounters between the
113
for example, the pair of bodies
same pairs of bodies;
simulation with a
= 0.025
the e
max
the period
3.77
max
<
period 4.38 < logl0t
4.60.
AU during
in
the pair of bodies
4.12 and
< log l0t
= 0.05 simulation with a
the e
; 0.52
in
%
1.2 AU during
the
One cannot help but wonder
what outcomes of these sequences
of close encounters
might result if tidal dissipation in the bodies were
in the simulation program.
included
In
the plots of system states at selected
(Figures 12,
15,
and 21)
18,
one can more easily follow
the progress of accretion in each simulation,
masses of the bodies,
represented by the box
shown along with the orbital eccentricities
rapid for smaller heliocentric distances
areas, are
and semi-
since the surface
is greater in that region.
tion of the eccentricity and
is
since the
As expected, accretion is more
major axes of the bodies.
density of bodies
times
again apparent in these
The evolu-
semi-major axis distributions
plots.
Also as expected, the
to have smaller orbital eccen-
more massive bodies tend
tricities since the accretion process averages the
eccentricity vectors of
The
final
lations
states of
the orbits of
the e
max
= 0.025,
the accreting bodies.
0.05,and 0.10
simu-
contain too many bodies with masses which are
too
114
small
relative to those of Venus and Earth.
hand, the e
max
= 0.15 simulation ends with a final state
which is very suggestive of
planets, and
Venus
On the other
two of these
the terrestrial
system of
bodies approach the Earth and
in mass.
The final
states of the four simulations are compared
with the terrestrial
bars indicate
system of planets in Figure
24.
the heliocentric range of each body.
The
numbers above the bars give the mass of the bodies
hundreths of an Earth mass and the numbers
The
in
between the
bars provide a measure of the stability of the heliocentric gaps between orbits.
expressed as fractions
where R is
The heliocentric gaps are
of A = 2.4 R[(m 1
the heliocentric
gap between m
and m 2.
numerical integrations
Birn
+ m2 )/M
1/3
distance to the center of the
(1973) has
carried out
and presented theoretical arguments
based on the Jacobian integral which purport to show that
bodies
in orbits separated by gaps greater
stable in the
sense that,
except for oscillations,
semi-major axes of all orbits remain
time
than A
are
the
constant over long
scales.
2. The Collision Statistics
The ratio of the
total energy at
impact for a pair of
115
colliding bodies
energy,
to their total gravitational binding
(T + U)/-Q, provides a convenient measure which
may be used in deciding whether a given collision
accretive or destructive.
When
is
(T + U)/-Q is greater
than one, a collision is expected to result in the
complete disruption and dispersal
both bodies;
however, when
a collision is expected
values of the ratio,
of the material of
(T + U)/-Q is less than one
to result in accretion.
(T + U)/-Q, for
Average
collisions occurring
in three of the four simulations discussed in this
chapter are tabulated
in Tables 2 to 4.
The numbers of
accretive and destructive collisions are
also tabulated
for each of the m.-m. values for which collisions occurred.
There are a number of interesting observations which
can be made from these statistics.
First, the average
(T + U)/-Q ratio for all of the collisions occurring
a given
simulation increases as e
max
the average value of
(T + U)/-
increases.
for the m I
collisions exceeds the average value of
in
Further,
= m2 =
0.02 M
(T + U)/-Q
when all of the collisions for a given simulation are
included.
0.02 M
It is no surprise
and m
=
0.04 M
, m
then that the m 1 = m 2 =
= 0.02 M
for all of the destructive collisions.
collisions account
Perhaps,
the
116
Table
Collision
m
2
Statistics
(e
max
= 0.05)
m.
1
(M /100)
2
2
2
2
2
2
2
2
2
4
4
4
4
4
4
4
4
6
6
6
6
8
14
(M9/100)
2
4
6
8
10
12
14
22
26
4
6
8
10
12
14
16
22
6
8
10
16
8
16
# (accretive)
34
14
4
3
2
3
2
1
1
5
3
2
1
1
1
1
1
1
1
1
2
1
1
Simulation average value
collisions = 0.103
# (destructive)
(T + U) / (-)
(accretive only)
0.158
0.073
0.065
0.065
0.012
0.030
0.030
0.021
0.039
0.084
0.063
0.046
0.004
0.012
0.031
0.022
0.693
0.169
0.033
0.020
0.031
0.037
0.027
1
of
(T + U)/(-Q) for
the
accretive
117
Table 3
Collision Statistics
m.
m.
