Continuum Processes

(in High-Energy A

strophysics)3rd X

-ray Astronom

y schoolW

allops Island May 12-16 M

ay

Ilana HA

RRUS

(USRA

/NA

SA/GSFC)

Bremsstrahlung

What is Brem

sstrahlung?

(and why this bizarre name of “braking radiation”?)

Historically noted in the context of the study of

electron/ion interactions. Radiation of EM waves because

of the acceleration of the electron in the EM field of the

nucleus.

And … when a particle is accelerated it radiates .

Bremsstrahlung

Important because for relativistic particles, this can

be the dominant m

ode of energy loss.

Whenever there is hot ionized gas in the U

niverse,there will be Brem

sstrahlung emission.

Provide information on both the m

edium and the

particles (electrons) doing the radiation.

The complete treatm

ent should be based on QED

===>in every reference book, the com

putations are made

“classically” and modified (“Gaunt” factors) to take

into account quantum effects.

Bremsstrahlung (non relativistic)

Use the dipole approxim

ation (fine for electron/nucleusbrem

sstrahlung)---

-e

Ze

v

b

Electron moves m

ainly in straight line-- Dv = Ze

2/me Ú(b

2+v2t

2) -3/2 bdt = 2Ze2/m

bv

Electric field: E(t) =Ze3sinq/m

e c2R(b

2+v2t

2)

R

Bremsstrahlung (for one N

R electron)

(using Fourier transform)

E(w) =Ze3sinq/m

e c2R p/bv e

-bw/v

Energy per unit area and frequency is:

dW/dA

dw = c|E(w)| 2

So that integrated on all solid angles:

dW(b)/d

w = 8/3

p (Z2e

6/me 2c

3) (1/bv) 2 e-2b

w/v

Bremsstrahlung

For a distribution of electrons in a medium

with iondensity n

i . Electron density ne and sam

e velocity v.

Emission per unit tim

e, volume, frequency:

dW/dVdtd

w = ne n

i 2pv Ú •

bmin dW

(b)/dw

bdb

Approxim

ation: contributions up to bm

ax

This implies (after integration on b)

dW/dVdtd

w = (16e6/3m

e 2vc3) n

e ni Z

2 ln(bm

ax /bm

in )

with bm

in ~ h/mv and b

max ~ v/w

Bremsstrahlung

If QED

used, the result is: dW

/dVdtdw = (16

pe6/3

3/2me 2vc

3) ne n

i Z2 g

ff (v,w)

Karzas & Latter, 1961, ApJS

, 6, 167

Bremsstrahlung

Now for electrons with a M

axwell-Boltzmann velocity

distribution. The probability dP that a particle has avelocity within d

3v is:

dP µ e

-E/kT d3v µ

v2e

(-mv2/2kT)dv

Integration limits: 1/2 m

v2 >> h

n (Photon discreteness effect)and using d

w=2

pdn

dW/dVdtd

n =

(32pe

6/3me c

3) (2p/3kTm

e ) 1/2 ne n

i Z2 e

(-hn/kT) <g

ff > ==

6.8 10-38 T

-1/2ne n

i Z2 e

(-hn/kT) <g

ff > erg s-1 cm

-3 Hz

-1

<gff > is the velocity average Gaunt factor

Bremsstrahlung

From Rybicki & Lightm

anFig 5.2 (corrected) --originally from

Novikov

and Thorne (1973)

Approxim

ateanalytic form

ulaefor <g

ff >

Bremsstrahlung

From Rybicki & Lightm

anFig 5.3 -- originally fromKarzas & Latter (1961)

u =hn/kT ;

g 2 = Ry Z2/kT

=1.58x105 Z

2/T

Num

erical values of <gff >.

When integrated over frequency:

dW/dVdt = (32

pe6/3hm

e c3) (2

pkT/3me ) 1/2 n

e ni Z

2 <gB >

== 1.4x10-27 T

1/2ne n

i Z2 <g

B > erg s-1 cm

-3(1+4.4x10

-10T)

Cyclotron/Synchrotron Radiation

First discussed by Schott (1912). Revived after 1945in connection with problem

s on radiation fromelectron accelerators, …

Very important in astrophysics: Galactic radio

emission (radiation from

the halo and the disk), radioem

ission from the shell of supernova rem

nants, X-ray

synchrotron from PW

N in SN

Rs…

Radiation emitted by charge m

oving in a magnetic

field.

Cyclotron/Synchrotron Radiation

As with Brem

sstrahlung, complete (rigorous) derivation is

quite tricky.

First for a non-relativist electron: frequency of gyration inthe m

agnetic field is wL =eB/m

c

= 2.8 B1G M

Hz (Larm

or)

Frequency of radiation == wL

Synchrotron Radiation

Because of relativistic effects: beaming and

Dq~1/g

Gyration frequency wB = w

L /gO

bserver sees radiations for duration Dt<<< T =2p/w

BThis m

eans that the spectrum includes higher harm

onicsof w

B .M

aximum

is at a characteristic frequency which is:

wc ~ 1/Dt ~ g 2eB

^ /mc

Angular distribution of

radiation (acceleration ^velocity).Rybicki & Lightm

an

Synchrotron RadiationTotal em

itted radiation is:

P= 2e4B

^ 2/3m2c

3 b2g 2 = 2/3 r

0 2 c g 2 B^ 2 when g>>1

Or P µ

g 2 c sT U

B sin2q (U

B is the magnetic energy

density)

P~1.6x10-15 g 2 B

2 sin2q

erg s-1

Life time of particle of energy t µ

E/P ~ 20/gB2 yr

Example of Crab-- Life tim

e of X-ray producing electron is

about 20 years.P µ

1/m2 : synchrotron is negligible for m

assive particles.

