Annu. Rev. Astron. Astrophys. 1979. 17: 135-87 Copyright © 1979 by Annual Reviews. All rights reserved

5. MASS-TO-LIGHT RATIOS OF BINARY GALAXIES

The derivation of masses of binary galaxies rests on several important assumptions. Let us first consider the simplest case of circular orbits. Assume that the galaxies are gravitationally bound, they interact as point masses, the distribution of orbital inclinations and phases is random, there is no intergalactic matter, and there is no dynamical interaction with matter outside the binary. A full discussion of this case is given by Page (1961), for example, but we present the rudiments here. Let M_T be the mass of each component (the two masses are assumed equal), R the spatial separation, and V the total orbital velocity. Then M_T = V² R / 2G. Owing to projection effects, V and R are unobservable. One measures instead v, the radial velocity difference, and r_p, the projected separation. These are given by v = V cos cos , and r_p = R cos , where is the angle between the spatial separation R and the plane of the sky and is the angle between the orbital velocity V and the plane determined by the two galaxies and the observer. Let us define an ``indicative mass'':

(5)

F (M_T) is always less than or equal to the true mass according to the projection factor cos³ cos F_p (, ). To obtain the true mass in the simplest possible way, we can average over some appropriate statistical sample to derive a mean projection factor. Then, in principle, for a large enough statistical sample,

(6)

Page (1952) assumed that for a collection of circular orbits, the angles and are randomly oriented, whence <F_p> = 3 / 32 = 0.295. This assumption is unrealistic because binary galaxies are selected and analyzed on the basis of their projected separation, r_p. Once r_p is specified, is no longer a random variable. The distribution of true spatial separations combines with the observed r_p to make certain values of more likely than others.

Depending upon the true distribution of spatial separations, <F_p> can therefore be a strong function of r_p. Page (1961) later amended his treatment to include this dependence. As a model for the spatial distribution function, he used

(7)

an expression empirically derived by Holmberg (1954). This distribution produces a marked increase in <F_p> as r_p approaches R_max, the maximum spatial separation of binary galaxies. Such a variation in <F_p>, if real, is quite inconvenient. It means we cannot compare indicative mass values F (M_T) at different projected separations without first correcting them for trends in <F_p>. Comparisons between independent data sets thus become much more cumbersome.

An important breakthrough in binary galaxy studies has been the realization that Equation (7) is not correct and that apparently D (R) is a power law over the observable range 20 to 500 kpc. Turner (1976a, b) and S. Peterson (1978), both working from samples of binary galaxies chosen by means of well-determined selection criteria, find that D (r) R^- where = 0.5±0.1. Since a power law has no scale-length, <F_p> is independent of r_p and is given by (Peterson 1978):

(8)

Fortunately, the sensitivity to is not great; over the range = 0.0 to = 1.0, <F_p> increases from 0.212 to 0.295. The fact that the projection factor is independent of radius for a power law makes the analysis much more transparent since radial trends in mass and M / L_B can now be determined immediately from trends in indicative mass.

The use of a mean correction factor <F_p> to rectify the observed values of F (M_T) is unwise for a number of reasons. First, regardless of what criteria are employed to select the sample of binaries, spurious pairs will creep in. Such pairs tend to have large values of v² and hence of F (M_T). Thus <M_T> for the sample is sensitive to the adopted cutoff in v. To avoid this difficulty, one should use a method based on the entire distribution F (M_T) and which weights the observations more or less equally.

Second, the use of a simple mean ignores all information contained in the shape of the distribution F (M_T) and, more particularly, in the bivariate probability density p(v, r_p). As pointed out by Noerdlinger (1975), p(v, r_p) is sensitive to the orbital eccentricities of galaxies in the sample. Turner (1976b) was able to rule out rather conclusively highly eccentric models for binary orbits on this basis.

Finally, the distribution of F (M_T) is highly skewed, owing to the skewness of F_p (, ) itself. F (M_T) peaks strongly at zero, and for circular orbits the median value of F (M_T) is only 11% of the value of <M_T> ultimately derived. Peterson argues that such a skewed distribution is not well represented by its mean.

