Radiative processes inhigh energy

astrophysical plasmasÉcole de physique des Houches

23 Février-8 Mars 2013

R. Belmont

Why radiation?

✓Why is radiation so important to understand?✓ Light is a tracer of the emitting media

✓ Geometry, evolution, energy, magnetic field...

✓ Light can influence the source properties✓ cooling/heating, radiation pressure...

✓ Light can be modified after its emission

✓Why are high energy plasmas so important to study?✓ High energy particles are the most efficient emitters✓ They emit over an extremely wide range of frequencies

2

Outline

✓Goals of this course: ✓ Review the main high energy processes for continuum emission

✓ Assumptions, approximations, properties

✓ Show how they can be used to constrain the physics of astrophysical sources

✓Main books/reviews: ✓ Rybicki G.B. & Lightman A.P., 1979, Radiative processes in astrophysics,

New York, Wiley-Interscience✓ Jauch J.M. & Rohrlich F., 1980, The theory of photons and electrons (2nd

edition), Berlin, Springer✓ Aharonian F. A., 2004, Very High energy cosmic gamma radiation, World

scientific publishing✓ Heitler W., 1954, Quantum theory of radiation, International Series of

Monographs on Physics, Oxford: Clarendon✓ Blumenthal G.R. & Gould R.J., 1970, Reviews of Modern Physics

3

Contents

✓ Introduction

✓ I. Emission of charged particles

✓ II. Cyclo-Synchrotron radiation

✓ III. Compton scattering

✓ IV. Bremsstrahlung radiation

4

Introduction

5

Introduction ●○

Non black-body radiation

✓Black-Body radiation is simple ideal limit✓ independent of internal processes, geometry...✓ Simple law

✓No black body radiation if:✓ Optically thin media

✓ <= finite source size and finite interaction cross sections✓ => absorption/emission/reflection features

✓Matter not at thermal equilibrium✓ <= low density, high energy plasmas

✓Then, radiation properties depend on✓ The particle distributions ✓ The microphysics: what processes?

6

Introduction ●●

Radiation processes

✓Lines (bound-bound):✓ Atomic, molecular transitions (radio to X-rays)✓ Nuclear transitions (γ-rays)

✓Edges (bound-free):✓ ionization/recombination

✓Nuclear reactions, decay and annihilations:✓ bosons, pions... ✓ electron-positron, photon-photon

✓Collective processes✓ Faraday rotation, Cherenkov radiation...

✓Free-free radiation of charged particles in vacuum:✓ Cyclo-synchrotron radiation✓ Compton scattering✓ (Bremsstrahlung radiation)

7

I. Radiation of relativistic particles

8

Emission of relativistic particles ●○○○

Emission from non-relativistic particles✓Electro-magnetic field created by charges in motion:

✓ Charged particles in uniform motion do not emit light✓ Only charged, accelerated particles can emit light

✓ Liénard-Wiechert potentials (1898): (A,Φ) => E-B✓ Power:✓ Spectrum:

✓Emission of low-energy particles:

✓ Total power:

✓ Dipolar field perpendicular to acceleration:

✓ Polarization depends on the direction:✓ Is also the emission in the particle rest frame...

@Pe

@⌦=

q2a2

4⇡csin2 ✓

Pe =2q2a2

3c

9

~E / ~n⇥ (~n⇥ ~a)

P / ~E2

P⌫ /���FFT( ~E)

���2

a

θ

Emission of relativistic particles ●●○○

Change of frame✓Relativistic particles:

✓ Velocity✓ Lorentz factor✓ Energy✓Momentum

✓Let’s consider a particle✓ moving at velocity β as seen in the observer frame K in the parallel direction✓ emitting radiation in its rest frame K’

✓Total emitted power: ✓ Is a Lorentz invariant:✓ Acceleration is not:✓ Emission of relativistic particles strongly enhanced: ✓ High energy sources are amongst the most luminous sources...

a0k = �3aka0? = �2a?

10

� = v/c� = 1/(1� �2)1/2

E = �mc2 Ek = (� � 1)mc2

p = (�2 � 1)1/2 = ��

Pe = P 0e =

2q2a02

3c

Pe =2q2

3c�4

⇣�2a2k + a2?

