12/18/18 1 8. Bremsstrahlung (contd) Electrodynamics of Radiation Processes http://www.astro.rug.nl/~etolstoy/radproc/ Chapter 5: Rybicki&Lightman Section 5.4 +examples Bremsstrahlung Produced by collisions between particles in hot ionized plasmas predominantly from collisions between electrons and ions In an electron-ion collision we can take the ion to be unaccelerated Precise results require quantum treatment, but useful approximate results can be obtained from classical calculation of the dipole radiation. 1. compute radiation power spectrum from a single collision with given electron velocity & impact parameter . 2. Integrate over impact parameter to get the emission from a single speed electron component 3. Integrate over a thermal distribution of electron velocities to obtain thermal bremsstrahlung emissivity 4. Consider thermal bremsstrahlung absorption & emission from a plasma with relativistic electron velocities free-free emission • Power spectrum from single collision (Lamor) dipole moment is d= -eR second derivative e a ¨ d = - • Integrate over impact parameter, emission from single speed electron • Integrate over thermal distribution of electron velocities optically thin optically thick Bremsstrahlung Spectral Energy Distribution radio X-ray - Relativistic Bremsstrahlung if the particles are moving with relativistic velocities, v ➝ c, then we have to compute the emission in the rest frame of the electron, which sees a fore- shortened and amplified electric field of a relativistic ion - and then transform back into the frame of the observer . Method of virtual quanta: classical treatment provides useful insight, even if a full understanding would require quantum electrodynamics. Relativistic Bremsstrahlung yʹ = b xʹ = vtʹ Lecture 6
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12/18/18
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8. Bremsstrahlung (contd)
Electrodynamics of Radiation Processes
http://www.astro.rug.nl/~etolstoy/radproc/
Chapter 5: Rybicki&LightmanSection 5.4 +examples
Bremsstrahlung
Produced by collisions between particles in hot ionized plasmas
predominantly from collisions between electrons and ions
In an electron-ion collision we can take the ion to be unaccelerated
Precise results require quantum treatment, but useful approximate results can be obtained from classical calculation of the dipole radiation.
1. compute radiation power spectrum from a single collision with given electron velocity & impact parameter.
2. Integrate over impact parameter to get the emission from a single speed electron component
3. Integrate over a thermal distribution of electron velocities to obtain thermal bremsstrahlung emissivity
4. Consider thermal bremsstrahlung absorption & emission from a plasma with relativistic electron velocities
free-free emission
• Power spectrum from single collision (Lamor)
dipole moment is d= -eR
second derivative e ad = �ev
• Integrate over impact parameter, emission from single speed electron
• Integrate over thermal distribution of electron velocities
optically thin
optically thick
Bremsstrahlung Spectral Energy Distribution
radio X-ray
-
Relativistic Bremsstrahlung
if the particles are moving with relativistic velocities, v ➝ c, then we have to compute the emission in the rest frame of the electron, which sees a fore-shortened and amplif ied electric f ield of a relativistic ion - and then transform back into the frame of the observer.
Method of virtual quanta: classical treatment provides useful insight, even if a full understanding would require quantum electrodynamics.
Relativistic Bremsstrahlung
y ʹ = b
x ʹ = vtʹ
Lecture 6
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Relativistic Bremsstrahlung
Given an electron moving in x-direction it will see f ield:
An electron at xʹ = 0; z ʹ = 0; yʹ = b
Lecture 6
Acceleration of a relativistic electron going past a nuclei
The electrons see a volume occupied by the foreshortened ions, and thus an oncoming stream of ions with v → c and density 𝛾ni (the ion density in the observer rest frame, or rest frame of the ions).
at higher frequencies Klein-Nishina corrections must be used
P =dW
dtdVd⌫=
16Z2e6neni3c4m2
ln⇣0.68�2c
⌫bmin
⌘~⌫ ⌧ �mc2
bmin ~ h/mc
For a thermal distribution of electrons a useful approximate expression for the frequency integrated power is:
relativistic correction
dW
dtdV= 1.4⇥ 10�27 T1/2 Z2 ne ni gB (1 + 4.4⇥ 10�10T)
Gamma-rays from the Milky WayGamma-ray emission is detected from our Galaxy which is thought to arise from relativistic Bremsstrahlung from high energy electrons.
the radiative energy is carried by photons with hν ~ Ee
energies in the range 30-100MeV, suggesting many relativistic electrons with 𝛾~100
High Energy (gamma-ray) spectrum of emission from the Milky Way
the likelihood of an energetic photon being emitted is small, however when it is emitted, it uses a signif icant fraction of the energy of electron.
probability distribution of energy packets being emitted
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An Example: Physical properties of a cluster
hot H gas in hydrostatic equilibrium
for a fully ionised plasma of pure hydrogen, Z=1 & ne = np
+
+ + ...
1/3 solar
h⌫ = 2⇥ 104eV
T =h⌫
k= 2.5⇥ 108K
I f the gas is in hydrostatic equilibrium, then the virial theorem applies:
p.e. = 2 k.e.an assembly of particles in stable equilibrium under their own gravity
equipartition of energy
R= 2 degree, d=20Mpc
for Virgo...
Mass of Cluster
R ⇠ 2⇥ 109⇣ M
M�
⌘cm
M = 1.1⇥ 1015M�
in the case of hydrostatic equilibrium - the mass of the cluster can also be inferred from the velocity dispersion of the individual galaxies, and the kinetic energy per unit mass will be the same for a galaxy as for an electron-ion pair.
density
R ⇠ 2⇥ 109⇣ M
M�
⌘cm
⇢ = 8⇥ 10�25 f1/2⇣ M
M�
⌘�3/2d
optically thin radiation
for a fully ionised plasma sphere of pure hydrogen, Z=1 & ne = np
d, distance to source; f, flux; V, volume
What can we learn from flux
Cooling TimeHow long can hot gas stay hot?
for a pure hydrogen gas, fully ionised:
HII regions: ne~102 - 103 cm-3; T~ 103 -104 K tc~ 100 - 1000 yr
Galaxy clusters: ne~103 cm-3; T~ 107 - 108 K tc~ 1010 yr
main cooling process at temperatures above T~ 107 K
as the density goes up, can cool more quickly
characteristic cooling time for H-plasma
energy content of the gas
rate at which energy is being radiated away=
Abell 3528 - Cooling Flows
ROSAT (X-ray)
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relative power from (L) line emission (R) radiative recombination &Bremsstrahlung (B)
X
X
X
Bremsstrahlung contribution Galactic Star Formation Region