**12.5.6. Inverse Compton Scattering**

The maximum brightness temperature
of any opaque synchrotron source is limited
by inverse Compton scattering to about
10^{12} ° 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

(12.23) |

where *L*_{c} = power radiated by inverse Compton scattering,
*L*_{s} = radio power radiated by synchrotron emission,
4 *r*^{2}
_{m}^{c}
*S d* ~
4 *r*^{2}
*S*_{m}
_{c},
*U*_{rad} = 3*L* /
4 *r*^{2} *c* =
energy density of the
radiation field, *U*_{B} = *B*^{2} /
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 *S*_{m} /
^{2}
^{2} is proportional to
the peak brightness temperature, *T*_{m}, and
including the effect of second-order scattering, we have

(12.24) |

where _{c} is the
upper cut-off frequency in MHz. Taking
_{c} ~ 100 GHz, then
for *T*_{m} < 10^{11} °K,
*L*_{c} / *L*_{s} << 1 and inverse Compton
scattering is not important; but for
*T*_{m} > 10^{12} °K, the second-order term
becomes important,
*L*_{c} / *L*_{s} ~ (*T*_{m} /
10^{12})^{10}, and
the inverse Compton losses become catastrophic.
The exact value of *T*_{m} corresponding to
*L*_{c} / *L*_{s} = 1 is somewhat dependent on
the specific geometry, the value of
, and the
spectral cut-off frequency
_{c}, but the strong
dependence of
*L*_{c} / *L*_{s} on *T*_{m}
implies that *T*_{m} cannot
significantly exceed 10^{12} °K, *independent* of
wavelength. This places a lower limit to the angular size of

(12.25) |

If the compact sources expand with
conservation of magnetic flux, then *T*_{m} varies
with radius as
*T*_{m}
^{-(-1) / (+4)}, so that
for ~ 1,
*T*_{m} remains constant and otherwise depends only weakly on
.