Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
Copyright © 1997 by . All rights reserved

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A red line in the history of astronomy is the extension of distance measurements into deep space as a result of efforts to take advantage of what Nature offers astronomers. Ptolemy (ca 150) stated that "none of the stars has a noticeable parallax (which is the only phenomenon from which distances can be derived)," which illustrates the situation of when we see a class of celestial bodies and also have in mind a sound method but cannot apply it because of insufficient observational means. Of course, only after Copernicus was there strong theoretical pressure to search for stellar parallaxes as a crucial cosmological test and as a method of distance determination. However, long before this triangulation method was successfully applied by Bessel, Henderson, and Struve in the 1830s (see Hoskin 1982), the photometric method, based on the inverse square law of light flux from a point source introduced by Kepler, had been recognized as a possible way of getting information on stellar distances. In 1668, James Gregory applied it to the distance of Sirius, using the Sun as the calibrator star. In Gregory's method of "standard candles," one had to assume that stars are other suns, identical to our Sun, and are observed through the transparent Euclidean space where Kepler's inverse square law is valid.

1.1. Stars and Galaxies are Gathered from the Sky and not from Space

The above notes illustrate how determination of distances is intimately related to our general astronomical knowledge and assumptions, and also to our abilities to measure the directions and fluxes of weak photon streams. Though knowledge and observational methods, and hence construction of the distance ladder, are steadily advancing, there are fundamental difficulties that will always haunt the measurers of the universe.

The astronomer makes observations from a restricted vantage point in time and space. In fact, he or she does not observe celestial bodies in space, but in the sky (as traces on photographic plates or CCD images). Fortunately, very luminous objects exist that may be detected even from large distances. This diversity in the cosmic zoo, which allows one to reach large distances and hence makes cosmology possible, also involves problematic aspects.

First, from large distances only highly luminous objects are detectable, and the photons usually do not carry information on how much the objects differ from their average properties: There are no genuine Gregory's standard candles. Second, objects in the sky that are apparently similar, i.e. have the same distance modulus, actually have a complicated distribution of true distances. Distances larger than those suggested by the distance modulus are favored because of the volume (r2 dr) effect. Third, at large distances there is much more space than within small distances from our position. Because very luminous objects are rare, these are not found in our vicinity. We can see objects that as a class might be useful indicators of large distances but perhaps cannot make the crucial step of calibration, which requires a distance ladder to reach the nearest of such objects (Sandage 1972). As an extreme example, even if one could find standard candles among luminous quasars (Teerikorpi 1981), there is no known method to derive their distances independently of redshift.

When one uses "standard candles" or "standard rods," calibrated on a distance ladder, systematic errors creep into the distance estimates. When related to the above problems, they are often collectively called Malmquist-like biases.

According to Lundmark's (1946) definition, a distance indicator is a certain group of astronomical objects that has the same physical properties in different galaxies. The principle of uniformity of natural law is assumed to be valid when one jumps from one galaxy to another. Slightly expanding Lundmark's definition, one may say that a distance indicator is a method where a galaxy is placed in three-dimensional space so that its observed properties agree with what we know about galaxies, their constituents, and the propagation of light. An ideal distance indicator would restrict the galaxy's position somewhere in a narrow range around the true distance if that indicator has an intrinsic dispersion in its properties, and a set of such indicators should lead to a consistent picture of where the galaxies are. In practice, even a "good" indicator is affected by sources of systematic error due to the intrinsic dispersion.

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