**4.2.3. Central Velocity Dispersions. II. Mass-to-Light Ratios**

Since elliptical galaxies contain little material in nearly circular
orbits, their masses are usually calculated using central velocity
dispersions and the virial theorem. A review of such mass determinations
is given in
Faber and Gallagher
(1979).
However, total masses
can be calculated only by assuming that the mass-to-light ratio
*M/L* is
the same everywhere in the galaxy, and that the velocity distribution
is everywhere isotropic with constant
. These assumptions are
probably not correct. Therefore, only core mass-to-light ratios will
be discussed in this section. Their determination requires fewer and
better-justified assumptions
(King and Minkowski
1972;
Faber and Gallagher
1979),
although not ones which are observationally verified.
If the velocity distribution is strongly anisotropic near the center,
then even these masses will be greatly in error
(Binney and Mamon 1982;
Tremaine and Ostriker
1982;
see since 3.3.8).

If we assume that cores are isothermal, then the virial theorem gives central mass-to-light ratios

(21) |

where *I*_{0} is the central surface brightness
(King 1966;
Schechter 1980).
Conveniently, the combination
*I*_{0}*r*_{c} is relatively
insensitive to
seeing. At present *M*/*L* values can be calculated for only a few
galaxies because there is not enough published photometry. For 17
giant ellipticals,
Schechter (1980)
finds an average value of
<*M*/*L*_{B}> = 7.8
(dispersion = 3.2) in solar units. (Here I have corrected the
values for NGC 4552 and 4636 for photometric zero-point errors in
King 1978; see
Boroson and Kormendy 1982).
The only disk-galaxy bulge for which there exists sufficient photometry
is M31;
Faber and Gallagher
(1979)
obtain *M*/*L*_{B} = 8.5. If we briefly relax our adopted
restriction and consider global *M/L* determinations, then there
exist slightly more data.
Faber and Gallagher
(1979)
obtain <*M*/*L*_{B}> = 7.8 ± 1.3 for five
additional (S0) bulges. The data are very sparse but there seems to
be considerable homogeneity in these stellar populations.

One question of considerable interest is the possible dependence
of *M/L* on *L*. With data on only a score of galaxies, a plot of
*M/L* versus *L*
(Schechter 1980,
Fig.2) has so much scatter that any weak
luminosity dependence is masked. However, we can look for a luminosity
dependence using mean relations between core parameters.
Faber and Jackson
(1976)
combined the *L*
^{4} relation
with other power-law
relations (not explicitly stated) between *I*_{0},
*r*_{c} and *L* to derive
*M*/*L*
*L*^{0.5}. On the other hand, S^{2}BS assumed that
all galaxies have
the same surface brightness, in which case the relation
*L*
^{4} and the
virial theorem
*M*
^{2}
*r*_{c} imply that
*M*/*L*
*L*^{0}. Other determinations
include *M*/*L*
*L*^{1/8}
(Schechter and Gunn
1978)
and *M*/*L*
*L*^{0.34}
(Michard 1980,
for global mass-to-light ratios).

The mean parameter correlations derived in this paper allow a new
determination of the luminosity dependence of *M/L*. The required
relations are (1)
*L*
^{5.4} from
section 4.2.2, (2)
*r*_{c}^{0.25} from Figure 22,
(3) a similar relation
*I*_{0}
*r*_{c}^{-0.87}, not illustrated
but derived in the same way as (2), and (4) the virial theorem,
*M*/*L*
^{2} /
*I*_{0}*r*_{c}.
Combining these gives

(22) |

intermediate between the results of
Faber and Jackson
(1976)
and of S^{2}BS and
Schechter (1980).
Note that if we assume that
*L*
^{4}, then
the above relation becomes
*M*/*L*
*L*^{0.49}, in agreement with Faber and Jackson.

The above weak dependence of *M*/*L* on *L* is consistent
with
Tinsley's (1978)
population models of elliptical galaxies. These model the
observed color-magnitude relations by varying the metallicity (cf.
Michard 1980).
Tinsley predicts that
*M*/*L*
*L*^{0.13}, which is certainly
consistent with equation (22) in view of the small sample of data.
This is a reassuring indication. However, there is a clear need for
data on more than ~ 20 galaxies, with a greater range of luminosity
than is represented in current measurements.