(M /100)
(M /100)
(e
max
= 0.10)
1
2
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
4
4
4
6
6
6
6
8
8
8
10
10
12
2
4
6
8
10
12
18
20
22
24
30
4
6
8
12
14
16
22
26
6
8
10
12
12
14
22
12
20
20
(T + U)/(-2)
# (accretive)
27
18
6
2
3
1
1
1
1
1
1
2
4
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
Simulation average value of
collisions = 0.171
# (destructive)
5
(T
+ U)/(-Q)
(accretive only)
0.325
0.153
0.102
0.256
0.096
0.005
0.024
0.009
0.002
0.009
0.101
0.168
0.073
0.061
0.012
0.001
0.002
0.040
0.021
0.094
0.021
0.009
0.043
0.072
0.027
0.033
0.044
0.005
0.281
for
the accretive
118
Table 4
Collision Statistics
m
m.
(M /100)
(M /100)
2
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
6
6
6
6
8
8
8
10
10
10
18
26
24
2
4
6
8
10
12
18
20
24
32
58
4
6
10
14
20
8
18
20
24
10
12
24
12
14
30
40
34
60
(T + U) / (-)
# (accretive)
# (destructive)
(accretive only)
9
2
0.438
0.253
0.266
0.292
0.052
0.208
0.065
0.006
0.022
0.039
0.030
0.163
0.165
0.094
0.051
0.050
0.025
0.0001
0.115
0.068
0.015
0.255
0.192
0.025
0.013
0.089
0.235
0.104
0.012
20
15
9
6
3
1
1
1
1
1
1
2
3
2
2
1
1
1
1
1
1
2
1
1
1
1
1
1
1
Simulation average value
collisions = 0.238
= 0.15)
(e
max
of
(T
+
U)/(-Q)
for
the
accretive
119
most significant observation to be made from these
statistics is that
the number of destructive collisions
increases beyond 0.10, so that
rapidly increases as e max
for simulations with initial values of
is no longer
e
max
> 0.15, one
justified in assuming that all collisions
result in accretions.
3. Eccentricity Evolution in the Simulations
In Table 5 statistics are
tabulated pertaining
to
the eccentricity changes occurring during the early
stages of each of the four representative
simulations
The quantity denoted in the
described in this chapter.
table by Ae is the average value of m.Ae. + m.Ae. over
simulation starting
those events occurring between the
time and the tenth accretion event, where Ae.
and Ae.
are the eccentricity changes of the two bodies in the
given event and the masses m.
the
initial mass,
0.02 M
and m.
are
In these statistics we
.
refer to those gravitational interactions
in collisions as
encounters.
simulations, at least for the
here,
in units of
not resulting
Note that in all of
the
early stage considered
the gravitational encounters on the average result
in eccentricity increases,
mass average
but as e
max
is
increased the
eccentricity increase per encounter decreases.
120
Table 5
Statistics of
the Eccentricity Changes Occurring For
Encounters
and Collisions
e
0.025
max
0.05
0.10
0.15
Ae/encounter
+0.00269
+0.00178
+0.00074
+0.00066
Ae/collision
-0.01333
-0.01434
-0.05069
-0.08463
IAe /encounter
0.01007
0.01040
0.00712
0.00569
Ae/encounter
TAe /encounter
0.27
0.17
0.10
0.12
+2.96
+1.59
+0.20
+0.41
collision-encounter
balance ratio
5.0
8.0
68.7
# encounters during
the first ten
collision events
147
140
137
ZAe (encounters)
IAe (collisions)
128.2
527
121
Collisions,
on the other hand,
tricity decreases, and
collisions
always result in eccen-
the eccentricity decreases
increase as e
is
max
increased.
due to
The mass
average eccentricity of the system of protoplanetary
bodies
is then determined by the balance between the
eccentricity increases resulting from gravitational
encounters and the eccentricity decreases resulting from
collisions.
The ratio EAe(encounters)/ILAe(collisions) I
provides a quantitative measure of this balance:
when this ratio
is greater than one,
eccentricity is increasing and,
less than one,
ing.
the mass average
conversely, when it is
the mass average eccentricity is decreas-
The collision-encounter balance ratio gives the
number of encounters required on the average to offset
the eccentricity decreases due to collisions.
A measure
of the average absolute eccentricity change per encounter
is provided by
ael/encounter
encounter and finally,
encounter) quantifies
=
the ratio
Im.ae.j)/
(ae/encounter)/(IAel/
encounters.
simulation starting
accretion event, the mass
quite rapidly for the
+
the rate of the eccentricity
increase due to gravitational
From the
(Imiaeil
e
max
time to the
tenth
average eccentricity increases
= 0.025
tions and decreases for the e
max
and
= 0.10
e
max
= 0.05 simula-
and 0.15
simula-
122
tions.
It
is expected
between 0.05
that an e
max
and 0.10 would
balance between the
value somewhere
on the average result in a
eccentricity developed in gravita-
tional encounters and that lost in collisions for
planar system of bodies
the co-
all the
chosen here to initiate
simulations.
