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

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Three basic methods can be used to determine the mass and mass-to-light ratio of a spheroidal stellar system. The first of these utilizes the global virial theorem, extensively discussed by Poveda (1958):

Equation 2 (2)

where I is the moment of inertia, T the kinetic energy, and Omega the gravitational potential energy (Limber 1959). For a galaxy in equilibrium, the left-hand side vanishes. Let us assume further that the galaxy is spherical and nonrotating and that the kinetic energy of each star per unit mass is independent of mass. Then

Equation 3 (3)

where the potential energy Omega is given by

Equation 4 (4)

M, R are the total mass and radius, M (r) is the mass contained within a sphere of radius r, and <V2> is the mass-weighted average of the square of the space velocities of the stars relative to the center of mass of the galaxy.

Equations (3) and (4) involve theoretical parameters which are far removed from observed quantities. <V2> for example is usually estimated from the observed line-of-sight velocity dispersion (sigma) in the nucleus by assuming that sigma2 is constant throughout the galaxy, an assumption usually not supported by any observational data. The estimate of the total potential energy Omega is likewise subject to great uncertainty. It is generally assumed that the light distribution is an adequate tracer of the mass and that the luminosity profiles of most ellipticals are similar and are adequately described by empirical expressions, such as the R1/4 law of de Vaucouleurs (1948) (see also Young 1976). The integral in (4) is then Omega = -0.33 G M2 Re, where Re is the isophotal radius containing half the light (and mass). This approach assumes that the outer structure of the galaxy obeys de Vaucouleur's law. In fact, there seem to be significant departures from de Vaucouleur's law in the outer profiles of elliptical galaxies which correlate with environment (Kormendy 1977b, Strom & Strom 1978). Furthermore, the total light in the envelopes of some cD ellipticals shows no sign of converging to a finite value (Oemler 1976, Carter 1977), making the determination of Re operationally impossible. Finally, if ellipticals contain appreciable amounts of dark material which is more extended than the luminous material, Re as determined for the stars alone may have no connection with the true mass distribution of the galaxy.

The virial theorem thus leads to uncertain results basically because it treats the whole galaxy, including the poorly understood outer regions. To circumvent this difficulty, King (King & Minkowski 1972 and in preparation) has devised a second method to determine M / L; this method is based on stellar hydrodynamical equations applied to the core only. The observational data required include the central surface brightness, core radius (the point where surface brightness drops to 1/2 of central value), and core line-of-sight velocity dispersion. From these one determines the core density and core mass-to-light ratio. Total mass is not derived. The method assumes only that the nuclear velocity distribution is Gaussian and isotropic with constant sigma over the core region, in agreement with the properties of model star clusters whose cores closely resemble the nuclear regions of elliptical galaxies (King 1966). Young et al. (1978) and Sargent et al. (1978) have developed a similar formalism which is applicable to regions outside the core.

King's formula contains an explicit correction for rotational motion based on the observed ellipticity. However, several studies (e.g. Bertola & Capaccioli 1975, Illingworth 1977, C. Peterson 1978) have shown that even flattened ellipticals rotate very slowly and are almost completely pressure supported; rotational corrections should therefore be small. Binney (1976) and Miller (1978) have presented alternative models for elliptical galaxies having anisotropic velocity dispersions. These models imply a correction to our assumption of an isotropic velocity distribution in the core, but the effect should again be small.

The last method for determining M / L in E and S0 galaxies is the most straightforward: find a test particle in circular motion about the spheroidal component. This approach is applicable to the stellar disks of S0 galaxies and to gas in orbit about an elliptical [NGC 4278 is apparently such a galaxy (Knapp et al. 1978)]. For S0 disks seen directly edge on, the observed rotational velocity must be increased by 30-40% to correct for stars at large spatial radii projected along the line-of-sight (Bertola & Cappaccioli 1977, 1978).

Burbidge & Burbidge (1975) summarized the results on M / LB in early-type galaxies through 1969. Their mean value was 19.7, for a variety of objects and techniques. King & Minkowski (1972) reported values of 7-20 for luminous elliptical galaxies based on King's method applied to the cores and utilizing Minkowski's velocity dispersions.

Since this early work, the trend in M / LB has been generally downward owing to two factors: remeasurements of velocity dispersions significantly smaller than earlier values, and the general adoption of core analyses in place of the global virial theorem. It is not easy, however, to summarize these recent results because there is still significant disagreement between various groups as to the correct measurement of sigm. The two largest sets of data available are those of Faber & Jackson (1976) (FJ) and Sargent, Young and coworkers (SY) (Sargent et al. 1977, 1978, Young et al. 1978). Although it seemed initially that the values of FJ exceeded those of SY by 28%, new measurements (S. Faber, unpublished, Schechter & Gunn 1979) now make it seem likely that the two systems agree within 10%. In comparison to these results, however, the measurements of Williams (1977), Morton and co-workers (Morton & Chevalier 1972, 1973, Morton et al. 1977), and de Vaucouleurs (1974) average about 35% smaller. This comparison is quite uncertain, however, because the number of objects in common is in all cases very small.

