With measurements of this type there is always the long-standing worry that we are missing many LSB galaxies and so are underestimating the cluster galaxy counts at each magnitude [7]. Then the LF that we derive could be seriously in error.
In all clusters I studied (the distant clusters in Figure 1), I found that on going to fainter magnitudes, I did not find systematically more galaxies that had surface-brightnesses close to my detection threshold. This is indicative that I am not missing many galaxies because they have surface-brightnesses too low to be detected, but is far from being a rigorous statement to this effect. There may be, for example, an entire population of extremely LSB galaxies with surface-brightnesses far lower than my detection threshold. A more formal analysis is suggested.
So let us define (as Phillipps and Disney
[26]
did for spiral galaxies)
a bivariate surface-brightness / luminosity distribution function
in the usual way:
(µ,
L) dµ dL is the number
density of galaxies with surface-brightnesses between µ and
µ + dµ and luminosities between L and
L + dL.
When we measure a LF (as in the rest of this paper), what we are
really measuring is the contraction of this
bivariate function over high surface-brightnesses:
(L)measured =
µc
(µ,
L)dµ where µc is the
detection threshold.
There is a constraint on the other side of this contraction (i.e. over
low surface-brightnesses):
0
0µc
L(µ, L) dµ dL must not
exceed the intracluster background light (e.g.
refs. 22 and
37,
and more recently
refs. 31,
44, and
46).
This is only a limit because
much of the intracluster background light might come from stars that
have been tidally stripped from the giant galaxies; such stars are
not associated with LSB galaxies.
This constraint is however severe enough to rule out extremely steep LFs
in clusters (e.g. the steep
(
~ - 2.8) LF recently
proposed by Loveday
[21]
for the field would not work in the Coma cluster because of this constraint).