Comparisons of the luminosity function in high and low-density regions are of great interest, both as a test of biased galaxy formation (see Sect. 7.2), and as a measure of environmental influences on galaxy evolution. One of the great difficulties here is in establishing the connection between observations and theory. On the theoretical side, there are many different variants of biasing (e.g. Rees 1985). While statistical bias in the distribution of fluctuation peaks (Kaiser 1984; Bardeen et al. 1986) may plausibly bias the spatial distribution of dark-matter halos, the connection between that distribution and the LF is not well understood (in large part because it involves star formation). On the observational side, the techniques used to probe the LF in clusters and outside of clusters are quite different, leading to different potential biases in the results (see above). Finally, there is the difficulty of relating the variations with position or local density within a virialized or partly virialized cluster (e.g. the morphology-density relation) to the expectations from linear theories of structure formation.
With these difficulties in mind, we shall briefly summarize the observations from the richest (but not necessarily the densest) environments, to the poorest, concentrating on the dE contribution.
The richest region that has been surveyed to faint limiting magnitudes is the
Coma cluster.
Thompson and
Gregory (1993)
catalogued dwarf galaxies in the Coma cluster, identifying faint cluster members on
the basis of
photographic b - r colors. They measured a faint-end slope 
=
-1.4, and a dwarf/giant ratio slightly lower than that of Virgo, when
converted to the same luminosity limit. There is a pronounced lack of
low surface-brightness dE galaxies (classified dSph by Thompson &
Gregory) within the central 0.3 degrees. They interpret this as the
result of tidal disruption in the core of the cluster of galaxies with
velocity dispersions less than 35 km s-1. While not
explicitly shown,
the lack of LSB dwarfs in the core of the cluster presumably leads to a
significantly flatter luminosity function. Outside the core of the
cluster, the early-type dwarfs follow the spatial distribution of E and
S0 galaxies.
The Virgo cluster was extensively studied by
Sandage et
al. (1985b).
They found an overall faint-end slope of
= -1.30 and a dE slope =
-1.35 to an absolute magnitude of -11.7. Comparison to the
Coma cluster
survey is a bit uncertain as the morphological classifications are
different and the galaxies are obviously not as well resolved at the
Coma distance. In contrast to
Coma, there is no strong deficit in the
number of faint dE's near the center of Virgo, but there is perhaps a
slight decline (see
Ferguson and
Sandage 1991).
Bothun et al. (1991)
re-examined the
Virgo cluster LF, adding in a sample of 24 extremely
low surface-brightness galaxies. They examine only the subset of galaxies
for which there is quantitative surface photometry, and apply
a rather indirect method of calculating the luminosity function,
assuming all galaxies have exponential profiles and deriving the
distribution of scale-lengths and central surface-brightnesses in a
region of parameter space essentially free from selection
effects. Modeling these distributions as power laws, they arrive at an
LF slope = -1.6. However, the
difference between this result and that of
Sandage et
al. (1985b) is
largely a result of the analysis technique, rather than the inclusion
of the additional LSB galaxies.
Ferguson and
Sandage (1991) compiled luminosity functions for seven
nearby groups
(including the previously-published Fornax and Virgo clusters), and
found faint end slopes -1.6 <
< -1.3, and turnover magnitudes
-23.2 < MB* < -21.2. A Schechter function was typically not a
very good fit to the data, but the systematic departures from the data
were different for different clusters and difficult to separate from
uncertainties in the magnitude estimates and completeness. The most
striking result from the survey is that the dwarf/giant ratio for
early-type (E, S0, dE, and dS0) galaxies varies by a factor of ~
5 from the richest to the poorest groups (see
Sect. 6.2).
This result has been extended to still poorer groups by
Vader and Sandage
(1991).
Surveys of the field typically find flatter luminosity functions, with
-1
(Efstathiou et
al. 1988;
Loveday et al. 1992;
Schmidt and Boller
1994),
and similar MB*.
Ferguson
(1992a) and
Lacey et al. (1993)
considered the effect of isophotal selection on the field-galaxy
luminosity function, and concluded that surface-brightness selection
alone could not account for the difference in the faint end slope
between the Virgo cluster
(Sandage et
al. 1985b)
and the field
(Loveday et
al. 1992).
The Loveday et al. survey was deep enough that relatively high
surface-brightness dE galaxies with -18 < MB < -16
would have been detected. However, if the dwarfs in the general field
follow the Virgo cluster dE luminosity function, it is quite
possible that the
total luminosity function turns up below the limits of the Loveday et
al. survey. The loose groups observed by
Ferguson and
Sandage (1991)
show such an upturn.
Although they did not explicitly compute a luminosity function,
the sample of
Eder et al. (1989)
provides some indication that
the LF in the field, even when probed to fainter surface-brightness
limits, is still dominated by HI rich galaxies, with a luminosity
function similar to that of Virgo. 122 galaxies from a well-defined
diameter-limited sample were observed at 21-cm and only 28 (22%)
were not detected. For those detected, the HI properties are
similar to the Virgo cluster irregular galaxy sample. However, in
the Virgo cluster dE's outnumber Sd+Im galaxies by at
least factor of 3 (the exact value depends on luminosity). The flat LF
of the field therefore reflects the flat LF of the Sd+Im types.
Jerjen et al. (1992)
arrived at a similar conclusion. For a
sample of groups within 10 Mpc, they found that the faint population
is dominated by irregulars (only 11% being dE+dS0) types,
and the resulting LF to MB = -15 was flat with = -0.98.
The Local Group provides information on the very faint tail
of the field-galaxy luminosity function. Van den Bergh (1992)
finds = -1.1, although the
number of galaxies is probably too small to argue that this is
significantly different from the steeper slope found in clusters.
Ferguson (1992a)
finds a much steeper slope =
-1.9 for the M81 group, although once again the statistics are very poor.
Compact groups in principle provide an interesting comparison to the
above studies. Such groups can have densities as high as the centers
of rich clusters, but have velocity dispersions and total numbers of
galaxies consistent with ``loose groups.''
De Oliveira &
Hickson (1991)
found a luminosity function significantly different from that of field,
loose-group, and cluster galaxies, with -0.2. More
recently,
Ribeiro et
al. (1994)
found = -0.82 ± 0.09 from a deeper survey of a large number
of groups. The Ribeiro et al. luminosity function appears to be
consistent with that seen in the field and in loose groups, but
significantly flatter than that of the Virgo cluster. The result is in
accord with the general variation of dwarf/giant ratio with richness (as
opposed to density), but is rendered somewhat uncertain by the rather
bright isophotal threshold of the Ribeiro et al. survey.
While there are still large uncertainties in the luminosity function and its environmental dependence at absolute magnitudes fainter than MB = -16, we believe that surveys to date have taught us the following:
| 1. | Selection effects are important, but not dominant. Dwarf galaxies are probably not a significant contributor to the mass-density of the universe. |
| 2. | The overall luminosity function changes with environment in a way that is consistent with a simple change in the relative proportions of dE and irregular galaxies. |
| 3. | The dE luminosity function is largely independent of environment,
with the exception of a possible depletion at the faint end in the
centers of rich clusters
(Thompson and
Gregory 1993).
The slope
in the range -16 < MB < -13 is -1.3. The
observations for clusters and groups, even when selection effects are
taken into account, do not support slopes as flat as = -1 or as
steep as = -1.8.
|
| 4. | LF variations are not consistent with simple biasing schemes. The relative abundance of low-luminosity galaxies is, if anything, lower in the field than in clusters. |