2.3.7. Correction for Bias in Extragalactic Samples

These five independent distance estimates to Virgo do not converge to a single value. Instead, there is a strong dichotomy. The TF, PNLF and SBF methods (which we regard as the most reliable based on the ability to calibrate their zeropoints using local galaxies) actually give an amazingly consistent distance to Virgo for the very first time. These distance estimates involve both samples of ellipticals and spirals. An unweighted mean of these three estimates gives (m - M) = 30.86 ± 0.15 (14.8 Mpc) if we use the Tonry (1991) and Pierce and Tully (1988) data sets. Using the Tonry et al. (1996) and Yasuda et al. (1996) data sets yields (m - M) = 31.01 ± 0.17 (17.02 Mpc). for the the distance modulus to the Virgo cluster. These estimates are significantly lower than the (m - M) = 31.4 - 31.8 distance that is derived via the other techniques. The lower of the two Virgo distance estimates is difficult to reconcile with H₀ = 50. Is there a resolution to these discrepant measures of the Virgo cluster distance modulus or are some of the techniques just wrong?

These discrepant distance estimates to Virgo are very difficult to sort out. Errors in distance can arise from either a) poor observations, b) systematic errors in zero point calibration, or c) lack of a fair sample. Sandage and collaborators in an extensive series of papers (see Sandage 1996a,b , Sandage et al. 1995, Federspiel et al. 1994, Sandage 1994a,b) vigorously argue that that the TF, PNLF and SBF methods all suffer from the lack of construction of a fair and equitable sample and that sample population biases have not been adequately treated. The general theme of the criticism is that Virgo samples have been constructed in such a way so as to yield samples which have the maximum brightness at fixed circular velocity (TF method), or fixed galaxy mass (PNLF method) or metallicity (SBF method). We regard this as remarkable coordination among different observing teams that have resulted in the same level of conspiracy and systematic error in the distance to Virgo. Moreover, if the criticism is true, then each of these three distance indicators we have described is worthless because they can never be applied in an unbiased manner to extragalactic distances. This position seems rather extreme and implies that we don't understand very much about the kinds of corrections that should be applied to extragalactic samples. Certainly mis-application of these corrections can lead to erroneous answers so one must be careful to fully understand the nature of these corrections.

There are basically three kinds of biases in extragalactic samples that can affect distance determinations. These biases are:

1. Sample incompleteness: at a fixed flux limit sources disappear beyond some distance. If you know the rough shape of the distribution in distance you can correct for the missing faint objects. In general, this form of bias is not present in the Virgo cluster samples of galaxies because Virgo is so nearby that even its faintest members can be detected (modulo the surface brightness arguments presented in Chapter 6).

2. The Scott Effect: This is frequently confused with the Malmquist bias which we discuss below. This effect comes about if some variable, call it S, correlates with intrinsic luminosity (for instance, S could be V_c). At fixed S there is some distribution in luminosity. The tendency to pick only the brightest objects in the distribution at fixed S is then a bias. This is frequently cited as a bias in the TF relation (at fixed line width, only the brightest objects are selected). One test of the existence of the bias is to see if the dispersion in the TF relation depends upon V_c. In the Virgo cluster samples it does not.

3. The Malmquist bias: This occurs when distance errors are involved. In a homogeneous distribution, more objects with intrinsically large distances will be scattered into the sample than will be scattered out of the sample (this is similar to the previously discussed Lutz-Kelker corrections). A correction for this bias can be made if the intrinsic scatter of the distance indicator is known. However, the correction only works well in the case where the sample is homogeneously distributed. As we will discuss in detail in Chapter 3, the galaxy distribution is anything but homogeneous, and application of the homogeneous correction is systematically wrong. For instance, if there is a large void at the back end of the sample then there are simply no distant galaxies to scatter into your sample and a homogeneous correction produces spurious results. The severity of the Malmquist effect is directly proportional to the intrinsic dispersion of the distance indicator (in the case of a Homogeneous Malmquist bias this is 1.38² mag). For the case of the TF relation, if the intrinsic dispersions is 0.4 mag then the correction is 20% in luminosity or 10% in distance. Estimates of the intrinsic dispersion in the TF relation are difficult to make because they are dependent on the range of M/L and SB in the sample under consideration. Estimates for the intrinsic scatter have ranged from 0.1 - 0.7 mag. However, most investigators find that this dispersion is 0.3 mag meaning that the Homogeneous Malmquist correction is small. However, as there is a void behind Virgo, the inhomogeneous Malmquist correction scheme as first proposed by Landay and Szalay (1992) is more appropriate but very difficult to apply in practice. This correction uses the actual distances estimated along the line of sight to formulate the volume sampling correction and hence needs a representative sample from which to form the distribution. For the case of Virgo, this correction is substantially smaller than the already small Homogeneous correction.

On the whole, we consider the various bias correction schemes which have been applied to the TF, PNLF and SBF samples by Sandage and co-workers that result in a 50% larger distance to be too large. For instance, Gonzalez and Faber (1997) have recently shown that Malmquist bias has only a 3.5 - 6.5% effect on the derived distance to Virgo - significantly smaller than has been used by Sandage and co-workers (see Sandage 1996c) in criticizing and correcting the short distance scale derived for Virgo. While, in general, these corrections are justified and necessary, the intrinsic scatter which is chosen, from which the correction is applied, is rather large. This is the key point. If, in fact, the scatter were as large as claimed for the TF, PNLF and SBF methods ( 0.5 mag), it is doubtful that the technique would have been used in the first place to estimate the distance to Virgo, let alone produce a consistent set of distances. Nevertheless, Sandage and Tammann (1995) are able to reconcile the TF, PNLF, and SBF distances to Virgo with their estimate of (m - M) 31.7 based on the GCLF or SN Ia luminosities. However, if the distance to the Virgo cluster is really 22 Mpc then there are two immediate problems: 1) the measured [O III] luminosity of the PN in the elliptical galaxies reaches a physically implausible value and 2) M31 is significantly under-luminous with respect to galaxies in the Virgo cluster with the same V_c.