N S x S

This was a familiar and important relation. Hubble (1936) had shown that counts of galaxies as a function of their optical brightness S fairly closely follow this relation. Since the relation assumes a uniform distribution Hub-ble's counts encouraged the assumption that the distribution of galaxies is close to homogeneous.

By the mid-1950s it was becoming clear that some galaxies are sources of radio radiation strong enough to be detectable by radio telescopes. Ryle (1955) presented the early estimate in Figure 3.5 of how the counts of these objects vary with their radio brightness. The data in this early study indicated that the counts increase with decreasing brightness S more rapidly than in equation (3.18). In a big bang cosmology this need not be a problem. Since distant sources are seen as they were in the past, because of the light travel time, one may account for the large number of faint sources by

39 In static flat space the energy flux density from a galaxy with luminosity L at distance r is S = L/(4nr2). The volume within this distance r is V = 4nr3/3. If all galaxies had the same luminosity, and their number density were n, the number of galaxies brighter than S would be N = nV = 4nnr3/3 a r3 a S-3/2, which is equation (3.18). Different galaxies have different luminosities. To take that into account separate the galaxies into luminosity classes. The law N a S—3/2 applies to each class, so it applies to the sum over all galaxies. The expansion of the universe, spacetime curvature, and galaxy evolution all change this relation when S is small enough (r is large enough).

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