⌘

Emission of relativistic particles ●●●○

Relativistic beaming✓Angles: an example

✓ Emitting body moving at velocity v✓ Photon emitted perpendicular to motion in the

body frame✓ Photon

✓ must fly at c in all frames✓ observed with parallel velocity v✓ observed with perpendicular velocity c/γ✓ observed with an angle

✓Angular distribution of emission:

✓ beaming✓ enhancement

11

c

body frame

(c2�

v2)1

/2=

c/�

✓

c

vobserver frame

sin ✓ = 1/�

@Pr

@⌦=

1

�4(1� � cos ✓)4

@P 0e

@⌦0

Emission of relativistic particles ●●●●

Relativistic beaming

✓The dipolar emission of charge particles:✓ The angular distribution depends on the

(a,v) angle

✓ Beaming is still present and

a v

av

a

v=0

12

✓ ⇡ 1/�

para

llel a

ccel

erat

ion

perpendicular acceleration

Cyclo-Synchrotron radiation

- Emitted power- Spectrum- Radiation from many particles- Polarization- Self-absorption

13

Cyclo-synchrotron radiation ●○○○○○○○○○○○○○○

Synchrotron in astrophysics

✓Applications to astrophysical sources started in the mid 20th

✓Most observations in radio but also at all wavelengths

14Galactic center

SNR

Radio galaxies

jupiter

PWN

⌫B =qB

2⇡mc

Cyclo-synchrotron radiation ●●○○○○○○○○○○○○○

Cyclo-synchrotron radiation

hν

✓Radiation from particles gyrating the magnetic field lines

✓Relativistic gyrofrequency

✓Assumptions:✓ Classical limit:

✓ Otherwise: quantization of energies, Larmor radii...✓ Observable cyclotron scattering features in accreting neutron stars...

✓ B uniform at the Larmor scale (parallel and perp)✓ ! no strong B curvature (pulsars and rapidly rotating neutron stars)✓ ! no small scale turbulence (at the Larmor scale)✓ ! no large losses (tcool >> 1/νB,r)

✓Emission/Absorption15

⌫B,r =1

�

qB

2⇡mc

h⌫B << mc2 B < Bc = 12⇥ 1012 G

Cyclo-synchrotron radiation ●●●○○○○○○○○○○○○

Emitted Power and cooling time

✓Circular motion:

✓Emission of an accelerated particle: ✓ Power goes as p2

✓ Power goes as B2

✓ Isotropic distribution of pitch angles:

✓Cooling time:

✓ ISM (B=1µG): tcool>tuniverse as far as γ<103

✓ AGN jets (B=10µG, γ=104): tcool = 107yr (=travel time!)

✓Maximal loss limit

16

a = a? =⌫B2⇡�

v?

P =2q2

3c3�4a2? = 2c�TUBp

2?

P =4

3c�TUBp

2

tcool

=�mc2

Pe⇡ 25yr

B2�

�2B <2q

r20 VLA, 6cmCentaurus A

30 000 lyr

tcool

>> 1/⌫B,r

Cyclo-synchrotron radiation ●●●●○○○○○○○○○○○

Emission Spectrum

✓ Sinus modulation of the electric field at νB:

✓ Spectrum = one cyclotron line at νB= νB,r

a

obser

ver

Very low energy particles:

17

Cyclo-synchrotron radiation ●●●●●○○○○○○○○○○

✓ Moderate beaming

✓ Asymmetrical modulation of the electric field at νB,r:

✓ Spectrum = many harmonic lines at kνB,r=kνB/γ

Emission Spectrum

obser

ver

Mid-relativistic particles:

18

�t ⇡ 1/(⌫B,r�3)

⌫c =3

2�2⌫B sin↵

Cyclo-synchrotron radiation ●●●●●●○○○○○○○○○

✓ Strong beaming: δθ=1/γ

✓ Pulsed modulation of the electric field at νB:

✓ Spectrum = continuum up to

Emission Spectrum

obser

ver

Ultra-relativistic particles:

19

γ γ2γ γ2

@

2P

@(⌫/⌫B)@⌦= 4�T cUB�

2?⌫

2

⌫

2B

1X

n=1

n

2 (cos ✓ � �k)2

(1� �k cos ✓)2J

2n(x)

x

2+ J

02n (x)

