V*"T. T .

1C/S5/258 J

INTERNAL REPORT(Limited distribution)

r -„ -International Atomic Energy Agency

a n d

_,JJrtited Nations Educational Scientific and Cultural Organization

INTERNATIONAL CEHTHE FOR THEORETICAL PHYSICS

N-BODY SIMULATIONS OF TIDAL ENCOUNTERS BETWEEN STELLAR SYSTEMS *

P. Devadas Rao and N. Ramamani

Centre of Advanced Study in Astronomy,Osmania University, Hyderabad, India

and

Saleh Mohammed All-tdin **

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

October 1985

ABSTRACT

M-Body sijnulations have been performed to study the tidal effects of a

primary stellar system on a secondary stellar system of density close to the

Roche density. Two hyperbolic, one parabolic and one elliptic encounters have

"been simulated. The changes in energy, angular momentum, mass distribution,

and shape of the secondary system, have been determined in each case. The

inner region containing about 4o!S of the mass was found to be practically

unchanged and the mass exterior to the tidal radiuB was found to escape. The

intermediate region shoved tidal distension. The thickness of this region

decreased as ve went from hyperbolic encounters to the elliptic encounter

keeping the distance of closest approach constant. The numerical results for

the fractional change in energy have been compared with the predictions of the

available analytic formulae and the usefulness and limitations of the formulae

have been discussed.

* To be submitted for publication.

** Permanent Address: Centre of Advanced Study in Astronomy,

Osmania University Hyderabad — 5OOOO7, India.

-1-

I. INTRODUCTION

The study of encounters between galaxies has received much attention

after the discovery of peculiar galaxies. Photographs of interacting

galaxies by Vorontsov-Velyaminov (1959, 1977), Arp (1966) and others show that

many pairs of galaxies have peculiar morphological features. Toomre and

Toomre (1972) using the restricted three-body approach, shoved that the tidal

forces play a vital role in the dynamics of interacting galaxies and lead to

the formation of such features. Ostriker, Peebles and Yahil (197^), Faber

and Gallagher (1979) and Gunn (i960) have discussed dynamical evidence for the

^alaxies being surrounded by large halos, often extending up to 100 kpc . This

1I:IE increased our esimate for the frequency of galactic collisions and ban

considerably stimulated research in galaxy interactions.

The effects of the tidal forces arc not merely peripheral. The over-

all .;tri;ct:.nr'.' of the galaxies is also uri'ected. The fractional increase in

the to Lai internal energy of a galaxy, ~T~T provides a good measure of the

1'tiu.rii'i: in its structure (Alladin 196'j). If this is equal to or greater than

unity the galaxy would be appreciably disrupted. This ratio has been

estimated by many workers using the impulsive approximation (IA) wherein the

motion of the stars in the test system is neglected In comparison with the

orbital motion of the systems. This approximation gives good results for

energy changes even in slow hyperbolic collisions (Toomre 1977, Dekel i>[. ;i.I

19&'0). however for a, detailed study of the change;.; in the shape and mass

distribution of a colliding stellar system, a self-conui stent method such as;

the muthoij of M-Body simulations i.s needed. Thir. jnethoi"! hai; been

extens i vcly used i,;/ :;t.ui!,yr mergers of galaxies (see Tremaine, 19^1, and

White, 1983 for reviews). Borne (lyflh) ha;; determined merger times of binary

galaxies using multiple three-body algorithm. In this paper we shall not

consider mergers. Here vc .'ih.'jJl restrict ourselves to the eases of

non-penetrating tidal encounters.

otudies of encounters between stellar systems may be divided into two

categories: (a) those in which the two systems are of comparable mass, and

(b) those in which there is great disparity in mass. In this paper we shall

study the second category. It is convenient in this case to designate the

more massive system aa the primary and the less massive as the secondary

(satellite). The primary lias disruptive effects on the secondary while the

secondary accelerates the stars in the outer parts of the primary and this leadc

to dynamical friction on the motion of the satellite. The former (disruptive)

effect "becomes more important than the latter (merging) effect as the ratio

p/p_ (where p is the density of the satellite system and p_ is the Rochen r\

density) decreases (Alladin et al 1905). The case of p >> p where

R

dynamical friction and merging are more Important than disruption has been

studied by Lin and Tremaine (1983) and others. In this paper we shall study

the case in which the density of the satellite is close to the Roche density.