The use of a simple mean projection factor nevertheless provides a quick and moderately accurate way to compare data sets from various workers and to estimate the effects of different assumptions concerning the geometry of the orbits. From the work of Turner (1976b), who analyzed his data using both a simple mean and a more elaborate technique (see below), we estimate that use of the simple mean yields results accurate to ±20%.

Because of the greater complexity of analyzing noncircular orbits, the general solution analogous to Equation (8) has not yet been derived. If the orbits are eccentric, the angles and as defined in this section are not applicable. Nevertheless, a mean projection factor still exists, call it <>, which reduces to <F_p> in the circular case. Karachentsev (1970) and Noerdlinger (1975) have calculated values of <> for collections of binaries with eccentric orbits. However, in their existing form these expressions share the same drawback as Page's original estimate: they do not take into account the bias introduced into the angle once r_p is specified. To treat the noncircular case, it would seem best at present to rely on Turner's model simulations of binaries with various eccentricities. These show that, for eccentric orbits, the projection factor is smaller and the derived masses therefore larger. If = 0.5, for example, the minimum value of <> is 0.105 for purely radial orbits, compared to 0.261 for circular orbits. Qualitatively this effect is easy to understand: if the orbits are eccentric, one views them preferentially near apogalacticon, where the kinetic energies are lower than average. Therefore the observed F (M_T) are systematically lower than in the circular case.

The success of the binary method clearly depends on being able to identify pairs that are real physical systems. The first discussion of this problem and the resulting catalog of doubles was prepared by Holmberg (1937). Other major lists having historical interest include Vorontsov-Velyaminov's Atlas of Interacting Galaxies (1959), Arp's Atlas of Peculiar Galaxies (1966), and Karachentsev's comprehensive list of 603 pairs (Karachentsev 1972). Recent velocity measurements have been published by Karachentsev et al. (1976), Karachentsev (1978), Turner (1976a), and S. Peterson (1978).

In all, there have been four major studies of masses in binary galaxies. Page's work (Page 1961, 1962) was the standard reference for many years. Assuming Holmberg's relation (Equation 7) and circular orbits, he obtained M / L_B = 0.7±0.9 for spiral-spiral pairs and M / L_B = 46±46 for pairs containing ellipticals and/or S0's (corrected to H₀ = 50 km sec^-1 Mpc^-1). The value for spirals is much smaller than the average of ~ 5 within the Holmberg radius derived in Section 2, while the value for early-type galaxies is significantly larger than our estimates for single galaxies presented in Section 3.

Turner (1976a, b) introduced a new standard of rigor into the study of binary galaxies by selecting a binary sample according to well-defined criteria and using, instead of the traditional mean projection correction <>, a rank-sum test that essentially compared the observed frequency distribution p (v, r_p) with simulated versions of p (v, r_p) for various orbital eccentricities. He was also able to model extended spherical halos surrounding the galaxies. Using the shape of the distribution p (v, r_p), he showed that binary orbits cannot be highly eccentric and that if massive halos are present, they must have radii 100 kpc.

S. Peterson (1978) applied a rank-sum procedure to the distribution F (M_T). His analysis was confined to the case of circular orbits, but his radial velocities, many of which came from 21-cm measurements, were more accurate than Turner's largely optical velocities. His selection criteria also differed from Turner's: his binaries were less isolated but extended to much larger radial separations because no outer angular cutoff was employed.

In Table 4 Turner's and Peterson's results for spiral-spiral pairs are compared. The original values of M / L_B given by these authors have been corrected to our system. For reference, Table 4 also includes median values of v and r_p for the Turner and Peterson samples. We have also reduced the results of Karachenstev (1977, 1978) to our system and included them in Table 4. Unfortunately only fragmentary accounts of Karachentsev's work were available to us, and we have had to combine results from several different papers. Details appear in the footnotes. We regret any inaccuracies introduced by this procedure.