��

⌫

⌫B(1� �k cos ✓)�

n

�

�

Cyclo-synchrotron radiation ●●●●●●●○○○○○○○○

Emission Spectrum

✓Exact spectrum depends on:✓ the particle energy✓ the pitch angle α:✓ the observation angle θ

with respect to B

✓Line broadening results from integration over:✓ Observation direction✓ Particle pitch angle✓ Particle energy

cos↵ = �k/�

x = (⌫/⌫B)�? sin ✓

20

Cyclo-synchrotron radiation ●●●●●●●●○○○○○○○

✓High energy plasmas have a continuous spectrum:

✓Peaks at the critical frequency:

✓Most of the emission is at νc* ✓ AGN (B=10µG, γ=104): 10cm

✓ Crab nebula (B=0.1mG, γ=107): 10 keV

✓Highest possible energy of photons:

Spectrum of Relativistic Particles@P

@(⌫/⌫B)= 12

p3�T cUBG

✓⌫

2⌫⇤c

◆

G(x) = x

2

K4/3(x)K1/3(x)�

3x

5

⇣K

24/3(x)�K

21/3(x)

⌘�

⌫⇤c =3

2⌫B�

2 / B�2

h⌫ = 3mec2/↵f ⇡ 70MeV

21

�2B <2q

r20

Cyclo-synchrotron radiation ●●●●●●●●●○○○○○○

Spectrum of many particles

✓Emission integrated over the particle distribution:

✓Thermal distribution:✓ Same as for a single particle for the mean energy..

✓Power-law distribution:✓ Integrated emission:

✓ Power-law spectrum with ✓ slope:✓ minimal energy:✓ maximal energy:

22

j⌫ =

ZP⌫(⌫, �)f(�)d�

f(�) / �2e��/✓

f(�) / ��s

↵ =s� 1

2

j⌫ /RG(3⌫/2⌫B�2)��s

d�

/ ⌫

�(s�1)/2Rx

(s�3)/2G(x)dx

/ ⌫

�↵

G(x)

j(ν)

⌫min / B�2min

⌫max

/ B�2

max

Cyclo-synchrotron radiation ●●●●●●●●●●○○○○○

The Crab Nebula

✓Pulsar wind nebula✓ Outflow of high energy electrons✓Magnetized medium (0.1 mG)

✓Synchrotron emission from radio to γ-rays

✓Broken power-law distribution✓ Two slopes: s1, s2

✓ Two breaks: γ1, γ223

γ1

γ2

s1

s2

Cyclo-synchrotron radiation ●●●●●●●●●●●○○○○

Polarization

✓One single particle produces a coherent EM fluctuation✓ Intrinsically polarized: elliptically✓ Depends on p, α and θ

✓Turbulent magnetic field: no net polarization

✓Ordered magnetic field:✓ Ensemble of particles with random pitch angles => partially linearly

polarized✓ Polarization angle perpendicular to observed B:✓ High polarization degree:

✓ depends on frequency and particle energy✓ averaged over all frequencies: Π=75% !✓ In average over PL distribution of particles: Π=(s+1)/(s+7/3)

⇧(⌫, p) =P? � Pk

P? + Pk

P? >> Pk

24

Cyclo-synchrotron radiation ●●●●●●●●●●●●○○○

Polarization

✓Such high polarization is characteristic of synchrotron radiation

✓Measure of the direction gives the direction of B

M87 - OPTICAL POLARIZATION

ARC

SEC

ARC SEC-10 -11 -12 -13 -14 -15

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

M87 - RADIO POLARIZATION

ARC

SEC

ARC SEC-10 -11 -12 -13 -14 -15

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

Figure 5

I A

B1

B2

LOP-1

HOP-3HOP-2

HOP-1

M87 - OPTICAL POLARIZATION

ARC

SEC

ARC SEC-10 -11 -12 -13 -14 -15

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

M87 - RADIO POLARIZATION

ARC

SEC

ARC SEC-10 -11 -12 -13 -14 -15

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

Figure 5

I A

B1

B2

LOP-1

HOP-3HOP-2

HOP-1

25

Crab nebula

M87

↵⌫(p, ⌫) =c2

2h⌫31

p�[�pj⌫ ]