We ignore the effects of dynamical friction and concentrate on how the tidal

field of the primary changes the mass distribution, energy and angular momentum

of the secondary. We confine ourselves to non-penetrating encounters and

treat the primary as a mass point. We investigate to what extent the results

of the analytic formulae derived under the assumption that the stars remain

stationary (Spitzer 1958), Alladin and Nai-asimhan I982; Narasimban and Alladin

19S3) agree with the numerical experiments.

In Section 2 we describe the method used and the initial conditions

employed in the present simulations. The numerical results are given and

discussed in Section 3. In Section h we give the main conclusions.

THE METHOD AND THE INITIAL COfTBITIOHS

2.1 The method

In K-Body simulations we solve a system of N second order

differential equations of motion of H particles in a system simultaneously.

The motions are governed by

N(1)

where r.. = r. - r , m, and r. are the mass and position vector of

particle i, G is the constant of gravitation. A problem arises in near

collision of two particles due to the singularity when r. . — ^ 0. Also on1 j

account of the restriction that N should be small to save computer time,

relaxation time is reduced and the system may not mimic a real system. The

relaxation time is increased by using a softened potential, given "by

-3-

- - c,(2)

The N-Body 1 Code supplied by S.J. Aarseth was used to compute the orbits of

the particles interacting via this potential.

2.2 Initial conditions

He studied the disruption of the secondary stellar system (satellite

system) of mass M due to the tidal effects of the primary system of mass

M = 10 M, The satellite system was modelled by 12l) particles each of unit

mass distributed randomly within a sphere of radius Ft = 20 units with the

mass distribution M(r) « r. Each particle was then given a velocity equal

to the circular velocity at the distance and the directions of the velocity

vectors vere chosen randomly. The system so generated had zero angular momentum.

Its half mass radius R was nearly 10 units and with G = 1, V vns 2.'',Ji J rrns

units and the crossing time was about 8 units. Following White (1978) we

chose £ - 1 in equation (2).

The system var, allowed to evolve by itself for about 7 crossing times

to give a well mixed dynamically stable system. This system had its half mass

radius almost unchanged. The radius containing mass was about 3H, as if,h

the case of a polytrope of index n = 3. The ratio of the rms radial and

L^.n^ont i P*l "VOlncitii f:: i" , TIU c, was 0.83 J ndicat, i nj~ that the moHol filno har elungay

orbits. t,rms

In order that the density of the system may be close to the Roche density

when it is at its closest approach p we chose ji = 100 RV

The orbital

plane was chosen as the xy plane with the x axis in the direction of the

closest approach. Four non-pentrating encounters were simulated with

encounter par;imeters chosen as in Table 1.

M was placed in the orbit at a distance r = 2p. Beyond this

distance the tidal effects are negligible. The initial relative velocity at

this distance was obtained from the two body formulae. The positions and

velocities of the particles in the test system and of the point mass perturber

were computed in the centre-of-masn frame of the entire system. These were

taken as the input for the N Body 1 Code. At various times, the essential

data containing the positions and velocities of all the particles were stored

for the latter processing. The computation was stopped when the system showed

no further change in its properties.

3. NUMERICAL RESULTS AND DISCUSSION

The position and velocity data obtained at various tiaes was used to

study the structural changes of the system during the encounter. We found the

Internal energy {binding energy), the angular momentum the mass loss and the ~

change in the shape, at the end of the encounter.

Due to the strong tidal forces acting on the system during the encounter

the system loses spherical symmetry. Thus the mass distribution and shape

cannot be conveniently studied with respect to the centre of mass of all the

particles in the system. For this purpose we took a reference frame with Its

origin at the centre of mass of the most dense region defined as the region

containing those particles with energy less rather than the average energy of the

bou-it! pariicles. A particle was considered to be bound or be an escaper

depending on whether

*" ' ^ v N U ; ? -J?2iL_ ,, (3)

was negative or positive. We determined the structure of the system in this

frame designated as the effective frame. The mass loss was estimated from the

number of escaping particles.

