Radiative processes in high energy astrophysical plasmas

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1 Radiative processes in high energy astrophysical plasmas École de physique des Houches 23 Février-8 Mars 2013 R. Belmont

2 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

3 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

4 Contents Introduction I. Emission of charged particles II. Cyclo-Synchrotron radiation III. Compton scattering IV. Bremsstrahlung radiation 4

5 Introduction 5

6 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

7 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

8 I. Radiation of relativistic particles 8

9 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: P / E ~ 2 Spectrum: P / FFT( E) ~ 2 a θ Emission of low-energy particles: Total power: P e = 2q2 a 2 3c Dipolar field perpendicular to = q2 a 2 4 c sin2 Polarization depends on the direction: Is also the emission in the particle rest frame... ~E / ~n (~n ~a) 9

10 Emission of relativistic particles Change of frame Relativistic particles: Velocity = v/c Lorentz factor =1/(1 2 ) 1/2 Energy E = mc 2 E k =( 1)mc 2 Momentum p =( 2 1) 1/2 = 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: P e = Pe 0 = 2q2 a 02 Acceleration is not: a 0 a 0 k = 3? = 2 3c a? a k Emission of relativistic particles strongly enhanced: P e = 2q2 3c High energy sources are amongst the most luminous sources a 2 k + a2? 10

11 Emission of relativistic particles (c 2 v 2 ) 1/2 = c/ 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 sin =1/ Angular distribution of = 0 4 (1 cos ) 0 c c body frame observer frame v beaming enhancement 11

v=0 a v parallel acceleration a v Beaming

12 Emission of relativistic particles Relativistic beaming The dipolar emission of charge particles: The angular distribution depends on the (a,v) angle a v=0 a v parallel acceleration a v Beaming is still present and 1/ perpendicular acceleration 12

13 Cyclo-Synchrotron radiation - Emitted power - Spectrum - Radiation from many particles - Polarization - Self-absorption 13

14 Cyclo-synchrotron radiation Synchrotron in astrophysics Applications to astrophysical sources started in the mid 20th Most observations in radio but also at all wavelengths PWN Radio galaxies SNR jupiter Galactic center 14

15 Cyclo-synchrotron radiation Cyclo-synchrotron radiation hν Radiation from particles gyrating the magnetic field lines Relativistic gyrofrequency Assumptions: B = qb 2 mc B,r = 1 qb 2 mc Classical limit: h B << mc 2 B<B c = G 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/Absorption 15

Cyclo-synchrotron radiation Emitted Power and cooling time Circular motion: a = a? = B 2 v? Emission of an accelerated particle: Power goes as p 2 Power goes as B 2 P = 2q2 3c 3 4 a 2? =2c T U B p 2?

16 Cyclo-synchrotron radiation Emitted Power and cooling time Circular motion: a = a? = B 2 v? Emission of an accelerated particle: Power goes as p 2 Power goes as B 2 P = 2q2 3c 3 4 a 2? =2c T U B p 2? Isotropic distribution of pitch angles: P = 4 3 c T U B p 2 Cooling time: t cool = mc2 P e 25yr B lyr ISM (B=1µG): tcool>tuniverse as far as γ<10 3 AGN jets (B=10µG, γ=10 4 ): tcool = 10 7 yr (=travel time!) Maximal loss limit t cool >> 1/ B,r 2 B< 2q r 2 0 Centaurus A VLA, 6cm 16

17 Cyclo-synchrotron radiation Emission Spectrum Very low energy particles: observer a Sinus modulation of the electric field at νb: Spectrum = one cyclotron line at νb= νb,r 17

18 Cyclo-synchrotron radiation Emission Spectrum Mid-relativistic particles: Moderate beaming observer Asymmetrical modulation of the electric field at νb,r: Spectrum = many harmonic lines at kνb,r=kνb/γ 18

19 Cyclo-synchrotron radiation Emission Spectrum Ultra-relativistic particles: Strong beaming: δθ=1/γ observer γ γ 2 γ 2 Pulsed modulation of the electric field at νb: t 1/( B,r 3 ) Spectrum = continuum up to c = B sin 19

20 Cyclo-synchrotron radiation Emission 2 / B )@ =4 2 2 T cu B? B 2 1X n=1 applen 2 (cos k) 2 (1 k cos ) 2 J 2 n(x) x 2 + J 0 2 n (x) apple B (1 k cos ) n x =( / B )? sin Exact spectrum depends on: the particle energy the pitch angle α: the observation angle θ with respect to B cos = k / Line broadening results from integration over: Observation direction Particle pitch angle Particle energy 20

21 Cyclo-synchrotron radiation Spectrum of Relativistic Particles High energy plasmas have a continuous spectrum: G(x) =x 2 applek 4/3 (x)k 1/3 (x) 3x 5 K 2 4/3 (x) K2 / B ) = 12p 3 T cu B G 2 c Peaks at the critical frequency: c = 3 2 B 2 / B 2 Most of the emission is at νc * AGN (B=10µG, γ=10 4 ): 10cm Crab nebula (B=0.1mG, γ=10 7 ): 10 kev Highest possible energy of photons: 2 B< 2q r 2 0 h =3m e c 2 / f 70MeV 21

