SCIENCE IN CHINA (Series A)
10
V O ~ . 42 NO.
October 1999
The formation of compact groups of galaxies * MA Er
(g
g s ) 1 ' 2 a n d YU Yunqiang ( & ; l f C ~ ~ ) 2 ' 3
( 1 . National Astronomical Observatory, Chinese Academy of Sciences, Beijing 100012, China;
2. 'Beijing Astrophysics Center, Chinese Academy of Sciences and Peking University, Beijing 100871, China; 3. Physics Department, Peking University, Beijing 100871, China) Received September 14, 1998
Abstract
In the compact group of galaxies the galaxies can merge into a few massive ones in a very short time, so they must be formed very recently. On the other hand, according to the theory of structure formation, the denser system should form earlier. By analyzing the apparent paradox, we suggest that the merging process of CDM halo plays an important role in the formation of the compact groups of galaxies: it delays the formation of compact groups of galaxies, and makes the p u p s of galaxies much denser.
Keywords:
structure formation, compact group of galaxies, merging of *es.
Recently, much attention has been paid to the multiple merges of galaxies in the compact groups of galaxies['-41 . The relevant model calculation showed that the time scale of merge would be only a few times of crossing time scale of the groupr51 Since the crossing time of compact groups is about 100 million years, the model calculation is consistent with the fact that we do often observe multiple merge in these groups. If the time scale of merge is only a few hundred million years, the look-back time of their formation should be less than a billion years, otherwise we would have no chance to see their appearance. But this conclusion seems to contradict the simple estimation from cosmology. According to the cosmological theory of structure formation, the small fluctuation in the early universe will increase by gravitational instability until it forms a gravitationally bound system in virial equilibrium. The formation time of this system t, (or the formation red shift Z , ) is determined, according to the spherically symmetric model calculation, only by the initial fluctuation amplitude 6 = ( 6p/ p ) ( 1 + zl ) , where zl is any early red shift and ( 6p/p ), is the fluctuation amplitude at that red
.
shift. The average density of the system is also determined only by this initial amplitude[61. Assuming that the system will remain unchanged after virialization, we can estimate the formation red shift of any system by their average density. The denser the system, the earlier its formation. The formation red shift of a compact group of galaxies should be Z, = 10 if its density remains unchanged after its virialization, which is an order of magnitude larger than the estimation from the merging time scale. This apparent contradiction is the motivation of our work. Based on the cold dark matter dominated model of structure formation, the formation process of compact group consists of two stages. In the first stage the clouds of dark matter form and the baryonic
* Project
partially supporter1 hy the Ministry of Science and Technology (of China)
.
1090
SCIENCE IN CHINA (Series A )
Vol. 42
matter is still mixed with cold dark matter. Later on there are two processes to compete. One is the cooling of baryonic matter and the formation of stars. After that, the cold dark matter cloud becomes the halo of the luminous galaxy. Another process is the merging of the dark matter clouds. For the first process, it is still difficult to deal with. However, the existence of groups of galaxies implies that the star formation should proceed before the merging of the dark haloes. After that, the merging of the dark haloes is unavoidable. Since the dark halo is much larger than the luminous galaxy inside it, the probability of halo merge should be much larger than the merge of luminous galaxy. No matter whether the haloes have merged, the system is always a group of galaxies. If a group of galaxies form earlier, there is enough time for dark matter clouds to merge into one big halo. Then, the galaxies in the group will have a common halo. In the merging of haloes, the average distance between galaxies will decrease, and after the halo merging, the system will re-virialize , and the density of the system will increase greatly. In the sense of cosmological evolution, today is only an arbitrarily assigned epoch. At that epoch, some groups of galaxies have had a common halo and some do not have. It might be the reason that we find two types of groups: normal groups in which galaxies have no common halo and compact groups in which galaxies do have a common halo. If it is so, the formation time of a compact group of galaxies should be understood as the time of re-virialization which is much later than that estimated by the simple description and the apparent contradiction disappears. Detailed calculation shows that the relatively large fluctuation will develop a common halo for the groups, so we can have the compact groups of galaxies as we observed and their formation time is quite late so the merging process is still going on. In the first section we will develop a mathematical modeling for the merging of haloes and the merging of galaxies in a common halo. In the second section the numerical solution of the model will be presented and shows the merging time for the haloes, the density variation of the group after the merge of haloes and the time needed for merging of galaxies in common halo. In the third section the formation of the compact group of galaxies will be discussed and the difference of normal and compact groups of galaxies will be stated. At last there are some comments and discussions on the formation of groups of galaxies.
