Annu. Rev. Astron. Astrophys. 1988. 26: 509-560
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2. DEFINITION

2.1. Luminosity Function and Density Function

Let nu(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

Equation 8 (8)

where

Equation 9 (9)

varphi(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. varphi 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, varphi(M) can be called the universal luminosity function of galaxies. This is clearly an approximation. In reality one expects that varphi does somehow depend on the location, i.e. on the environment from which the galaxies are sampled. The question of universality of varphi(M) is discussed in Section 4.3ff. Systematic differences of varphi(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 varphi(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 varphi(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 varphi(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 neq constant is obvious, could strictly not have a luminosity function. In order to distinguish the function varphi(M) <D> from varphi(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 varphi and D (see Section 3.2, Table 1). A redefinition of varphi(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 varphi(M) makes any sense at all). The discussion of D is consequently left out of the present review.

The normalization of varphi(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 Mlim. The ideal case, where the faint end of varphi(M) goes to zero at a magnitude M' leq Mlim is at present applicable only to certain types of galaxies that are sampled nearby (cf. Section 5). In general, varphi not only is nonzero but is growing exponentially at Mlim, making such a normalization infeasible; any extrapolation of varphi(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 varphi(M) into varphi(L) and setting

Equation 10 (10)

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

Equation 11 (11)

D in Equation 8 is then the density of galaxies that are brighter than tilde M. tilde M may be different for different samples. Future work will push tilde M 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 varphi(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 varphi(M), such as in the context of the morphology-density relation (see Section 6).] For the discussion of varphi(M) alone, no normalization is required because it is a probability distribution. Therefore, any varphi(M), whether normalized or not, is called here a luminosity function. The luminosity functions of different samples can then be compared by their shape (i.e. the shape is the luminosity function). What matters only is that varphi(M) is decoupled from the density function.

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

Equation 12 (12)

Phi(M) is less frequently used than varphi(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 varphi(M).

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