galaxies in clusters: alignments, formation from pancakes, and tidal forces

Vistas in Astronomy, Vol.26.pp.225-241,1982 0083-6656/82 $0.00 +. 50 Printed in Great Britain. All rights reserved. Copyright © 1983 Pergamon Press Ltd

GALAXIES IN CLUSTERS: ALIGNMENTS, FORMATION FROM PANCAKES, AND TIDAL

FORCES

Paul S. Wesson

Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1

ABSTRACT

Evidence relevant to the question of whether or not galaxies in clusters are aligned is reviewed. It is concluded that there is evidence in favour of alignment in some clusters. The most co,only discussed mechanism for the origin of such alignments is that of galaxy formation from cluster pancakes, as proposed by Doroshkevich, Sunyaev and Zel'dovich. How- ever, an alternative mechanism which can in principle lead to alignments, and which in any case is relevant to their secular stability (however produced), is that of tidal forces, as recently discussed by Miller and Smith. These two mechanisms are reviewed. It is concluded that the first is more plausible as an account of the origin of alignments; but that the second has several implications for the dynamics of galaxies in clusters which deserve further study.

i. INTRODUCTION

The three subjects mentioned in the title of this article are related aspects of the

study of galaxies in clusters. The alignment of spiral and elliptical galaxies in a given

cluster, both with other nearby galaxies and the cluster centre, is a subject on which there

is a small but valuable literature. It is valuable because it represents information about

the origin of the galaxies in the cluster and the cluster environment. This applies both

ways: the presence or absence of alignments are both useful data. If, as concluded below,

there is evidence of alignment in some clusters, then two questions arise: Firstly, what

is the most plausible mechanism for the origin of such alignments? Secondly, what can be

learned about the cluster environment from considering the secular stability of such align-

ments? These two questions are interrelated (see the next paragraph), and it is the objec-

tive of the present work to review data relevant to them and hopefully answer them.

The most conunonly discussed mechanism which can lead to alignments between galaxies in

a cluster is (a) galaxy formation from a cluster 'pancake'. [A pancake is a sheet of gas

which fragments to form a population of non-randomly oriented galaxies.] This has been pro-

posed as a model of galaxy formation by Doroshkevich, Sunyaev and Zel'dovich (1974; see also

Zel'dovich, 1978). However, an alternative mechanism which can in principle lead to align-

ments, and which in any case is relevant to their secular stability (however produced), is

that of (b) tidal forces in combination with braking. [In general, an aspherical galaxy

tilts in an effort to minimize the tidal torque acting on it, and if braking is present

this can produce a population of non-randomly oriented galaxies.] This has been discussed

recently as a means of studying the cluster environment by Miller and Smith (1982). The

two hypotheses (a) and (b) may appear to be distinct when considered as mechanisms for the

origin of alignments between the galaxies in a given cluster. But actually they have to be

considered together for the following reason : (a) can be proved valid only if it can be

shown that (b) does not lead to the destruction of any initial alignment over the history

225

226 P.S. Wesson

of the cluster. [However, if galaxy alignments can be explained by tidal forces as in (b),

this does not in itself prove that galaxy formation from a cluster pancake as in (a) is

invalid.] Thus, both (a) and (b) have to be considered when trying to explain the data on

galaxy alignments.

The plan of the present work is as follows. Section 2 is a review of work relevant to

alignments of galaxies in clusters. Section 3 is a short review of the formation of gala-

xies from cluster pancakes. Section 4 is a review of tidal forces on galaxies in clusters,

and represents the bulk of the work. [The discussion in Section 4 is mainly qualitative in

nature, but some of the topics which come up are discussed quantitatively in the Appendix.]

Section 5 is a summary.

2. ALIGNMENTS OF GALAXIES IN CLUSTERS

Previous work has mainly been concerned with whether or not galaxies in clusters are

aligned in any significant way, the latter phrase being understood to refer to the applica-

tion of a specified statistical test.

Hawley and Peebles (1975) have found no evidence of correlation in the orientation of

neighbouring galaxies, but marginal evidence for the alignment of galaxies in the Coma clus-

ter in the radial sense (i.e., along the direction to the cluster centre). Thompson (1976)

has studied the orientation of galaxies in 8 rich clusters, and has found that elliptical

galaxies in Coma seem to be aligned radially, and that galaxies in A2197 are also systema-

tically aligned. The latter cluster, it should be noted, is markedly aspherical in shape,

and this has suggested to Thompson that the alignments in that case should be explained in

terms of pancake formation rather than the action of tidal forces. Carter and Metcalfe

(1980) have found that eD galaxies are aligned with the axes of the clusters in which they

are located, a fact which was first noted by Sastry (1968). Adams, Strom and Strom (1980)

have found that the axes of galaxies located in linear clusters tend to be either parallel

or perpendicular to the axes of the clusters. MacGillivray et al (1982) have found that

nearby galaxies have a tendency to be aligned with the plane of the Local Supercluster.

Kapranldis and Sullivan (1983), on the other hand, have found no systematic alignments of

spiral galaxies in the Local Supercluster, though they admit there are some significant

departures from randomness. Helou and Salpeter (1982) have reached a similar conclusion

concerning galaxies in the Virgo cluster, where they have found that galaxy spin vectors

are not aligned, but neither are they completely random.

Observational work on alignments, as summarized above, leads to the conclusion that

there is evidence in favour of weak alignment in some clusters. The evidence is most

convincing in the case of aspherieal clusters, and only marginal at best in the case of

spherical clusters like Coma. Where they are present, alignments appear to be of the type

where the galaxies are oriented preferentially either along or perpendicular to the direc-

tion to the cluster centre.

3. FORMATION OF GALAXIES FROM CLUSTER PANCAKES

Reviews of this theory of galaxy formation have been given by Doroshkevich, Sunyaev

and Zel'dovich (1974) and Zel'dovich (1978). A short account would therefore seem to be

appropriate here. The theory is based on the hypothesis that large clouds of gas in the

early Universe became unstable and collapsed anisotropically, producing more or less flat

Galaxies in Clusters 227

structures like pancakes. In a given pancake, smaller scale instabilities later produced

galaxies. The latter would clearly have been formed with orientations which, at least

initially, were non-randomly distributed.

