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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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Bremsstrahlung


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bremsstrahlung

Bremsstrahlung

Rybicki & Lightman Chapter 5

slide2

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

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

slide3

Method of Attack:

(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-

 thermal bremsstrahlung

or non-thermal bremsstrahlung

(3) Relativistic bremsstrahlung (Virtual Quanta)

slide5

Emission from Single-Speed Electrons

e-

v

Electron moves past ion, assumed to be stationary.

b= “impact parameter”

b

R

Ze

ion

- Suppose the deviation of the e- path is negligible

 small-angle scattering

The dipole moment is a function of time during

the encounter.

- Recall that for dipole radiation

where

is the Fourier Transform of

slide6

After some straight-forward algebra, (R&L pp. 156 – 157), one can derive

in terms of impact parameter, b.

slide7

Now, suppose you have a bunch of electrons, all with the same

speed, v, which interact with a bunch of ions.

slide8

Let ni = ion density (# ions/vol.)

ne = electron density (# electrons / vol)

The # of electrons incident on one ion is

d/t

# e-s /Vol

around one ion, in terms of b

slide9

So total emission/time/Vol/freq is

Again, evaluating the integral is discussed in detail in

R&L p. 157-158.

We quote the result 

slide10

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)

slide11

Naturally, in most situations, you never have electrons with just

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

slide12

where

In cgs units, we can write the emission coefficient

ergs /s /cm3 /Hz

Free-free emission coefficient

slide13

Integrate over frequency:

where

In cgs:

Ergs sec-1 cm-3

slide14

The Gaunt factors

- Analytical approximations exist to evaluate them

- Tables exist you can look up

- For most situations,

so just take

slide17

Important Characteristics of Thermal Bremsstrahlung Emissivity

(1) Usually optically thin. Then

(2) is ~ constant with hν at low frequencies

(3) falls of exponentially at

slide19

Examples:

Important in hot plasmas where the gas is mostly ionized, so

that bound-free emission can be neglected.

T (oK)

Obs. of

slide20

Bremsstrahlung (free-free) absorption

Brems emission

photon

e-

e-

ion

photon

Inverse Bremss.

free-free abs.

e-

collateral

Recall the emission coefficient, jν, is related to the absorption

coefficient αν for a thermal gas:

is isotropic, so

and thus

in cgs:

slide21

Important Characteristics of

(1)

(e.g. X-rays)

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)

slide22

(2)

e.g. Radio: Rayleigh Jeans holds

Absorption can be important, even for low ne

in the radio regime.

slide26

Spherical source of X-rays, radius R

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

slide27

(b) Assume the cloud is in hydrostatic equilibrium around a

central mass, M.

Find M, and the density of the cloud, ρ

Vol. emission coeff.

Vol.

1/r2

slide28

- Since T=109 K, the gas is completely ionized

- Assume it is pure hydrogen, so ni = ne, then

ρ=mass density, g/cm3

Z=1 since pure hydrogen

(1)

slide29

- Hydrostatic equilibrium  another constraint upon ρ, R

Virial Theorem:

For T=109 K

(2)

- Eqn (1) & (2) 

Substituting L=10 kpc, F=10-8 erg cm-2 s-1