Inspection of Table 4 shows that values of M / L_B derived by various authors differ substantially. Is this discrepancy due to differences in the data sets or to differences in statistical treatment? To answer this question, it is helpful to consider just those values based on the assumption of point masses, circular orbits, and = 0.5. (We have provided an additional entry for Karachentsev's data which converts his results to this case.) We then obtain M / L_B (Turner) = 17±4, M / L_B (Peterson) = 32±11, and M / L_B (Karachentsev) = 5.9±2.7, values which still differ by more than the observational errors. Therefore the differences must be due to the data themselves.

Table 4. Binary mass-to-light ratios in spiral-spiral pairs

Histograms of v (uncorrected for observational errors) for all three samples are quite similar. The median values of v (for v 750 km sec^-1) in Table 4 confirm this fact. Thus, the observed differences in the v distributions do not seem large enough to account for the discrepancy in M / L_B (see below).

On the other hand, the three samples differ markedly in their distribution of projected separation, r_p. Evidently Karachentsev's sample primarily includes rather close pairs, a conclusion supported by the high percentage of interacting galaxies in his sample (Karachentsev 1977). Turner's sample is intermediate, while Peterson's, chosen without any arbitrary cutoff in angular separation, contains many very wide binaries.

It seems to us that these results are consistent with an increase in M / L_B with radius, as would be expected from massive envelopes. This view is supported by the fact that M / L_B for Peterson's outermost pairs (with r_p > 112 kpc) is in good agreement with M / L_B for Turner's sample if massive halos having radii of 100 kpc are assumed. Peterson furthermore finds that M_T increases linearly with r_p out to ~ 100 kpc and then levels off, whereas v is constant out to ~ 100 kpc, and then begins to decline. Although of limited statistical significance (Peterson's radial bins overlap), this behavior is also consistent with the existence of massive envelopes having a limited extent. On the other hand, the same data also show that although M_T increases with r_p out to 100 kpc, M / L_B appears to be approximately constant from 20 kpc to 500 kpc, a result which Peterson takes as strong evidence against massive envelopes.

In our opinion the data are not yet strong enough to take any of these radial trends too seriously, and the global result must be given highest weight. Taken as a whole, the binary data of Turner and Peterson imply that M / L_B 35 at large separations. As the average value for spirals within the Holmberg radius is only ~ 5, this result would seem to argue convincingly for additional mass beyond R_HO.

Before continuing, it is of interest to inquire why Page with essentially similar data obtained a much lower value of M / L_B for spiral-spiral pairs. According to Peterson, the difference is due to Page's weighting scheme, which set weights inversely proportional to the square of the variance in (v)² due to observational error. Small observed values of v therefore are given very high weight, which acts to reduce the calculated M_T and hence M / L_B. Page's method also gives highest weight to pairs with small separations (large 1/r_p). If M / L_B does increase with radius, this effect would further shift M / L_B to systematically smaller values. Both Turner and Peterson have subjected their data to Page's scheme of analysis and confirm the fact that the method yields spuriously low values.

The high M / L_B values determined from binary galaxies are subject to several potential sources of uncertainty. The first arises from observational errors in the velocity differences. Let us suppose that there were no actual increase in M / L_B beyond R_HO. Then velocities would decline approximately as r^-1/2, and at a distance of 100 kpc, we would predict orbital velocity differences of roughly 140 km s^-1. The observed velocity difference v is further reduced by the projection factor cos cos , the mean value of which is roughly 0.46 (for circular orbits). The expected v on this hypothesis is therefore only ~ 65 km s^-1. Since this is comparable to the precision of typical optical velocities, it has therefore been argued that existing data are biased against low mass-to-light ratios.