1 Mathematical model for multi-merging system We have discussed the multiple merging process in a gravitationally self-bound system[71. Now we will generalize it to the galaxy system in which galaxies are located within a common halo of dark matter. Denote the number of galaxies as N , the total mass of the system as M , the total kinetic energy as K and the virial radius as R . To simplify the question we assume that the halo is uniformly distributed. Then, we have the equation of dynamical evolution as follows:
where Ro stands for the initial virial radius and Mh is the total mass of the dark halo within Ro. a , is a form factor in determining the gravitational energy. It has been proved that the dissipation force has no contribution to the virial term in the appendix of ref. [ 7 ] . Assuming the ratio of kinetic energy loss in each merge to the average kinetic energy of galaxies as a', the conservation of energy gives
No. 10
THE FORMATION OF COMPACT GROUPS OF GALAXIES
1091
The variation of the number of galaxies in the merging process is
in which a is the cross section of galaxy merging, a3R3 is the total volume of the system, a4 is the ratio of relative velocity between two galaxies to the average velocity of galaxies. Since we consider the are a complete set of equations for the merging process. factor a , to a 4 constant, eqs. ( 1 )--(3) While we deal with the merging process of the dark clouds in the earlier stage, this equation set will still be valid by omitting the last term in ( 1 ) and ( 2 ) There are many input parameters involved in the equation set. As in ref. [ 7 ] , we make the variables dimensionless. Introducing a time scale to, the dimensionless variables are defined as r = =/to, (4)
.
y1 = ( R / R , ) ~ ,
(5)
y2 = d y l / d r ,
(6)
y3 = N/No,
(7)
y4 = 4( K / M ) ( to/R0)', in which No is initial number of galaxies. The corresponding equation set is rewritten as dyl/dr = dy2/dr =
Y4
~
- Pyi
2
(9)
9
1/2 -
=
-
(10)
Y1,
2 -3/2 1/2 2 0 ~ 3 ~ Y4 1 9 3/2 -3/2 a 2 2 0 ~ 3 ~ Y1 4 PY;~/~YZ
d~3/dr
-
(11)
=while deducing eq . ( 10) , Ro and to are asked to satisfy the relation 2 t ; a , G ~ , = R:. d~4/dr
(12)
Y29
The t o defined by ( 13 ) is the crossing time of the system. The input parameters are where p = alM/M,, is the ratio of luminous mass to the dark mass. The
(8)
(13)
P,
a,,
20,
(14)
Zo is defined as
2o - (2-3/2a4/a3) ( N , ~ / R ; ) .
(15) It is the ratio of sum of the galaxies' merging cross sections to the area of the system. a 2 is the statistically average loss of kinetic energy by one merge, if two galaxies with the same average velocity hit each other face on, then a2= 2 . 0 , if two galaxies move in the same direction a, will be small. We assume that the initial state of the system is in virial equilibrium, then the initial conditions read : y l ( 0 ) = y3(0) = y4(0) = 1,
(16) y2(0) = 0 . While discussing the merging of dark clouds in the earlier stage, the dimensionless variable definition (4)-(8) and (14) will still be used, but to should satisfy 2t;alGM = R:. Here to is again the crossing time. Setting Mh = 0 , eqs. ( 1)-(3)
(17) are still valid.
SCIENCE IN CHINA (Series A)
1092
2
Vol. 42
Time scale for merging and the variation of the compactness
We have solved the equation set numerically. The first result we want to show is the time needed for forming a common halo by the merging of the dark clouds. Since the equation set is not suitable for small N we calculated the time rl,, for reducing N to N o / 5 . The input parameters Z o and a2 are difficult to estimate. We tentatively tried a few numbers in a wide range. Figure 1 shows the results of rl,, versus Bo , where a2 is taken to be 0.25, 0 . 5 , 1. 0 and 1.5. Roughly speaking, if az 3 0.5 and Zo3 0 . 2 , the merging time scale is less than 6 t o . If a2 = 0.25 and .Zo is also small, the merging time scale will increase greatly. After merging of dark clouds, the system shrinks and the density increases. We calculate the shrinking of the system radius ( R / R 0 ) 1 / 5while the number of clouds decreases to one fifth its initial
xo.
value. Fig. 2 shows the results of density ( Ro/R);,, versus It can be seen from the figure that after the formation of a common halo the average density of the system increases by a few tens times if a2 is relatively large, and by a few times only if a2 is small.
lor
Fig. 1 .
Time needed to decrease the number of galaxies of
the system to fifth the initial value for different
Bo.
Fig. 2.
System average density ( shown by ( Ro/R ):,5 )
when the number of galaxies of the system decreases to fifth the initial value for dierent So.
After the formation of the common halo, the galaxy system will re-virialize and galaxies will start to merge in the common halo. We have calculated the time needed for this merging process, and the variation of the average density of galaxy system for a2 = 0 . 5 and 1 .0 . In cold dark matter scheme we assume = 1 and O B-0. 1 , which implies3!, = 0 . 1 . Fig. 3
no
No. 10
Fig. 3.
THE FORMATION OF COMPACT GROUPS OF GALAXIES
Time needed to decrease the number of galaxies
Fig. 4 .
1093
System radius ( R / R o ),,, when the number of
of the system to half the initial value for different Z o when
galaxies of the system decreases to half the initial value for
common halo exists.
different Bo when common halo exists.
shows the time needed to decrease the number of galaxies of the system to half of the initial value for different
Z o . Fig. 4 shows the shrink of the system radius R / R o . The results show that the merging
time scale is still a few times of crossing time, but the radius shrinking is much slower than that in the common halo, because the gravitational force of the halo is playing an important role. It implies that the density increasing happens mainly in the earlier stage of clouds merging. On the other hand, since in the merging process the system is not in virial equilibrium, the crossing time decreases even slower.
