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

6.2 Mass-to-Light Ratios

Limber (1959) derived the virial theorem as applied to small groups while Materne (1974) examined the treatment of observational errors. A number of uncertainties plague the analysis, as discussed by Aarseth & Saslaw (1972). Potential problems include observational errors, extrapolation to include the luminosity of faint members, and incomplete data. Furthermore, the virial theorem is based on the time-averaged energies, whereas we observe the group at just one moment in its history, when it may be out of equilibrium. Finally, we must employ projection corrections to convert the radial velocity dispersion and angular separations to their three-dimensional values. These mean projection corrections may be valid averages over a long period moment. Derivation of the of time but are incorrect at any given mean correction factors also involves assumptions as to the character of the motions, in particular that the projection corrections in velocity and radius are uncorrelated. This assumption is untrue for binaries, for example, and the standard virial formulation can substantially over- or underestimate binary masses, depending on the eccentricity of the orbits.

Taken together, these are serious difficulties. However, they are of minor importance compared to the twin problems of group membership and group definition. The group catalogs referred to above differ substantially in their membership assignments; these differences introduce uncertainties of a factor of 3-4 in the resultant mass-to-light ratios (see below).

As a result of a heightened awareness of these sources of error, a new consensus is emerging that the earlier piecemeal approach which attacked groups one at a time is unlikely to succeed. Since the analysis of any one group is subject to large statistical uncertainty, discussions of group properties must be based on a representative sample. On the observational side, we require accurate redshifts for a large, magnitude-limited sample of galaxies. Although not complete as yet, the available redshift sample is growing. Recent lengthy lists include redshifts in the RC2, velocities of galaxies in Gott-Turner groups measured by Kirshner (1977), accurate 21-cm redshifts for spiral galaxies (Dickel & Rood 1978, Shostak 1978, S. Peterson 1978, Thuan & Seitzer 1979), and many new optical redshifts by Sandage (1978). Several lengthy unpublished lists also exist (Tully & Fisher 1978, G. Knapp and W.L.W. Sargent, private communication, M.S. Roberts and co-workers, unpublished). A complete program to obtain a magnitude-limited sample of redshifts has been initiated by Davis and Huchra (Huchra 1978), but the final results are still a few years away. Taking an alternative approach, Gregory & Thompson (1978) have collected a redshift sample complete to a very deep limiting magnitude over a small region of sky.

On the theoretical side, we strongly believe that it will never be possible to assign individual galaxies to groups or fields in a definitive way. Any approach which relies solely on such group assignments is inevitably subject to insurmountable bias and cannot possibly yield reliable results. New theoretical methods are required which either avoid this bias altogether or correct for it in a statistical way.

Before discussing recent work along these lines, we summarize the results of traditional virial analyses of small groups. As was noted in previous sections it is of paramount importance to employ a consistent system of masses and luminosities when comparing mass-to-light ratios derived by various workers. For example, the magnitude correction factor for Gott-Turner M / L_B's to our system is 0.50, and for Rood-Dickel values is 1.25. The local luminosity density on our system is ~ 1.0 x 10⁸ L Mpc^-3 (Gott & Turner 1976, Davis et al. 1978). With this value, M / L_crit the mass-to-light ratio for a universe having critical density, is ~ 700. In such a universe, / _crit ( ) is unity.

We confine ourselves to those investigations in which a sizeable number of groups have been treated simultaneously in homogeneous fashion. Results are collected in Table 5. Consider first the TG sample. The median M / L_B for the original TG groups including bogus members is 70. Gott & Turner (1977) later presented a list of revised groups, culled of obvious nonmembers. The median M / L_B for this sample is 30, a significantly lower value. Thus although the median M / L_B for the unculled groups is of theoretical interest because of its value as an unbiased estimator of some statistical property of the ensemble, by itself it has no obvious connection with the true M / L_B of the sample. In order to obtain M / L_B from the TG catalog alone, one must resort to the usual strategy of choosing members, accepting the inevitable bias therein. The great value of the unculled TG catalog is that it is an unbiased sample which can properly be compared with numerical simulations of galaxy clustering. In this role it appears to be quite powerful (see below).

Rood & Dickel (1978a) have determined M / L's for the culled TG groups and for STV groups, all required to have at least three members with measured velocities. Their median value for the TG groups is 40, not significantly different from the Gott-Turner value despite the inclusion of 40% more radial velocities. The median value for the STV groups, however, is 140. According to Rood and Dickel, this difference is due to different definitions of what constitutes a group: many of the STV groups break up into subgroups using the TG prescription. This is one more illustration of the extent to which the final value of M / L depends on how the groups are defined.

