Past exam of the frontiers of Observational Astrophysics course of the University of Cambridge / 2014 / Paper 1 / 3 / d

Table of contents
- Solution d

Solution

thermal bremsstrahlung is continuum radiation from electrons accelerated in ion electric fields. For Maxwellian electrons,

j_{ν} \propto n_{e} \sum_{i} Z_{i}^{2} n_{i} T_{e}^{- 1/2} e^{- h ν / (k_{B} T_{e})} g_{ff} .

(1)

At high frequency the exponential cutoff constrains

T_{e}

; X-ray fitting must also model foreground absorption, line emission, detector response and possible multitemperature or nonthermal components. At radio frequencies,

h ν ≪ k_{B} T_{e}

, an optically thin free-free spectrum is almost flat, with the weak frequency dependence set mainly by the Gaunt factor.

For pure ionized hydrogen, radio free-free opacity gives

τ_{ν} \propto T_{e}^{- 3/2} ν^{- 2} g_{ff} EM, EM = \int n_{e}^{2} d l, I_{ν} = B_{ν} (T_{e}) (1 - e^{- τ_{ν}}) .

(2)

The turnover constrains emission measure; a resolved, optically thick source has brightness temperature approaching

T_{e}

. An unresolved source has an unknown beam filling factor, so its observed brightness need not equal its electron temperature. Angular size and an independent distance give a geometric depth

L

; a uniform model then yields $n_{e} = EM / L$ . Clumping instead measures a density-squared average, not the mean density.^[ref]

A free-free spectrum alone does not uniquely determine density and size. The density-depth degeneracy of free-free radiation is explicit: for a resolved uniform column,