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

**6.3. A Case for Simplicity**

Apart from this last-mentioned suggestion that the *brightest* galaxies
lie preferentially in dense environments (probably as a result either
of stochastic sampling or of dynamical evolution), there is no
evidence that the shape of the *type*-specific LFs,
_{T}(*M*),
depends on
the local density (see Section 5 and
Figure 1). Perhaps the strongest
case against a density dependence of
_{T}(*M*)
comes from the Virgo and
Coma clusters. Despite the very different type
mixtures of these clusters (namely E: S0: S+Irr = 12:26:62% for Virgo, but 44:49:7% for
Coma), the LFs of these three bins of galaxies are
essentially identical
(Binggeli 1987).

Adopting the hypothesis that
_{T} is
independent of *D*, i.e.
_{T}(*M*,
*D*) =
_{T}(*M*),
one can treat the total
(*M*) as a
function of type and
density by expanding the formalism in Section 2 (Equations 8-11) as

(21) |

where

(22) |

This means that the separability of
and *D*
(Equation 8) is recovered for any given *T*.

If *f*_{T}(*D*) is the relative frequency of a given
type over all magnitudes as a function of the density, one can write

(23) |

where

(24) |

From Equations 21 and 23, the sum over all types becomes

(25) |

and the LF over all types is therefore

(26) |

where (from Equations 22 and 24),

(27) |

Equation 26 clearly reflects that the general LF is *not universal.*
Instead, it is the sum over all type-specific LFs weighted by the
density-dependent type fraction.

The difficulty with the normalization in Equations 22 and 27 is that
the integration of to
infinity is impractical because at least the LF
of dEs is exponential. A solution to the problem was suggested with
Equation 11, where
was integrated to a specified absolute
magnitude .
If this procedure is again adopted here, the type fraction
*f*_{T}(*D*) also
becomes a function of .
Therefore, as stressed already by
de Souza et al. (1985),
the *same magnitude cutoff
must be used for
the morphology
density relation*. Equation 26 can then be reformulated as

(28) |

where

(29) |

and

(30) |

A remaining question is to what detail the type binning should be
carried. The minimum detail required for spirals is the
differentiation into subtypes, because the subtypes have different LFs
(cf. Figure 1) and follow different
morphology-density relations, as
discussed in Section 5.3. Therefore, in order that
_{T} be universal,
the morphological binning should be at least as fine as that used in
Figure 1. But it is not clear whether
this degree of differentiation is sufficient. It was conjectured in
Section 6.1 that the
morphology-density relation may extend down into the finest
morphological separation. However, the morphological binning must not
reach the point where every galaxy is treated as an individual;
otherwise the hypothesis that
_{T} is
universal would be trivially
true. A fine *T*-binning is often also impractical e.g. for the LFs of
distant clusters. Nevertheless, even with a coarse binning the
hypothesis of constant
_{T} may be a
valid and useful *approximation.*

The precise universality of
_{T} is not to
be expected for other
reasons as well. Dynamical interaction with the intracluster medium
and between galaxies in the central parts of clusters has some
influence on the evolution
(Dressler 1984)
and also on
(*M*)
(cf. Section 4.1). But, following the
introduction of a general LF over all types by
Abell (1962,
1964),
its consolidation by
Schechter (1976),
and its canonization by
Felten (1977,
1985),
the present hypothesis that the type-specific
_{T}(*M*)
is universal, independent of
the environment, is probably the next logical step for an
understanding of the LFs of galaxies. Of course, its validity must be
tested further.