The formation of clusters and galaxies from pancakes is mathematically viable. There

is also some evidence in favour of it from recent observational work. Firstly, Leir and

van den Bergh (1977) and Binggeli (1982) have found that many rich clusters are aspherical.

This is highly suggestive, and incidentally creates a headache for theoretical workers, who

have traditionally assumed that rich clusters are spherical in order to avoid the serious

analytical complications which arise otherwise. Secondly, JSeveer and Einasto (1978) have

reviewed data on the large-scale distribution of galaxies, and have concluded that the real

Universe has a cell-like structure. The structure can be described roughly as consisting of

elongated, supercluster-sized aggregates which intersect in a kind of lattice, with large

voids in between. The superclusters themselves contain clusters and groups of galaxies.

There is now considerable observational activity in progress which appears to be confirming

that the real Universe does indeed have a cell-like structure, a situation which conflicts

somewhat with the uniformity traditionally assumed in deriving cosmological models.

The two results mentioned in the preceding paragraph are clear-cut data in support of

the pancake theory of Doroshkevich, Sunyaev and Zel'dovich, and their importance should not

be underestimated. Galaxy formation from cluster pancakes is thus a very plausible mechan-

ism for producing alignments, at least initially, between galaxies in clusters.

4. TIDAL FORCES ON GALAXIES IN CLUSTERS

It is useful to recall that there are two main reasons for studying tidal forces on

galaxies in clusters as they relate to alignments. It is widely believed that alignments

of galaxies in clusters are the results of a pancake mode of formation. But before observed

alignments can be interpreted as evidence in support of the pancake theory, it is necessary

to show that initial alignments would not be destroyed or altered significantly by tidal

forces acting over the history of the cluster. Another reason is that recent computer simu-

lations of tidal forces between galaxies in a cluster and the cluster core have shown that,

under some circumstances at least, tidal forces in combination with braking can themselves

produce alignments (see below). Thus, tidal forces are relevant to the secular stability

and maybe also the origin of alignments.

Tidal forces are inherently complicated, with aspects that are often difficult to

differentiate. In order to give a review which is as intelligible as possible, the present

section is mainly qualitative in nature, while more quantitative and detailed results are

deferred to the Appendix. Simplifying assumptions, which are believed to be reasonable, are

made use of in several places. The material of this section is split up into several sub-

sections. These are arranged in an order which is designed to lead as logically as possible

to an understanding of how effective tidal forces are as they relate to alignments.

4.1 Types of Tidal Force

An aspherical galaxy in a cluster may be acted upon by tidal forces of different types.

A given 'subject' galaxy may in fact experience one or more of three different types of in-

teraction, which can be categorized as follows.

228 P.S. Wesson

(i) Close encounters with other galaxies at low relative velocity. Such encounters can

lead to significant changes in both the spin and shape of the subject galaxy. [For example,

the subject galaxy may lose part of its spin angular momentum and be 'fattened up' from a

disk to an ellipsoid.] Many different sorts of model galaxy have been produced in computer

simulations of such encounters. However, they are quite rare: ~r the core of the Coma clus-

ter, the probability of a collision or near miss occurring at a relative velocity of

500 km s -I or slower is less than 1% (Farouki and Shapiro, 1981, p.38). Thus, these encoun-

ters are so rare that they may be neglected in a discussion of alignments.

(ii) Remote encounters with other galaxies at high relative velocity. In a typical

rich cluster, the mean distance between galaxies is about i00 kpc, and they move with ran-

domly-directed velocities whose mean magnitude is about i000 km s ~I (see below). Remote

galaxy/galaxy encounters at high relative velocity are therefore common.

Farouki and Shapiro (1981; see also Farouki and Shapiro, 1980) have studied such en-

counters bymeans of computer simulations. In their work, a spiral galaxy is modelled as

a thin disk embedded in a halo of equal mass. [It should be noted that the halo used in

these simulations serves to suppress a bar-like instability of the disk, and is not to be

confused with the massive halo discussed in Section 5 below.] When the passage of such a

galaxy through the core of a rich cluster and the associated encounters with other galaxies

are simulated, two things become apparent. Firstly, the spin of the subject galaxy is not

significantly affected either in magnitude or direction: a galaxy which experiences 20

encounters over a period of about 109 yr has its spin axis deflected by about 3 ° or less.

Secondly, the shape of the subject galaxy is not significantly affected: the encounters

cause the disk to thicken progressively, especially in the outer regions; but even after

109 yr, the ratio thickness/radius is only I/i0 at a distance of 15 kpc from the centre of

the subject galaxy. Thus, neither the spins nor the shapes of spiral galaxies are signifi-

cantly affected by such galaxy/galaxy encounters over a period (= 109yr) comparable to the

time it takes a given galaxy to cross the cluster of which it is a member.

Encounters of this type also cannot have significant effects in producing alignments

even over longer times of the order of the age of a cluster (~ 1010yr). The reason is sim-

ple, in that while such encounters are common, they are random in direction. They therefore

cannot lead to any systematic alignments along the members of a population of galaxies, and

in particular cannot produce alignments with respect to the cluster centre as discussed in

Section 2.

(iii) Interactions of galaxies with the cluster core considered as a whole. A given

galaxy at some distance from the centre of the cluster in which it is located will interact

with all the galaxies interior to that distance as if they constituted a core, with a mass

equal to the combined masses of the galaxies concerned. Although it is not a good appro-

ximation for some clusters (see Section 3), it is necessary in order to make the problem

tractable to assume that the cluster concerned is spherical. This assumption implies that

the interaction of a given galaxy with the cluster core can be modelled as an interaction

with a point mass located at the centre of the cluster. Of course, the subject galaxy, like

the other galaxies in the cluster, is moving. This implies that the strength of the inter-

action with the cluster core varies with time depending on the distance from the cluster

centre and the mass enclosed within this distance. However, irrespective of the location of

the subject galaxy in the cluster of which it is a member, the interaction with the cluster


core is always present. Also, the direction of the interaction is effectively constant over

periods less than the crossing time (z 109yr); and may be approximately constant over even

longer periods in certain circumstances (see below). In other words, these interactions

may be of importance in explaining galaxy alignments which are with respect to the cluster

centre.

