The flux density S as a function of frequency and the angular size of a radio source are the observables directly relevant to most cosmological problems. They are related to the intrinsic source luminosity L and projected linear size d as described below.
Consider an isotropic source at redshift z with spectral luminosity L at frequency (measured in the source frame). Its spectral flux density S measured at the same frequency (in the observer's frame) will be
(15.1) |
where A is the area of the sphere centered on the source and containing the observer and - ln(S / S0) / ln( / 0) is the two-point spectral index between the frequencies and 0 = / (1 + z) in the observer's frame. (Note that the negative sign convention for is used throughout this chapter.) The (1 + z)1+ term expresses the special relativistic Doppler correction; the geometry and expansion dynamics of the universe appear only in A. An "effective distance" D (Longair 1978) can be defined by A 4 D2. Since the area of the sphere centered on the observer and containing a source at redshift z is always A / (1 + z)2, the relation between (projected) linear size d and measured angular size is
(15.2) |
The "angular size" distance is defined by D d / = D / (1 + z). The "bolometric luminosity distance" Dbol defined by Sbol = Lbol / (4 Dbol2) is given by Dbol = D(1 + z).
In Friedmann models (cosmological constant = 0) with zero pressure, density parameter = 2q0, and current Hubble parameter H0 the effective distance is traditionally given (Mattig 1958) as
(15.3) |
However, this formula is numerically unstable for small z, the transformation (based on Terrell 1977)
(15.4) |
is better for numerical calculations. For particular values of , D reduces to the simpler forms:
(15.5a)
|
To describe the distributions of sources in space and time, we also need the comoving volume dV of the spherical shell extending from z to z + dz. It is dV = 4 D2 dr, where the comoving radial coordinate element is dr = - (1 + z)c dt. In a Friedmann universe the expansion rate is
(15.6) |
so
(15.7) |