3.1

lAMr

1'lnergy increase and mass loss

Fig. 1 gives the fractional energy increaseAU

and the mass loss

as a function of time for the four models. C indicates the time of

closest approach. It may be noted that thr energy increase Is somewhat

symmetric about the closest approach. This is expected from Spitzer's (1958)

theory based on impulsive approximation which leads to

-5- (>0

On the other hand Ahmed and Alladin (1981) found that the energy transfer in

a head-on collision between two identical galaxies is asymmetric with respect

to the closest approach, the transfer in the second half being double that of

the first half. This was because of the dynamical friction, the relative

velocity of the galaxies was slower in the second half. In our case the

deceleration due to dynamical friction Is very small and was not considered.

In the case of parabolic and elliptic encounters we found that most

of the energy increase occurred when - ~ < 6 •: ^ (taking 6 = 0 at

closest approach). This does not agree with the analytic formulae obtained

under IA (Alladin and Harasimhan (1982), Harasimhan and Alladin (I983))which

predicts that the energy increase in this range is half the total (-1: < B < TT

SinceAU predicted by these formulae for the total encounter is reasonable,

as will be discussed subsequently, it follows that IA underestimates the

tidal effects near the perigalacticon and overestimates on the far side.

AU obtained In the various mode]sIn Table 2, we compare the values of r—j-

with those predicted from the analytical formulae derived under IA. For

fast encounters, assuming the relative orbit to lie rectilinear, Dekcl et al (1980)

obtained from Spitzer's (1958) theory

(5)

From Alladin and Narasimhan (1982) and Harasimhan and Alladin (1983), we obtain

for the conic orbit

AU ^ 8 G, M

AU predicted by the analytical formulae (5) and (7) agree well with the results

of numerical experiments for models HI and H2, For model P, Equation (5)

underestimates AU while Equation (6) overestimates. The computed value lies

almost exactly in-between these two estimates. For model E the numerical

value is In good agreement with that predicted by Equation (6). Equation (5)

underestimates by a factor of about 3 In this case.

3

In Table 2 we also give -ryp— whereU is the binding energy of the

final bound part, M the fractional mass loss, and'0.9 the ratio of

the radii containing 90% mass of the bound system to that of the total system.

We find a linear relation of the form (Fig. 2)

H L * Jo-1The first part of the result was earlier noted by Dekel et al (1980) for their

A model.AU

The B AUrange in "n;T is considerably smaller than the range in TuT-

r-'his

fixed, TJTT- and

is because a large part of the transferred energy is carried away by the

escaping particles, a result earlier noted by Richstone (1975). An interesting

point is that as we go from the hyperbolic to the parabolic case keeping pI AHM

parabolic to the elliptic ease ; "'J' remains almost unchanged but

increases drastically. This is because the escaping particles in the elliptic

encounter carry away much larger energy than in the case of the parabolic

encounter. White (1978) found that more particles escape from parabolically

colliding systems than from initially bound systems.

both go on increasing. But when we go from the

|AM| . , AUM

3.2 Gain in angular momentum

As a result of the tidal interaction, the satellite system acquires

angular momentum from the orbital angular momentum of the pair. In Table 3

we give the orbital angular momentum (L?j'or-t, o f t h e Pair> the spin angular

momentum for the total systems S ) t o t a l and for the bound part

-7-

after the encounter. It can be seen that only a very small fraction of the

orbital angular momentum is transferred, and that most of the transferred

angular momentum is carried avay by the escaping particles as earlier noted

by White (iy','9) in simulations of merging galaxies. We also note that the

escapers carry avay small angular momentum in large angular momentum encounters

and vice-versa. Thus the angular momentum transfer is inversely related to

the orbital angular momentum.

3.3 ['article orbits

An idea of the change in the nature of the stellar orbits can be had

from the values of (V ) and (V,) after the encounter. These arer rms t rms

given for the total system and for the bound part in Table It. Both the

radial and the transverse velocities of the stars are increased as a result of

the tidal interaction. Hyperbolic and parabolic encounters favour the radial

motion more than the transverse motion. In bound orbit encounters both are

almost equally affected. The bound part has more transverse orbits than

radial orbits even compared with the initial system. It follows that the

escaping stars are characterised by radial orbits.

