Using the impulsive approximation to study the velocity changes of stars during disk-sphere collisions and a method due to Bottlinger to study the post collision orbits of stars, the formation of various types of interacting galaxies is studied as a function of the distance of closest approach between the two galaxies

Chatterjee, Tapan K.

English

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A DYNAMICAL PROXIMITY ANALYSIS OF INTERACTING GALAXY PAIRS 
Tapan K. Chatterjee 
Facultad de Ciencias, Fisico-Matematicas, 
Universidad A. Puebla, (A.P. 11521, Puebla, Mexico 
ABSTRACT. Using the impulsive approximation to study the 
velocity changes of stars during disk-sphere collisions and a 
method due to Bottlinger to study the post collision orbits of 
stars, the formation of various types of interacting galaxies is 
studied as a function of the distance of closest approach between 
the two galaxies. 
INTRODUCTION 
The interpretation of multiple interacting systems as the 
relics of gravitational interaction between galaxies is now 
strongly favored (see Toomre, 1974, for a review). Observations 
indicate that the proximity of the constituent galaxies in 
different types of interacting galaxies (eg., mildly perturbed 
galaxies, galaxies with bridges and tails, ring galaxies) is 
different. There seems to be a relation between the proximity of 
the constituent galaxies and the intensity of interaction between 
them. To study this relation it is necessary to study the 
formation of peculiar interacting systems, on the basis of the 
collision theory, as a function of the distance of closest 
approach between the constituent colliding galaxies. 
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An inspection of photographs of interacting galaxies from 
catalogues (eg. Voronotsov-Velyaminov, 1959; Arp, 1966) show that 
the majority of these systems consist of a disk and a spherical 
galaxy lying close to each other. Moreover the basic criteria 
determining the type of the interacting galaxy bears a strong 
relationship to the degree of perturbation encountered by the 
disk galaxy. Sphere-sphere collisions are important in a study 
of merging galaxies and do not lead to a variety of interacting 
galaxies, but rather to a particular variety. Hence, in 
subsequent work, we have studied the formation of peculiar 
interacting galaxies on the basis of collisions involving a disk 
and spherical galaxy, as our aim is a functional analysis of the 
/ 
type of peculiar interacting galaxy and the proximity of the 
constituent galaxies. 
THEORY 
In a previous paper (Chatterjee 1984; hereafter referred to 
as Paper I) the conditions of the formation of ring galaxies, 
during head-on collisions between disk and spherical galaxies, 
was studied. Using the same method we study in this paper both 
head-on as well as off-center and non-vertical collisions between 
disk and spherical galaxies and determine approximately the range 
of impact parameters favorable €or the formation of different 
types of interacting systems. 
The spherical galaxy is modeled as a polytrope of index 
n=4. The spiral galaxy is modeled as an exponential model disk, 
570 
t h i c k e n e d  by a method i n d i c a t e d  by R o h l f s  and Kre i t schmann 
(19811, w i t h  a p o l y t r o p i c  n=4 b u l g e  s u p e r p o s e d  on t h i s  d i s k .  The 
model  is d i s c u s s e d  i n  d e t a i l  i n  C h a t t e r j e e  1989. 
The t h e o r y  is e s s e n t i a l l y  t h e  same as i n  P a p e r  I. Using t h e  
i m p u l s i v e  a p p r o x i m a t i o n ,  t h e  change  i n  v e l o c i t y  (x8'  of t h e  
r e p r e s e n t a t i v e  s ta r  i n  t h e  d i s k  g a l a x y ,  due t o  t h e  c o l l i s i o n  is 
g i v e n  by, (case RS=RD=R),  
Iz = 0 ( 4 )  
where  t h e  symbols  have t h e  same meaning as i n  Pape r  I ,  except: f o r  
p, which is t h e  d i s t a n c e  of closest a p p r o a c h  between t h e  
