44 Special Relativity
and they are bright enough to be seen out to about 300kpc. A very important class
of standard candles are theCepheidstars, whose absolute luminosity oscillates with a
constant period푃in such a way that log푃∝ 1 .3log퐿. The period푃can be observed
with good precision, thus one obtains a value for퐿. Cepheids have been found within
our Galaxy where the period–luminosity relation can be calibrated by distances from
trigonometric parallax measurements. This permits use of the period–luminosity rela-
tion for distances to Cepheids within theLarge Magellanic Cloud(LMC), our satellite
galaxy. At a distance of 55kpc the LMC is the first important extragalactic landmark.
The resolution of individual stars within galaxies clearly depends on the distance to
the galaxy. This method, calledsurface-brightness fluctuations (SBFs),isanindicator
of relative distances to elliptical galaxies and some spirals. The internal precision of
the method is very high, but it can be applied only out to about 70Mpc.
Globular clusters are gravitationally bound systems of 10^5 –10^6 stars forming a
spherical population orbiting the center of our Galaxy. From their composition one
concludes that they were created very early in the evolution of the Galaxy. We already
made use of their ages to estimate the age of the Universe in Section 1.5. Glob-
ular clusters can also be seen in many other galaxies, and they are visible out to
100Mpc. Within the Galaxy their distance can be measured as described above, and
one then turns to study the statistical properties of the clusters: the frequency of stars
of a given luminosity, the mean luminosity, the maximum luminosity, and so on. A
well-measured cluster then becomes a standard candle with properties presumably
shared by similar clusters at all distances. Similar statistical indicators can be used to
calibrate clusters of galaxies; in particular the brightest galaxy in a cluster is a stan-
dard candle useful out to 1Gpc.
The next two important landmarks are the distances to the Virgo cluster, which
is the closest moderately rich concentrations of galaxies, and to the Coma cluster,
which is one of the closest clusters of high richness. The Virgo distance has been
determined to be 17Mpc by the observations of galaxies containing several Cepheids,
by theHubble Space Telescope[4]. The Coma is far enough, about 100Mpc, that its
redshift is almost entirely due to the cosmological expansion.
The existence of different methods of calibration covering similar distances is a
great help in achieving higher precision. The expansion can be verified by measuring
the surface brightness of standard candles at varying redshifts, theTolman test.If
the Universe does indeed expand, the intensity of the photon signal at the detector
is further reduced by a factor( 1 +푧)^2 due to an optical aberration which makes the
surface area of the source appear increased. Such tests have been done and they do
confirm the expansion.
TheTully–Fisherrelation is a very important tool at distances which overlap those
calibrated by Cepheids, globular clusters, galaxy clusters and several other methods.
This empirical relation expresses correlations between intrinsic properties of whole
spiral galaxies. It is observed that their absolute luminosity and their circular rotation
velocity푣care related by
퐿∝푣^4 c. (2.65)