42 Special Relativity
Redshift and Luminosity Distance. Consider an astronomical object emitting pho-
tons isotropically with power or absolute luminosity퐿.Attheluminosity distance푑L
from the object we observe only the fraction퐵s, its surface brightness, given by the
inverse-square distance law
퐵s= 퐿
4 휋푑L^2. (2.58)
Let us now find푑Las a function of푧in such a way that the Euclidean inverse-square
law [Equation (2.58)] is preserved. If the Universe does not expand and the object is
stationary at proper distance푑P, a telescope with area퐴will receive a fraction퐴∕ 4 휋푑P^2
of the photons. But in a universe characterized by an expansion푎(푡), the object is
not stationary, so the energy of photons emitted at time푡eis redshifted by the factor
( 1 +푧)=푎−^1 (푡e). Moreover, the arrival rate of the photons suffers time dilation by
another factor( 1 +푧), often called theenergy effect.Theapparent brightness퐵ais then
given by
퐵a= 퐿
4 휋푑P^2 ( 1 +푧)^2. (2.59)
Equating퐵a=퐵sone sees that푑L=푑P( 1 +푧), and making use of the expression in
Equation (2.57) one obtains
푑L(푧)≈푐
퐻 0(
푧+
1
2
( 1 −푞 0 )푧^2
)
. (2.60)
In Figure 2.5 we plot the function푑L(푧)for small values of푧.
Astronomers usually replace퐿and퐵by two empirically defined quantities,abso-
lute magnitude푀of a luminous object andapparent magnitude푚. The replacement
rule is
푚−푀=− 5 +5log푑L, (2.61)where푑Lis expressed in parsecs (pc) and the logarithm is to base 10. For example, if
one knows the distance푑Lto a galaxy hosting a supernova, its absolute magnitude푀
can be obtained from observations of its apparent magnitude푚.
Parallax Distance. Some measurements of퐻 0 depend directly on the calibration of
local distance indicators which form the first rung of a ladder of distance measure-
ments. The distances to relatively nearby stars can be measured by thetrigonometrical
parallaxup to about 30pc away (see Table A.1 in the appendix for cosmic distances).
This is the difference in angular position of a star as seen from Earth when at opposite
points in its circumsolar orbit. Theparallax distance푑pis defined as
푑p=푑P∕√
1 −푘휎^2. (2.62)
It has been found that most stars in the Galaxy for which we know the luminosity
from a kinematic distance determination exhibit a relationship between surface tem-
perature or color and absolute luminosity, theHertzsprung–Russellrelation. These
stars are calledmain-sequence starsand they sit on a fairly well-defined curve in the