Table 5 also includes M / L_B values from Materne & Tammann (1974), Tammann & Kraan (1978), and Tully & Fisher (1978), all based on a selection of nearby, rather poor groups. The values in the table are very approximate because these authors did not give M / L_B directly. Thus, M / L_B had to be inferred from M_VT / M_L, the ratio of virial mass to luminous mass.

Gott & Turner and Rood & Dickel have emphasized the existence of missing mass outside of galaxies, while the remaining three authors have minimized its importance. Yet as Table 5 indicates, the observational data seem to be similar in all cases. The disagreements among authors stem in part from differences in adopted magnitude conventions, and have been removed by placing all values of M / L_B on a common system, as in Table 5. To a large extent, however, the disagreement is philosophical: Gott & Turner and Rood & Dickel have placed greatest weight on the median values, which appear to support the existence of unseen matter. The remaining authors instead have emphasized those groups with small M / L_B and have criticized the rest as being contaminated or not in equilibrium.

Derived M / L's have also been questioned because of observational errors in the radial velocities (Karachentsev 1978, Materne & Tammann 1975, Tully & Fisher 1978). Group velocity dispersions are in many cases less than 100 km s^-1 (Tammann & Kraan 1978), conceivably too small to be accurately measured using conventional optical velocities. which often have errors of the same magnitude. However, we believe that velocity uncertainties are not likely to grossly inflate the median M / L_B for the following reasons: first, the velocities used by Rood & Dickel are substantially more accurate than those used by Gott & Turner, yet their median M / L_B is slightly higher: second, an increasingly large number of velocities are very accurate 21-cm redshifts and 21-cm checks of optical velocities in the mean show good agreement (Rood & Dickel 1976): and third, the really high values of M / L_B are virtually all associated with groups having large velocity dispersions of several hundred km s^-1, much greater than the measuring errors. The radial velocities are therefore not a likely source of error. Nevertheless, we are still left with the membership question, which cannot be resolved in any convincing way on a group-by-group basis.

Attempting to break this deadlock. Aarseth and co-workers (Aarseth et al. 1979, Turner et al. 1979) have recently introduced a new method based on N-body simulations of galaxy clustering. The resultant models are based on a variety of initial conditions, obtained by varying the mass-to-light ratio for galaxies, initial density fluctuation spectrum, starting redshift, and peculiar velocities of galaxies. Group catalogs are constructed for the model universes in a manner identical to that of the original TG catalog, and the two sets of group properties are compared. Since no culling is performed in either case, contamination enters equally in both analyses. By comparing models and data in exactly parallel fashion, one ought to obtain a useful measurement of M / L_B for groups free of bias. Turner et al. point out that an N-body model with equal to 0.1 (M / L_B = 70) exhibits group membership characteristics very similar to those of the real universe. In particular, the distribution of galaxies in redshift space is quite different from the true spatial distribution, and group assignments based on radial velocities would often be in error.

For the same model, Turner and collaborators have emphasized that the median M / L_B for the simulated unculled groups agrees well with the true (i.e. model) M / L_B, both being equal to 70. However, this good agreement is largely a coincidence. The binary galaxies in this model have highly eccentric orbits, and their mass is badly underestimated as a result. The median M / L_B for binaries is only 23, considerably less than the model value of 70. On the other hand the groups with three or more members are strongly affected by contamination. Their median M / L_B is 200, much larger than the model value. The two errors combine fortuitously so as to make the median M / L_B of all groups together equal to the true value of the model. Moreover, nearly all TG groups in the real universe have three or more members. Therefore, if the model can be taken as a guide, it corroborates our previous conclusion that contamination in the real unculled TG groups is serious and significantly biases the median M / L_B to higher values. Furthermore, if we restrict our attention to groups with three or more members, the spread in the model mass-to-light ratios is significantly smaller than is observed in the real universe. This result supports the suggestion of Rood & Dickel (1978b) that there exists an intrinsic spread in M / L_B among groups which is larger than can be attributed to the various sources of error.

These model experiments are in their infancy and can surely be improved. Future calculations should include a realistic mass spectrum for galaxies, the effect of massive envelopes on the interactions of galaxies, and a more complete examination of initial conditions. The method of comparison between the real and simulated data might also be refined. Nevertheless, as an unbiased statistical approach to the problem, the present computations represent an original and promising line of attack which should be vigorously pursued.