Synchrotron emissivity and self-absorption

For an ultrarelativistic electron, let gamma be its Lorentz factor and alpha its pitch angle to magnetic field B. A conventional critical frequency is $\nu_c=3eB\sin\alpha\,\gamma^2/(4\pi m_e)$ in SI units. The spectrum is broad and its peak is not exactly at this critical frequency. The scaling with gamma squared and B is more useful here than treating the emitted radiation as a monochromatic line. A broad observed spectrum requires integration over electron energies and pitch angles.

For an isotropic pitch distribution the total synchrotron power is four thirds times the Thomson cross section, c, magnetic energy density, and gamma squared beta squared. Here $U_B=B^2/(2\mu_0)$ in joules per cubic metre. The same formula must not be used as a single fixed-pitch expression without its angular correction. At very high fields or photon energies quantum synchrotron corrections can matter; our classical ultrarelativistic treatment assumes they are negligible. The cooling estimate also neglects competing losses and ongoing acceleration unless explicitly included.

Let the number of electrons per volume per Lorentz-factor interval be $N(\gamma)=K\gamma^{-p}$ between finite lower and upper cutoffs. Gamma is dimensionless, so K has inverse-volume units. For frequencies well inside the corresponding power-law synchrotron range, the thin emissivity scales with frequency to the minus one half of p minus one. The normalization also depends on K, magnetic field and the pitch distribution. A measured thin slope can constrain p only if a single component and the stated regime dominate.

The same population can absorb synchrotron photons. For a homogeneous source the absorption coefficient has a different magnetic and frequency dependence from emissivity, giving a source-function scaling proportional to B to the minus one half times frequency to the five halves. At large positive self-absorption depth, intensity approaches this source function. The resulting five-halves slope assumes fixed apparent area and an appropriate power-law population. It is not a universal low-frequency spectrum: cutoffs, spatial gradients, external absorption and mixtures of source sizes can change it.

In an opaque power-law synchrotron source, different frequencies mainly interact with electrons of different characteristic energies. The effective brightness temperature can therefore grow with frequency even though no single Maxwellian temperature describes the whole population. The thick five-halves intensity slope differs from the frequency-squared Rayleigh-Jeans slope of an isothermal thermal emitter. Both follow transfer toward a source function, but the underlying population statistics and source functions differ. Calling all opaque radiation blackbody emission would erase that distinction.

Synchrotron aging can alter the electron distribution, while continued injection, escape, inverse-Compton losses, and adiabatic expansion can change the spectral history. Polarization and source geometry provide additional diagnostics beyond one continuum slope. Our hand estimates hold those ingredients fixed and use ratios in well-separated thin or thick regimes. They do not locate a real source's turnover from a slope alone, prove a unique magnetic field, or reconstruct a time-dependent electron acceleration mechanism. A full model needs an electron distribution, transfer geometry and independent observational constraints.