22 Cyclo-synchrotron radiation Spectrum of many particles Emission integrated over the particle distribution: j = Z P (, )f( )d Thermal distribution: f( ) / 2 e Same as for a single particle for the mean energy.. / Power-law distribution: Integrated emission: f( ) / s j / R G(3 /2 B 2 ) s d / (s 1)/2 R x (s 3)/2 G(x)dx / j(ν) Power-law spectrum with slope: 2 minimal energy: min / B min 2 maximal energy: max / B max 2 = s 1 G(x) 22

23 Cyclo-synchrotron radiation The Crab Nebula s2 s1 γ1 γ2 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, γ2 23

24 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: (,p)= P? P? + P k 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 k P? >> P k 24

0 HOP-1 LOP-1 B1 5.5 5.5 ARC SEC 5.0 4.5 ARC SEC 5.0 4.5 B2 4.0 4.0 3.5 3.5 HOP-3 HOP-2 3.

25 Cyclo-synchrotron radiation Polarization Such high polarization is characteristic of synchrotron radiation Measure of the direction gives the direction of B M87 Crab nebula HOP-1 LOP-1 B ARC SEC ARC SEC B HOP-3 HOP ARC SEC M87 - RADIO POLARIZATION Figure I A ARC SEC M87 - OPTICAL POLARIZATION 25

26 Cyclo-synchrotron radiation Synchrotron self-absorption hν hν hν hν Spontaneous emission True absorption Stimulated emission (negative absorption) emission coefficient: j @ absorption coefficient: (p, ) = c2 1 2h 3 p [ pj ] +h /mc 2 1 2m 2 1 ( pj ) True absorption Stimulated emission (negative absorption) Absorption decreases with frequency high energy photons are weakly absorbed low energy photons are highly absorbed 26

27 Cyclo-synchrotron radiation Synchrotron self-absorption Radiation Equation for specific intensity Iν: = j I dl Solution for a uniform layer of thickness L: I = j (1 e L ) τν=ανl is the optical depth at energy hν When τν<<1: thin spectrum When τν>>1: thick spectrum transition for: = L 1 The transition energy increases with optical depth, i.e. with: Physical thickness of the layer Density of the medium 27

/ 2 thin thick F / 5/2 thin Tends to BB radiation in the

28 Cyclo-synchrotron radiation Self-absorbed Spectra Thermal distribution Power-law distribution Rayleigh-Jeans thick F / 2 thin thick F / 5/2 thin Tends to BB radiation in the thick part Does not tend to BB radiation in the thick part 28

29 Compton Scattering - Thomson/Klein-Nishina regimes - Spectrum, angular distribution - Particle cooling - Multiple scattering 29

30 Compton scattering In the particle rest frame Scattering of light by free electrons Result described by 6 quantities 4 Conservation laws: Energy h 0 + mc 2 = h + mc 2 Momentum 1 symmetry One quantity is left undetermined, e.g. the scattering angle θ Direction/energy relation hν0 γ hν backward scattering (θ=π) Photon loose energy in the particle rest frame Two regimes: h 0 c ~n 0 = h ~n + mc~p c h h 0 = The Thomson regime (hν0<mc 2 ): coherent scattering: hν=hν0 hν0 The Klein-Nishina regime (hν0>mc 2 ): incoherent scattering: hν<hν h 0 m e c 2 (1 cos ) h 0 1+2h 0 /mc 2 apple h apple h 0 hν γ θ hν hν0 forward scattering (θ=0)

31 Compton scattering Thomson scattering Scattering of linearly polarized waves: Harmonic motion of the particle Emission of light in all directions hν0 γ θ hν Scattering of unpolarized waves: Average of linearly polarized waves with random directions @ F Thomson = cos 2 8 T 2 Total cross section: Partially polarized: T = cm 2 = 1 cos2 1 + cos 2 Spectrum: a line at the incident frequency 31

0 =1 + ln (1 + 2! 0) 2! 0 1+3! 0 (1 + 2!