In a system where non-zero orbital inclinations are
allowed, fewer close gravitational
ted to result in collisions.
are expec-
interactions
The number of
encounters
per collision for a three-dimensional system of bodies
with the same members as the system used
to initiate the
two-dimensional simulations is expected to be at
one order of magnitude greater than the number
two-dimensional case.
to above is
e
max
in the
The eccentricity balance referred
then expected to occur for higher values
of
in the three-dimensional case than in the two-
dimensional one.
The necessary eccentricity to produce
terrestrial-like final configurations of bodies
those
least
in the e
max
=
such as
0.15 simulation might then be built
up by the additional gravitational
encounters occurring
in the three-dimensional case.
A quantitative
eccentricity
estimate of the effect of orbital
averaging of the many bodies accumulating
123
into a planet has been made by Ziglina and Safronov
(1976) .
A
number of
assumptions were made in this
estimate including a mass distribution for the bodies,
the total
number of bodies per unit area as a function of
the distance to the Sun, and the distribution of orbital
eccentricities of the bodies, among
The estimate
others.
they obtained for the orbital eccentricities of the
just-
formed terrestrial planets may be written as
1/2
e
where m 1
C(-
acc
1
m
)
V
is the upper limit of
impacting bodies and m
5-1
e- 1
c
the mass distribution of
is the present mass of the planet.
surface of the planet is
The parabolic velocity at the
denoted by Ve and the circular Keplerian velocity at the
heliocentric distance of the planet by V c.
C,
constant,
includes parameters used to specify the mass distribu-
tion of bodies and
their velocity relative to the
circular Keplerian velocity;
assumed by Safronov, C =
estimated by Safronov
rotation axes of
acc
taking m
1
for the parameter values
0.11.
(1969)
Values of ml/m were
from the
inclinations of
the terrestrial planets.
-3
e
The
/m =
1.8 x 10
-3
the
The values of
for Mercury, Venus, and
124
than one order of magnitude smaller than the current
values observed in the
terrestrial system.
Ziglina and
Safronov, without any further arguments or estimates,
conclude that the modern orbital eccentricities of
the
planets must be determined to a greater extent by their
prolonged gravitational
interaction since the time their
accumulation was complete, rather than directly by the
process of
formation.
On the basis of their estimates of ml/m from the
rotation axis
inclinations of the terrestrial planets,
Ziglina and Safronov rule out the possibility of
collisions and accretions of bodies with moderate mass
ratios
in the final
terrestrial planets.
ml/m occurring
stage of accumulation of the
On the other hand, the values of
in our simulations
are a little more than
two orders of magnitude greater than those estimated
by Safronov and yield values
with
of e
acc
which are
the present orbital eccentricities
trial planets.
This brings
for the terres-
up the intriguing possibility
that the present orbital eccentricities of
planets might perhaps serve
consistent
the terrestrial
to distinguish between those
scenarios for the final stage of accumulation of the
terrestrial planets characterized by very small values of
125
ml/m, e.g.
Safronov
(1969) and Weidenshilling
(1975) , and
scenarios characterized by values of ml/m often greater
than 1/10.
A
careful review of Safronov's estimates of
ml/m along with quantitative estimates of upper
limits
on the orbital eccentricity changes of the terrestrial
planets due to their mutual gravitational interactions
and interaction with Jupiter are required in order to
determine whether this discrimination between scenarios
is possible.
4. Numerical Checks
on the Simulations
One numerical simulation was run to check the
validity of the encounters model for those close
encounters typically occurring
Values
in the present simulations.
for the orbital element changes for each of
close encounters occurring
in the
the
simulation were calcu-
lated both by use of the model and by integration of
regularized
equations of motion.
the
Comparison of these two
sets of values revealed that the model, in agreement
with the conclusions stated
in Chapter II,
provide a useful approximation, E e , E a
a
orbital element changes occurring
ters for which E
r
4 and c
0.80.
does indeed
50%,
for the
in those close encounEven
with E < 4 the model was often a useful
for encounters
approximation,
126
especially for those encounters with 2 < E < 4.
since
there were some encounters
However,
for which the predictions
of the model were in radical disagreement with the results
obtained by numerical integration,
it was thought to be
more prudent to compute orbital element changes
encounters having C < 4 by numerical
for the
integration of the
regularized
equations of motion for the duration of the
encounter.
It was
further noted that the model, almost
without exception, predicted larger values
for the minimum
separation between bodies during the encounters with E >
than was
obtained by numerical
integration.
4
Therefore,
the model consistently somewhat underestimates
the orbital
element changes relative to the changes obtained by
integration for encounters with E >
4.