3. Radial mass distribution

To study how the radial distribution of mass changes due to the

encounter we find the radii containing 10$, 20%, etc. of the total mass in the

effective frame. This is done for the initial system and the final system

including the escapers. The results are given in Table 5.In Figure 3

h is plotted against — where M and R. are the initial mass and

Fi \ h

half-mass radius of the system.

We distinguish three regions in the final system : (1) the inner region

containing about U0/5 of the mass which remains practically intact (2) the

intermediate tidally distended region which extends up to the tidal radius,

and (3) the region exterior to the tidal radius which is found to escape.

Dekel et al (1980) also observed that the mass distribution vithin R. almosth

remained intact and there was expansion for r > R, . The figure shows that thereh

is not much difference in the final mass distribution between models P and E.

3.5 Tidal radius

The tidal radius obtained from numerical experiments is compared with

those predicted from analytic formulae in Table 6. (E )

the initial radius of the test system, beyond which the mass escapes in the

numerical experiment. (ft ) is obtained from King's (1969) formulat King

f lL (9)

and (R ) is taken as the value of B in equation (6) or (7) for which

= I• We note that ' Rt' N, B o d y agrees well with (R

t)jA f°r hyperbolic

and parabolic encounters and with {FL)K- for parabolic and bound orbit

encounters as expected. We also give in the table the radius of the intact

sphere measured in the unit of the tidal radius. It may be noted that this

fraction is smaller for the hyperbolic encounters which implies that the size

of the intermediate region which produces tidal distension increases as we go

from hyperbolic encounter to elliptic encounter keeping the distance of closest

approach constant. In this sense the hyperbolic encounter strikes deeper as

suggested in Alladin et al (1985). If we keep p constant, the intermediate

region becomes thinner with decreasing e.

It is of interest to compare the result of the present work with that

of Angeletti et al (1983). Angeletti et al (1983) studied analytically the

tidal linear stability of a homogeneous star cluster considered as a galactic

satellite. They indicated the regions of stability and instability in the v,

e plane where v = — (r—) , Me, Mg, a, Re, being respectively, the mass

of the cluster, the mass of the Galaxy the semi-major axis of the orbit and

the radius of the cluster. For a given e, three regions of v may be

distinguished (a) the region of instability when v < v , , (b) the region of

stability, where v > v , and (c) the transition region v > v > V .max max m m

containing zones of both stability and instability. The work shows that in

the case of the circular orbit, there is a sharp transition between regions of

stability and instability and the width of the transition region increases with

e.

-9-

3.6 Change in shape

We can make a crude quantitative estimate for the shape of a system

of particles from the values of < ]x| >, < |y| >, < |z.| >. These are

given for the total system and for the 'bound part in Table 7. We also give

in table 7, the expansion parameters E , E , F__ defined "by

jx. | etc,

where f and i denote final and initial values. We note that the maximum

expansion is always in the x direction and least in the z direction. At

the encounter we have a system flattened in the orbital plane, consistent with

the gain in angular momentum in the z direction. The final bound part is

however, almost spherical in shape.

It may be noted that for the hyperbolic encounters, the expansions in

the y and z directions are nearly equal. But in the case of parabolic and

elliptic encounters the expansion in the z direction is considerably smaller

than that in the y direction. We can summarize our results as

We expect E , E , E to he proportional to the energy increments in the•/. pc y s

x, y, z, directions. According to Spitzer's (19J8) formula if e > > 1 then

AV = AV2 and AVS = 0. According to Alladin and Narasimhan (1982) forx z y

2 2 1 2e ,< 1, AV = AV ' = T- AV A comparison of the prediction of the analytic

formulae with the numerical results indicates that IA overestimates (AV )

in every case. Since the overall estimate Tor fill predicted by IA agrees•"-> 2

with the numerical results , i t follows that IA underestimates AV« = AV +2

AV The IA estimate agrees with the numerical results in predicting that

E > > E for e = 3 but does not agree in predicting that E > E for

O.!> < e ,< 1.0.

