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``Gerard de Vaucouleurs on the one hand, and Allan Sandage and Gustav Tammann on the other, arrived at estimates of the size of the universe, as measured by the Hubble constant, differing from each other by a factor of two. Moreover, when I asked the protagonists what was the range outside which they could not imagine the Hubble constant lying, these ranges did not overlap. Given that they were studying more or less the same galaxies with rather similar methods, often using the same observational material, I found this incredible.''

Michael Rowan-Robinson, in The Cosmological Distance Ladder (1985)


Aristarchus of Samos, in the third century B.C., may have been the first person to try measuring the size of his universe when he estimated the ratio of the distances between the Sun and Moon. His efforts, which were later followed by the work of such well-known scientists as Eratosthenes, Hipparchus, Ptolemy, Copernicus, and Kepler, led to a set of reasonably good relative distances within the solar system. With the advent of radar measurements in the mid-20th century, these relative values were placed on an absolute scale with unprecedented accuracy.

Once outside the solar system, however, there is an enormous loss in the accuracy of distance determinations. Measurements of nearby stars and galaxies typically carry uncertainties of 10 to 20%, and are thus 6 orders of magnitude less accurate than solar system measurements. (But the latter, of course, are more than 6 orders of magnitude closer!) For the more distant objects (up to 11 orders of magnitude more distant than solar system objects), even this seems remarkable, especially when one considers the number of rungs on the distance ``ladder'' (Figure 1), and the fact that each rung has its own ``10% errors''. Nevertheless, a number of steps are sufficiently redundant and secure that the accurate measurement of extragalactic distances seems a real possibility.

Figure 1. In this diagram we illustrate the various modern routes which may be taken to arrive at H0 and the genealogy and approximate distance range for each of the indicators involved. Population I indicators appear on the left-hand side and Population II on the right-hand side. The distance increases logarithmically toward the top of the diagram. The following abbreviations have been used to conserve space: LSC - Local Super Cluster; SG - Supergiant; SN - Supernovae; B-W - Baade-Weselink; PNLF - Planetary-Nebula Luminosity Function; SBF - Surface-Brightness Fluctuations; GCLF - Globular-Cluster Luminosity Function; pi - parallax.

On the other hand, it is over 2000 years since Aristarchus, and yet we are still unable to determine the scale of our universe to the satisfaction of the astronomical community. By itself, this failure is not a serious transgression; it takes time to solve difficult problems. It is, however, a major embarrassment that the leading proponents in the field have historically failed to agree within their stated errors. If we dismiss the possibility of repeated oversights in the analyses, then the most likely cause of the discrepancy is that the measurement uncertainties, internal and/or external, have continually been underestimated.

It is this line of reasoning that led Rowan-Robinson (1985, 1988) to survey the field of extragalactic distance determinations, and we strongly encourage anyone interested in this topic to consult these reviews. Other recommended reading on the subject includes Balkowski and Westerlund (1977), de Vaucouleurs (1982), Aaronson and Mould (1986), van den Bergh and Pritchet (1988), and van den Bergh (1989). In this paper, we examine very recent developments in the field. More importantly, we present detailed critiques of the principal methods currently used for measuring the extragalactic distance scale. In particular, we pay careful attention to the uncertainties in each method with the hope that the reader will derive a greater appreciation for the true range of acceptable values for galaxy distances.

We begin with a discussion of the Cepheid variable stars which are used to calibrate distance indicators within the Local Group and to measure distances out to ~ 5 Mpc. Despite major advances in technology during the past decade, Cepheids are too faint to use directly in distance determinations of cosmological interest (but see Pierce et al. 1992). These objects are, however, the primary calibrator for the secondary standard candles that are applied at much greater distances. Because of their vital role in establishing the distance scale, we review the basis for using Cepheids and the uncertainties incurred by adopting them as distance standards. Unless otherwise stated, we adopt the Cepheid distance to M31 of 770 kpc (Freedman and Madore 1990) and a differential extinction to M31 of E (B - V) = 0.08 (Burstein and Heiles 1984) throughout this review. In fact, we will adopt the Burstein and Heiles (1984) reddening values for all galaxies discussed in this review. The various approximation techniques (cf. Sandage and Tammann 1976, 1981; de Vaucouleurs 1981) necessarily impose a smoothing function to the Galactic extinction, whereas the empirical approach of Burstein and Heiles (1978, 1984) appears to offer significantly improved individual values.

We then turn our attention to other methods currently used to measure the extragalactic distance scale. Although many techniques have been proposed over the years, we restrict our discussion to those which are applicable to at least 20 Mpc and for which a detailed error analysis can be performed. For example, we reject arguments based on the relative size of M31 compared to that of spirals in distant clusters (Sandage and Tammann 1990) since this method in its most recent use provides neither a resultant distance nor an uncertainty (but see Stenning and Hartwick 1980). We also reject methods based on an extreme property, such as the brightest or largest object in a class (e.g., red supergiants, HII regions, emission rings) since these techniques are susceptible to Malmquist effects, population sample size effects (the ``Scott'' effect; Neyman et al. 1953), and usually provide no quantitative assessment of errors. [Humphreys (1988) has demonstrated that brightest red stars, corrected for parent galaxy absolute magnitude, are relatively good standard candles. However, like all methods based on a few extreme objects in a galaxy, it needs to be developed further to provide good statistical estimates of the errors arising from small sampling statistics and selection effects.] Instead we concentrate on the remaining methods (in no particular order) discussed by Sandage and Tammann (1990): the globular cluster luminosity function, novae, Type Ia supernovae, the HI line width - luminosity relation for spiral galaxies (Tully-Fisher), the planetary nebula luminosity function, the amplitude of surface brightness fluctuations, and the fundamental plane relationships for elliptical galaxies (Dn - sigma).

A separate section of the paper is devoted to each technique. Within each section, we discuss the physical basis of the method, its assumptions and calibration procedures, its observational requirements, and, most importantly, its strengths and weaknesses. We try to bring all internal and external errors, both random and systematic, to the surface so that the reader can judge the relative merits of each technique. We also attempt to identify additional observations and tests that would further strengthen or discredit the method.

For the purposes of this review, we define internal errors to be those which are intrinsic to the method, such as measuring magnitudes of galaxies, or fitting a function to a set of data points. These should be random since all systematic effects are removed during the reduction and calibration phase of the data analysis. Thus, the net internal error can usually be reduced by making additional measurements. External errors are those which accompany a measurement that is needed to apply the method, such as errors in Galactic extinction and absolute calibrations. These generally carry a higher probability of introducing a systematic error into the result than do the internal errors, and are more difficult to assess accurately. Note that for distance measurements beyond the point where geometrical methods apply, there is no way to define external uncertainties accurately since the true distances are never known exactly. We can assess the validity of the error estimates to some degree, however, by comparing the results from the different methods. We discuss our analysis of such a comparison in Sec. 11.

We also present a table of the distances to the Virgo cluster, the one common landmark for all the methods. The critical column in the table is not actually the distance, but rather the uncertainty in the distance. If the derived distances do not overlap within one or two times the stated errors, then something is very wrong. As will be seen from this table and the general comparison, we find good agreement among the methods, and we find that the errors are generally, but not always, well understood. If a distance scale controversy exists at all, it is in the difficult task of estimating the uncertainties.

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