Bremsstrahlung. Rybicki & Lightman Chapter 5. Bremsstrahlung. “Free-free Emission” “Braking” Radiation. Radiation due to acceleration of charged particle by the Coulomb field of another charge. Relevant for (i) Collisions between unlike particles: changing dipole emission
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Rybicki & Lightman Chapter 5
Radiation due to acceleration of charged particle by the Coulomb
field of another charge.
(i) Collisions between unlike particles: changing dipole emission
e-e-, p-p interactions have no net dipole moment
(ii) e- - ions dominate: acc(e-) > acc(ions) because m(e-) << m(ions)
recall P~m-2 ion-ion brems is negligible
(1) emission from single e-
pick rest frame of ion
calculate dipole radiation
correct for quantum effects (Gaunt factor)
(2) Emission from collection of e-
or non-thermal bremsstrahlung
(3) Relativistic bremsstrahlung (Virtual Quanta)
Electron moves past ion, assumed to be stationary.
b= “impact parameter”
- Suppose the deviation of the e- path is negligible
The dipole moment is a function of time during
- Recall that for dipole radiation
is the Fourier Transform of
After some straight-forward algebra, (R&L pp. 156 – 157), one can derive
in terms of impact parameter, b.
speed, v, which interact with a bunch of ions.
ne = electron density (# electrons / vol)
The # of electrons incident on one ion is
# e-s /Vol
around one ion, in terms of b
Again, evaluating the integral is discussed in detail in
R&L p. 157-158.
We quote the result
Energy per volume per frequency per time due to bremsstrahlung
for electrons, all with same velocity v.
Gaunt factors are quantum mechanical corrections
function of e- energy, frequency
Gaunt factors are tabulated (more later)
one velocity v.
Maxwell-Boltzmann Distribution Thermal Bremsstrahlung
Average the single speed expression for dW/dwdtdV
over the Maxwell-Boltzmann distribution with temperature T:
The result, with
In cgs units, we can write the emission coefficient
ergs /s /cm3 /Hz
Free-free emission coefficient
Ergs sec-1 cm-3
- Analytical approximations exist to evaluate them
- Tables exist you can look up
- For most situations,
so just take
Handy table, from Tucker: Radiation Processes in Astrophysics
(1) Usually optically thin. Then
(2) is ~ constant with hν at low frequencies
(3) falls of exponentially at
Important in hot plasmas where the gas is mostly ionized, so
that bound-free emission can be neglected.
Recall the emission coefficient, jν, is related to the absorption
coefficient αν for a thermal gas:
is isotropic, so
Because of term,
is very small unless ne is very large.
in X-rays, thermal bremsstrahlung emission can be
treated as optically thin
(except in stellar interiors)
e.g. Radio: Rayleigh Jeans holds
Absorption can be important, even for low ne
in the radio regime.
distance L=10 kpc
flux F= 10 -8 erg cm-2 s-1
R&L Problem 5.2
(a) What is T? Assume optically thin, thermal bremsstrahlung.
Turn-over in the spectrum at log hν (keV) ~ 2
central mass, M.
Find M, and the density of the cloud, ρ
Vol. emission coeff.
- Assume it is pure hydrogen, so ni = ne, then
ρ=mass density, g/cm3
Z=1 since pure hydrogen
For T=109 K
- Eqn (1) & (2)
Substituting L=10 kpc, F=10-8 erg cm-2 s-1