1.2. Distance Indicators

Measuring the distance to a galaxy almost always involves one of the following properties of the propagation of light: (1) The apparent brightness of a source falls of inversely with the square of its distance; (2) The angular size of a source falls off inversely with its distance. As a result, we can determine the distance to an object by knowing its intrinsic luminosity or linear size, and then comparing with its apparent brightness or angular size, respectively. If all objects of a given class had approximately the same absolute magnitude, we could immediately determine their distances simply by comparing with their apparent magnitudes. Such objects are called standard candles. Similarly, classes of objects whose intrinsic linear sizes are all about the same are known as ``standard rulers.'' True standard candles or rulers are, however, extremely rare in astronomy. It is much more often the case that the objects in question possess another, distance-independent property from which we infer their absolute magnitudes or diameters. For example, the rotation velocities of spirals galaxies are good predictors of their luminosities (Section 3), while the central velocity dispersions and surface brightnesses of ellipticals together are good predictors of their diameters (Section 4). Whether standard candles or rulers, or members of the more common second category, objects whose absolute magnitudes or diameters we can somehow ascertain are known as Distance Indicators, or DIs.

Absolute calibration of most DIs is not straightforward. One discovers that a particular distance-independent property is a good predictor of absolute magnitude because it is well correlated with the apparent magnitudes of objects lying at a common distance - in a rich cluster of galaxies, for example. Such data may be used to determine the mathematical form of the correlation (e.g., linear with a given slope). However, the cluster distance in most cases is not accurately known. Thus, the predicted absolute magnitude corresponding to a given value of the distance-independent property - the ``zero point'' of the DI - remains undetermined up to a constant, assuming one has no rigorous, a priori physical theory of the correlation, as is usually the case (but see below). Any distances obtained from the DI at this point will be in error by a fixed scale factor. This situation is obviously unacceptable for the Hubble constant problem, in which absolute distances are required. The remedy is to determine the zero point of the DI by applying it to galaxies whose true distances have been determined by an independent technique (e.g., Cepheid variables), as discussed above. Such a DI is said to be ``empirically'' calibrated.

For peculiar velocity surveys, the situation is simpler because absolute calibration is not required. However, the DI must still be calibrated such that it yields distances in km s-1, the radial velocity due to Hubble flow. For this, one must apply the DI to many galaxies, widely enough distributed around the sky and at large enough distances that peculiar velocities tend to cancel out. Only then can redshift be taken as a good indicator on average of distance in km s-1, and a calibration in velocity units thereby obtained (Willick et al. 1995, 1996). Empirical DI calibration, in this sense, is needed even for peculiar velocity work.

DIs of this sort tend to make some people nervous. They argue that a good distance estimation method should be based on solid, calculable physics. There are, in fact, a few such techniques. One involves exploitation of the Sunyaev-Zeldovich effect in clusters, in which comparison of Cosmic Microwave Background distortions and the X-ray emission produced by hot, intracluster gas yields the physical size of the cluster (cf. Rephaeli 1995 for a comprehensive review). Another method involves modeling time delays between multiple images of gravitationally lensed background objects (see the Chapter by Narayan and Bartelmann in this volume). Other DIs for which theoretical absolute calibration may be possible are Type II Supernovae, whose expansion velocities may be related to luminosities (Montes & Wagoner 1995; Eastman et al. 1996), and Type Ia Supernovae, whose luminosities may be calculated from theoretical modeling of the explosion mechanism (Fisher et al. 1993). Such approaches are indeed promising, and will undoubtedly contribute to the measurement of H0 over the next decade. However, at present these methods should be considered preliminary. Some of the underlying physics remains to be worked out, and many of the underlying assumptions will need to be tested. Furthermore, the data needed to implement such techniques are currently rather scarce. With the exception of Type Ia Supernovae (discussed in Section 6 in their traditional, empirical context), I will not discuss these methods further in this Chapter.

I will focus instead on methods that require empirical calibration. These DIs arise from astrophysical correlations Nature was kind enough to provide us with, but mischievous enough to deny us a full understanding of. The canonical wisdom, which states that we need hard physical theory that explains a DI in order to trust it, is a bit too exacting given our present theoretical and observational capabilities. We should conditionally trust our empirical DIs while recognizing the uncertainties involved. In particular, we must remember that since they possess no a priori absolute calibration, they must (for measuring H0) be carefully calibrated locally. We must also remain open to the possibility that they may not behave identically in different environments and at different redshifts. Our belief in their utility should be tempered by a healthy skepticism about their universality, and the distance estimates we make with them subjected to continuing consistency checks.