�+h⌫/mc2

�

⇡ 1

2m⌫21

p�@� (�pj⌫)

j⌫ = n@P

@⌫@⌦

Cyclo-synchrotron radiation ●●●●●●●●●●●●●○○

Synchrotron self-absorption

✓ Absorption decreases with frequency✓ high energy photons are weakly absorbed✓ low energy photons are highly absorbed

True absorption

Stimulated emission (negative absorption)

hν

Spontaneous emission

hν

True absorption Stimulated emission(negative absorption)

hν

hν

26

emission coefficient: absorption coefficient:

Cyclo-synchrotron radiation ●●●●●●●●●●●●●●○

Synchrotron self-absorption

✓Radiation transfer:✓ Equation for specific intensity Iν:✓ Solution for a uniform layer of thickness L:

✓ τν=ανL is the optical depth at energy hν✓When τν<<1: thin spectrum✓When τν>>1: thick spectrum✓ transition for:

✓The transition energy increases with optical depth, i.e. with:✓ Physical thickness of the layer✓ Density of the medium

27

@I⌫dl

= j⌫ � ↵⌫I⌫

I⌫ =j⌫↵⌫

(1� e�↵⌫L)

⌧⌫ = ↵⌫L ⇡ 1

Cyclo-synchrotron radiation ●●●●●●●●●●●●●●●

Self-absorbed Spectra

✓Thermal distribution ✓ Power-law distribution

Raylei

gh-Je

ans

thick

thin

28Tends to BB radiation in the thick part Does not tend to BB radiation in the thick part

thin

thic

kF⌫/⌫5/2

F ⌫/⌫2

Compton Scattering

29

- Thomson/Klein-Nishina regimes- Spectrum, angular distribution- Particle cooling- Multiple scattering

Compton scattering ●○○○○○○○○○○○

In the particle rest frame✓Scattering of light by free electrons

✓Result described by 6 quantities

✓4 Conservation laws:✓ Energy✓Momentum

✓1 symmetry

✓One quantity is left undetermined, e.g. the scattering angle θ

✓Direction/energy relation

✓Photon loose energy in the particle rest frame

✓Two regimes:✓ The Thomson regime (hν0<mc2): coherent scattering: hν=hν0

✓ The Klein-Nishina regime (hν0>mc2): incoherent scattering: hν<hν0 30

hν0

hν

γ

θ

h⌫0 +mc2 = h⌫ + �mc2

h⌫0c

~n0 =h⌫

c~n+mc~p

h⇥

h⇥0=

1

1 +

h�0mec2

(1� cos �)

hν0hν γ hν0hνh⌫0

1 + 2h⌫0/mc2 h⌫ h⌫0

backward scattering (θ=π) forward scattering (θ=0)

Compton scattering ●●○○○○○○○○○○

Thomson scattering✓Scattering of linearly polarized waves:

✓ Harmonic motion of the particle✓ Emission of light in all directions

✓Scattering of unpolarized waves:✓ Average of linearly polarized waves with random directions

✓ Scattered power:

✓ Thomson cross section:

✓ Total cross section:

✓ Partially polarized:

✓Spectrum: a line at the incident frequency 31

hν0

hν

γ

θ

@P

@⌦=

@�

@⌦F

@�

@⌦=

3

8⇡�T

1 + cos

2 ✓

2

�T = 6.65⇥ 10�25cm2

⇧ =

1� cos

2 ✓

1 + cos

2 ✓

� = �T3

4

1 + !0

!30

✓2!0(1 + !0)

1 + 2!0� ln (1 + 2!0)

◆+

ln (1 + 2!0)

2!0� 1 + 3!0

(1 + 2!0)2

�

Compton scattering ●●●○○○○○○○○○

Klein Nishina scattering ✓Requires quantum mechanics but still

analytical formulae

32

� ⇡ �T !0=

1

Thomson Regime

Klein-Nishina Regime

h⌫=

mc2

Total cross section:

Ang

ular

dis

tribu

tion:

Spec

trum

Differential cross sections:

hν0/mc2

10-5

10-1

1

10

102

103

d�

d⌦= �T

3

16⇡

✓h⌫

h⌫0

◆2 ✓h⌫0h⌫

+h⌫

h⌫0� sin2 ✓

◆

⌫ = 2�0⌫0

Compton scattering ●●●●○○○○○○○○

In the source frame✓ In the particle frame: one incident parameter (hν0)