2IA is not satisfactory for predicting AV in slow hyperbolic or

7-

bound orbit encounters. Agreement with numerical results may be achieved in

a fast hyperbolic encounter. It may be noted that in IA, the term V.& V is

neglected and that AVr is negative for a star on the positive z axis. The

negative sign of AV^ acts in the direction of reducing the expansion in the

z direction. That the tidal expansion in the z direction is small is

1J-

consistent with Chandrasekhar's (19^2) analysis for the tidal disruption of a

star cluster, which shows that the disruption does not occur on account of tidal

instability in the z direction.

3.7 Escaping particles

In Figure h we plot the number of escaping particles in each quadrant

at different times. The circle on the relative orbit represents the position

of the perturber. The number in each quadrant is the number of particles

beyond a distance of 30 units. Almost all of them are escapers. The system

first tends to get elongated towards and away from the perturber. This

direction slowly rotates following the perturber's motion. A similar scenario

was also observed by Miller (1981+) in his study of the tidal effects on a satellite

system . The satellite system was seen to get elongated towards and away from

the perturber and the particles were seen to escape from the end of the bulges.

Dekel et al (lQ8o) observed that in interpenetrating encounters, the escaping

particles had a preferred direction which was opposite to the direction of

closest approach, whereas in distant encounters particles escaped both towards

and away from the perturber.

We found from our experiments, as in Miller, (1981+) that in the case of

hyperbolic and parabolic encounters, the escaping particles did not get

bound to the perturber. These would become a part of the intergalaetic

material. Because of its loose distribution, this mass would become a part of

the invisible mass.

k. CONCLUSIONS

Many results of the present experiment are consistent with those

reported by earlier workers. Detailed comparison between the results of the

experiments with the predictions of the analytic formulae appear to be new.

The main conclusions of this work are as follows:

1. The thickness of the shell between the inner intact sphere and the outer

• • 1 1 -

stripped region decreases a VJO p;o from !-i hyperbolic: orbit to "be n Ijound oroit.

encounter keeping the distance of closest approach fixed.

2. The estimate for ITJT- obtained under IA for a conic orbit (Equations

6 and 7) is in fairly good agreement with that obtained by the numerical method

for e >, 0.5- Spitzer's formula holds for e 1.

3. Although IA gives good estimates for the overall energy transfer it

does not predict correctly the components of the stellar velocity perturbations.

Unless e > > 1, (SV '' is always overestimated. In a bound orbit encounter

the error is large. The expansion parameters obtained from the numerical

experiments satisfy the relations V > > E - F< for e > '•> 1 andx y -i.

> > K Tor e .< 1. On the other hand, the IA formulae suggestz

> >y

for e > > 1 and I1.v.

= I'y

for e.< 1.

It. In the case of tound orbit encounter, IA underestimates the energy

increase during the encounter on the perip;nlacticon side and ovprestimates on

the u.po}fjil act i con side.

5- As we go from hyperbolic to parabolic encounters keeping p constant the

energy increase and the mass loss increase, hut as ve RO from parabolic to

elliptic encounter, the energy increase continues but the mass loss remains

constant.

6. An effect of the encounter is to impart spin angular momentum to the system

in the some sense as that of the relative orbit. Most of this angular momentum

is carried away by the escapers.

7. Tidal encounters elongate the stellar orbits in the system. Escaping stars

leave the system in radial orbits. The stellar orbits in the bound part after

the encounter have more circular motions than the initial system.

ACKNOWLEDGMENTS

The authors are grateful to Dr S,J. Aarseth for kindly supplying the

W-Body 1 Code and to Dr. E.K. Bhatia for useful discussions,PDE thanks

University Grants Commission, Hew Delhi, for providing him with a Teacher

Fellowship and to M.G.M. College, Udupi, for permitting him to make use of

this opportunity. SMA Is grateful to the University Grants Commission,

Mew Delhi, for providing him with a National Fellowship. He also thanks

Professor Abdus Salam, the International Atomic Energy Agency and the UHESCO

for the hospitality extended to him at the International Centre for

Theoretical Physics, Trieste, where part of the work was done. He also

henefitted from discussions with Professor D.W. Sciama and other scientists

at the International School for Advanced Studies, Trieste.