Miller and Smith (1982) have studied such interactions by means of computer simulations.

In their work, a galaxy is modelled as a bar, which has zero translational velocity but

rotates in the gravitational field of the cluster as a whole. They have found that the bar

rotates fastest when oriented along the line to the cluster centre (where the gravitational

potential energy is a minimum) and slowest when oriented perpendicular to the line to the

cluster centre (where the gravitational potential energy is a maximum). Also, if dissipa-

tion is present, the motion is braked and the bar comes to rest in the former position. It

should be noted that the varying speed of rotation of the bar is a response to tidal forces,

which act on the bar because it is an object of finite size located in the gravitational

field of the cluster as a whole. If dissipation is neglected, the bar tends to spend less

time near the radial position than near the perpendicular position, so a collection of such

objects would show a degree of alignment in the perpendicular direction. However, dissipa-

tion cannot be neglected over long periods of time (~ 109yr), since there is a tendency for

kinetic energy of rotation of the bar as a whole to be converted to kinetic energy of random

motions of its constituent particles, so a collection of such objects would be braked and

eventually show a degree of alignment in the radial direction. [The tendencies to produce

perpendicular and radial alignments in the simulations of Miller and Smith refer to a situa-

tion where the bar rotates initially about an axis which lies at right angles to the llne

to the cluster centre: other positions for the rotation axis would introduce precession

effects and complicate the motion, as discussed in Section 4.3 below.] Thus, it appears

that galaxy/cluster core interactions can significantly alter the dynamics of aspherical

galaxies, and in particular cause their rotations to be braked, over times of order 109 yr.

The alignments which result from this mechanism are of the types observed in real clusters

and discussed in Section 2.

Having discussed the three maln types of tidal interaction in clusters, some interim

conclusions may be drawn about their relevance to alignments. The galaxy/galaxy interac-

tions of types (i) and (ii) are ineffective in producing alignments; and although it has

not been demonstrated explicitly, it is reasonable to infer from the computer simulations

that have been carried out that they are also ineffective in destroying any initial align-

ments. The galaxy/cluster core interactions of type (iii), on the other hand, can in

principle produce alignments; and the results of Miller and Smith are of great potential

significance as regards this topic.

These conclusions depend largely on computer simulations. This is certainly a powerful

technique, but it should be appreciated that in the context of tidal forces on galaxies it

has so far only been applied to a very restricted class of problems. [A review of tidal

interactions between galaxies (Wesson, 1982a) and a discussion of unsolved problems to do

with them (Wesson, 1982b) are available.] In particular, the simulations of Miller and

Smith described above have given results on a very restricted class of motions. It is

clearly desirable to have a more general result which can be used to investigate the impli-

cations of galaxy/cluster core tidal forces for alignments. Such a general result may be

obtained by considering the magnitude of the tidal torque involved, and the response of a

230 P.S. Wesson

galaxy to this torque.

4.2 The Tidal Torque

It is prohibitively difficult to calculate the tidal torque on a galaxy of arbitrary

shape due to a perturbing mass of arbitrary type. Therefore a simple case can be considered

as an approximation, as illustrated in Fig. i.

This simple case involves a subject galaxy of mass M 1 which is acted upon by a torque

of magnitude z due to a perturbing mass M2, where the distance between the two masses is

assumed to be large compared to the size of the subject galaxy so that the perturbing mass

may be taken to be a point. For the sake of definiteness, the subject galaxy is taken to be

a spiral, which is modelled as a disk of radius R I that is thin. [This latter restriction

is actually not necessary in order to carry out a calculation like that outlined in the

caption to Fig. i. The torque may be calculated for any subject galaxy that has a shape

specified by two semi-axes a, b which are unequal. In that case, (i) below holds up to a 2

factor of order unity, but with R I replaced by Ja 2- b21. See Peebles (1969) and Wesson

(1982a,b). Thus, the torque may be calculated for any flattened galaxy, be it spiral or

elliptical. However, the calculation is simpler for the former than for the latter, so

the result for a spiral galaxy is used here.] The distance from the centre of the subject

galaxy to the perturbing mass is r; and the angle between the plane of the galaxy and the

line from its centre to the perturbing object is e. The magnitude of the torque is

BGMIM2R~sin20 T = (i)

r 3

Here, G is the Newtonian gravitational constant, and B is a numerical constant which for the

model being considered here has the approximate value B = 0.18 (see the caption to Fig. I).

It can be noted that Peebles (1969) has used a relation similar to (i) to study the origin

of the angular momenta of proto-galaxies. Peebles' relation agrees with (i) except that

he has B = 0.3 instead of the ~ = 0.18 noted above. This difference is presumably due to

differences in the models assumed; but it can be mentioned that (i) with B = 0.18 leads to

an equation of motion for a subject galaxy with negligible spin angular momentum which, as

shown in the Appendix, is numerically in agreement with that of Miller and Smith (1982).

The value 8 = 0.18 would therefore appear to be reliable; though a small uncertainty in its

value of the size just noted is in any case of little consequence considering the idealistic

and therefore inexact nature of other assumptions involved in the calculation leading to (i).

While (i) has been derived as a relation describing the interaction of a disk with one

perturbing object, it is possible to consider a sum of torques as describing the interaction

of a disk with several objects. Also, it is possible to consider the torque due to one per-

turbing object which has different positions with respect to the disk at different times.