c o l l i d i n g  g a l a x i e s ,  measured i n  u n i t s  of t h e  r a d i u s  of e i t h e r  
g a l a x y ;  and x D s f  which a measure of t h e  m u t u a l  potent ia l  due t o  
t h e  g r a v i t a t i o n a l  i n t e r a c t i o n  between t h e  d i s k  and t h e  s p h e r i c a l  
g a l a x y  and is d e f i n e d  as t h e  m u t u a l  po ten t i a l  f u n c t i o n  €or t h e  
d i s k - s p h e r e  pa i r  ( B a l l a b h ,  1975). 
? and x$' are  f u n c t i o n s  of b o t h  t h e  d i s tance  of closest 
a p p r o a c h  between t h e  two g a l a x i e s ,  p,  and t h e  a n g l e  of 
57 I 
inclination, i, of the disk to the direction of relative motion 
between the two galaxies (see Paper I). Using equations (1) to 
( 4 1 ,  we calculate the velocity perturbations of the 
representative stars for different collisions, each collision 
being characterized by a valbe of p and i. We then apply a 
method due to Bottlinger (Bottlinger, 1932, 1933) as discussed in 
Paper I, to study the post-collision orbits of the stars, the 
orbits of the innermost and outermost stars being studied by 
numerical integrtion due to the validity of the Bottlinger force 
law within a limited range of distance only. For convenience we 
define the angle a = 90' - i, so that for a=O we have a collision 
whose trajectory is normal to the plane of the disk. 
\?e increase p and a in small steps (of 0.05 and 5'1, and 
find AV' for the representative stars for each collision and 
scale its value for different values of the collision parameters 
(MS, MD, R, V). 
of disk galaxy for each collision, approximately, and test 
whether the rearrangement of stars leads to the formation of a 
peculiar galaxy (and if so, its shape and form, which determine 
its type). 
Then we determine the final density distribution 
NUMERICAL RESULTS AND DISCUSSION 
We find that, for central or near central impacts, ring 
galaxies are formed provided the angle of inclination does not 
exceed about 45O. If a N < 30' and the impact is central, or 
nearly so, we get very symmetrical and bright rings whose 
572 
i n t e n s i t y  d i s t r i b u t i o n  is almost uniform. I f  a is i n c r e a s e d  
beyond 4 S 0 ,  w e  get messy, i l l - d e f i n e d  s t r u c t u r e s .  If p is 
i n c r e a s e d  t h e  r i n g s  become asymmetrical .  and a c q u i r e  non-uniform 
i n t e n s i t y  d i s t r i b u t i o n s .  Beyond p - 3, f a i r l y  w e l l  d e f i n e d  and 
prominent  r i n g s  do n o t  form. Messy i l l - d e f i n e d  r i n g s  form up 
t o  p - 2/3. 
Beyond p - 2/3, w e  g e t  one-armed s t r u c t u r e s  which do n o t  
close on themse lves ,  b u t  end a b r u p t l y .  Beyond p - 1 w e  g e t  ve ry  
f a i n t  b r i d g e s  and t a i l s  and e x t r e m e l y  f a i n t  r i n g - l i k e  s t r u c t u r e s  
which do no t  close on themse lves ,  r e sembl ing  embryonic s p i r a l  
arms, and are embedded i n  t h e  background of t h e  v i c t i m  d i s k  (see 
C h a t t e r j e e ,  1986, for d e t a i l s ) .  
Prominent  br idges and t a i l s  s t a r t  forming o n l y  when 
p - 1 3/4. B r i d g e s  and t a i l s  form f o r  a l l  o r i e n t a t i o n s  of the 
c o l l i d i n g  g a l a x i e s  t o  t h e  t r a j e c t o r y  of t h e  c o l l i s i o n ,  o n l y  they  
are better formed when t h e  t r a j e c t o r y  of t h e  co l l i s ion  does n o t  
make a l a r g e  a n g l e  w i t h  t h e  p l a n e  of t h e  d i s k ,  be ing  most 
prominent  when t h i s  a n g l e  is q u i t e  small  b u t  n o t  z e r o  ( i .e . ,  i > 
0' b u t  G 1 5 0 ) .  
As t h e  impact pa rame te r  is i n c r e a s e d ,  w e  f i n d  t h a t  b r i d g e s  
and t a i l s  cease to  form when p is s l i g h t l y  g r e a t e r  t han  2 ( a b o u t  
2 . 1 ) .  However the  s h a p e s  of t h e  g a l a x i e s  are a f f e c t e d  due to  
t i d a l  f o r c e s  up t o  p N 2 1 /2 .  Beyond t h i s  v a l u e  of p t h e  
