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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

Strong et al. 2004 ApJ, 613, 962

Relativistic Bremsstrahlung Spectrum

Pion decayRelativistic BremsstrahlungInverse Compton ScatteringExtra-galactic Background

non-rel

Ionisationloses

Bremsstrahlung loses

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

Bremsstrahlung AbsorptionFrom Bradt: Astrophysics Processes

HII star-forming regions W3(A) mainW3(OH)

Final remarks on Bremsstrahlung

a photon is emitted when a moving electron deeply penetrates the Coulomb f ield of positive ion and then is decelerated

the energy of the photon can never exceed the kinetic energy of the electron

generally the ''classical'' description of the phenomenon is adequate, the quantum mechanics requires an appropriate Gaunt Factor only

we have either ''thermal'' or ''relativistic'' bremsstrahlung, depending on the population (velocity/energy distribution) of the electrons.