-13-

REFERENCES

Ahmed F. and Alladin S.M. I981, Bull. Astr. Soc. India 9.1 1*0-

Aliudin S.M. 1965, Astrophys. J, l]a, 768.

Alladin S.M. and Narasimhan, K.S.V.S. 1982 Phys. Rep. £2, 339.

Alladin S.M., Ramamani N. and Mei n/a Singh T, 1985, J. Astrophys. Astr-oi §_, 5.

Angeletti L,, Capuzzo-Dolcetta R.and Giannone P, 1983, Astroi'. Astrophys. 121, I

Arp H.C. 1966, Astrophys. J. Suppl. 1^, 123,

Borne K.D. .19flJi, Astrophys. J. 2Q]_, 503

"liiiiiiirnsekar \1. 19^2, -°TI i"1--,''-'' ••'- o'~ '<'-\o\'\ •'••• TiynnF.ic.s, (University o f Chic.-i/'o

Press, Chicago)

Dekel A., Lecar M. and Gl-.lifirti J, I960, Astrophyn. J 2h^, 9^6-

Fabtr S.M. and Gallagher J.G, 1979, Ann Rev. /isl.ron. Asti'orliJ'R. JJ., 135.

Ounn J.E. 1980, Phil. Trims. R. Ikic. A 29^, Mi.

King I.R. 1962 AstTOn. J. &T, I47I.

Lin D.B.C. and Tremfiine S. 1983, Astrophyis. J 2_6[J_, 36U -

Filler R.H. 19Bh, ESO preprint Wo. 3'i3, Af;Lron. Aatrophys. (submitted).

Narasimhan K.S.V.S. and Alladin S.M. 1983, Bull. Asti. Soc. India, 1^, 221.

Oct.rJki.fi- J.I1., I'ecblOLi .J.i.h;. and Yahil A., I9'[h, Astrophys. J. Lett. 1_9_2, ''•.

Richstone D.O. 1975, Astrophys. J. £00, 535.

Sjiitzer L, l'.''''! Astrophys. J. 127, 17.

'i'oomre A. and Toomre J. 1972, Astrophys. J. 178, 623.

Toomfe A. 1977 > The Evolution of Galaxies and Stellar Populations, [MB. :i.P-I.

Tinsley and R.B, Larson (Yale Dr.ivers !.ty Observatory) p. k'Jl. -^

Tremaine S.D. 1901, The Structure and Evolution of TiormaJ Galaxies, Eds. fi.M.

!''all and D. linden (Bull C'ambridiTfi University Press), p. 67.

VoronLsov-VGlyaminov B.A. 1959, Alia;; and Catalogue of IntorHchinpr Galaxies,

^ Astronomical TusLit.ut c, Moscow).

Vorontsov-Velyaminov B.A. 1977, Astron. Astrophys. Suppl. 28, 1.

White S.D.M. 1978 Mon. riot. Roy Soc. 16^, l85-

White S.D.M. 1979 Mon. Hot. Roy Soc. l8£, 831,

White S.D.M. 1983, IAU Symp. 100: Internal Kinematics and Dynamics of Galaxies,

Ed. E- Athanassoula,(D. Reidel, Dordrecht,) p. 337

- l i i -

-15-

Model

III

H2

P

E

e

3-0

2 . U

1.0

0 . 5

P

\

50

1*2

50

50

Table 1

VPV

rnu

101

101

12

62

M

M(

10"

10"

10'

10

-5

-5

Table 2

odel

HI

H2

P

E

(:'

0

1

1

5

AulilT

'JH-Body

. 6 1

.12

.96

.1)8

(r0

1

l

l

' Op.