Furthermore,
since almost all collisions result from encounters with
6 < 4,
the model frequently predicts collisions between
bodies for encounters
in which integration
of the equations
of motion produces no collision.
A further check on the accuracy of
was made by calculating
the total
angular momentum of the
system of protoplanetary bodies for
states of the e
max
the
total
= 0.15
the simulations
the initial and
simulation.
final
These values of
system angular momentum and total
system energy
are compared to the corresponding values for the current
127
terrestrial
The total angular
system in Table 6.
as it should be, while
momentum is closely conserved,
the total
system energy decreases by
initial and
final states of the e
max
1.23%
= 0.15
between the
simulation.
Table 6
Comparisons of Total System Angular Momentum and Energy
L
2
(g-cm /sec)
system
E
(g-cm2/sec 2 )
system
initial (100 bodies)
5.152 x 10
-6.020 x 1040
final (6 bodies)
5.152 x 10
-6.094 x 1040
terrestrial system
4.95
Finally, the numerical
x 10
-6.20
integration of
x 1040
the individual
encounters was checked by integrating backwards
to make
sure that the initial conditions
in time
could be re-
produced.
B.
An Estimate of the Evolution Times Expected for ThreeDimensional Systems
The evolution time expected for a three-dimensional
system is
estimated using the ratio of the two-body
128
collision probability for a three-dimensional system to
a two-dimensional system along with the evolution
that for
times obtained in the two-dimensional computer simulations.
The collision probability per unit time for
one body may
be written as
5-2
U n G
T
n is
where U is the mean relative speed of the bodies,
the number of bodies per unit surface area or volume, and
a is the collision cross-section.
section
in two dimensions is
The collision cross-
taken to be
2R and in three
2
where R is a multiple
dimensions to be fR
physical radius of the bodies,
r.
The
f
times the
factor
f, which
provides a measure of the increase of the collision
cross-section over the geometric cross-section due to
"gravitational focusing,"
is
tions of circular encounters
mately 28
0.02 M
.
for the
integra-
in Chapter I to be approxi-
two-dimensional case with m I = m
2
=
The surface area available to the 100 bodies in
the simulations
S
estimated from the
=
is given by
11(1.52
T'lhe correspondinq
-
0.52)
=
21r (AU) 2.
volume assumed
5-3
to be available to the
129
100 bodies in a three-dimensional
system is
taken to be
equal to that of the solid of revolution formed by rotating
= i
a regular trapezoid, bounded above by
=
-imax,
max
= 0.5 AU and r
and by r
max
=
below by
1.5 AU in
full revolution.
heliocentric distance, one
,
Making use
of the Pappus-Guldinus theorem we obtain for this
volume,
V =
2r/5 tan
i
5-4
(AU) 3.
max
The ratio of the three-dimensional to the
relaxation
two-dimensional
is then given by
times
T
3-D
T -D
2/5
S-
tan
i
1 AU
max
8f
5-5
R
test bodies having a dimensionless
For a set of
relative velocity,
U = u/Vk
k
V
2
0
= -GM
a
K
5-6
with respect to a field body in a circular orbit of radius,
a,
U
there exist
(Opik, 1951).
when
expressions
for e
max
The maximum possible
and
i
max
in terms
of
eccentricity occurs
the orbits are co-planar and when the perihelion of
the test body's orbit
is at the field body's orbit and
130
may be written in terms of U as
e
= U2 + 2U.
max
For any U,
the maximum
5-7
inclination
orbital plane of the field body)
i
max
= 2 sin
For small values of U,
e
max
-1
(relative to the
is according
to Opik
(U/2).
= 2U and
5-8
i
= U are good
max
approximations.
With the values
m
I
= m
2
= 0.02 M , and
determined
e
max
e
max
= 0.15
f =
28,
=
0.075 radians),
the ratio
to be approximately 280.
T3 - D /
2- D
is
This ratio with the
= 0.15 simulation time of 61,400 years yields an
estimate of 2 x
107 years for
stage of accretion considered
case of a three-dimensional
i
= 0.075
max
radians.
Chapter
the time required for
in our
simulations
system with e
max
the
in the
= 0.15 and
This estimate is well within the
limits for the formation time
C.
(i
max
interval discussed in
I.
Heliocentric Distance Sampling Of Material for
the
Final Bodies of the Simulations
It is informative to examine the
range of helio-
centric distance sampling of material for the bodies
in
131
the simulations at several stages of their evolution.
Figures 22
and 23
In
the original semi-major axis distribu-
tions of component bodies are shown for those bodies
with mass greater than 0.1 M
the final one,
of the e
max
at six stages,
= 0.05 and
e
max
simulations, respectively.
