Numerical simulation of a three-dimensional gravitational system with collisions

Chinese Astronomy & Astrophysics 5_ (1981) 71-76 Act. Astron. Sinica -21 (1980) 104-111.

NUMERICAL

SIMULATION

GRAVITATIONAL

Hong-nan

OF A THREE-DIMENSIONAL

SYSTEM

Zheng Jia-zhuang ~ZB Zhou

Pergamon Press. Printed in Great Britain 0146-6364/81/0301-0071-$07.50/O

WITH

COLLISIONS*

~un~a~~ C&ser0atory,Acadmia Sinica;

and

Astronomy &qxrtment, kmjing hiversity

Received 1979 October 22

ABSTRACT

A system of particles subject to mutual inelastic collisions and

moving in the gravitational field of an ellipsoidal central body is numerically stimulated on computer. The results confirm Poincare's (1911) assertion and show that, after rapidly becoming flattened, the system enters a quasi-equilibriumstate in the form of a disk of finite thichness. The disk slowly expands outwards and condenses towards the centre.

1. INTRODUCTION As early as 1911, Poincare 113 pointed out that a system of particles subject to inelastic collisions will evolve as follows:

1) a central condensation will gradually form, and at

the same time the system will extend outwards due to conservation of angular momentum; 2) the particles will tend to settle into a plane perpendicular ro the initial angular moments vector i.e. the system will flatten out;

3) the particle orbits will become

more and more circular. Poincare put forward these propositions without giving any proof, and they have not been rigorously proved so far. Inelastic collisions among macroscopic particles is one of the mechanisms of forming a disk-like structure, and have been widely studied. In the early 70's, Monte Carlo studies of inelastic collisions were made in connection with the evolution ofgalacticnuclei [Z] and of jet streams 13, 41. More recently, Brahic put forward a general numerical method for studying a system of colliding particles and set up a "standard model" and a "v model" including transverse viscosity [5 - 93. Brahic [6] proved that the evolutionary speed is proportional to the number of particles in the system and to the geometrical cross-section of each particle. This means that a small number of large particles is equally effective as a large number of small particles while requiring less computing. In this paper we shall follow a numerical simulation of a system in the gravitational field of a central rotational ellipsoid.

* Condensed translation

Colliding

72

2.

ELLIPSOID MODEL

In Brahic’s would

the

works,

be a spheroid

1. of

System

of

We consider

a rotational

rigid,

body

uniform

is

taken

ellipsoid

of

and of

We neglect

all

density.

system

uniform

the

mutual

to

be a point-mass.

We therefore

a three-dimensional

frictionless 2.

central

of

density.

take

particles

We suppose

The next

approximation

our model

as follows:

moving

the

all

in

the

gravitational

particles

field

to be atomic,

same mass and radius.

gravitational

attractions

between

the

particles,

and their

own spin. 3.

Following

the particles consider

the

According

the

are

usual

taken

to

first-order

to

symbols

is

the

have

produced

shape

their

3

2

parameter

usual

sin*;

all

subsequent

perturbating

(1 -

considered

the

initial

perturbed of

function

the

(1)

as small

quantity

of

the

perturbations

first-order.

in

the

a -

AC =

c -

A; =

i -

Am

=

AM, all

aO, cO,

all

by radii

each

step.

5.

Collision

collisions

the

2 capture

are

R,

and R,

order

taken

of

= 0.5, by

normal taken with

meanings

and Suffix

to be Kepler

inclinations calculate2

take

Between

to

located

in

assigned

according

can

be instantaneous.

o refers

ellipses randomly

any two particles

are of

the

taken

the to

we decided

i,,

= n(t

perturbations

to

component

radius

r,)

the

(2)

are

anywhere

collisions,

values

spherical

between

to

place

initial

at

to.

shell

0 and a specified incorporated

on their

a particle

at

orbits. is

the

end

The

subject

only

perturbations.

collisions

calculations,

= 3,

orbits

With due regard

N = 100, R

their

(21

>

n(t -

--g

(M&

between

are

radial 7.

coo =

have

first-order All

-

M, -

=

%rnax . The first

6.

elements

i,,

m

symbols

Atto,

bounded

to the

The other

orbital

P2

of

we only body.

is

AQ=Q-Q9,=--~(r--ro)cosi,

4.

motion, central

of

are Aa =

where

orbits

e*)-f

The first-order

meanings.

[lo],

due to non-sphericity

secular

‘-1

i

Mechanics In their

perterbation

first-order

R=R,=i$ where A,

in Celestial ellipses.

secular

the

[II],

procedure be Kepler

particle time central

to

be inelastic

relative

the

on the rp step

velocity

capabilities

= 0.07, h = 0.4,

body R,

= 0.3,

need of

following

and frictionless.

the

model (in

shape

of

only

the

change

be considered.

computer

used

parameters:

units

radial

to

Hence

Total

a standard

collision parameter

(Type

the

number of

radius

coefficient of

TQ-6)

of

test

particles

system),

k = -0.3, central

and after

R,

radius

body A,

= 1, of

= lo-*.

in

Colliding

3.

REALIZATION OF COMPUTERPROGRAMME

We have written dimensional body,

a programme

gravitational

and subject 1.

to

Remembering

in BCY language

system

mutual

that

six

random numbers

follows

a =

X in

ri

2%X,,,,

by the

we can

find

the

positions

Mechanics

Brahic’s

(to,

= (ti -

for

rj)* -

rj are position

are

examined

4. related

5.

