Synchrotron and Compton Spectra from a Steady-State Electron
  Distribution
  Energy densities of relativistic electrons and protons in extended galactic
and intracluster regions are commonly determined from spectral radio and
(rarely) $\gamma$-ray measurements. The time-independent particle spectral
density distributions are commonly assumed to have a power-law (PL) form over
the relevant energy range. A theoretical relation between energy densities of
electrons and protons is usually adopted, and energy equipartition is invoked
to determine the mean magnetic field strength in the emitting region. We show
that for typical conditions, in both star-forming and starburst galaxies, these
estimates need to be scaled down substantially due to significant energy losses
that (effectively) flatten the electron spectral density distribution,
resulting in a much lower energy density than deduced when the distribution is
assumed to have a PL form. The steady-state electron distribution in the
nuclear regions of starburst galaxies is calculated by accounting for Coulomb,
bremsstrahlung, Compton, and synchrotron losses; the corresponding emission
spectra of the latter two processes are calculated and compared to the
respective PL spectra. We also determine the proton steady-state distribution
by taking into account Coulomb and pion production losses, and briefly discuss
implications of our steady-state particle spectra for estimates of proton
energy densities and magnetic fields.
