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14. Compton Scattering &Plasma Effects

Electrodynamics of Radiation Processes

http://www.astro.rug.nl/~etolstoy/radproc/

Chapter 7: Rybicki&Lightman Section 7.7

Chapter 8: Rybicki&Lightman

Compton y parameter

In general when y >>1 the total photon energy in the spectrum will be significantly altered, but not for y << 1.

Spectral Regimes for Compton spectra

For intermediate cases (unsaturated Comptonisation) more detailed treatment of Kompaneets equation is needed.

A detailed analysis of Compton spectra requires a solution of the Kompaneets equation with a photon source term.

y<<1: modified black-body spectray>>1: saturated Compton (Wien spectra)

Approximate analysis are usually OK.

For frequencies where:

We consider thermal media, in which absorption and emission arise from Bremsstrahlung (free-free) processes. The effect of absorption changes with frequency, where the importance of absorption is greatest at lowest frequencies.

Spectral Regimes

Characteristic Frequencies:

ν0 : scattering and absorption coefficients are equal es = ↵(⌫0)

νt : frequency at which medium becomes effectively thin es = ↵(⌫t)⌧2es

es =�T

mp= ↵ =

↵↵⌫

⇢

νcoh : frequency at which incoherent scattering (inverse Compton) can be important.

es

=⇣mc2

4kT

⌘↵

(⌫coh

)

↵↵⌫ = AT�1/2⇢2⌫�3(1� e�h⌫/kT)

thermal bremsstrahlung absorption mass absorption (opacity) coefficient

y<<1: modified black-body spectra

x << x0 absorption dominates over scattering, Iν → Bν

At x0< x < 1 IνMB ∝ ν instead of the steeper Rayleigh-Jeans slope of IνRJ ∝ ν2

x >> x0 scattering is important, Iν → IνMB modified black-body

Only coherent scattering is important

emergent intensity

x = hω / kTe

y>>1: wien spectrainverse compton may becomes important depending on xcoh

e–α rate at which photons are produced

xcoh >> 1 inverse compton maybe neglected, electrons not sufficiently energetic

xcoh << 1 x

coh

=

⇣mc

2

4kT

⌘1/2

x

0

for x << xcoh modified blackbody spectrumfor x ≥ xcoh must consider inverse compton effects

if xcoh << 1 inverse Compton will saturate, and Wien intensity will result

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Emergent Intensity Spectrum

spectrum of thermal, non-relativistic medium, characterized by free-free emission and absorption & saturated inverse compton scattering

blackbody

modified blackbody wien

The spectrum falls off roughly exponentially at photon energies much above the electron temperature, as expected for a thermal spectrum

Unsaturated Comptonisation y >>1 but xcoh∼1, inverse Compton important but does not saturate to Wien spectrum - comptonisation with soft photon input

Modified steady-state Kompaneet’s equation

0 =

⇣kT

mc

2

⌘1

x

2

@

@x[x

4(n

0+ n)] + Q(x)� n

Max(⌧es, ⌧2es)

n / e�xFor x>>1

For xs << x << 1 n maybe neglected in comparison with n’ and n / x

m

y >> 1 leads to low frequency limit of Wien law

Unsaturated Comptonisation

sensitive to y

The shape of the unsaturated Compton spectrum with a soft photon source determines both the electron temperature and the scattering optical depth of the source. The emergent intensity in the power-law regime satisfies

I⌫ ⇠ I⌫⇣ ⌫

⌫s

⌘3+m

The input energy is significantly amplified for m > -4, which is y > 1

Sunyaev-Zeldovich EffectAn important application of the Kompaneets equation concerns spectral distortions of the Cosmic Microwave Background Radiation if the radiation traverses extensive regions of hot ionised gas with electron temperature Te much greater than the radiation temperature Trad. (Zeldovich & Sunyaev 1969)

The predicted spectrum is found by solving the Kompaneets equation without the terms describing the cooling of the photons, as the heating took place relatively recently and so there has not been time to cool.

Assuming the distortions are small, ZS inserted the trial solution n = (ex − 1)−1

where the Compton optical depth, yy =

Z(kTe/mec

2)�TNedl

The effect is to shift the spectrum to higher energies with the result that the intensity of radiation in the Rayleigh–Jeans region of the spectrum, x<<1, whilst that at x>>1 increases.

Sunyaev-Zeldovich Effect Synchrotron-Self Compton (SSC)Case of special interest: where the relativistic electrons that are the source of low energy photons (Synchrotron emission) are also responsible for Compton scattering these photons to X- and γ-ray energies:

SYNCHROTRON–SELF-COMPTON RADIATION.

Need to measure the synchrotron radio flux density and the X- and γ -radiation from the same source region to estimate the magnetic flux density. The difficulties are the upper and lower limits to the electron energy spectrum and in ensuring that electrons of roughly the same energies are responsible for the radio and X-ray emission

Ratio of energy loss rates

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Cen A Spectral Energy Distribution

Plasma Effects

If propagation medium is not an empty vacuum…

…a globally neutral ionised gas is called a plasma

The Plasma FrequencyFor a fully ionised plasma (np = ne), suppose a layer of electrons of thickness x is displaced by a distance δx relative to the ions, then the net effect is to setup two oppositely charged sheets with a surface charge density σ = e neδx and the system forms a parallel plate capacitor with opposite surface charges σ on the plates. The electric field created is