The radial by the

locating

of

the

r,,

ri -

and the

cl-

time

of

to be -the

terminates

after of

=

the

ii -

the

time

the

initial

particles

For small

+ii =

earliest of

j$

-

42

-

jk.

collision,

collision

elements

as

using

the

usual

h,

(4)

(zr,)*

<

can be replaced

Tq,

-

0,

1

(6)

if

there

and will

are

more than

be used

as

6

one,

in the

next

3000 collisions. relative

velocity

(‘i)’

=

lI(I”i)

’

*

(7)

the

and

(2)

orbital If

escaped.

by the

taken

all

rii - +ii)(f

j.;,

’ (1 -

with

of

two particles.

* (1 + 4) + (U,)

randomly

define

(5)

[cVi)

new one,

to

a collision.

=

5.

as captured

=

components

formulae.

having

computer

after

and before

collision

are

formulae

usual If

(3)

mlx

and velocities

cVi>’

Using

conditions

(2rp)* SZ 0.

vectors

taken

The programme

step.

the

approximation

+ h ) is

to

to

[lo].

condition

r,i pairs

subject

(4)

b*(t) = r3j + 2rij * ii; - (t - tl) + (t$ +

in

as follows:

are

spheroidal

2-X+5,

second-order

All

are

a three-

a central,

.

Celestal

and

simulate

of

+ Xj+,)/2,

D =

We use

field

Xi+~l/Za,

>

of

i

the

I

+(I)

where

from

2rxi+,,

From (2)

3.

IX, -

called

TQ-6 to

in the

O
w =

(Mo>o =

2.

R,>(Xj

e,

a,

i = Xj+l * i max>

e =a-f

formulae

-

RI).

computer

The details

fe),

are

the

moving

elements

R,>U(l

l]

for

particles

initial

[0,

RI + (R,

e=(R,-

of

collisions.

the

RIGu(l-ee),

the

73

System

the

-

the

speed its

elements [0,

4)

+

elements

reaches

pericentric

central

from

(1

body. a,

2111.

e,

(Vi)

of

the

4)]/2,

the

collising

parabalic

distance

becomes or

i equal

to

pair

less

all

a collision

particle

than

captured

the mean of

after

the

speed,

Each escaped

(7)

I

(1 + 4)1/2.

Rc,

particle present,

will then will

are

found

be regarded

it

will

as

be regarded

be replaced

and the

from

angular

by a elements

Colliding System

74

4. COMPUTER RESULTS 1.

After

decreases, After

subject

system

and the curve

times

orbits

refers

our

the

the

becoming to

inelastic
decrease

assumes

a few units

to

mean eccentricity

725 collisions,the

and the is

being the

shape

in ci> of

particle

decreases

is

model”,

After

3000

Poincare’s thus

Fig.

after

slower

dashed

rapidly increase.

and collisions

of

See Figs curve,



a short-lived

collisions,the

conclusion

confirmed.

and the

mean inclination

rapidly

and become

a disk,

radius.

more circular

“ellipsoidal

the

collisions, also

the

less

thickness the

gradual

frequent, of

the

disk

flattening

1 and 2,

where

standard

“spherical

the

solid model”.

1

NN

Collisional

evolution

Fig.

of

.

2

NN

Collisional

2.

tinder

collisions,

the

evolution

system

extends

of

ci>.

outwards.

Compare

Figs.

3 and

5.

Colliding

75

System

Fig. 4

Fig. 3

q3

.

)‘3

.

1..

I 2

2-3 *.. _;]

. . :

:*;

.. . .

’

..

I.--’

.

=“I . .I _..

*.

.-2.

:

.

- . . .

-3

Initial distribution in the r$fnlane

Initial distribution in the .~anlnne

Fig. 6

Fig. 5

Distribution in thercyplane after 3000 collisions

Distribution in the yz plane after 3000 collisions

3. Under collisions, the system forms a central condensation. This IS shown in the number of captures as collision proceeds:

Number of collisions Number of captures

500

1000

1500

2000

2500

3000

1

6

8

13

16

19

4. Frequency of collision decreases as system evolves, from 2.25 collisions per unit time at the beginning to 0.9 at the end of 3000 collisions. 5. The frequency distribution of the number of collisions suffered by a particle shows

Colliding

76

a Gaussian 6. the

faster

distribution.

A small

decrease

System

test in

computers

program

has

shown that

and +i> becomes become

even

if

we take

a smaller

more pronounced.

This

value should

of

rp(’

be kept

available.

REFERENCES Hoinca&, H., J&eaa anr 1~ HYPthmm Ces~~~egeniquesp. 86, Hermann, Paris, 1911. H&a. 8. M, Bull. Astron. Ser., SIII(1968), 265. ~,.&.en, J., AslropLys. Space. Sd, 17(1972), 241. Trulseq J.. Astrephys. Space, Sk., lB( 1972), 3. Brahie. A., I. A. 27. Sympoium. No. 58(1975). 287. Brahic, A., J. Camp. Phys., 22(1976). 171. Brahie, A., ICWUS. ?5(1975), 452. Brahic, A.. Ast~n. Astrophys., 54(1977), 895. Brahic. A. H&on, M.. A8trm. Astrophy&, 57(1977). “Introduction to Celestial Yi Zhao-hua et al. Kexue Chubanshe 1978 Liu Lin et al. “Theory of Flotion of Artificial Kexue Chubanshe 1978

Mechanics’ Satellites”

(in

Chinese) (in

Chinese)

.07)

then

in mind when