The original
of the component bodies are
indicated by a
including
= 0.15
semi-major axes
(+) and
current semi-major axis and mass of each body are
in the figures by an
As expected
sampling
the range of heliocentric distance
in bodies with smaller mass tends to be narrower
in bodies with larger mass.
As also seems reasonable, the range of radial
is greater in the e
max
max
= 0.15
sampling
simulation than in the
0.05 simulation for bodies of
=
located
(x).
than the range of sampling
e
the
a given mass.
The semi-major axis distributions of component bodies
are
sometimes not at all symmetric
about the current semi-
major axis of the body being considered.
This is often
the case when two bodies are accreting material from
overlapping
"repel"
heliocentric ranges:
the two bodies tend to
each other so that the body closer
to the Sun
samples more material outside its orbit than inside, and
the reverse is of course true for the body farther from the
Sun.
It must be emphasized, however, that this behavior
132
is by no means always observed.
is only a tendency and
The radial sampling function observed for the bodies
in these simulations may generally be described as follows.
from the
i.e.
the material comes
75% of
For a given body, approximately
zone bounded by the orbits of the next inner,
8% comes from
toward the Sun, and next outer bodies,
the zone bounded by
the next inner and
the orbits of
second inner bodies,
17%
and the remaining
comes from the
zone bounded by the orbits of the next outer and
outer bodies.
second
However, we must caution that this radial
from a small
sampling function is based on statistics
Furthermore,
number of simulations.
it undoubtedly
reflects the edge effects resulting from our choice of
The
the initial radial distribution of material.
similarity of the final state of
tion to the system of
the e
max
sampling to be expected for
Much more complex "edge
and Mars.
simula-
terrestrial planets does permit us
to form the above working hypothesis
the radial sampling of
= 0.15
for the
radial
at least the Earth and Venus.
effects" have no doubt
influenced
the material comprising Mercury
It must be pointed out that although the radial
sampling observed for bodies
in the simulations represents
only the sampling occurring during
accretion, the range of radial
the last stage of
sampling
is expected to be
133
much narrower for bodies with small values of
Therefore,
their mass.
it is apparent that the range of radial
sampling exhibited in the final bodies is primarily
determined by
the last few accretion events
involving
the given body.
Hartmann (1976) has used Wetherill's
tions
(1975)
calcula-
of the gravitational dispersal of planetesimals from
different orbits
about the Sun to obtain data on the
sources of different late-accreted materials added to the
terrestrial planet surface layers.
the final impact destinations of
particles
impacts
statistical samples of
starting in nine different orbits in the terres-
trial region of the solar
(p. 555)
Wetherill tabulated
Hartmann has
system.
In his Table III
converted the relative numbers of
obtained by Wetherill to obtain the percentage
of mass striking target planets from each source orbit.
These results
indicate that large fractions of the mass
added to the terrestrial planets in their later stages of
growth probably formed nearer other planet's
orbits,
in
agreement with conclusions drawn from the present
simulations.
However, our radial
sampling function
is
not directly comparable to that given by Hartmann, but
seems
to be qualitatively similar.
134
D. Conclusions
Drawn From the Simulations Regarding
the
Formation of the Terrestrial Planets
Perhaps the primary conclusion is
like planetary systems
that terrestrial-
can be formed as a result of the
gravitationally-induced collisions and
accretions occur-
ring between bodies in co-planar systems of protoplanets
such as those
provided the
chosen here to initiate the
initial average orbital eccentricity of the
system of bodies
is
in the range 0.05 < <e>
Simulations with initial values
are found
simulations,
to result
of <e>
in final-state
<
0.10.
smaller than 0.05
configurations having
many bodies of nearly equal mass, none of which approach
the Earth or Venus
in mass, Figure 24.
On the other hand,
initial values of <e> greater than 0.10 result in substantial numbers of catastrophically disruptive collisions
between bodies and
it is
expected that the accretion
process for this case will at
least be dramatically
slowed, if not reversed.
The numerical
eccentricity values
simulations indicate
in the range 0.025 < <e>
be sustained by the gravitational
the eccentricity
that mass average
lost
encounters offsetting
in collisions,
at least for
early stages of evolution of the co-planar
to begin these simulations.
< 0.05 can
the
systems used
These values of <e> are
135
somewhat too
low to produce terrestrial-like systems,
but higher values of <e> may result from the higher
ratio of close gravitational
encounters to collisions
holding for three-dimensional systems.
Estimates of
the time interval required for this
stage of accretion in a reasonably thin hypothetical
three-dimensional protoplanetary system were made earlier
in this chapter.
