Free-free emission and absorption

For one ion species of charge number Z, electron density n e, ion density n i, and temperature T, the emissivity scales as $j_\nu\propto Z^2n_en_iT^{-1/2}e^{-h\nu/(k_BT)}g_{\mathrm{ff}}$. The Gaunt factor accounts for quantum corrections to the approximate encounter calculation and varies slowly in common regimes. The omitted coefficient carries the SI units needed for our per-solid-angle emissivity. Using proportionalities for ratios avoids copying a cgs numerical coefficient into an SI formula with an incompatible volume or angular normalization.

The exponential cutoff reflects the shortage of thermal electrons energetic enough to radiate high-energy photons. At radio frequencies much lower than k T over h, this factor is nearly one and the emissivity depends only weakly on frequency through the Gaunt factor. Relativistic electron temperatures, strong plasma-propagation effects, and non-Maxwellian distributions require a different treatment. At sufficiently high temperatures a plasma's frequency-integrated free-free cooling has the familiar square-root-temperature scaling, which is different from the inverse-square-root dependence at a fixed low frequency.

In the Rayleigh-Jeans limit the net absorption coefficient scales approximately as density product times T to the minus three halves times frequency to the minus two, with the same Gaunt-factor qualifications. A homogeneous isothermal slab without incident background then has intensity B times one minus the attenuation factor. At low enough frequency its own free-free absorption can make it thick, yielding a Rayleigh-Jeans intensity proportional to frequency squared for fixed apparent area. A thin radio spectrum is much flatter and is often approximated by a small negative spectral index when the Gaunt factor is included.

The turnover frequency where optical depth is near one depends on temperature, density structure and path length, not just on the emission process's name. A varying source size, ionization stratification, or foreground absorber can change the observed spectrum. Thermal free-free emission can obey its local emission-absorption relation without every atomic transition being in LTE, since Maxwellian electron encounters are the relevant thermal population here. Conversely, detecting a nearly flat radio continuum alone does not prove that all observed radiation is free-free; a superposition of other components can imitate a simple slope.

For fixed composition and temperature, a thin surface brightness weights the density product along the sightline. In pure ionized hydrogen it therefore measures a density square, not simply the square of a mean density. Dense subregions can dominate emission while occupying little volume. A path length or filling factor is needed to turn an emission measure into a characteristic density. An inference based on a uniform model returns a root-mean-square density, which can exceed the actual path-averaged density of a clumpy medium.

This density sensitivity also explains why compressing both electron and ion densities enhances local free-free emission strongly. Temperature changes must be included independently: at a fixed low frequency increasing T lowers the emission coefficient but lowers absorption even more, leaving a larger thermal source intensity. These scalings offer useful consistency checks before a detailed calculation. They assume the same ion species and ionization fractions remain appropriate and that the chosen ray is still thin if a thin formula is used. A density scaling cannot replace those regime checks.