Such cases occur below, and the torques involved will be referred to as having different

directions (meaning that they tend to tilt the disk in different directions: see the caption

to Fig. i). In the next subsection, (I) will be applied to the torque on a subject galaxy

due to the core of a cluster. It is therefore convenient to introduce here a model for the

cluster.

The cluster model consists basically of a spherical collection of mass points. This

is a reasonable approximation for certain rich clusters. For the present, consider the


jIR S 2

Mll ~ F'

F

M 2 --0

Fi$. i. To illustrate the derivation of the magnitude of the torque T acting on a disk galaxy of mass M 1 due to a perturbing mass M 2. The distance from the centre of the disk 0 to the mass M 2 is r, and e is the angle between the plane of the disk and the line from 0 to M 2. [Owing to the symmetry of the problem, only two coordinates, namely the three-dimensional distance r and the angle 0, are needed to specify the magnitude of the torque. The action of the latter is such as to tend to cause rotation about a diameter at right angles to the line from 0 to M2, and this can be regarded as defining the direction of the torque. A third coordinate, namely an azimuthal angle, would be needed to specify this direction, but this will not be introduced here.] The diagram is not to scale, and represents a disk galaxy seen in cross-section. It is assumed that the galaxy is thin, with a geometrical radius RI; and that it can be modelled dynamically as a dumbell consisting of two masses MI/2 each located a distance R, from O. It is also assumed that r is so large that the forces F and F' on the masses MI/2 can be taken to act parallel to the line from 0 to the perturbing mass. By Newton's law, F =GM2(MI/2) (r- R, cos@) -2 and F'= GM2(Ml/2)(r+R,cose~ 2. Resolving these forces perpendicular to the plane of the galaxy, taking their difference, and multiplying this by R, gives the magnitude of the torque tending to cause rotation about 0 as T =GMiM2RSsin28/r3. By calculating the centre of ~ravity of half a disk of uniform density, one finds R, =4RI/3~. Thus T =BGMiM2R~sin2e/r~, where B = 16/9~ 2 =0.18 is a constant.

232 P.S. Wesson

situation where the subject galaxy is located at a certain distance R c from the centre of

the cluster, and where the cluster mass interior to this distance is M c. [For the present,

attention is being focussed on calculating the torque on the subject galaxy at a specific

place and time. In the Appendix, the situation is considered where a galaxy with negligible

spin angular momentum moves from place to place in the cluster over a long period of time.]

If R c and M c refer to the core of a typical rich cluster, appropriate numerical values are

R c =0.5 Mpc, Mc= ix 1014M e. The mean density of the core is $~ 3Mc/4~R ~= 1.5 ×10 -26 g cm -3.

Let the mass of the core be in the form of galaxies where the mean galaxy mass is Mg. If

Mg= 2 x 1011M~, there are approximately Mc/Mg =500 galaxies in the cluster core. Their mean

number density is i000 Mpc -3, and their mean distance apart is £ E (1000 Mpc-3) ~/3 =i00 kpc.

In the next subsection, this cluster model will be used in combination with (I), and

M c will be substituted for M 2. Before proceeding, however, it is as well to check that it

really is the torque due to the cluster core, rather than the torque due to other nearby

galaxies, which exerts the larger influence on the subject galaxy. [The computer simula-

tions described in the previous subsection do not show this explicitly. The torques asso-

ciated with interactions of type (ii) change direction rapidly and randomly: the timescale

is £/v = 108yr, where v = i x 103 km s -I is the mean dispersion velocity of galaxies in the

cluster. The torque associated with the interaction of type (iii) changes direction slowly

and in a sense determined by the position of the cluster centre as seen by the subject

galaxy: the timescale is tcr =2Rc/V = 109 yr, where tcr is the crossing time. However,

while these two different types of interaction clearly involve different effects, it is not

apparent which involves the larger tidal torque.] That is, it is as well to check that the

torque rgc acting on the subject galaxy due to the cluster core really is larger than the

torque Tgg acting on the subject galaxy due to other nearby galaxies.

The torque Tg c is given by (i) with M I = Mg, M 2 = M c and r = R e . The torque Tgg can

be estimated by considering the nearest neighbouring galaxy to the subject galaxy. [This is

Justifiable: the torque (i) falls off rapidly with distance, so much of the torque is in

fact due to the nearest neighbour. However, the torques due to other nearby galaxies will

act in directions different to that associated with the nearest neighbour, so considering

only the latter actually overestimates the magnitude of the torque due to other galaxies.

Taking this last comment into account increases the right-hand side of (2) below, and bols-

ters the remark made at the end of this paragraph. See Wesson (1982a) for a discussion.]

The torque Tgg is then given by (i) with M I = Mg = M 2 and r = £. Although the two torques

will not in general agree in direction, an estimate of their relative magnitudes is given by

= = 4 (2) Tgg

Here, the first relation is from (I) and the second uses numerical data from above. This

calculation is admittedly approximate. It neglects, for example, the possible effect of a

nearby clump of galaxies, and therefore is only applicable to clusters like Coma that have

a relatively smooth distribution of galaxies. But while approximate, (2) does show that

it is indeed the cluster core rather than other nearby galaxies which exerts the larger

influence on the subject galaxy.

4.3 The Response to a Tidal Torque

Since the torque (I) is largest for the cluster core, an aspherical galaxy which


responds to such a torque might be expected to alter its orientation in space in a way de-

termined by the direction to the cluster centre. This holds for any aspherical galaxy, since

while (i) was derived for a spiral, a similar expression can be derived for an elliptical,

as noted above. However, this does not automatically imply that alignments will be set up,

since the establishment of these latter depends in a complicated way on the response of the

galaxy to the tidal torque.

