12.5.6. Inverse Compton Scattering
The maximum brightness temperature of any opaque synchrotron source is limited by inverse Compton scattering to about 1012 ° K. This is the brightness temperature corresponding to the case where the energy loss by synchrotron radiation is equal to the energy loss by inverse Compton scattering and may be derived as follows (Kellermann and Pauliny-Toth, 1969).
For a homogeneous isotropic source
where Lc = power radiated by inverse Compton scattering, Ls = radio power radiated by synchrotron emission, 4 r2 mc S d ~ 4 r2 Sm c, Urad = 3L / 4 r2 c = energy density of the radiation field, UB = B2 / 8 = energy density of the magnetic field, R = the distance to the source, = angular size, and the radius = R / 2. Then using Equation (12.22) and recognizing that Sm / 2 2 is proportional to the peak brightness temperature, Tm, and including the effect of second-order scattering, we have
where c is the upper cut-off frequency in MHz. Taking c ~ 100 GHz, then for Tm < 1011 °K, Lc / Ls << 1 and inverse Compton scattering is not important; but for Tm > 1012 °K, the second-order term becomes important, Lc / Ls ~ (Tm / 1012)10, and the inverse Compton losses become catastrophic. The exact value of Tm corresponding to Lc / Ls = 1 is somewhat dependent on the specific geometry, the value of , and the spectral cut-off frequency c, but the strong dependence of Lc / Ls on Tm implies that Tm cannot significantly exceed 1012 °K, independent of wavelength. This places a lower limit to the angular size of
If the compact sources expand with conservation of magnetic flux, then Tm varies with radius as Tm -(-1) / (+4), so that for ~ 1, Tm remains constant and otherwise depends only weakly on .