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For an adiabatic Bondi accretion flow, $\dot{M}\propto M^2{\rho }_{\infty }{c}_{s,\infty }^{-3}$. Inside the Bondi radius the speed and ion temperature are approximately virial, so $v\propto (M/r{)}^{1/2}$ and $T\propto M/r$, while mass conservation gives $n\propto \dot{M}{M}^{-1/2}{r}^{-3/2}$. The frequency-integrated thermal bremsstrahlung emissivity is proportional to $n^2{T}^{1/2}$. Its volume integral is dominated by the inner flow and consequently scales asSince the Eddington luminosity is proportional to $M$, this may be written $L/L_{\mathrm{Edd}}=C(\dot{M}{c}^2/L_{\mathrm{Edd}}{)}^2$. With the standard fully ionized-plasma constants and the Bondi profiles, $\sqrt{C}=9\times 10^{-3}$. Eliminating $\dot{M}$ from $\eta =L/(\dot{M}{c}^2)$ then givesThe cancellation of $M$, ${\rho }_{\infty }$, and $c_{s,\infty }$ expresses the scale-free character of the ideal flow. More physically, two-body emission scales as density squared, so an increasingly dilute flow radiates a progressively smaller fraction of its available accretion power.

Thermal bremsstrahlung is radiation from the acceleration of electrons in the Coulomb force of ions. At fixed temperature, the encounter rate per electron is proportional to ion number density $n_i$, and the number of radiating electrons per unit volume is $n_e$. Therefore $j_{\nu }\propto n_e{n}_i$ in a dilute, optically thin plasma. This is a binary-encounter rate, not a one-particle emission process.

For a fixed encounter geometry, the ionic electric field and hence the electron acceleration are proportional to ionic charge $Ze$. The Larmor formula is quadratic in acceleration, giving $j_{\nu }\propto Z^2$ at fixed species number densities. The dimensionless Gaunt factor $g(\nu ,T)$ corrects the simple classical result for the detailed quantum scattering and radiation calculation, after averaging over the thermal electron distribution. It depends on frequency and temperature and, in a more complete notation, ionic charge.

The primordial atomic cooling curve describes hydrogen-helium gas without heavy elements. Near the hydrogen excitation threshold it rises sharply, then shows hydrogen and helium excitation/ionization features. Bound-state losses weaken when both elements become highly ionized. At sufficiently high temperatures thermal bremsstrahlung gives a slowly rising tail. Composition, density convention and ionization assumptions fix the numerical normalization.