The times required for
the two-dimensional numerical
systems
simulations from initial
of 100 equal mass bodies with m = 0.02 M
state configurations
final
completion of
to the
are used with theoretical
estimates of the ratio of the three-
to the two-dimensional
relaxation time, based on the two-body gravitational model,
to predict a time interval of
the
the order
stage of accretion considered here.
well within the limits of a few times
2 x
107
years for
This time falls
108 years for the
formation of the terrestrial planets, based
on the oldest
ages measured for rocks from the Earth's crust along with
40
40
the K 40-A40 and U,
Th-He
4
gas-retention ages for various
meteorites as discussed in Chapter
I.
Finally, the original heliocentric distance distribution of component bodies making up the bodies
final
mately
states of the simulations
75% of
in the
indicate that approxi-
the material of a given body comes
from
136
zone bounded by the orbits of the next inner and
the
next outer bodies.
remaining 25%
The
comes from two
zones bounded by the orbits of the next inner
and second
inner bodies and by the orbits of the next outer and
second outer bodies,
respectively.
This description
of the heliocentric distance dependence of the
sampling
function for a given body
is,
however, only intended as
a general rule of thumb.
For
certain bodies a number of
factors such as edge effects or possible bombarding
material perturbed into the
terrestrial region by an
early-formed Jupiter may significantly modify this
sampling function.
E.
Limitations of the Present Numerical Simulations
these simulations
Many of the desirable features of
were described in the final section of Chapter I.
this
section we will turn our attention
limitations.
First,
In
to some of their
actual systems of protoplanets,
if they are quite thin,
even
are nevertheless three-dimensional,
and this fact presents a whole new set of difficulties
attempting
to simulate
such systems.
earlier that the number of
It was argued
encounters per
collision for
a three-dimensional system of bodies having
the same
in
137
bodies as those in the system used to initiate the twodimensional simulations is
expected to be at least one
order of magnitude greater than the number
dimensional case.
Therefore,
for the two-
significantly more computer
time would be required to simulate the three-dimensional
systems.
The eccentricity balance might as a result
occur for higher values of <e>
in the three-dimensional
case than in the two-dimensional one, but
since the
character of the encounters presumably also changes in
three-dimensions, one cannot be
is
sure that this
expectation
justified in fact.
Several simulations which were carried out have not
been discussed here, because the orbital spacings between
bodies were not stable
close approaches,
(Birn, 1973),
(Ar) m
< R ,
min
s
simulations terminated.
when no further
were predicted and the
In the two-dimensional case this
situation is most often the result of an alignment of
some number of
system bodies.
the orbital eccentricity vectors of the
Inclusion of the effects
encounters in some of these "final"
states would give
rise to some additional collisions and
of these systems.
expected to be
of distant
subsequent evolution
"Avoidance" mechanisms
such as this are
an even greater problem in three-dimensional
138
simulations which include only the effects of close
encounters.
Another limitation of
present form is the
in their
the simulations
small number
of bodies
(N = 100)
This
which can currently be handled by the programs.
makes an assessment of
the influence of our
initial system of one hundred bodies of
0.02 M , on
choice of an
equal mass, m
=
the final configurations produced in the
simulations difficult.
Two-dimensional
simulations
beginning with protoplanetary systems having more bodies
and a variety of mass distributions
the influence of the
initial conditions on the final
configurations produced.
upon
should elucidate
Such simulations seem
feasible
some improvement and/or modification of the way in
which encounters are handled in the simulation programs.
We have attempted here to simulate the
last stage
of terrestrial planet formation neglecting any outside
influences.
If,
for example, Jupiter has
attained a substantial part of
already
its mass before the stage
of accretion considered here, the
influence of bombarding
material perturbed by Jupiter into orbits crossing the
terrestrial region must be
scenario.
included
in a realistic
~_I___
____1~_1~_
139
Finally, some collisions do occur in these simulations which, based on their expected relative velocities
at impact, should be completely disruptive, and yet the
simulations assume that all collisions result in accretion.
This assumption appears to be well justified for collisions occurring in systems with mass average eccentricities
<e> < 0.05.
It cannot be ruled out, however, that bodies
were accreted and
several times
planets.
subsequently dispersed by collisions
in the process of forming
the terrestrial
In order to simulate properly this formation
process, it would be necessary to construct models of
the disruptive
collisions commonly occurring
systems with <e> > 0.05.
in those
140
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Opik, E.J.
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Astroaeologic
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Proc.
Pechernikova, G.V.
(1974),
Astron. AJ 18,
Russell, H.N.
Roy. Irish Acad.
English
54A,
165.
translation in Soviet
1305.
(1935),
The Solar System and Its Origin,
Macmillan, New York.
Rybicki, G.B.
(ed.
(1972), in Gravitational N-Body Problem
by M. Lecar),
Safronov, V.S.
Cloud
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D.
