Relativistic Distance Measures 43temperature–luminosity plot. Temperature can be determined from color—note that
astronomers define color as the logarithm of the ratio of the apparent brightnesses in
the red and the blue wavelength bands. Cool stars with surface temperature around
3000K are infrared, thus the part of their spectrum which is in the visible is domi-
nantly red. Hot stars with surface temperature around 12000K are ultraviolet, thus
the part of their spectrum which is in the visible is dominantly blue. The Sun, with a
surface temperature of 5700K, radiates mainly in the visible, thus its color is a blended
white, slightly yellow. Most main-sequence stars like our Sun are in a prolonged state
of steady burning of hydrogen into helium.
Once this empirical temperature–luminosity relation is established, it can be used
the other way around to derive distances to farther main-sequence stars: from their
color one obtains the luminosity which subsequently determines푑L. By this method
one gets a second rung in a ladder of estimates which covers distances within our
Galaxy.
Angular Size Distance. Yet another measure of distance is theangular size distance
푑A. In Euclidean space an object of size퐷that is at distance푑Awill subtend an angle
2 휃such that
2 휃=tan(퐷∕푑A)≈퐷∕푑A,where the approximation is good for small휃. This can serve as the definition of푑Ain
Euclidean space. In general relativity we can still use this equation to define a distance
measure푑A. From the metric in Equation (2.32) the radius of a source of light at
comoving distance휎is퐷=푎휎휃,so
푑A=퐷∕휃=푎휎=푎푆푘푑P. (2.63)This definition preserves the relation between angular size and distance, a property
of Euclidean space. But expansion of the Universe and the changing scale factor푎(푡)
means that as proper distance푑Por redshift푧increases, the angular diameter distance
initially increases but ultimately decreases. Light rays from the object detected by the
observer have been emitted when the proper distance to the object, measured at fixed
world time, was small. Because the proper distance between observer and source is
increasing faster than the speed of light, emitted light in the direction of the observer
is initially moving away from the observer.
The redshift dependence of푑Acan be found from Equations (2.57) and (2.37) once
푘is known. In Figure 2.5 we plot푑Afor the choice푘=0when
푑A=푎푑P=푑P
1 +푧
. (2.64)
The푘dependence makes it a useful quantity to determine cosmological parame-
ters. In particular,푘is sensitive to certain combinations of well-measured parameters.
Distance Ladder Continued. As the next step on the distance ladder one chooses
calibrators which are stars or astronomical systems with specific uniform properties,
so calledstandard candles.TheRR Lyraestars all have similar absolute luminosities,