6.3.1. Diameter-limited Catalogs

The two selection parameters which need to be specified for a galaxy catalog selected by diameter are the diameter limit _l and the isophotal level µ_l at which the diameter is measured. To show these it is only necessary to calculate the relative volume sampled as a function of the disk parameters, so the explicit value of _l is irrelevant. DP derive the visibility as a function of surface brightness at fixed luminosity, then state that this can be scaled by luminosity (i.e., V L^3/2). For a diameter limited catalog, they find that the volume sampled goes as

(13)

at fixed luminosity (see their equation 31). In order to show the full functional dependence on two parameters which describe the disk, we scale by luminosity as they suggest, arbitrarily normalized at M* = -21 as they chose to do. Their full visibility, including the effects of both central surface brightness and absolute magnitude, is thus

(14)

This is shown in Fig. 6-5 for the case µ_l = 25 mag arcsec^-2.

Figure 6-5:

The DP visibility function has a broad peak which always occurs at µ₀ = µ_l - 2.17. For the case of µ_l = 24.0, this peak would occur at µ₀ = 21.8. If there is an intrinsic distribution (M, µ₀) which populates a range of µ₀ as well as M, then the visibility function will cause catalogs to preferentially sample galaxies with µ₀ near this peak. That this is very close to the Freeman value is the root of Disney's argument. However, a robust prediction of the DP visibility formalism is that the central surface brightness typically found in diameter limited surveys will grow fainter as surveys are pushed deeper. Contrary to this expectation, the peak in the apparent distribution is observed not to vary with µ_l (Phillipps et al. 1987), leading some to conclude that surface brightness selection effects are not important (van der Kruit 1987). Also, as noted by DP, the peak is too broad to explain the narrow observed distribution.

An alternative derivation of these effects can be done as follows. For diameter limited surveys, the requirement is that = 2r _l when µ(r) = µ_l. From equation 5 it follows that

(15)

The maximum distance at which a galaxy can lie and meet the selection criteria occurs when = _l, so

(16a)

For arbitrary _l the relative volume sampled V d_max³ is

(16b)

This is plotted in Figure 6-6.

Figure 6-6:

There is a very significant difference between Figures 6-5 and 6-6 (and between Figures 6.7 and 6.8). The "visibility" in Fig. 6-5 does not have a peak at some preferred surface brightness, but increases without bound as µ₀ becomes brighter. In contrast, Figure 6-6 shows the variation of V with µ₀ is extremely rapid, so we expect that the apparent surface brightness distribution should always be very strongly peaked around the brightest value which exists in the intrinsic distribution, regardless of the value of µ_l. In this treatment, galaxy size and surface brightness are orthogonal properties of a galaxy for purposes of selection, and it is really necessary to consider their separate effects fully. For instance, very luminous galaxies can be missed if their central surface brightness is very low. To date, available data strongly show that µ₀ and _l are uncorrelated (see Figure 6-14).

Figure 6-7:

Figure 6-8: Various representations of the Visibility function of galaxies adapted from McGaugh et al. (1995). Representations where scale length and central surface brightness are used to define the visibility function are substantially different than we central surface brightness and absolute magnitude are used.

Mathematically, one can use equation transfrom _l in equation 16b with M to recover equation 14. While algebraically correct, this makes little physical sense. Absolute magnitude and central surface brightness are not independent quantities, so the axes of Fig. 6-5 have a high degree of covariance. This is the reason that this representation of the selection effect in this manner is so misleading. In order to keep M fixed as µ₀ varies, it is necessary to also change _l. This is unphysical. At issue is how any given galaxy will be selected. Imagine a galaxy of scale length _l. As µ₀ varies, M varies with it but _l remains fixed. In order to also hold M fixed we must alter the size of the object so that it is no longer the same object. In this sense, the characteristic size and surface brightness of a galaxy are more fundamental properties that its luminosity, a quantity which loses information by lumping together these two distinct pieces of information (through equation 11) in a degenerate manner.