When an aspherical galaxy is acted upon by a tidal torque, it tends to reorientate it-

self so as to try to reduce the magnitude of the torque it feels. There are two positions

for which the torque is zero: 0 =0 and ~/2 in (i) above. The first position is radial

alignment, along the direction to the cluster centre. The second position is alignment

perpendicular to the direction to the cluster centre. Of these, the former involves a

lower gravitational potential energy (see Section 4.1) and presumably is the position of

stable equilibrium. If a galaxy is free to orientate itself, it will therefore move towards

a position of radial alignment; and if dissipation and braking are present (so it does not

overshoot the equilibrium position), the galaxy will eventually attain a state of radial

alignment. However, there is an important condition which has to be satisfied in order that

this alignment process occur in a real cluster.

A galaxy in a real cluster moves about, and this means that the effective direction of

the torque due to the cluster core changes on a timescale of the crossing time tcr. Let the

galaxy's timescale of response to the torque be tresp. Then clearly, tresp< tcr is a con-

dition which must be satisfied if alisnment is to be produced and maintained. This is a

general condition, and the rest of this subsection is devoted to an investigation of whether

or not it is satisfied in real clusters.

The question of whether or not a tidal force can align a galaxy with respect to the

direction to the cluster centre really boils down to calculating the response time. Good

alignment can be expected if tresp < tcr , marginal alignment if tresp = tcr , and bad or no

alignment if tresp> tcr. Unfortunately, while it is not difficult to state these criteria,

it is very difficult to calculate tresp analytically for a galaxy with arbitrary spin, shape

and orientation. To make progress, it is useful to consider first a simple case, and second

a more complicated (though still special) case.

A simple case is that of a galaxy with negligible intrinsic spin and a shape specified

by two semi-axes of which one is significantly longer than the other. This is a good appro-

ximation for an elliptical galaxy. [It now appears then elliptical galaxies have consi-

derably smaller spin angular momenta than spirals. See, e.g., Richstone (1980) and

Schwarzschild (1982).] The torque which acts on the galaxy due to the cluster core then

has a form similar to that given by (i) above. Let it be assumed for the sake of simpli-

city that the galaxy does not deform under the action of the torque (this assumption will be

commented on below). Then the equation of motion which describes the response of the galaxy

to the torque is of the form

d28 -kGMcsin20

at 2 R3 (3) c

Here, k is a positive constant of order unity, and the minus sign indicates that the torque

represents a restoring force towards 0 = 0. The mean density of the cluster core is

Mc/R ~. Therefore, (3) suggests that the timescale of response of the galaxy to the

234 P,S. Wesson

torque due to the cluster core is tresp = (G~)-I/2=tcr, to order of magnitude.

Equation (3) describes a kind of tumbling motion. It is the kind of motion which an

elliptical galaxy in a cluster is expected to undergo, and the kind of motion studied by

Miller and Smith (1982). It is straightforward to derive equations of the form of (3) valid

for realistic models of clusters, and integrate them numerically (see the Appendix). When

this is done, the approximate result derived in the preceding paragraph is confirmed: namely,

tresp = tcr. Also, the results of Miller and Smith (1982) for times shorter than 109 yr are

confirmed. Of course, the latter authors implicitly included braking in their numerical

simulations, an effect which is significant over times longer than 109 yr, but which is not

included in the analytical relation (3). However, the presence of braking can only increase

the effective response time. Thus, for an elliptical galaxy in a cluster, tresp ~tcr. The

conclusion can therefore be drawn that tidal forces are expected to produce marginal or no

alignment of elliptical galaxies in a rich cluster.

A more complicated (though still special) case is that of a galaxy which spins about

an axis of symmetry and has the form of a disk. This is a good approximation for a spiral

galaxy. The torque which acts on the galaxy is then given by (i) above. But while (i) is

a simple and exact expression for the torque, the resulting motion is very complicated and

cannot be computed exactly. The reason is, of course, that the intrinsic spin of the galaxy

'resists' the action of the torque, resulting in the phenomenon of precession. The motion

in the general case consists of three parts: nutation (a nodding motion, which may be prime-

val in origin, but which is not discussed here); free precession (a wobbling motion, which

again may be primeval in origin, and which can have significant observational consequences);

and forced precession (also a wobbling motion, which is due to the galaxy/cluster core

interaction). It is instructive to consider briefly the implications of the last two parts

of the motion.

Free precession is a wobbling motion under the action of no external torques. It is a

motion in which the total angular velocity vector precesses around the axis of symmetry. The

period of the free precession is 2~/(~cos~), where ~ is the total angular velocity and ~ is

the offset angle between the total angular velocity vector and the axis of symmetry (the two

would coincide if there were no precession). Thus, if the offset is small as expected for a

real spiral galaxy, the free precession period is close to the spin period, which for the

Milky Way is approximately 3× 108 yr. Even a small offset can have observable consequences,

though, since it introduces a velocity component for the disk of the galaxy perpendicular to

its plane. The mean magnitude of this velocity may be estimated approximately by dividing

the amplitude of the motion perpendicular to the plane of the disk by the free precession

period. For reasonable parameters (R I = i0 kpc, ~ = I0 ° and a free precession period of

3 ×108 yr), its size is of order i0 km s -I. Velocities of this order can be measured by

radio observations, and it is tempting to try to identify the effects of free precession in

observational data on certain galaxies (see the following and references therein: Baldwin,

1978; Huchtmeier and Witzel, 1979; Krumm and Salpeter, 1979; Newton, 1980a,b). However, this

is a risky procedure because effects such as disk warping, which exist for real galaxies,

have been neglected in the superficial discussion just given. Lynden-Bell (1965) has dis-

cussed in more detail the effects of free precession on the structure of the Milky Way.

Forced precession is a wobbling motion under the action of the torque due to the clus-

ter core. Following Thompson (1976), an approximate estimate of its period may be made by


considering the torque T due to the core of a rich cluster, and a galaxy with a spin period

t s and a moment of inertia I. Then by dimensional analysis, the period of forced precession

is expected to be of order I/(Tts). Substituting for T from (i), assuming e = ~/4 and replac-

ing M2/r3 by the mean density of the cluster core ~, and substituting I = MIR~, shows that

the period is of order (Ggts)-l. For ~ = ix i0-27g cm -3 and t s = 3 xl08 yr, this gives the

period of forced precession as 5 ×1010yr. This result is only approximate, but does show

that the period of the motion associated with the cluster core torque is comparable to the

age of the cluster itself.

