On astronomical distances
It isn't hard to imagine how distances to stars can be measured. The fundamental method upon which all other methods rely consists in measuring the trigonometric parallax. The parallax of a star is the angle at the star subtended by the mean radius of the earth's orbit (assuming the radius is perpendicular to the view line). How can this angle be measured? Well, you have to follow for about half a year the changes in the star's position with respect to some other objects known to be at incomparably larger distance (you may assume they are at infinite distance!). The angle turns out to be so small that the method only yields good results for a few thousand of the nearest stars. To measure much larger distances, like those corresponding to galaxies and quasars, quite different methods are used and the results obtained are just approximations of the real, unknown distance. The precision of these methods is essential for the design and verification of cosmological models. At such huge distances the use of meters or parsecs is not convenient any more. The distance to the nearest cluster of stars, the Hyades, is 44 pc, while the distance to the huge galaxy M101 is 7.2 Mpc and the distance to the farthest observed objects is almost 3 Bpc. Therefore let's introduce another unit based on the notion of the magnitude of a star.
Let's recall first that the absolute magnitude M of a star is its brightness m reduced to the standard distance of 10 pc. Thus M = m + 5 - 5 log r. The value m-M (= 5 log r - 5) is called the module of the distance. Its physical meaning is simple: it represents the increase in brightness if the object were placed at the distance of 10 parsecs. The module of the distance to the sun is equal to -31.57 and the module of the distance to the galaxy cluster Virgo is +31.7.
Assume that the absolute magnitude of a star is known. Let it be the white dwarf in the Hyades. All its known parameters suggest that its absolute magnitude should be M=+11m.00. On the other hand observational data yield a magnitude m=+14^m.23. Thus the module of its distance is equal to +3.23.
Now, let's try to determine the module of the distance to a remote galaxy cluster in the Virgo constellation.
The most convenient parameter related to the distance between galaxies is the dependence between the period and the luminosity of the cepheids. The cepheids, very bright variable stars (M ~-2m up to -5m) have an interesting property: there is a simple relation between the period of changes and the absolute star magnitude. By determining the period and the mean visible luminosity we get the module of distance. However, the cluster in Virgo is so distant that we cannot see any cepheids in it. Since the weakest stars observed have luminosity m=+21m, we deduce that m-M>26.
The brightest red supergiants, being among the brightest stars to be found in galaxies, are also quite good indices of distance. Their absolute magnitude is M ~-8m. Neither have such red giants been found in the Virgo cluster, so m-M>29.
The brightest blue variable stars have absolute brightness up to M ~-10m, so if such stars could be observed in a galaxy, we might infer that the module of distance is smaller than 32. Unfortunately, blue variable supergiants need not occur in every galaxy, so we can draw no conclusion from the fact that they are not seen in Virgo.
Generally speaking, the brightest stars to be found in nature are the yellow supergiants of type F. The brightest of them all have magnitude M ~-11m. Such stars have been observed in one of the galaxies of the cluster in Virgo. The brightest of them has magnitude m=20m.8. Thus we get a first evaluation of the module of distance. It is equal to 31.7, with a considerable error of the order of 0.5.
Another method of determining large distances consists in measuring the maximal brightness of novae and supernovae. The cluster of our consideration is so remote that no explosions of novae could ever be seen, but explosions of supernovae were registered 8 times in the last 60 years. The maximal brightness of the brightest supernovae is M~ 20m.8 (the same value as for the entire galaxy). Knowing the properties of supernovae in closer galaxies and registering the brightness of supernovae in Virgo ranging from m=+12m to m=+14m, we can determine the module of distance for the cluster: m-M=32.2 (up to 0.6). This result does not contradict our previous conclusions. By investigating closer galaxies we can also analyse other relations, like the dependence between the brightness and the radii of ionized hydrogen clouds on the one hand and the spectra of stars that illuminate them on the other. The observation of one of such clouds found in the Virgo cluster yields m-M=31.9 (up to 0.3).
Galaxies contained in a cluster include a great number of globular star clusters. The comparison of the brightness of globular clusters with their occurrence frequency (scaled on close objects) yields a module of distance equal to 31.65 (up to 0.5).
Taking the mean value of all these and also of some other results and taking into account the grade of precision of the measurements we finally get the following module of the distance to the galaxy cluster in Virgo:
m - M = 31.7 (up to 0.08), i.e. r = 21.9 Mpc (up to 0.9).
The mean radial velocity of the system of 102 galaxies contained in the cluster is VR=1100 km/s with mean relative velocity equal to dVR= 68 km/s.
With the above data we can now determine the value of Hubble's constant:
H0= r-1 VR =50.3 (up to 4.2) km s-1 Mpc-1
The cluster in Virgo is one of the most distant systems with the distance determined by various independent methods. Distances of still more remote systems can only be computed under very restrictive assumptions, like e.g. that all spiral galaxies have the same diameter or that they all have the same luminosity.
On the other hand, using Hubble's law we can determine distances to even more distant objects (bright enough to allow measurement of the displacement of spectral lines towards the red) until the curvature of the space makes the law non-linear. Then we must choose a determined geometry for the universe in order to apply Hubble's law in a non-Euclidean version.