✓ In the source frame: new dependance on✓ The particle energy✓ The collision angle

✓Photon can now gain/loose energy

✓An example:✓ Head-on collision:

✓ Cold photon hν0 and relativistic electron γ0>>1✓ Photon energy in the particle frame:

✓ Thomson backward scattering:✓ Emitted photon energy in the electron frame:✓ Photon energy in the source frame:

✓ In the end:✓ Compton up-scattering

✓Often: isotropy assumption and average over angles33

⌫00 = 2�0⌫0

⌫0 = ⌫00

⌫ = 4�20⌫0

h⌫00 << mc2

Compton scattering ●●●●●○○○○○○○

In the source frame

✓For isotropic distributions:

✓The Thomson limit:

✓The scattered spectrum:

✓Average energy of scattered photons

✓ Down scattering:

✓ Up-scattering:✓ Amplification factor:

✓Scattering by relativistic plasmas produces high energy radiation

✓Particle cooling:

34

�0h⌫0 << mc2

(�0 � 1)mc2 < h⌫0

(�0 � 1)mc2 > h⌫0

A =< h⌫ >

h⌫0⇡ �2

@Ep

@t=

4

3c�T p

2Uph

Compton scattering ●●●●●●○○○○○○

Blazar Spectra

✓E > TeV! Comptonization?

✓Model = Synchrotron Self-Compton (SSC) + Doppler boosting✓ Seed photons = synchrotron photons✓ The same particle emit though synchrotron and scatter these photons

✓Here: ✓ A=108 => γ=104

✓ hν0 = 0.1 keV => KN regime✓ Synchrotron peaks at Bγ2, Compton amplifies with A=γ2 => B! 35

optical X-RaysUV !-Rays

Compton scattering ●●●●●●●○○○○○

Single scattering by many electrons✓Emission integrated over the particle

distribution

✓Thermal distribution:✓ Same as for a single particle for the

mean energy..

✓Power-law distribution:✓ Power-law spectrum with

✓ slope:✓ minimal energy: hν0 γ2min

✓ maximal energy: hν0 γ2max

✓Scattered photons distributions should also be integrated over the source seed photons

36

f(�) / �2e��/✓

f(�) / ��s

↵ =s� 1

2

Compton scattering ●●●●●●●●○○○○

Multiple Scattering

✓Photons can undergo successive scattering events

✓Medium of finite size L: Thomson optical depth:

✓Competition scattering/escape:✓ τ (or τ2) = Mean number of scattering before escape ✓ τ<1: single scattering✓ τ>1: multiple scattering

✓y parameter = <photon energy change> before escape✓ y= <Energy change per scattering> * <scattering number> ✓ For mono-energetic particles: y = τγ2

✓ For thermal distributions: y = 4τθ(1+4θ)

37

⌧T = ne�TL

Compton scattering ●●●●●●●●●○○○

The SZ effect

✓Typical distortion whose amplitude gives: y ≈ τθ ≈ 10-4

✓Bremsstrahlung gives T

✓=> density...

38

kBTph=2.7 KkBTe=1-10 keV ("e=kBTe/mec2≈10-2, p≈#≈0.1)

!=Ne"TL ≈ 10-2

Compton up-scatteringCold photons

Hot electrons

Compton scattering ●●●●●●●●●●○○

Compton orders

hν0/mc2=10-7

γ0=10 A =100

bumpy spectrum

cutoff at the particle energy

τ => spectrum hardness

A A AA

39

Compton scattering ●●●●●●●●●●●○

Compton regimesSub-relativistic particles: hν0/mc2 < p < 1

Thomson regime hν0#0<< mc2

Inefficient scattering: A = 1

Ultra-relativistic particles: #0>> 1KN regime: hν0#0>mc2

Efficient scattering: A >> 1

w0=10-4, p0=1 (w0 g0 < 1)A = 2

Power-law spectrum- cutoff at the particle energy- Slope = ln(τ)/ln(A)

one-bump spectrum= single scattering !

