To address the question of how much mass different galaxies contribute
to the total, we begin by examining their luminosities, which are more
directly observable. The emission from a set of galaxies in a
particular waveband or in a spectral line can be characterized by a
luminosity function . This is
a distribution function
(essentially a histogram) of the number of galaxies in each range of
luminosity. Luminosity functions have some generally common
characteristics that we will examine in this section, and we will
begin to look at how a luminosity function is translated into a mass
function - which is our ultimate goal.
The function is often written
as a density so that integrating over all luminosities
(L)dL gives the total
number of galaxies per some volume unit like 1 Mpc3. Since
dL gives the total
number of galaxies of each luminosity, it can be multiplied
by other galaxy properties and integrated to derive population totals
(or totals within narrower luminosity ranges) for other properties.
For example, the total luminosity of a population is:
![]() | (1) |
In principle, if we know the relationship between mass and luminosity we can integrate
![]() | (2) |
to determine the total mass. This however assumes some simple relationship between mass and luminosity, which may not be realistic as we shall discuss further below.
Classically, is determined
in a well-studied region where the
galaxy distances are known and deep integrations have generated quite
complete samples. This has usually confined such studies to nearby
galaxy clusters, which is a point to keep in mind in case there are
biases in the galaxy properties in such high density regions.
One of the features of galaxy luminosity functions that appears to be
true at most wavelengths is that the number counts grow larger at
fainter luminosities like the luminosity to some negative power. This
power law behavior, (L)
L
where
< 0, is a
common sort of distribution in nature, describing the size distribution
of asteroids and turbulence for example.
The power law behavior of appears
to persist over several
orders of magnitude in galaxy luminosity. At very high luminosities
the number counts of galaxies drop off much more rapidly than the
power law, and at very low luminosities the definition of a galaxy
must break down as the luminosity approaches the range of globular
clusters. The faint end is not very well observed though, so this
low-luminosity turnover or cutoff has not been characterized.
One frequently employed description of the luminosity function is that developed by Schechter (1976). The Schechter function is a power law with an exponential drop-off at high luminosities:
![]() | (3) |
This function is plotted in Figure 3. The value L* characterizes the highest luminosities before the number counts start to drop off significantly from the power law distribution. L* is the luminosity of the most common type of bright galaxy, which are therefore some of the better known, and is approximately the luminosity of the Milky Way and M31.
![]() |
Figure 3. The distribution of galaxy luminosities or masses as modeled by the Schechter luminosity function assuming different power law indices. |
We can use the Schechter function to explore which galaxies contribute
most to the total luminosity of galaxies. In
Figure 3, the
shape of the luminosity function is shown for values of the power law
index from -1 to -2. The
shapes of these functions do
not look very different, but the resultant fraction of the luminosity
contributed by different decades of luminosity is quite different, as
shown in Figure 4. If the power law is shallower,
the main contribution to the total
luminosity is from galaxies with luminosities near
L*. However,
if the power law were as steep as -2, the lowest luminosity galaxies
contribute the largest fraction of light.
![]() |
Figure 4. The total contribution to the luminosity function from different luminosity decades when assuming different power law indices in a Schechter luminosity function. |
Estimates of the Schechter function generally indicate
~ -1.25. The power law
appears to be even shallower than
this outside of clusters, perhaps because the dwarf galaxies
corresponding to the cluster dwarf elliptical populations have not
been well-cataloged. (See
Binggeli,
Sandage, & Tammann 1988
for a review of luminosity function determinations.) Such shallow power
laws would indicate that there is no major luminosity contribution
from faint objects.
However, the more important question is what this relationship
signifies for the total mass of galaxies. For example, low
luminosity galaxies may have a larger mass-to-light ratio than
brighter galaxies since their interstellar mediums are more weakly
confined by their gravity and are probably less efficient at forming
stars. If it were possible to characterize the mass-to-light ratio as
another power law, M/L
L
, then the
mass per
luminosity interval would look like Figure 4 for a
Schechter exponent of
' =
+
. (Note that this is not
the same as the power law exponent in the mass function - the
number per mass interval - which would have a distribution function
obeying
(M) dM =
(L) dL.)
If
is negative
as I have argued, then
'
would be closer to -2 than
, and low luminosity galaxies
would be more significant
contributors to the total mass than they are to the total luminosity.