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 mM (= 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=+11^{m}.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 ~2^{m}
up to 5^{m}) 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=+21^{m},
we deduce that mM>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 ~8^{m}. Neither
have such red giants been found in the Virgo cluster, so mM>29.
The brightest blue variable stars have absolute
brightness up to M ~10^{m}, 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 ~11^{m}. Such stars have been
observed in one of the galaxies of the cluster in Virgo. The brightest
of them has magnitude m=20^{m}.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~
20^{m}.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=+12^{m} to m=+14^{m},
we can determine the module of distance for the cluster: mM=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 mM=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:
The mean radial velocity of the system of 102
galaxies contained in the cluster is V_{R}=1100 km/s with
mean relative velocity equal to dV_{R}= 68 km/s.
With the above data we can now determine the value
of Hubble's constant:
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 nonlinear. Then we must choose a determined
geometry for the universe in order to apply Hubble's law in a nonEuclidean
version.
Tomasz Chlebowski