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in Evolution of
and Formation of
the Protoplanetary
the Earth and Planets,
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1972.
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(1965),
The Origin and Evolution of the
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Ter Haar,
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lation in Soviet Astronomy AJ
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135.
English trans-
244.
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English translation in Soviet Astronomy AJ
144
FIGURE CAPTIONS
Figure 1.
Orbital eccentricity and semi-major axis
changes are shown along with minimum separation distances
for a sequence of
encounters between bodies which are
initially in co-planar, circular orbits about the Sun
with various orbital spacings,
of the
R
s
"sphere of
= R[(M
2
(M
1
influence" radius of
+ M ))/(M
2
initial orbit of M2 was
M 1 and M
2
given here as multiples
2
0
)]
1/5
.
the larger body,
The radius of the
taken to be 1.0 AU.
In addition,
were assumed to orbit the Sun in the same sense
and only those encounters for which the orbit of M 1 lay
outside
that of M 2 were considered.
The
integrations of
the regularized equations of motion for all
began and ended with separations of 500 R
s
encounters
.
It proved
more convenient to plot logl0 (RMIN/(RI + R2))
rather
the minimum separation, RMIN, where R 1 and R 2
are the
radii of M 1 and M 2 assuming
that they are homogeneous,
spherical bodies of lunar density.
regions where log
be
M
1
=
10
Therefore,
the two
0(RMIN/(R 1 + R2)) = 0.0 represent
collisions between the bodies.
encounters
In this
sequence of
the masses of the two bodies were taken to
-5
M
and
M
2
=
10
-2
than
M .
$
145
Figure 2.
Same as Figure 1, but with M1
Figure 3.
Same as
M
1
= 10
-3
M
and M
Figure 1 and Figure 2,
= M2
= 10
-2
M .
but with
= M
2
Figure 4.
Same as Figure 1, but with M1
Figure 5.
The parameters
= M2
= M .
specifying encounters
in the
model.
Contours of ae and Aa
Figure 6.
by the model for m
E
in a,
E.
all these
2
encounters was
1 AU.
taken to be
(-)
I
, m
I
= m
Figure 7.
= m
2
0
center of mass for
2
taken to be along a circular
The sense of motion of m
I
about
and the orientation of the
relative hyperbolas was given by
were m
6 obtained
along with contours of Ee and
The motion of the m
orbit with RO =
m2 was
and m
(AU) in 0,
=
/10. The masses
with m0/M = 0.01 M /M.
The parameters specifying
these encounters are
identical to those in Figure 6 except for the orientation
angle
= 77/10.
Figure 8.
Dependences of Ae and Aa
tion angle
of E
e
, E
a
of motion
(AU) on the orienta-
given by numerical integration.
,
E
e
,
(+).
was assumed
and E
a
Dependences
on c are also shown for both senses
The motion of the ml, m 2
center of mass
to be along a circular orbit of radius R
1 AU.
The remaining
0.16,
and mi1 = m 2 = m 0 .
fixed parameters
were
E = 4,
a
=
I~U~1~
146
Figure 9.
Dependences of Ae and Aa
(AU) fn e 0
of Ee and
given by numerical integration.
Dependences
Ea on e0 and R 0 are
The remaining
parameters were
S=
= m0.
m2
to be
The
= 4, a = 0.16, e0 =
sense
the bodies
and m 2
in the e
max
For clarity in plotting an
was assigned to each gravitational
resulting in a collision or not.
times are given as
and
m1
= 0.0,
and
was
taken
The time evolutions of the orbital eccentrici-
are plotted for
tion.
of motion of
T/2,
fixed
(-).
Figure 10.
ties
s
also shown.
and R 0
= 0.025
simula-
equal abscissa increment
encounter, whether
The associated
logl0 of the time in years.
event
Collisions
subsequent accretions of bodies are denoted by a
vertical line connecting the eccentricity values
two colliding bodies
just prior
and by a small box locating
of the
to the current encounter
the orbital eccentricity of
the newly accreted body.
Figure 11.
The semi-major axis
to the eccentricity evolutions
Equal
evolutions corresponding
in Figure 10 are plotted.
abscissa increments were again assigned to each
gravitational
encounter and accretions
the same way as in Figure
10.
are denoted
in
147
Figure 12.
The semi-major axes,
masses of the system bodies are
stages in the e
max
eccentricities, and
shown for nine different
= 0.025 simulation,
initial and final ones.
The number of
including the
system bodies
for the nine stages is
indicated at the top of each
frame.
of the initial
The boundaries
and eccentricity distributions are
rectangles in each frame.
semi-major axis
indicated by the
The areas of the small
marking the location of each b ody are proportional
boxes
to
the masses of the bodies.
Figure 13.