=> AGN/Blazars=> X-ray binaries 40

Relativistic particles: #0>>1 Thomson regime: hν0#0<< 1

Efficient scattering: A >> 1

Compton scattering ●●●●●●●●●●●●

X-ray binaries

✓Soft states:✓Multi-color black body at 1 keV from the accretion disk✓ Non-thermal comptonization in a hot corona (τ=1)

✓Hard states:✓ Soft photons from the accretion disk or synchrotron✓ Inefficient thermal Comptonization in a hot corona (100 keV, τ=0.01)

✓What heating acceleration mechanism?41

Zdziarsky 2003

Bremsstrahlung radiation

42

Bremsstrahlung ●○○○

Bremsstrahlung

✓Radiation of charged particles accelerated by the Coulomb field of other charges

✓Astrophysical sources:✓ Some modes of hot accretion disks✓ Hot gas of intra-cluster medium (1-10

keV)✓ ...

43

~v

Bremsstrahlung ●●○○

Easy bremsstrahlung

✓ Assumptions:✓ Classical physics✓ Sub-relativistic particles✓ Far collision (small deviation, no recombination)✓ Small energy change (Δv << v i.e. hν<<mv2/2)

✓Single event✓ No p+/p+, e-/e-, e+/e+ Bremsstrahlung ✓ p+/e, iZ+/e Bremsstrahlung✓ Approximation: static heavy iZ+

✓Approximated motion: ✓ Typical collision time: τ ≈ 2b/v✓ Typical velocity change: Δv ≈ τ a ≈ τ F/m ≈ τ Z e2/(mb2) ≈ 2Ze2/(mvb)✓ τ and Δv characterize the motion = enough to compute the spectrum

44

� << 1

2b

b

~v

Ze

h⌫

Bremsstrahlung ●●●○

Easy bremsstrahlung✓Spectrum:

✓Fourier transform:

✓One single particle produces a flat spectrum:

✓Many particles with a range of impact parameters:✓ with bmin from the small angle approximation✓ with bmax from ντ<1

✓ Produce a flat spectrum that cuts at the electron energy✓ Total losses:

45

@P

@⌫/

���FFT ( ~E)���2/ |FFT (~a)|2

TF [~̇v] =

Z +1

�1~̇v(t)e�2i⇡⌫tdt ⇡

⇢0 if ⌫⌧ >> 1�~v if ⌫⌧ << 1

⌫

@E@⌫

1/⌧1 1/⌧2

b1

b2 < b1

@P

@V @⌫= ninev

Z bmax

bmin

@E@⌫

2⇡bdb =32⇡e6

3m2c3vnineg↵(v, ⌫)

@E@⌫

(v, b) ⇡⇢

0 if ⌫⌧ >> 116e6Z2

3c3m2v2b2 if ⌫⌧ << 1

@P@V @⌫

h⌫mv2/2Pcool

/ ni

ne

Z2v

1/v

✓Emission integrated over the particle distribution

✓Power-law distributions produce power-law spectra

✓Thermal distributions:✓ Emission coefficient: ✓ Total losses:

✓Relativistic and quantum correction can be added to give more general spectra

✓ In media of finite size: bremsstrahlung self-absorption at low energy✓ c.f. synchrotron

j⌫ / nineZ2T�1/2e�h⌫/kBT

Pcool

/ ni

ne

Z2T 1/2

Bremsstrahlung ●●●●

Emission from many electrons

46

h⌫

j⌫

kBT1 kBT2

T2 > T1

Summary✓Particle cooling:

✓ Synchrotron: P ∝ σT p2 UB

✓ Compton in the Thomson regime: P ∝ σT p2 Uph

✓ Bremsstrahlung: P ∝ σT αf p Ui (with Ui= ni mec2)✓Photons:

✓ Synchrotron: ✓ Thin spectrum of 1 particle peaks at νc ∝ γ2 B✓ Thin spectrum of a power-law distribution is a power-law✓ Absorption => Thick spectrum at low frequency

✓ Compton✓ Amplification factor in the Thomson regime: A = γ2

✓ Mildly relativistic particles: power-law spectrum✓ Comptonization by a relativistic power-law distribution is a PL spectrum

✓ Bremsstrahlung✓ Flat spectrum✓ up to the particle energy

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