**12.3.3. The Mass Distribution in Spiral Galaxies**

By combining optical photometry data with the mass derived from 21-cm
observations via Equation (12.4), we can determine total mass-to-luminosity
ratios - although one should note that values thus obtained depend
inversely on the assumed distance scale. Arbitrarily choosing some
limiting radius, e.g., *r*_{25},
measured at the blue isophote of 25.0 mag arcsec^{-2}, the
integral of Equation (12.4)
and the blue luminosity within *r*_{25} yield a measure of
the global mass-to-light ratio. Using this system,
Rubin et al. (1985)
find that ratio to be *6.2, 4.5, and 2.6 for Sa,
Sb, and Sc galaxies*, respectively, with an error on those mean values of
about 10%. Since the mass is proportional to
*r*_{25} *V*_{max}^{2}, and since it
is found that, in the blue, the relationship between luminosity and
radius is independent of morphological type, the variation in
(*M*_{T} / *L*_{B}) with type is essentially one
of *V*_{max} with type. The mean
values of *V*_{max} as a function of type bear out that
conclusion. The values of (*M*_{T} / *L*_{B})
are virtually constant within a given Hubble
type, over a range of several
magnitudes. Now, what fraction of the total dynamical mass does the
luminous mass constitute? To answer that question, we must first obtain
an estimate of the fraction of the total luminous mass represented by
*L*_{B}. Work based on model
stellar populations indicates that reasonable values for the ratio
between the total mass in all stars and LB are 3.1, 2.0, and 1.0 for Sa,
Sb, and Sc types, respectively.
Although these numbers are relatively uncertain, they indicate that the
ratio of total to luminous mass within the 25-mag arcsec^{-2},
isophote is likely to be on the order
of 2 for spirals of all types. As the mass grows more or less linearly
and the light fades exponentially, the mass-to-light ratio grows rapidly
outwards from *r*_{25}.

The analysis of rotation curves yields information on the mass
distribution as a
function of distance from the galactic center, but none whatsoever on
the distribution as a function of distance from the plane of the
disk. One would like to know
which fraction of the total mass resides in the disk and which in the
halo, and also how the total mass-to-light ratio of the disk alone
varies with radius. In order to
answer these questions, it is necessary to measure the characteristics
of some tracer of the gravitational potential in the *z*-direction
(i.e., perpendicular to the disk);
handily, such is the HI. It is possible to obtain the distribution and
the velocity
dispersion, in the *z*-direction, of the galaxian HI. The former can be
measured from high-resolution maps of edge-on galaxies, the latter from
spectral profiles of isolated regions in face-on disks.

Following
van der Kruit and
Shostak (1983)
and references therein, let us assume
that a spiral disk can be approximated by a self-gravitating sheet which
is locally isothermal (i.e., the velocity dispersion of any of its
components is independent
of *z*). Then the total mass density can be expressed as

(12.7) |

where *z*_{0} may be a function of the distance from the
center (in the plane), *r*. The HI
disk can be assumed to be effectively massless, in comparison with other
dynamically important components; it can then be shown that the HI
density decreases to half of its midplane value at a height
*z*_{H},

(12.8) |

where
<*v*_{z}^{2}>_{H}^{1/2} is
the *z*-velocity dispersion of the gas and *G* is the
gravitational constant. We can rewrite Equation (12.8) as

(12.9) |

where *M*_{d}(*r*) is the disk mass within radius
*r* and the luminosity profile *L*(*r*) is
obtained from major-axis photometry. Measurements of
<*v*_{H}^{2}>_{H}^{1/2} have
been made directly in a few face-on galaxies; all yield values in the
range of 7 to 10 km s^{-1} and
appear to vary relatively little with *r*. One can thus apply Equation
(12.9) to a well-mapped edge-on object, such as NGC 891 or NGC 7814, for which
*z*_{H}(*r*) is then known. By assuming a value of
<*v*_{z}^{2}>_{H}^{1/2} as
measured in face-on spirals, and by
photometrically determining *L*(*r*), we can obtain the
details of the (*M*_{d} / *L*) ratio within
the disk. Such an operation yields the following results:

The mass-to-luminosity ratio of the disk is independent of

*r*, and hence the luminosity profile of the spiral disk is also a profile of the disk's mass.Only one-third of the total mass of NGC 891 within the distance to the edge of the optical disk actually resides in the disk itself, the rest being distributed in a much thicker component, the halo.

At

*r*= 10 kpc, the mass density of the disk at*z*= 0 exceeds that of the halo (assumed spherical) by a factor of 4; at*r*= 21 kpc, that ratio decreases to 2.In the edge-on Sab galaxy NGC 7814, only a small fraction of the total mass, interior to the optical radius of 22 kpc, is in the disk; its light distribution, furthermore, is dominated by the spheroidal bulge. The rotation curve then can be used to sample the bulge's mass-to-light ratio; it is found to increase by a factor of 10 between the inner regions (

*r*< 10) and 22 kpc.

In conclusion, the dynamical masses of spiral galaxies are found to be still growing linearly beyond the edges of their optical disks, at least out to the greatest radii at which gravitational potential tracers like HI are detectable. Within the optical radius, no more than one-third to one-half of the mass resides in the disk. The ratio between luminous and dynamical mass within the disk is about constant with radius and independent of morphological type, while that of the spheroidal component grows rapidly with distance from the galactic center.