Synchrotron RadiationThe com

putation of the spectral distribution of thetotal radiation from

one UR electron: (com

putation done in bothpolarization directions -- parallel and perpendicular to the direction of them

agnetic field).

P^(w) = (÷3e3/4

pmc

2) B sinq [F(x)+G(x)] ; x=w/wc

P;

(w) =(÷3e3/4

pmc

2) B sinq [F(x)-G(x)]

Where F(x)==x∫

∞x K 5/3 (y)dy ; G(x)==x

K 5/3 (x) and Km

odified Bessel function

and wc = 3/2

g 2 wL sinq

Total emitted power per frequency:

P(w)=(÷3e3/2

pmc

2) B sinq F(w/ w

c )

0.29

Synchrotron RadiationH

ypothesis: Energy spectrum of the electrons between

energy E1 and E2 can be approximated by a power-law --

N(E)=KE

-r dE (isotropic, homegeneous).

Num

ber of e- per unit volum

e, between E and E+dE (inarbitrary direction of m

otion)

Intensity of radiation in a homogeneous m

agnetic field:

I(n,k)=(÷3/r+1) G(3r-1/12) G(3

r+19/12) e3/m

c2 (3e/2

pm3c

5) (r-1)/2K [B sinq] (r+1)/2 n

-(r-1)/2

Average on all directions of m

agnetic field (for astrophysical applications).L is the dim

ension of the radiating region

I(n)= a(r) e3/m

c2 (3e/4

pm3c

5) (r-1)/2 B(r+1)/2 K L n

-(r-1)/2 erg cm-2 s

-1 ster-1 H

z-1

wherea(r) = 2

(r-1)/2 ÷(3/p)G(3r-1/12) G(3

r+19/12) G(r+5/4)/[8(r+1) G(r+7/4)]

Synchrotron Radiation

If the energy distribution of the electrons is a powerdistribution

Synchrotron Radiation

y1 (r) and y

2 (r) are tabulated (or you can compute them

yourself..).

Estimating the two boundaries energies E

1 and E2 of electrons

radiating between n1 and

n2.

E1 (n) ≤ m

c2 [4

pmcn

1 /3eBy1 (r)] 1/2 = 250 [n

1 /By1 (r)] 1/2 eV

E2 (n) ≤ m

c2 [4

pmcn

2 /3eBy2 (r)] 1/2 = 250 [n

2 /By2 (r)] 1/2 eV

If interval n2 / n

1 << y1 (r)/y

2 (r) or if r<1.5 this is only rough estimate

Synchrotron Radiation

Expected polarization:

(P^(w) - P;

(w))/(P^(w) + P;

(w)) == (r+1)/(r+ 7/3)

can be very high (more than 70%

).

Synchrotron Self-Absorption

A photon interacts with a charged particle in a

magnetic field and is absorbed (energy

transferred to the charge particle).This occurs below a cut-off frequency

The main result is:

For a power-law, the optically thick spectrum is

proportional to B-1/2v

5/2 independent of thespectral index.

====> break frequency

Compton/Inv Com

pton Scattering

Es =h

ns

For low energy photons (hn << m

c2), scattering is classical

Thomson scattering (E

i =Es ; s

T = 8p/3 r

0 2)

More general: E

s =Ei (1+E

i (1-cosq)/mc

2) -1 or

ls -l

i =lc (1-cosq) (l

c =h/mc)

This means that E

s is always smaller than E

i

Even more general: (Klein-N

ishina)

ds/d

W = 1/2 r

0 2 y2(y+1/y -sin

2q) with y=Es/ E

i

Pe , E

Ei =h

ni

q

Compton/Inv Com

pton Scattering

If electron kinetic energy is large enough, energytransferred from

electron to the photon: InverseCom

pton

One can use previous form

ula (valid in the restfram

e of the electron) and then Lorentztransform

.

So : Ei foe=E

i labg(1-bcosq) then Ei foe becom

es Es foe

and Es lab=E

s foeg(1+bcosq')

This means that E

s lab µ E

i lab g 2

The boost can be enormous!

Inverse Com

pton Scattering

Inverse Compton

Scattering

Compton/Inv Com

pton Scattering

The total power emitted:

Pcom

pt = 4/3 sT cg 2b

2Uph [1-f(g,E

i lab)] ~ 4/3 sT cg 2b

2Uph

And U

ph is the initial photon energy density

We had P

sync µ g 2 c s

T UB

In fact : Psync /P

compt = U

B /Uph

Synchrotron == inverse Compton off virtual photons in

the magnetic field.

Both synchrotron and IC are very powerful tools ===>direct access to m

agnetic and photon energy density.

Not covered

ßThermal brem

sstrahlung absorption (energy absorbedby free m

oving electrons)

ßBlack body radiation

ßTransition radiation (often not mentioned)

Books and references

ßRybicki & Lightman "Radiative processes in A

strophysics"

ßLongair "High Energy A

strophysics"

ßShu "Physics of Astrophysics"

ßTucker "Radiation processes in Astrophysics"

ßJackson "Classical Electrodynamics"

ßPacholczyk "Radio Astrophysics"

ßGinzburg & Syrovatskii "Cosmic M

agnetobremm

strahlung"1965 A

nn. Rev. Astr. A

p. 3, 297

ßGinzburg & Tsytovitch "Transition radiation"