3
The difference between the compact and normal groups of galaxies From the calculation we see that the groups of galaxies are observationally different no matter
whether a common halo of galaxy has formed. The average density can be a few times to a few tens times higher for groups in which galaxies have a common halo. We identify the denser ones as compact groups of galaxies. And the groups without common halo should be identified as normal groups of galaxies. A small fluctuation with amplitude 6 = ( 6 p / p ) ( 1 + zl ) will form a virialized system at time t, . As a system the group of galaxies consists of many dark clouds. By the spherical model, the formation red shift Z, , the look-back time at formation t l b and the crossing time are all determined by its fluctuation amplitude
only, and we have
1 + 2, = 6 / 1 . 6 9 ,
(18)
to = 3.19/6'.~,
(19)
SCIENCE IN CHINA (Series A )
1094
t , , = 13.5 x (1
Vol. 42
- 2.20/6'.~).
(20) In eqs . ( 19) and (20) , Gyr is used as the unit of time. Afterwards the dark clouds will start to merge and form a common halo for luminous galaxies. The time needed for the process is determined by Bo or a 2 , whose values are difficult to determine and we can only try different values in a wide range to see their influence. If we take So = 0.2-0.8, and a 2 = 0 . 5 - 1 . 0 , then rIl5is about 2 . 5 - 5 . 5 . So a common halo will form if t l b / t O 3 (2.5-5.5). Fig. 5 shows the results of t , , / t 0 versus 6 . From the figure we see, the demand of t l b / t o 3 2.5-5.5 implies 6 3 2.1-2.6. Based on standard cold dark matter scheme, using the formula for the initial fluctuation spectrum given by Bardeen et a1. , assuming biasing factor b8 = 1 .68 and Mh =
8-
I
1014 solar mass, the formation of the compact groups of galaxies need a fluctuation amplitude 6 3 1. 62 . 0 s where a is the average amplitude for that mass Fig. 5 . t , ,/to as a function of 6 . scale. Since the compact groups came from high peak fluctuation, their number density should be smaller than that of normal groups. Figure 2 shows that the density of the compact groups of galaxies is a few tens times larger after the formation of common halo, and the crossing time is 6-10 times shorter if a2 is relatively large. The crossing time for system with 8 = 2.1-2.6 is roughly in the range of 0.7-1.1 Gyr, so the crossing time for compact groups of galaxies can be 0 . 1 Gyr as observed. Fig. 3 shows that the merging time scale for galaxies within compact groups of galaxies is a few hundred million years. All these model predictions are consistent with the observation. The normal groups and clusters of galaxies should have much lower density. If a fluctuation 6 < 1 . 9 ( 1 . 5 ~ ,) the look-back time for their virialization is 2 Gyr which is 1 . 8 times its crossing time. In these cases, the common halo cannot have formed. The density nearly remains its initial value and no multiple merger happens. 1.7
2.7
2.2
312
6
4
Discussion
In the last twenty years the progress in the theory of structure formation is tremendous. People realize the importance of the dark matter domination in making a satisfactory consistence of the theory with observation. Our work is a further support to this point of view. Though we still do not know how to compare the theory of matter clustering to the real observation, we want to emphasize that in the first stage of matter clustering the dark matter clouds will inevitably merge and change the observational property of the system greatly. The key here is whether there is enough time for the merger to happen. For the systems as large as clusters of galaxies, they should barely form right now and have no time for merge. Groups are just
No. 10
THE FORMATION OF COMPACT GROUPS OF GALAXIES
1095
the right size to study the merging effect, for some of them formed early enough and have time to merge, and some formed late so no merge happens. Though uncertain parameters are involved, we want to emphasize that the observed groups may not be the direct product of the initial fluctuation and merging process plays an important role.
References 1
2 3 4 5 6 7 8
Hickson, P . , De Ollveira, C . M . , Huchra , J . P . et a1 . , Dynamical properties of compact groups of galaxies, ApJ , 1992, 399 : 353. Zepf, S . E . , The frequency of mergers in compact groups, ApJ , 1993, 407 : 448. Xia, X . Y . , Boller, T. , Deng , Z. G . et a1 . , Soft X-ray properties of ultra-luminous IRAS galaxies, Astrophysics Reports , 1997, 21: 166. Zou, Z . I , . , Xia, X . Y . , Deng, Z . G . et al. , IRAS 23532 + 2513 : a compact group including a Seyfert 1 and a Starbunt Galaxy, A&A , 1995, 304 : 369. Barnes, J . , Evolution of compact groups and the formation of elliptical galaxies, Nature , 1989, 338 : 123. Evrard, A . E . et a1 . , Testing origins of hubble sequence, ApJ , 1989, 341 : 26. Ma, E . , Chen, S . , Yu, Y . , An analytic model for multiple merging of galaxies, Acta Astrophysics Sinica , 1998, 18 : 361 . Bardeen, J . M . , Bond, J . R . , Kaiser, N . et al. , The statistics of peaks of Gaussian random fields, ApJ, 1986, 304: 15.