c o l l i d i n g  g a l a x i e s  do n o t  a f f e c t  each  o t h e r  enough t o  produce 
o b s e r v a b l e  d i s t o r t i o n s .  
573 
CONCLUSIONS 
As e x p e c t e d  a dynamica l  s t u d y  of  t h e  f o r m a t i o n  of  p e c u l i a r  
i n t e r a c t i n g  g a l a x i e s ,  on t h e  bas i s  of  t h e  c o l l i s i o n  t h e o r y ,  
i n d i c a t e s  t h a t ,  as t h e  d i s t a n c e  of  closest a p p r o a c h  between t h e  
c o l l i d i n g  g a l a x i e s  i n c r e a s e s ,  t h e  i n t e r a c t i o n  f a d e s  o f f  r a p i d l y  
b u t  what is i n t e r e s t i n g  is t h e  e x i s t e n c e  of  an  almost c o n t i n u o u s  
series of s u b t y p e s  of  d i f f e r e n t  t y p e s  o f  p e c u l i a r  i n t e r a c t i n g  
g a l a x i e s  as a f u n c t i o n  of  t h e  p r o x i m i t y  of  t h e  c o n s t i t u e n t  
g a l a x i e s .  Each s u b t y p e  seems t o  have a c h a r a c t e r i s t i c  r ange  o f  
v a l u e s  of  t h e  d i s t a n c e  of c losest  a p p r o a c h ,  which l e a d s  to  t h e  
c o n c l u s i o n  t h a t  d i f f e r e n t  s u b t y p e s  s h o u l d  have  d i f f e r e n t  r a n g e s  
of v a l u e s  f o r  t h e  p r o x i m i t y  of  t h e  c o n s t i t u e n t  g a l a x i e s .  T h i s  
may p l a y  an  i m p o r t a n t  role i n  the s u b j e c t i v e  c l a s s i f i c a t i o n  of 
i n t e r a c t i n g  g a l a x i e s .  More o b s e r v a t i o n a l  s u r v e y s  s h o u l d  be 
c o n d u c t e d  i n  t h e  l i g h t  of these f a c t s ,  t o  e n a b l e  compar i sons  w i t h  
t h e  theore t ica l  c o n c l u s i o n s .  
F i n a l l y  w e  would l i k e  t o  men t ion  t h a t  t h e  v a r i o u s  parameters 
which d e t e r m i n e  t h e  degree of  i n t e r a c t i o n  between g a l a x i e s  c a n  be  
e x p r e s s e d  i n  terms of  t h e  f r a c t i o n a l  change  i n  i n t e r n a l  e n e r g y  
i n c u r r e d  by e i t h e r  g a l a x y ,  AU/lUl, where U is t h e  i n i t i a l  
i n t e r n a l  e n e r g y  and AU is its change  due  t o  t h e  c o l l i s i o n .  I f  w e  
d e f i n e  AU=(AU/(U( )/(AU/lUl)fil, where ( A U / ~ U ~  I M  is t h e  f r a c t i o n a l  
i n t e r n a l  e n e r g y  change  f o r  which merger  is a f f e c t e d  f o r  t h e  same 
v a l u e  of p ,  t h e n  t h e  r a t i o  6U h a s  c h a r a c t e r i s t i c  v a l u e s  which 
d e t e r m i n e  t h e  d e g r e e  o f  i n t e r a c t i o n  between t h e  c o l l i d i n g  
g a l a x i e s  and hence  t h e  p e c u l i a r i t y  i m p a r t e d  t o  t h e  sys t em.  I t  is 
i n t e r e s t i n g  t o  n o t e  t h a t  t h e s e  c h a r a c t e r i s t i c  v a l u e s  of  6U d e f i n e  
574 
the interaction in terms of a two parameter family, e.g., p and 
V, for a given pair of galaxies (for details see Chatterjee, 
1989b). 
REFERENCES 
Arp, H. C. 1966, Astrophys. J. Suppl., 14, 1. 
Ballabh, G. M. 1975, Astrophys. Space Science, 38,  307. 
Bottlinger, K. F. 1932, Erqebn. Exacten Naturwissenschaften, 11. 
31 . 
Bottlinger, K. F. 1933, Vercff. Univ. Sternw. Berlin-Babelsberg, 
10, 2. 
Chatterjee, T. K. 1984, Astrophys. Space Science, 106, 309. 
Chatterjee, T. K. 1986, Astrophys. Space Science, 121, 212. 
Chatterjee, T. K. 1989a, Astrophys. Space Science,l63, 127. 
Chatterjee, T. K. 1989b, Proceedings of the ''VIth Latin-American 
Regional Meeting in Astronomy'' in press in Rev. Mexicana 
Astron. Astrof. 
Rohlfs, K. and Kreitschmann, J. 1981, Astrophys. Space Science, 
79, 289. 
Toomre, A. 1974, in J. R. Shakeshaft (ed.), "The Formation and 
Dynamics of Galaxies," IAU Symp. 5 8 ,  365. 
Vorontsov-Velyaminov, B. A. 1959, "Atlas and Catalogue of 
Interacting Galaxies," Sternberg State Astronomical 
Institute, Moscow. 
57 5 


A dynamical proximity analysis of interacting galaxy pairs

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