Although simple, this argument approaches the problem in backwards fashion. The proper question is whether the full width of the observed histogram of v's is essentially all due to observational error, for only if this is the case can we substantially reduce the measured values of M / L_B. Peterson's 21-cm velocities are most useful in answering this question because of their high accuracy, typically better than ±20 km s^-1. M / L_B for this subsample (30 pairs) is actually slightly larger than for his sample as a whole. Furthermore, the v distribution for pairs with 21-cm velocities is very similar to that for pairs having at least one optical velocity. Both these tests indicate that the optical velocities are substantially correct. L. Schweizer (in preparation) and Karachentsev (1978) are currently collecting new, highly accurate velocities for binary galaxies which should fully resolve this question. For the moment, however, we are inclined to believe that the velocities are not at fault.

The second problem which might affect the results is contamination by spurious pairs. Turner was able to show conclusively that his sample is not appreciably contaminated by objects in the distant foreground or background. However, a great many of Turner's and Peterson's binaries are members of small groups of galaxies, in which the problem of contamination by foreground and background group members could be serious. Statistical estimates of the frequency of spurious pairs made to date are unsatisfactory because they do not include a probable spatial correlation between the target galaxy and contaminating galaxies. Furthermore, the relative velocities of group members, typically a few hundred km s^-1, are just in the range where much of the information in the v distribution resides.

Yahil (1977) has pointed out a disturbing fact which may be related to contamination problems. He has searched for a positive correlation between F (M_T) and the combined luminosity of the pair. Even though variations in <> produce a large spread in F (M_T), Yahil predicts that there should be a marked correlation between F (M_T) and luminosity, provided binaries have uniform M / L_B. For Turner's sample, no correlation is found, indicating that M / L_B must vary over at least one order of magnitude. The correlations between F (M_T) and luminosity for the Peterson and Karachentsev samples appear similar, supporting this conclusion. To Yahil, this result suggests that the large-scale distribution of matter in the universe is not strongly coupled to the distribution of luminous matter, and that the concept of mass-to-light ratio is not useful on scales much larger than 10 kpc. Alternatively, one might conclude that the lack of correlation is caused by errors in F (M_T) introduced by the inclusion of spurious pairs. Yet the observed distribution of radial separations, D (R), suggests that the majority of pairs must be real. If they were chance alignments, D (R) would increase roughly as R for small separations, whereas the observed distribution is peaked near zero, suggesting real physical association.

To investigate the contamination problem further, it would be extremely useful to compile a binary sample having significantly more stringent isolation criteria than those used heretofore. At the very least, one might test whether M / L_B is noticeably smaller for those binaries in existing samples having only distant neighbors but sizeable spatial separations. Note that an analysis confined to just those binaries that show obvious Signs of interaction will not help to test the existence of dark material, since such pairs have separations not much larger than the Holmberg radius. They therefore should have rather small M / L_B.

For the moment we continue to assume that the masses for spiral-spiral pairs as measured by binary galaxies are real, but the exact value of the mass-to-light ratio remains somewhat doubtful until the problem of contamination is conclusively cleared up.

Turning now to binary mass determinations for early-type galaxies, we recall Page's finding that E and S0 galaxies have much larger M / L_B than spirals. Both Turner and Peterson obtained a similar result, although their measured differences are smaller: Turner finds the ratio to be 2.0±0.5, while Peterson obtains 1.7 with larger errors.

Actually it seems more probable that most of the mixed E-spiral pairs identified to date are not in fact physically bound to one another. The evidence for this assertion can be found in Figure 4, which presents distributions of v for Turner and Peterson's data. For simplicity let us assume circular orbits, although our conclusion does not rest on this assumption. For a sample of binary galaxies that are real physical pairs, we expect that the distribution of v's will always be peaked at zero owing to the effect of the projection factor cos cos . This prediction is verified for spiral-spiral pairs (lower histograms). But the upper distributions, in which at least one member is an E galaxy, are nearly flat, with little or no peak at zero. The Kolmogorov-Smirnov test (Hollander & Wolfe 1973) shows quantitatively that the E and spiral distributions are unlikely to be drawn from the same population. The probability for the Turner sample is only 8% and that for the Peterson sample is only 3%. This test therefore confirms the fact that the samples really are quite different.

