13.3.2. Inverse Compton Radiation

In very compact sources, in which the radiation energy density is comparable to the magnetic energy density, inverse Compton scattering will cause additional electron energy losses. For a homogeneous isotropic source

(13.16)

where L_c is the power radiated by inverse Compton scattering, L_s ~ 4 R² S_{m c} is the radio power radiated by synchrotron emission, U_rad = 3L / 4 d² c is the energy density of the radiation field, U_B = B² / 8 is the energy density of the magnetic field, R is the distance to the source, the angular size, and d = R is the source diameter. Then, from Equation (13.16), recognizing that S_m / ^{2 2} is proportional to the peak brightness temperature, T_m, and including the effect of second-order scattering, we have

(13.17)

where _m is the spectral upper cutoff frequency. Taking _m ~ 100 GHz, when T < 10¹¹, L_c / L_s << 1, inverse Compton scattering is not important. But when T > 10¹² K, the second-order term becomes important, L_c / L_s ~ (T_m / 10¹²)¹⁰, and inverse Compton losses become catastrophic. The exact value of the peak brightness temperature which corresponds to the case where L_c ~ L_c is somewhat dependent on the specific geometry, the value of p, and the spectral cutoff frequency, _m, but the strong dependence of L_c / L_s on T_m means that the maximum brightness temperature, T_m, cannot significantly exceed 10¹² K, independent of wavelength. This places a lower limit to the angular size of

(13.18)

Observations show that the peak brightness temperature of compact radio sources measured by VLBI is almost always in the range of 10¹¹ to 10¹² K. Thus, the angular size of an opaque source can be estimated from the peak flux density, S_m, and the self-absorption cutoff frequency, _c to give

(13.19)

The observed angular size is generally in good agreement with that expected from Equation (13.19) and the measured peak flux density and cutoff frequency, and there is no evidence that the peak brightness temperature ever exceeds 10¹² K. This is strong evidence that the compact radio sources indeed radiate by the synchrotron process, and that the radio emission is limited by inverse Compton cooling.

The inverse Compton scattered flux density, S_c, at an energy E is given by Marscher (1983) as

(13.20)

where _m is the upper cutoff frequency of the synchrotron radiation spectrum.

Near the E = 1 keV band of the Einstein Observatory, this becomes (Biermann and Zensus 1984)

(13.21)

where the effective brightness temperature, T_B, is approximately ^{2 2} S / 1.22 when T_B is expressed in units of 10¹² K.

Observations at millimeter wavelengths, where the effect of self-absorption is small, do indeed show a correlation between measured radio and X-ray flux density in the sense expected if the X-ray emission is due to inverse Compton scattering from the radio photons (Owen et al. 1981).