E = �/✏0 = ene�x/✏0

⌫p =⇣ e2ne4⇡2✏0me

⌘1/2= 8.97⇥ 103 n1/2 Hz

The equation of simple harmonic motion with angular frequency

!p =⇣ e2ne✏0me

⌘1/2= 5.63⇥ 104 n1/2 s�1

n is given in cm-3

!p =⇣ e2ne✏0me

⌘1/2= 5.63⇥ 104 n1/2 s�1!p =

⇣4⇡e2ne✏0me

⌘1/2

The equation of motion per unit surface area for electrons in the layer is

x = � eE

me⇠ �4⇡e2ne�x

me✏0

r ·E =1

✏4⇡⇢ r ·B = 0

r⇥E =1

c

@B

@tr⇥B =

4⇡

cj+

✏

c

@E

@t

Maxwell’s Equation in a medium

charge density ρ

current density j = �E conductivity,

dielectric constant

� =nee2

!me

✏ = 1� 4⇡�

!me= 1� 4⇡nee2

!2m2e

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This is the plasma cut-off frequency below which there is no stable em-propagation through an ionised gas.

e.g., Earth’s ionosphere, typical ne~104 – 106 cm-3, this means radio waves ≤ ~ few MHz are blocked, and reflected back e.g., ISM, ne~10-2 – 10-3 cm-3

and so νp < 300-1000Hz

Plasma Frequency

c2k2 = ✏ !2 = !2 � !2p

for ω < ωp then k is imaginary.

k =

q!2 � !2

p

c

wave still oscillates but rapidly fades away with an exponential decay

dispersion relation for a plane wave

⌫p =⇣ e2ne4⇡2✏0me

⌘1/2= 8.97⇥ 103 n1/2 Hz

Phase & Group Velocities

nr ⌘p✏ =

r1�

!2p

!2Index of refraction nr =

c

vp

Different frequencies travel at different speeds, the high frequencies arrive earlier and the pulse gets broadened or dispersed

Phase Velocity vp ⌘ !

k=

c

nr= (✏0µ0)

�1/2 speed with which a point of fixed phase on a sinusoidal wave travels through space

k =

q!2 � !2

p

cGroup velocity vg ⌘ @!

@k= c

r1�

!2p

!2

1 =1

2c(!2 � !2

p)�1/22!

@!

@k

vg =@!

@k=

c(!2 � !2p)

1/2

!= c

q1� !2

p/!2 when nr < 1 then vg < c

when ω > ωp

Group Velocity

This means that a pulse observed at frequency 105Mhz arrives roughly 2.5s later when observed at 95MHz

normal frequency dependent term introduced by plasma DISPERSION MEASURE

Group velocity is useful when studying pulsars, as each individual pulse contains a range of frequencies and thus the pulse is dispersed going through the ISM and this dispersion measure provides an accurate distance to a pulsar assuming an average density of intervening ISM.

tp ⇠ d

c+

1

2c!2

Z d

0!2pds !p =

⇣4⇡e2ne✏0me

⌘1/2

Pulses from the Vela pulsar arrival time versus frequency

Faraday Rotation:

The interstellar medium is permeated by magnetic field lines and hence is a magnetised plasma. Typically this magnetic field will be stronger than the E-fields of the propagating photons, and so the position angle of the E-vector is rotated to the magnetic field direction. This is known as Faraday rotation.

�✓ =2⇡e3

m2c2!2

Z d

0nBkds

as Δθ varies with frequency (ω-2), we can determine the value of the integral by making measurements at several frequencies, to deduce the the strength of the interstellar magnetic field along a line of sight. However if the field changes direction along this line of sight then this will only provide a lower limit.

!B =eB0

mc!B = 1.67⇥ 107 B0 s�1

!p = 5.63⇥ 104 n1/2 s�1�✓ =

1

2

Z d

0

1

c2!2!2p!Bds

mex = �e[E+

v

c

⇥B]

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FARADAY ROTATION IN THE JET OF THE RADIO GALAXY 3C 120

Gómez et al. 2011 ApJ

Having a dispersion measure and rotation measure to same pulsar (at known distance, e.g. Crab) can give magnetic field strength

Measuring B towards the Crab pulsar Cherenkov RadiationSince the group velocity in an unmagnetised plasma

The potentials will differ from those in a vacuum.

� =

"p✏q

R� =

p✏q

R

#A =

"qu

cRA =

qu

cR

#Liénard-Wiechart potentials (in a medium with dielectric constant ε)

= 1� �nr cos ✓

As κ can vanish for an angle θ such that cosθ = (nrβ)-1 the potentials can become infinite at certain points and thus the particle can now radiate.

It is possible for a highly relativistic charged particle to travel with a speed v > vg

This forms a conical photonic shock wave, producing Cherenkov light with a frequency much greater than ωp

Cherenkov RadiationWhen v > vg the potentials will be determined by TWO retarded positions of the particle rather than just one.

Cherenkov cone

Cherenkov radiation confined within cone, velocity c/nr.

Razin EffectWhen nr < 1, as it is in cold plasma, then Cherenkov radiation is not possible.However in this case the beaming effect, which is important to Synchrotron radiation, is supressed. This is again related to the change to the κ factor in the Liénard-Wiechert potentials in the case of a medium. This is because there is now no velocity and angle combination for which κ is small.

In a vacuum the critical angle defining the beaming process was previously shown to be:

✓b ⇠ 1

�=

p1� �2 thus is a medium: ✓b ⇠

p1� n2r�

2

Need to understand whether nr or βdominates in keeping θb from being small.

If nr~1 then same as vacuum case.

If nr is very different from unity ✓b ⇠p

1� n2r =!p

!Thus at low frequencies the medium will dominate beaming properties. At higher frequencies θb decreases until it becomes of order vacuum value (1/γ), and then vacuum results apply. Thus, medium is unimportant when ! � �!p

Below frequency ω << γωp the synchrotron spectrum will be cut-off because of the suppression of beaming: RAZIN EFFECT