32 Compton scattering Klein Nishina scattering Requires quantum mechanics but still analytical formulae apple 3 1+!0 2!0 (1 +! 0 ) = T 4! 0 3 ln (1 + 2! 0 ) 1+2! 0 Total cross section: T Thomson Regime!0 =1 + ln (1 + 2! 0) 2! 0 1+3! 0 (1 + 2! 0 ) 2 Klein-Nishina Regime Angular distribution: Differential cross sections: d d = T 2 3 h h 0 16 h 0 h + h h 0 sin 2 hν0/mc h = mc 2 Spectrum

33 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 0 0 =2 0 0 h 0 0 << mc 2 0 = 0 = Often: isotropy assumption and average over angles 33

Compton scattering In the source frame For isotropic distributions: The Thomson limit: 0h 0 << mc 2 The scattered spectrum: Average energy of scattered photons Down scattering: ( 0 1)mc

34 Compton scattering In the source frame For isotropic distributions: The Thomson limit: 0h 0 << mc 2 The scattered spectrum: Average energy of scattered photons Down scattering: ( 0 1)mc 2 <h 0 Up-scattering: ( 0 1)mc 2 >h 0 Amplification factor: Scattering by relativistic plasmas produces high energy radiation Particle cooling: A = <h > h 0 = 4 3 c T p 2 U ph 34

35 Compton scattering Blazar Spectra opticaluv X-Rays!-Rays 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=10 8 => γ=10 4 hν0 = 0.1 kev => KN regime Synchrotron peaks at Bγ 2, Compton amplifies with A=γ 2 => B! 35

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.

36 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: 2 minimal energy: hν0 γ 2 min maximal energy: hν0 γ 2 max = s 1 f( ) / 2 e f( ) / s / Scattered photons distributions should also be integrated over the source seed photons 36

37 Compton scattering Multiple Scattering Photons can undergo successive scattering events Medium of finite size L: Thomson optical depth: T = n e T L Competition scattering/escape: τ (or τ 2 ) = 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 = τγ 2 For thermal distributions: y = 4τθ(1+4θ) 37

38 Compton scattering Cold photons The SZ effect Hot electrons Compton up-scattering kbtph=2.7 K kbte=1-10 kev ("e=kbte/mec , p # 0.1)!=Ne" T L 10-2 Typical distortion whose amplitude gives: y τθ 10-4 Bremsstrahlung gives T => density... 38

39 Compton scattering Compton orders A A A hν0/mc 2 =10-7 γ0=10 A =100 bumpy spectrum cutoff at the particle energy τ => spectrum hardness 39

Compton scattering Compton regimes Sub-relativistic particles: hν0/mc 2 < p < 1

#0>>1 Thomson regime: hν0#0<< 1 Efficient scattering: A >> 1 Ultra-relativistic

p0=1 (w0 g0 < 1) A = 2 Power-law spectrum - cutoff at the particle energy - Slope

40 Compton scattering Compton regimes Sub-relativistic particles: hν0/mc 2 < p < 1 Thomson regime hν0#0<< mc 2 Inefficient scattering: A = 1 Relativistic particles: #0>>1 Thomson regime: hν0#0<< 1 Efficient scattering: A >> 1 Ultra-relativistic particles: #0>> 1 KN regime: hν0#0>mc 2 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! => X-ray binaries => AGN/Blazars 40

Compton scattering X-ray binaries Zdziarsky 2003 Soft

accretion disk Non-thermal comptonization in a hot

accretion disk or synchrotron Inefficient thermal

41 Compton scattering X-ray binaries Zdziarsky 2003 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

42 Bremsstrahlung radiation 42

43 Bremsstrahlung Bremsstrahlung Radiation of charged particles accelerated by the Coulomb field of other charges ~v Astrophysical sources: Some modes of hot accretion disks Hot gas of intra-cluster medium (1-10 kev)... 43

44 Bremsstrahlung Easy bremsstrahlung Assumptions: Classical physics b Sub-relativistic particles Far collision (small deviation, no recombination) Small energy change (Δv << v i.e. hν<<mv2/2) ~v 2b Ze h << 1 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 e 2 /(mb 2 ) 2Ze 2 /(mvb) τ and Δv characterize the motion = enough to compute the spectrum 44

45 Bremsstrahlung Easy / FFT( ~ E) 2 / FFT(~a) 2 Fourier transform: TF[ ~v] = Z +1 1 ~v(t)e 2i t dt 0 if >> 1 ~v if << 1 One single particle produces a flat (v, Many particles with a range of impact parameters: with bmin from the small angle approximation with bmax @ = n in e v 0 if >> 1 16e 6 Z 2 3c 3 m 2 v 2 b 2 if << 1 Z bmax 2 bdb = 32 e6 3m 2 c 3 v n in e g (v, ) Produce a flat spectrum that cuts at the electron energy Total losses: P cool / n i n e Z @ 1/v b 2 <b 1 b 1 1/ 1 1/ 2 mv 2 /2 h 45

46 Bremsstrahlung Emission from many electrons Emission integrated over the particle distribution Power-law distributions produce power-law spectra Thermal distributions: Emission coefficient: j / n i n e Z 2 T 1/2 e h /k BT Total losses: P cool / n i n e Z 2 T 1/2 j T 2 >T 1 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 k B T 1 k B T 2 h 46

47 Summary Particle cooling: Synchrotron: P σt p 2 UB Compton in the Thomson regime: P σt p 2 Uph Bremsstrahlung: P σt αf p Ui (with Ui= ni mec 2 ) Photons: Synchrotron: Thin spectrum of 1 particle peaks at νc γ 2 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 = γ 2 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 47

Radiative Processes in Astrophysics

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