Compton scattering in a converging fluid flow - II. Radiation-dominated shock

The problem of Compton scattering in an optically thick fluid flow in which bulk motion is the dominant source of photon heating is illustrated by analysing a radiation-dominated, plane-parallel shock of speed u with photon to electron ratio greatly exceeding $$\sim(m_\text p/m_\text e)$$. In traversing the shock (of thickness $$\sim(c/u)$$ Thomson optical depths), a typical photon experiences $$(c/u)^2$$ scatterings, each one giving a secular fractional energy increase $$\sim(u/c)^2$$ and a total average increase of order unity. In a converging fluid flow, an exponentially small number of photons are accelerated to an exponentially large energy. Thus, a power-law spectrum will be transmitted at high frequencies. For a shock of Mach number M, bulk acceleration produces a spectral index α = (M² − ½)(M² + 6)(M² – 1)⁻², which tends to unity for a strong shock. The applicability of these results to quasars and the microwave background is briefly discussed.