Annu. Rev. Astron. Astrophys. 1988. 26:
509-560
Copyright © 1988 by . All rights reserved |

**2.1. Luminosity Function and Density Function**

Let
(*M*, *x*, *y*,
*z*) denote the number of galaxies lying in volume *dV* at
(*x*, *y*, *z*) that have absolute magnitudes between
*M* and *M* + *dM*. On the
assumption that galaxian magnitudes are not correlated with spatial
location, one can write

(8) |

where

(9) |

(*M*) gives the
fraction of galaxies per unit magnitude having absolute
magnitudes in the interval
(*M*, *M* + *dM*) and is called the *luminosity
function*.
*D*(*x*, *y*, *z*) gives the number of galaxies (of
all magnitudes)
per unit volume at (*x*, *y*, *z*) and is called the *
density function*.
and
*D* should be viewed as probability densities, which in practice are
approached and represented either by (nonparametric) histograms or by
(parametric) analytical forms.

If Equation 8 is valid for a sufficiently large portion of the
Universe, or for sufficiently many samples of galaxies,
(*M*) can be
called the *universal* luminosity function of galaxies. This is
clearly an approximation. In reality one expects that
does somehow
depend on the location, i.e. on the environment from which the galaxies are
sampled. The question of universality of
(*M*) is discussed in
Section 4.3ff. Systematic differences of
(*M*) with respect to
type and environment, which are discussed in
Sections 5 and 6, lead us to
reject universality in the above sense; Equation 8 is subsequently
revised in Section 6.3.

The present definition of the luminosity function of galaxies, as
expressed by Equations 8 and 9, is identical to that used in stellar
statistics
(von der Pahlen 1937,
Mihalas & Binney 1981).
It should be noted that the conventional definition of the galaxian
(*M*)) was
different during the previous decades. The usual ("classical") method
to determine the luminosity function of field galaxies (outside of
rich clusters) was based on the assumption that
*D*(*x*, *y*, *r*) = <*D*> = constant
(see Section 3.2), which allows the product
(*M*)^{
. }<*D*> to be
discussed as *one* function; ever since
van den Bergh (1961)
and Kiang (1961),
this product has been tagged with the label "luminosity function
(*M*)," most
recently in the review of
Felten (1985).

Consequently the "luminosity function" has been given the units of
density (number of galaxies per magnitude per cubic megaparsec). The
drawback of this definition is the creation of an artificial dichotomy
between field and cluster samples. Clusters of galaxies, where *D*
constant is obvious, could strictly not have a luminosity function. In
order to distinguish the function
(*M*)
<*D*> from
(*M*),
Schechter (1976)
has introduced for the latter the term "luminosity distribution."
Starting with
Sandage et al. (1979)
and Kirshner et al. (1979),
the assumption that *D* = constant for field galaxies has been
dropped. The general inhomogeneity of the distribution of galaxies is
now widely acknowledged, and all new methods used to derive the luminosity
function of field galaxies aim at a clear separation of
and
*D* (see *M*) along the
lines of stellar statistics (Equations 1, 2) is therefore most desirable at
present. The mean density *D*, averaged over a significant portion of
the observable universe, remains of course a most important quantity
for cosmology, but there is no reason why it should be built into the
luminosity function (provided that the notion of a universal shape for
(*M*) makes any
sense at all). The discussion of *D* is
consequently left out of the present review.

The normalization of
(*M*) to unity by
integrating over all
magnitudes (Equation 9) is difficult in practice because any sample of
galaxies is complete, or has good statistical weight, only to a
certain limiting magnitude *M*_{lim}. The ideal case, where the
faint end of
(*M*) goes to
zero at a magnitude
*M*'
*M*_{lim} is at present applicable only
to certain types of galaxies that are sampled nearby (cf.
Section 5). In general,
not
only is nonzero but is growing exponentially at
*M*_{lim}, making such a normalization infeasible; any
extrapolation of
(*M*)
to fainter magnitudes by an analytical model will diverge. A way to
avoid this divergence would be to go to the luminosity (*L*)
representation of the luminosity function, transforming
(*M*) into
(*L*) and setting

(10) |

which for physical reasons must always converge. However, we wish to
keep the magnitude representation, since
(*M*) is closer
to the observations than is
(*L*). An obvious
and practicable way to normalize
(*M*) is to
restrict the discussion to galaxies brighter than a certain
arbitrary absolute magnitude
, in which case Equation
9 is replaced by

(11) |

*D* in Equation 8 is then the density of galaxies that are brighter
than .
may be different for
different samples. Future work will
push
toward fainter and fainter limits until the ideal normalization of
Equation 9 can be realized.

It should be noted that the normalization is not a principal problem
for the present concept. A normalization of
(*M*) is needed
only for
the discussion of the density function *D*, which, by virtue of the
adopted separation, is not a subject of this review. [Densities are
only discussed where they have fundamental consequences for
(*M*), such
as in the context of the morphology-density relation (see
Section 6).]
For the discussion of
(*M*) alone, no
normalization is required because
it is a probability distribution. Therefore, *any
( M), whether
normalized or not, is called here a luminosity function*. The
luminosity functions of different samples can then be compared by
their

(*M*) is
sometimes called the differential luminosity function, which
should be distinguished from the integrated (or cumulative) luminosity
function
(*M*), defined as

(12) |

(*M*) is less
frequently used than
(*M*); it tends
to conceal an
intuitive interpretation of the information available for the fainter
galaxies. In what follows, unless otherwise stated, LF is always meant
to designate the *differential* luminosity function
(*M*).