The general motion of a spinning disk galaxy under the action of a torque due to the

cluster core is governed by Euler's equations. These are three relations involving the

angular velocities about three axes, their rates of change and the three components of the

torque (see, e.g., Fowles, 1962). It is not possible to find an analytic solution of

Euler's equations in the general case. But a computer program which integrates them numeri-

cally has been developed by Smedley and Wesson, and some preliminary results of this may be

mentioned here (fuller details will appear elsewhere). The program models the subject

galaxy as a collection of mass points; and the cluster core as another collection of mass

points (representing galaxies) which are distributed within a sphere according to a random

scheme. One thing which emerges from the program is that the subject galaxy tends to deform

over periods longer than 109 yr. This might have been expected, and is presumably due in

part to the fact that precession affects the elements of the disk individually rather than

coherently. Another thing is that, while it has not been possible to follow the motion

through a complete cycle, it is clear that the response time of a spinning galaxy is longer

than the crossing time for the cluster core on whose edge it is located. This might also

have been expected, since the response time is presumably comparable to the period of forced

precession. Thus, for a spiral galaxy in a cluster, tresp > tcr. The conclusion can there-

fore be drawn that tidal forces are expected to produce no alignment of spiral galaxies in

a rich cluster.

5. SUMMARY

There is evidence in favour of weak alignments in some clusters, notably of the type

where the galaxies are oriented preferentially either along or perpendicular to the direc-

tion to the cluster centre (Section 2). This can in principle be explained by two hypothe-

ses. (a) Galaxy formation from cluster pancakes. This is supported by the observations that

many clusters are aspharical, and that the real Universe has a kind of cell structure

(Section 3). (b) Tidal forces on galaxies in clusters. This is supported by the facts that

the computer simulations of Miller and Smith (1982) have shown that tidal forces can result

in alignments (Section 4.1), and that the galaxy/cluster core interaction is the dominant

one (Section 4.2). However, it is not supported by an analysis of the response of a galaxy

to the torque due to the cluster core (Section 4.3). The latter shows that the computer

simulations of Miller and Smith, while useful, refer to a very special case in which dis-

persion velocities are neglected and the galaxy is restricted to a tumbling type of motion.

In more realistic cases, it has been seen that tidal forces are expected to produce marginal

or no alignment of elliptical galaxies, and no alignment of spiral galaxies, in a rlch clus-

ter. Thus, galaxy formation from cluster pancakes is a more plausible explanation than

tidal forces for alignments.

The reliability of this conclusion is good but not perfect, and it is only fair to

remark on several points which require clarification. Firstly, observations concerning

236 P.S. Wesson

alignments of galaxies in clusters are still sketchy. In view of the comments made above,

it may be that there are two different types of cluster: aspherical ones, which formed from

pancakes, and in which galaxies were aligned at formation, and have maintained their align-

ment because dispersion velocities are small and disruption has therefore been avoided; and

spherical ones, which formed by symmetric collapse, and in which galaxies were not aligned

at formation, and have not become so because dispersion velocities are large and tidal forces

are relatively ineffective. If this suggestion is correct, alignments should only be notice-

able in aspherical clusters, while the marginal evidence for alignments in spherical clusters

must be spurious. The latter possibility has indeed been mentioned by Hawley and Peebles

(1975). Secondly, the argument for the ineffectiveness of tidal forces as a means of pro-

ducing alignments rests largely on a discussion of timescales, and is therefore indirect.

While this is acceptable, it would be interesting to set up a computer simulation of a more

realistic sort than those developed hitherto, which could be used to study tidal and other

processes in a cluster in a direct manner. Thirdly, it has been assumed implicitly above

that the dynamical shapes of galaxies are similar to their geometrical shapes as determined

by optical observations. If galaxies have dark halos with masses an order of magnitude

greater than their visible parts, this assumption becomes invalid. In particular, if gala-

xies have massive halos which are spherical, all discussion about tidal forces automatically

becomes null and void. Conversely, if galaxies have massive halos which are aspherical,

looking for the effects of tidal forces in principle provides a way of investigating the

nature of such halos.

The three problems outlined in the preceding paragraph are the major ones requiring

further work. It ishoped that the review given here will encourage other workers to tackle

these and related problems.

ACKNOWLEDGEMENTS

Thanks are due to W.Y. Chau and R.N. Henrlksan for comments. Thanks are also due to

members of the 6th Kingston Meeting on Theoretical Astrophysics for showing me that over-

simplification can lead to confusion. This work was supported by the Natural Sciences and

Engineering Research Council of Canada.

APPENDIX: TUMBLING GALAXIES IN CLUSTERS

In what follows, the motion of an aspherlcal galaxy with negligible intrinsic spin

under the action of the tidal torque due to a cluster core is studied, a motion which may

be termed tumbling. There are two specific objectives to this study. Firstly, to confirm

the results of Miller and Smith (1982) quoted in Section 4.1 of the main text. Secondly, to

confirm by numerical integration of the equations of motion the result tresp = tcr used in

Section 4.3 of the main text.

