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The flux density S as a function of frequency nu and the angular size theta 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 nu (measured in the source frame). Its spectral flux density S measured at the same frequency nu (in the observer's frame) will be

Equation 15.1 (15.1)

where A is the area of the sphere centered on the source and containing the observer and alpha ident - ln(S / S0) / ln(nu / nu0) is the two-point spectral index between the frequencies nu and nu0 = nu / (1 + z) in the observer's frame. (Note that the negative sign convention for alpha is used throughout this chapter.) The (1 + z)1+alpha 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 ident 4pi 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 theta is

Equation 15.2 (15.2)

The "angular size" distance is defined by Dtheta ident d / theta = D / (1 + z). The "bolometric luminosity distance" Dbol defined by Sbol = Lbol / (4pi Dbol2) is given by Dbol = D(1 + z).

In Friedmann models (cosmological constant Lambda = 0) with zero pressure, density parameter Omega = 2q0, and current Hubble parameter H0 the effective distance is traditionally given (Mattig 1958) as

Equation 15.3 (15.3)

However, this formula is numerically unstable for small Omegaz, the transformation (based on Terrell 1977)

Equation 15.4 (15.4)

is better for numerical calculations. For particular values of Omega, D reduces to the simpler forms:

Equation 15.5 (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 = 4pi D2 dr, where the comoving radial coordinate element is dr = - (1 + z)c dt. In a Friedmann universe the expansion rate is

Equation 15.6 (15.6)


Equation 15.7 (15.7)

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