Same as Figure 10,
but for the e
= 0.05
Same as Figure
but
for the e
= 0.05
but for the e
= 0.05
but for the
e
= 0.10
Same as Figure 11,
but for
the
e
=
Same as Figure 12,
but for
the
e
= 0.10
max
simulation.
Figure 14.
11,
max
simulation.
Figure
15.
Same as
Figure 12,
max
simulation.
Figure 16.
Same as Figure
10,
max
simulation.
Figure
17.
max
0.10
simulation.
Figure
18.
simulation.
max
148
19.
Figure
Same as Figure
10, but for the e
max
= 0.15
simulation.
Figure 20.
Same as Figure 11,
but for
the e
= 0.15
Same as Figure 12,
but for
the e
= 0.15
max
simulation.
Figure 21.
max
simulation.
Figure 22.
The original semi-major axis distributions
of component bodies are shown for those bodies with mass
greater than 0.1 M
one, of the e
max
major
"+"
at
six stages,
= 0.05
including the final
simulation.
axes of the component bodies
The original semiare indicated by a
and the current semi-major axis and mass
of each
body are located by an "x".
Figure
23.
Same as Figure 22,
but for the
e
max
= 0.15
simulation.
Figure 24.
A comparison of the final
representative
simulations with the present terrestrial
system of planets.
range of each body.
mass of
states of the four
the bodies
The bars indicate
the heliocentric
The numbers above the bars give the
in hundreths of an Earth mass, and
the heliocentric distance gaps between orbits are
sed as multiples of As
As
=
is a gap argued to be
2.4 R[(ml + m2 )/M
stable by Birn
1/3
1/3
(1973).
expreswhere
0.015
0.010
M, =M /10'
Ma =M /100
0.005
0.015
0.010
0.005
1.0125
1.0100
1.0075
1.0050
1.0025
1.0000
0.9975
0.9950
0.9925
0.9900
0.9875
3.0
2.0
2.5
3.0
3.5
4.0
4
.5
5.0
5.5
6.0
6.5
ORBITAL SPACING(R.)
7.0
7.5
8.0
8.5
9.0
9.5
10.0
0,0
SLQ6 *0
0066*0
SZ66*0
0S66*0
SL66*0
00001
3D
to
S200. I
0s00.!
5L00 I
0010*
S2101
s00.0
01010
SIO0
s00.0
01010
slbl
0.060
0.040
Ml =M* / 10, M,, =M*
0.020
0.060
O.040
0.020
1.080
1.070
1.060
1.050
1.040
1.030
1.020
1.010
S1.000
0.990
0.980
0.970
0.960
0.950
0.940
0.930
0.920
3.0
*
.
....
......
2.0
1.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
ORBITAL SPACING(Rs)
7.0
7.5
8.0
8.5
9.0
9.5
10.0
0.060
0.0oo0
0.020
0. 060
0.0k0
0.020
1.080
1.070
1.060
1.050
1.0O0
1.030
1.020
1.010
1.000
0.990
0.980
0.970
0.960
0.950
0. 90
0.930
0.920
3.0
( ao ,eo)
Parameters m
I
, m2
a, E, c,
Sao,e,
Snn
Sun
± (ml and m2 about their CM)
\ CM
80 (CM about the Sun)
m
154
oc
a.a, (AU)
0
a (AU)
E0
Z.o I-
11
4.0
0
8.o
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,
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.
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0
08
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6
OC
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Figure
6
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so
160
31 0
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04
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31
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173
BIOGRAPHICAL NOTE
Larry Paul Cox was born and reared
Alabama.
in Birmingham,
from Ensley High School,
He was graduated
He attended the Massachusetts
Ensley, Alabama in 1962.
Institute of Technology and received an S.B.
in 1969.
Research for his
in Physics
senior thesis led to publica-
tion with J.V. Evans of "Seasonal Variation of the F Region Ion Composition"
(J.G.R., 75,
Following
159-164).
graduation he continued to do research with J.V. Evans
as a staff member of the Millstone Hill Ionospheric
Observatory, Lincoln Laboratory.
This research
led to
a further publication with J.V. Evans of "Seasonal
Variation of
75,
the O/N
2
Ratio
in the F -Region"
(J.G.R.,
6271-6286).
As
a graduate student
in the Department of Earth
and Planetary Sciences the author has held a National
Defense Education Act Fellowship, a Teaching Assistantship
in planetary physics, and Research Assistantships
in the Department of Mathematics as well as
Department of Earth and Planetary Sciences.
work has been in the fields of
in the
His research
ionospheric physics,
galactic dynamics, and planetary accretion dynamics.
174
He married Candace Lenoir Kelton of Denver, Colorado
in August, 1969.
His wife has taught English in junior
high school since
1970.