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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

Equation 12.23 (12.23)

where Lc = power radiated by inverse Compton scattering, Ls = radio power radiated by synchrotron emission, 4pi r2 integnumnuc S dnu ~ 4pi r2 Sm nuc, Urad = 3L / 4pi r2 c = energy density of the radiation field, UB = B2 / 8pi = energy density of the magnetic field, R = the distance to the source, theta = angular size, and the radius rho = Rtheta / 2. Then using Equation (12.22) and recognizing that Sm / theta2 nu2 is proportional to the peak brightness temperature, Tm, and including the effect of second-order scattering, we have

Equation 12.24 (12.24)

where nuc is the upper cut-off frequency in MHz. Taking nuc ~ 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 gamma, and the spectral cut-off frequency nuc, 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

Equation 12.25 (12.25)

If the compact sources expand with conservation of magnetic flux, then Tm varies with radius rho as Tm propto rho-(nu-1) / (nu+4), so that for gamma ~ 1, Tm remains constant and otherwise depends only weakly on rho.

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