It was mentioned in Section 4.2 of the main text that the torque due to a perturbing

mass M 2 has the form (i) for any galaxy that has a shape specified by two semi-axes a, b

which are unequal. For simplicity, it can be assumed that one axis is significantly longer

than the other, so (i) can be used as it stands. This means, in real terms, that the galaxy

being studied is a flattened elliptical. [Actually, (i) is a reasonable approximation for

most elllpticals, since as mentioned in the caption to Fig. i, it is based on the assumption

that the galaxy can be modelled as a dumbell; and a dumbell is a good approximation to the

dynamical, if not the geometrical, shape of most elllpticals.] The torque and associated


equation of motion can be evaluated for two realistic cases of the cluster model outlined in

Section 4.2 of the main text. These two cases are those of a homogeneous and an inhomogen-

eous matter distribution. In either case, the subject galaxy is allowed to be located any-

where within the cluster core. This reflects the fact that a real galaxy moves in the

cluster of which it is a member by virtue of its dispersion velocity. [The dispersion

velocities of galaxies in the core of a cluster should not be confused with possible infall

velocities in the outer regions of a cluster, as reported for the case of Coma by Capelato

et al (1982). The former may take galaxies through or near the centre of a cluster, but

are overall randomly directed, and arise because of a virial balance between the kinetic and

gravitational energies in the core. The latter are systematically radial, and arise because

of the non-virial nature of the dynamics in the outer region, where cluster formation pro-

cesses may not be complete. Only dispersion velocities are considered in the present account,

since even if infall velocities exist, they are of significance only outside the core region

of interest here.] It should be mentioned that dispersion velocities were neglected in the

work of Miller and Smith (1982). On the other hand, dissipation and braking are neglected

in the present account, whereas they were included in the work of Miller and Smith (1982).

A more realistic investigation would, of course, include both effects. However, this would

be a major undertaking in terms of setting up a computer model which could accommodate them,

whereas the objective of this Appendix is to obtain an approximate idea of the motion in

short order.

The action of the torque given by (i) of the main text causes the angle 8 of Fig. i to

change with time. (The angle e is that between the plane perpendicular to the axis of lar-

gest moment of inertia of the galaxy, and the line from the galaxy centre to the cluster

centre.) While the plane of the galaxy may at any instant be moving towards or away from

e =0, the torque always tries to return it to the e =0 position. The motion is thus like

that of a pendulum; and since it is a motion about a diameter, it may be termed tumbling.

Like a pendulum, the galaxy may have one of two different modes of motion: oscillation,

meaning a bounded motion (about e =0); and rotation, meaning unbounded motion. For both

modes, the equation of motion may be obtained from (i) in the usual way, giving

d28 -46GM2sin28

dt 2 r3 (AI)

Here, the minus sign indicates that the torque represents a restoring force towards @ =0.

The constant 6 will henceforth be taken to have the value mentioned in the main text, namely

6 = 0.18. The equation (AI) may be integrated numerically for a galaxy in a cluster when M 2

is identified with the mass interior to radius r, so 0 <M 2 <M c and 0 <r <R e . As mentioned

above, parameters as specified for the cluster model outlined in Section 4.2 of the main

text will he employed, so M e =i x l014M e and R c =0.5 Mpe. The integration will be carried

out for two cases of interest for real clusters, namely those where the total mass M e is

distributed within the volume of radius R c in a homogeneous and an inhomogeneous manner.

A homogeneous cluster has a density $ = 3M /4~R 3, which means (AI) can be written as c c

d2e - - = -3.0 G~ sin 28 (A2) dt 2

This holds for any homogeneous cluster which is spherical; and shows that the angular accel-

eration of the disk does not depend on location as the galaxy moves inside the cluster. For

the purpose of numerical integration, it is convenient to write (A2) in another form by

JPVA 26:3 - F

238 P.S. Wesson

"/T 3.0

2.5

~ 2 .0

"TF/27 '-- 1 . 5 - -

1.0

0 .5

0 0

_ I 5 .5 i _

- 3 . 0 9 8 -

0 . 5 1.0 1.5 2 . 0

T

Fi 8. 2. A numerical integration of equation (A3) in the form d2e/dT2=-4.8 sin2e. Here, T Et/tcr is the time (t) measured in units of the cluster crossing time (tcr). The noted equation is unchanged if e is replaced by -e, so lel is plotted. The two constants required to determine. 101 = Ie(T)lexplicitly are fixed by the boundary conditions 10(r=0) I E e o and le(T=0)] ~ 0o, where the illustrated curves all have 0 o =0 but different values for e o as noted in the plot. [This latter parameter is just the angular velocity when the plane of the galaxy passes through the line from the galaxy centre to the cluster centre: see Fig. I.] The curves fall into two classes, separated by the critical curve eo =3.098. The critical curve represents a galaxy whose plane passes through e =0 with just enough kinetic energy to reach [0 I = ~/2, where it comes to rest with this plane perpendicular to the line from the galaxy centre to the cluster centre. Curves lying below the critical curve represent cases where the energy is less than the critical energy, so lel < 7/2, and there is oscillation about e = 0. Curves lying above the critical curve represent cases where the energy is greater than the critical energy, so lel is unbounded and there is rotation.


assuming the cluster is in virial equilibrium. The kinetic term in the virial theorem is

2T =Mc v2, where v =2Re/tcr is the mean dispersion velocity and tcr is the crossing time of

galaxies in the cluster. The gravitational term in the virial theorem is W =-3GM~/SR c.

Substituting these terms into 2T+ W •O gives v 2 =3GMe/5R e or GMc/R ~ =20/3t~r , which means

(A2) can be written as

d2O -4.8 sin28 2 (AS)

dt 2 = tc r

This equation, it can be noted, agrees numerically with that of Miller and Smith (1982,p.68),

verifying the work of the latter authors. (A3) holds for any homogeneous cluster which is

spherical and obeys the virlal theorem. [For parameters as specified in the cluster model,

v = 740 km s -I = 1.4 ×109 yr and the interaction time between galaxies is ~/v= 1.4×108yr.] , tcr

The assumption of virial equilibrium is, of course, open to dispute. It is not the inten-

tion to argue that clusters are indeed in virlal equilibrium; but this assumption is conven-

ient, since it leads to an equation which is simple to integrate numerically, especially

when the time (t) is measured in units of the cluster crossing time (tcr). Fig. 2 illus-

trates a numerical integration of (A3) for a range of reasonable boundary conditions.