. 7 0

.1.0

. • u o

. 9 0

[r0

I

2

6

.72

.60

.60

. 00

0.19

0.2T

0.39

O.llO

0.21*

0.27

0.36

0.1*3

I K i o

0.30

0.1*1*

0.58

0.71

Model

HI

H2

P

E

2.8(+7)

2.0(+7)

Table 3

'/.' Total

l*.0C + 3)

7.6(+3)

6.o(+lt)

?, bound

S.31+2)

-16-

Table 1*

Model

Intial

HI

H2

P

E

Vrrms

1.1*1

1.82

2.37

2.81*

2.58

TotalVtrms

1.70

1.81

1.01

2.38

1*.69

.. ,v . . ,

rms

trma0.83

1.00

1.31

1.91

0.76

Bound

rms

1.1*1

1.05

1.03

1.19

1.06

trms

1.70

1.81*

1.85

1.93

2.00

Vrrins

trms

0.83

0.57

0.56

0.62

0.53

TatlH 5

0.2 0.3 0.1* 0.5 0.6 0.7 0.8 1.0

Initial

III

H2

P

E

0.12

0.18

0.16

O.ll*

0.15

0.26

0.30

0.30

0,28

0.29

0.1*5

o.i*6

0.1*6

0.1*0

0.56

0.6T

0.70

0.75

0.71

0.79

0-95

1.06

1.19

1.39

1.93

1.35

1.59

2.28

3.26

3.38

1.83

2.6'+

3.75

9 . 2

7.9

2.66

5-91

8.6

13.2

16.1

-3.15

9.1*

13.1

17.5

21.3

7.8

35

1*0

36

35

Table 6

Model

HI

HS

t'lI-Body

26

18

13

13

H

13

15

15

-17 -

<VlA

2l*

16

12.5

8

intact/ t

0.1*

0.5

0.6

0.6

Table TFIGURE CAPTIOUS

Model

Initial

HI

H2

P

E

<u6.f

,k

32

1*2

i.8

Total

> <|y|>

3 6.6

11

15

26

32

<\y,\>

6.6

12

15

11

6

2

3

6

EX

.6

• 9

.14

.3

Total

Ey

0.6

1.2

3.0

3.9

F

0.

1.

0.

0.

1

8

7

< x|>

6.6

7.^

6.5

it.5

Bound

6.6

7.3

5.2

3.1*

I*.5

< 1 Z '

6.6

6.2

5.5

It.5

14.5

Fig. l(a) Hyperbolic orbit (H.).

Fig. l{b) Hyperbolic orbit (H £).

Fig. l(c) Parabolic orbit (P)

Fig. l(d) Bound orbit (E),

AM . fS} %Fig, 2 —— and I""*I vs T—r *

M i r J 0 . 9 lul

Fig. 3 Mass distribution M(r).

rig. !*(a) Hyperbolic (Hn) No. of particles outside a distance 30 in each

quadrant are also given at various times T.

Fig. M b ) Hyperbolic (H ) No. of particles outside a distance 30 in each

quadrant are also given at various times T,

Fig. M e ) Parabolic (P) No. of particles outside a distance 30 in each

quadrant are also given at various times T.

Fig. U(d) Bound (E) N O . of particles outside a distance 30 in each quadrant

are also given at various times T,

-18--L9-

«'. - * "

2.0

ID

Illl(

A MM

10 20T

30 40

- 2 0 -

2.0

1.0

C ^ - ^: /; /;/

/(•I':

I i/ :

~~ —~- -_. ___

^^ - - _ _ , . 1

1 1 j

10 20T

A U

lul

d MM

30 40

-21-

2.0

1.0

10 20T

30

MM

40

6.0

5.0

4,0

3.0

2.0

1.0

C /

IIi

: I

A MM

10 20 30 40 SO

- 2 2 - -23 -

01

1.0

M(r) 0.8M

0.6

0.4

02

Initial

Hvperbolic (H,)Hyperbolic (H2)Parabolic (P)Bound orbit (E)

10 12 14 16 18 20 22 24

A UR

nor

-24--25 -

T=100 T=400

-26 -

T=10.0 T»14.0 'T-40.0

- 27 -

6 1 0 1 10 i 15 \ o I 27

T=14.0 T-18.0 T-28.0 T-38.0 T=40.0

0 | 11 \ 14 I 0 \ 9 I 17 \ 0 I 22— M — / M J- M M

3 / 0 I 16 / 4 I 12 / 25

T- 18.0 T-28.0 T-38.0 T-52.0

-28- - 29 -

(1i