If these systematic differences in sigma are taken into account, one finds that the agreement between investigators is good, with M / LB typically 5-10. Sargent et al. (1977) found substantially higher M / LB for elliptical galaxies but did so by applying the virial theorem to the entire galaxy assuming constant sigma. Since sigma was available only for the core, we believe that an analysis based only on the measured core quantities is preferable.

Determinations of velocity dispersions for normal elliptical galaxies have revealed two possible regularities. Both FJ and Sargent et al. (1977) found that sigma increased with total galaxy luminosity approximately as LB1/4. For a sample of ellipticals with core radii and central surface brightness determined by I.R. King (unpublished), FJ derived power-law correlations between luminosity, core radius, and central surface brightness (see also Kormendy 1977b). Using these correlations plus the L1/4 law for velocity dispersions, FJ found that M / LB increased with luminosity as L1/2. Schechter & Gunn (1979) found no such correlation, based on a published subset of King's data (King 1978). However, the published data were not corrected for seeing effects, unlike the unpublished list used by FJ. This difference apparently accounts for the discrepancy.

Using a method rather different from those above, Ford et al. (1977) have estimated the mass of M32 from motions of planetary nebulae far from the nucleus. Their result is 4.3 x 108 Msun, from which M / LB = 1.8. This result would seem to be consistent with the possibility noted above that M / LB is smaller in less luminous ellipticals.

The nucleus of M87 differs significantly from other ellipticals in having a bright central luminosity spike and a rapid decline in sigma just outside the spike (Young et al. 1978, Sargent et al. 1978). A mass with very high M / L, perhaps even a black hole, apparently exists in the middle of the core. Since very few elliptical galaxies have been studied with such high spatial resolution, it is not known whether such cases are common.

Table 3. Mass-to-light ratios in early-type galaxies
Table 3

Recent results for M / LB among early-type galaxies (corrected to the M / L system of the preceding section) appear in Table 3. The first group contains objects for which nuclear values of M / LB have been measured using King's method or a related treatment. The mean value obtained by Faber and Jackson is 8.5 for 10 galaxies, while the value of 8.5 for the inner bulge of M31 is new in this review. The second group consists of galaxies for which the circular rotation of test particles can be measured. The mass and M / LB within radius R have been computed assuming a spherically symmetric mass distribution as in the preceding section.

Taken at face value, these data suggest that there is no gross increase in M / LB from the core to the Holmberg radius. This conclusion is supported by the tendency of velocity dispersions to decrease away from the nucleus in M32 (Ford et al. 1977), NGC 3379 and NGC 4472 (FJ), and NGC 4486 (Sargent et al. 1978), leading to constant M / L in the inner regions. On the other hand, sigma does not decline with radius in NGC 4473 (Young et al. 1978). Schechter and Gunn find that sigma is basically constant in 12 more ellipticals, but their measurements extend to only a few core radii.

In summary, rotation curve data indicate that M / LB within the Holmberg radius is approximately 10 for S0's. This number is entered in Table 2. No comparable estimate for E's can be given at this time owing to inadequate data on the velocity dispersions away from the nuclei.

Of great importance is the question whether a strong increase in M / L occurs beyond the Holmberg radius, as seems to be the case with spirals. The evidence on this point is fragmentary but highly suggestive of dark envelopes around early-type galaxies as well. NGC 4278 is the only galaxy in Table 1 for which rotation measurements extend beyond RHO, and its M / LB seems significantly higher than the others. Faber et al. (1977) found no decrease in the velocity dispersion in the halo of cD galaxy Abell 401 at a radius of 44 kpc, but the accuracy of the measurement was not high. Dressler & Rose (1979) have detected an actual increase in sigma out to 100 kpc in the halo of the cD galaxy Abell 2029 with much better data, implying a strong increase in the local mass-to-light ratio. Finally, we mention the novel mass determination of M87 based on the assumption that the X-ray emission centered on M87 is due to thermal bremsstrahlung from isothermal gas in hydrostatic equilibrium within the potential well of the galaxy (Bahcall & Sarazin 1977, Mathews 1978). Mathews finds in this case that the total mass of M87 exceeds 1013 Msun and the total M / LB is several hundred.

All these data point strongly to the existence of dark matter around at least some elliptical galaxies. With the recent increased availability of efficient two-dimensional detectors for spectroscopy, additional information on the dynamics of the outer regions of elliptical galaxies should soon be forthcoming. Ultimately, radial velocities of globular clusters will be used to probe the structure of spheroidal systems at very large radii, but these observations seem to lie just beyond the capabilities of present equipment.

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