Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 347 / 3 / b

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.