A fully ionised hydrogen plasma has electron and ion number densities (in ) and…

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Radiation, radiative transfer and stellar atmospheres

Stars from the Inside, Ch Hot and fast electrons: bremsstrahlung, Compton scattering and synchrotron emission

Problem

A fully ionised hydrogen plasma has electron and ion number densities

n_e=n_i=n

(in

\mathrm{m^{-3}}

) and electron temperature

T

. For electrons of a single speed

v

, the free-free (bremsstrahlung) energy radiated per unit volume, per unit time and per unit angular-frequency interval, from encounters with ions of charge

Ze

\frac{\mathrm{d}W}{\mathrm{d}\omega\,\mathrm{d}V\,\mathrm{d}t} = A\,n_en_iZ^2\,v^{-1}\ln\Lambda,

where

A

is a constant (independent of

n

v

T

and

\omega

) and

\ln\Lambda

, the logarithm of the ratio

\Lambda

of the largest to the smallest impact parameter that contributes, may be treated as a constant throughout this problem. A photon of frequency

\nu

can only be produced by an electron whose kinetic energy is at least

h\nu

. ((a)) Electron speeds follow a Maxwell-Boltzmann distribution

f(v)\,\mathrm{d}v = 4\pi v^2\left(\dfrac{m_e}{2\pi k_BT}\right)^{3/2}e^{-m_ev^2/2k_BT}\,\mathrm{d}v

. Using the energy threshold above, integrate

v^{-1}f(v)

from

v_{\min}=\sqrt{2h\nu/m_e}

\infty

to show that the volume emission coefficient is

j_\nu \propto n_en_iZ^2\,T^{-1/2}\,e^{-h\nu/k_BT}

. You may use

\int_{v_{\min}}^{\infty} v\,e^{-\alpha v^2}\,\mathrm{d}v = \dfrac{1}{2\alpha}e^{-\alpha v_{\min}^2}

. ((b)) Kirchhoff's law for thermal emission states

\alpha_\nu = j_\nu/B_\nu

, with

B_\nu = \dfrac{2h\nu^3}{c^2}\left(e^{h\nu/k_BT}-1\right)^{-1}

the Planck function. Use this and part (a) to show

\alpha_\nu^{ff} \propto n_en_iZ^2\,T^{-1/2}\,\nu^{-3}\left(1-e^{-h\nu/k_BT}\right)

. ((c)) A solar-like coronal loop has

n_e=n_i=10^{15}\,\mathrm{m^{-3}}

T=2\times10^{6}\,\mathrm{K}

and

Z=1

, and is observed at

\nu=10^{9}\,\mathrm{Hz}

. Show that

h\nu\ll k_BT

here, and give the power-law index

b

such that

\alpha_\nu^{ff}\propto\nu^{b}

in this limit.

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