An inhomogeneous cluster can be taken to have a density 0 =a/r 2 where a is a constant,

since this is a good approximation for a typical cluster of this type over most of the

radius r (van den Bergh, 1977; see also Press and Davis, 1982). This means (AI) can be

written as

d20 -9.1 Gasin28 = 2 (A4)

dt 2 r

This holds for any inhomogeneous cluster which is spherical and has a density of the noted

type; and shows that the magnitude of the angular acceleration of the disk becomes larger

as the galaxy moves nearer the cluster centre. For the purpose of numerical integration, it

is necessary to know r =r(t) in (A4). Consider the situation where the galaxy is momentar~y

at rest at the edge of the cluster core (r =Rc) , and moves radially inwards from there as

the time t increases. [It is of course unrealistic to assume that the galaxy moves through

the exact centre of the cluster, or that the density law noted above holds all the way to

the centre; but the situation being considered is nevertheless a good approximation for the

motion of the galaxy when the latter is not too near the centre of the cluster.] By

Newtonian mechanics, the velocity is given by

dtd-~r =-I8~Ga l°ge(~)l I/2 (A5)

This again is general. For parameters as specified previously, a =i.i ×i022g cm -I. Substi-

tuting this into (A4) and (AS) gives these equations in numerical (c.g.s.) form as

d20 -6.4 ×1015sln28 (A6)

dr2 = r2

[lOge(l'5 ~i024)11/2 d--Lr = -1.3 ×i08 - (AT) dt

These two equations have to be integrated simultaneously. For the purpose of doing this

numerically, it is convenient to change the units from cm and s to Mpe and 109 yr. The

latter unit is chosen for the time because, if virlal equilibrium is assumed, the crossing

time is approximately 109 yr. [The kinetic term as before is McV2 and the gravitational 2 2

=4R~/GM c. For the same term is -GM~/Rc, so the vlrial theorem gives v =GMc/R c or tcr

240 P.S. Wesson

L _

v

0.5 "/T

3 . 0 -

2 . 0 -

1.0

0 M

0 0.1

0.4 !

0.2

R

T

0.3 I

0.3

0.2 I

0.4

0.1 0

40

8.5

2.5

20~

05

Fig. 3. A numerical integration of equations (A6) and (A7) in the form d20/dT 2 = -0.64 x (sin 2e)/R 2 and dR/dT =-l.3[loge(0.5/R)] I/2. Here, T ~ t/lO 9 yr is the time (t) measured in units of the cluster crossing time (i xlO 9 yr); and R Er/l Mpc is the distance from the cluster centre (r) measured in units of the cluster diameter (i Mpc). The noted equations are integrated simultaneously. They are unchanged if e is replaced by -e and are singular for R =0 (T =0.5), so [0l is plotted for half a crossing time. The three constants required to determineJ0 i =le(T) I explicitly are fixed by the boundary conditions [0(T=0)[ ~ 9o' ]e(r=O)[ ~ 0o and R(T=O) ~ Re, where the illustrated curves all have %0 =0 and R o =0.5 but different values for 6 o as noted in the plot. [A test galaxy therefore starts at the edge of the cluster with a plane which momentarily has an angular velocity 6o and is oriented along the line from the galaxy centre to the cluster centre.] As T increases from 0 towards 0.5, a test galaxy moves inwards from the edge of the cluster (R =0.5) towards its centre (R=O). The T-values of the lower abscissa correspond to the R-values of the upper abscissa, and together these define R =R(T) and therefore the translational velocity of the galaxy along a radial line from the edge of the cluster to its centre. Due to the singular nature of the equations for R =0 (T= 0.5), the integrations are stopped at T =0.48. This corresponds to R =0.06, or a distance of approximately 60 kpc from the centre of the cluster. The integrations may be artificially resumed on the other side of the centre by starting them off at T =0.52, R =0.06 with the same slopes as they had at T =0.48, R =0.06. If this is done, the curves lying below and above that for 0o =4 diverge sharply, indicating that galaxies attain large negative or positive angular velocities respectively. This device of stepping over R =0 (T= 0.5) and resuming the integrations may be useful as a model for galaxies that pass near the centre of a cluster.


parameters as before, v = i000 km s -I, tcr = 1.0 ×109 yr and the interaction time between

galaxies is £/v ~i.0 ×108 yr.] With the time (t) measured in units of the cluster crossing

time (tcr), Fig. 3 illustrates a numerical integration of (A6) and (A7) for a range of

reasonable boundary conditions.

The calculations of the two preceding paragraphs and their associated Figs. 2 and 3 are

useful for the following reasons. Firstly, they confirm the work of Miller and Smith (1982).

Secondly, they show that the tumbling motion of an aspherical galaxy with negligible intrin-

sic spin (such as an elliptical) has a timescale comparable to the crossing time of the

cluster in which it is located.

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433. Carter, D. and Metcalfe, N. (1980) Mon. Not. Roy. Astr. Soc. 191, 325. Doroshkevich, A.G., Sunyaev, R.A. and Zel'dovich, Y.B. (1974) Confrontation of Cosmological

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Soc. 198, 605. Miller, R.H. and Smith, B.F. (1982) Ap. J. 25__~3, 58. Newton, K. (1980a) Mon. Not. Roy. Astr. Soc. 190, 689. Newton, K. (1980b) Mon. Not. Roy. Astr. Soc. 191, 169. Peebles, P.J.E. (1969) Ap. J. 155, 393. Press, W.H. and Davis, M. (1982) Ap. J. 259, 449. Richstone, D.O. (1980) Ap. J. 238, 103. Sastry, G.N. (1968) Pub. Ast. Soc. Pac. 80, 252. Schwarzschild, M. (1982) Ap. J. 263, 599. Thompson, L.A. (1976) Ap. J. 209, 22. van den Bergh, S. (1977) Vistas Astr. 21, 71. Wesson, P.S. (1982a) Vistas Astr. 25, 411. Wesson, P.S. (1982b) Astrophys. Lett. 23, i. Zel'dovich, Y.B. (1978) The Large Scale Structure of the Universe (I.A.U. Symposium 79,

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galaxies in clusters: alignments, formation from pancakes, and tidal forces

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