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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: phi(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: phi (L)measured = integµcinfty phi(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): integ0infty integ0µc Lphi(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 (alpha ~ - 2.8) LF recently proposed by Loveday [21] for the field would not work in the Coma cluster because of this constraint).

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