Figure 4. Histograms of v for binary galaxies in Turner and Peterson's samples, divided according to morphological type. Lower histograms represent pure spiral or S0 pairs; upper histograms represent pairs in which at least one member is an E. Shaded area refers to spiral pairs in which at least one member is an S0.

Histograms for S0 pairs are shown for comparison in Figure 4 as the hatched areas. The S0 distributions apparently resemble those of spirals more closely, so that S0-spiral binaries probably exist. It is this fact which is responsible for the queer nature of the E pairs having escaped discovery before now: since E and S0 pairs have traditionally been lumped together, the peculiar histogram of the ellipticals was diluted by the more normal one for the S0's.

If these mixed E pairs are indeed not physical associations, there exists at present virtually no reliable information on masses of early-type galaxies in binary systems. Jenner (1974) studied the motions of the companions of cD galaxies, but only ten pairs were included. Using his mean mass to obtain M / L_B on our system, we find M / L_B ~ 50 at 60 kpc spatial separation. However, if one system with extremely large v is omitted, M / L_B drops to only ~ 15. Smart (1973) obtained results consistent with Jenner's.

We found in Section 3 that data on M / L for S0 galaxies within R_HO are scanty, while those for ellipticals are nonexistent. The binary data are likewise fragmentary for these early morphological types. The results of Jenner and Smart, however, suggest that the mass-to-light ratio of E and S0 galaxies are broadly similar to those of later-type spirals.

Although the binary data seem to imply the existence of dark matter, we encounter a possible problem when trying to estimate the extent of the dark envelopes from these data. Such an estimate can be made in two ways. First, we have the results of Turner's model simulations, which ruled out envelopes larger than 100 kpc in extent. Second, we have the estimates of global M / L_B from Turner's halo model and also from Peterson's widely-spaced pairs. These both yield M / L_B 35 at large separations (the value is a lower limit because both estimates assume circular orbits). According to the usual version of the massive halo hypothesis, M / L_B (R) R. Since the average value of M / L_B within R_HO is ~ 5 for spirals, the extent of the envelopes must be greater than or equal to ~ 7 R_HO, or 150 kpc.

These estimates are in fair agreement with one another, but are marginally at variance with the conclusions of White & Sharp (1977), who pointed out that spherical halos around close binaries must interpenetrate strongly and that the effects of dynamical friction will be severe. In fact, two binaries ought to merge completely within an orbital period if their distance of closest approach is less than three times the half-mass radii (r_1/2) of the halos. For Turner's sample, this implies that the mean value of r_1/2 is less than 58 kpc, and hence that r, the outer boundary of the halo, is less than 116 kpc for the usual halo model. This limit must be reduced even further if the orbits are appreciably eccentric.

Even though this limit is barely consistent with Turner's estimate, many individual binaries must have true spatial separations much smaller than 100 kpc, and their envelopes should interpenetrate strongly. How are they then able to persist? Perhaps the outer envelope radius varies widely from galaxy to galaxy. Close pairs might then simply be those objects with initially small envelopes the others having already merged long ago. This reasoning would suggest that the observed radial distribution function for binaries, D (R), is strongly determined by the initial distribution function for the envelope radii themselves and that the binaries we see today are just those which were able to survive over a long period of time. In this regard, we recall the suggestion that merged galaxies become ellipticals (Toomre 1977, White 1978); this effect might then explain the rarity of E binaries.

As White and Sharp point out, the dynamics of binary galaxies, if analyzed from this more general point of view, might well place severe constraints on the distribution of unseen matter. N. Krumm (private communication) has emphasized the advantages of studying interacting pairs because of the information they afford in disentangling projection effects, which make the study of ordinary binaries so difficult. Tidal tails in interacting galaxies might have significantly different shapes if the gravitational effect of dark envelopes were included. In short, dynamical modelling of binary galaxies including the effects of halos seems a fruitful area for observer and theorist alike in the near future.

In summary, for spatial separations greater than 100 kpc, the binary data indicate M / L_B 35-50, where the higher value applies if the orbits have moderate eccentricity ( 0.7).