HIGH ENERGY ASTROPHYSICS - Lecture 7. PD Frank Rieger ITA & MPIK Heidelberg Wednesday

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1 HIGH ENERGY ASTROPHYSICS - Lecture 7 PD Frank Rieger ITA & MPIK Heidelberg Wednesday 1

2 (Inverse) Compton Scattering 1 Overview Compton Scattering, polarised and unpolarised light Di erential cross-section d /d andtotalcross-section Compton kinematics f ( i, ) Thomson ( f ' i )andklein-nishinaregime Inverse Compton scattering Energy change in Inverse Compton scattering 2

3 2 Thomson Scattering of Polarized Radiation by an electron Consider radiation from a free electron in response to incident linearly polarised electromagnetic wave. Force on charge (neglecting magnetic force for v c): ~F = m e ~r = ee0 ~ sin! 0 t with ~ denoting E-field direction. With dipole moment ~ d := e~r: ~d = e2 E 0 m e ~ sin! 0 t ) ~ d = e2 E 0 m e! 2 0 ~ sin! 0 t electron 3

4 Power of radiating dipole (cf. Larmor s formula, lecture 4): dp d = q2 4 c 3 ~v 2 sin 2 = ~d 2 4 c 3 sin2 and by intergration over solid angle (Larmor s formula): P = 2q2 3c 3 ~v 2 = 2 ~ d 2 3c 3 Time-averaged power: Averaging ~ d 2 = e4 E0 2 m 2 e R T =! sin 2! 0 tdt=1/2 gives: 1 T sin 2! 0 t over time t,with< sin 2! 0 t>= dp d = e4 E m 2 ec 3 sin2 and P = e4 E 2 0 3m 2 ec 3 (Note: := angle between ~ d and ~n!) 4

5 Incident (time-averaged) radiation flux on electron: Di erential cross-section d With dp d = e4 E 2 0 sin2 8 m 2 ec 3 : < ~ S >= c 8 E2 0 for scattering into d, is defined as dp d :=< ~ S > d d = ce2 0 d 8 d d d polarized = e4 m 2 ec 4 sin2 =:r 2 0 sin 2 with classical electron radius r 0 := e2 m e c 2 = cm Visualization: d gives area presented by electron to a photon that is going to be scattered in direction d. 5

6 Total cross-section by integrating over solid angle, or immediately from P =< ~ S > ) = P/< ~ S > where P =(e 4 E 2 0)/(3m 2 ec 3 )=E 2 0r 2 0c/3 and< ~ S >= ce 2 0/(8 ), giving: with Thomson cross-section T : = 8 3 r2 0=: T Note: T = cm 2 1. T applies only to non-relativistic regime; for higher energies, Klein-Nishina cross-section KN must be used (see later) 2. Scattered radiation is linearly polarised in direction of incident polarisation vector, ~, anddirectionofscattering, ~n. 3. Single particle (time-averaged) Thomson power P = T < ~ S >= T c (E 2 0/8 ) = TcU rad with radiation energy density U rad = E 2 0/8 =time-averaged energy density in incident wave. 6

7 3 Thomson Scattering of Unpolarized Radiation Unpolarised radiation: can have E-field in any direction = independent superposition of two linearly polarized beams with perpendicular axes. To scatter non-polarised radiation propagating in direction k into direction n, need to average two scatterings through angle and /2:! d = 1 d ( ) + d ( /2) d unpol 2 d pol d pol = r2 0 2 (sin2 +1)= r2 0 2 (cos2 +1)= 3 T 16 (1 + cos2 ) where = 6 (k, n). Forward-backward symmetry (! + )! Total cross-section, integrated over, is again = T. 1 n 2 /2 k 7

8 4 Thomson Optical Depth =Probability for a photon to experience Thomson scattering while traversing region containing free electrons with density n e Z T := n e T ds For T 1 source optically-thin to Thomson scattering, for T 1 optically-thick. Example - Thomson scattering in black hole accretion flows: Ion density = electron density, in terms of dimensionless accretion rate ṁ := M/ M Edd,whereM Edd ' 2.2 (M BH /10 8 M ) M /yr ' M BH,8 g/sec M 10 n(r) = = M rs 3/2 ṁ cm 3 4 r 2 m p v r r M BH assuming radial inflow velocity v r = (GM/r) 1/2 = c(r s /r) 1/2 / p 2(fraction <1offreeinfall);Schwarzschildradiusr s := 2GM/c 2. ) Thomson optical depth: T ' n e (r) T r ' 5 1 ṁ r s r 1/2 ) optically-thin for low accretion rates. 8

9 5 Kinematics of Compton Scattering Scattering of photon o an electron at rest; withmomentum4-vector(e/c,~p) notation: Initial and final four momentum of photon: Pi = h i c (1,~n i), P f = h f c (1,~n f) Initial and final four momenta of electron: Qi = m e (c, 0), Q f = f m e (c, ~v f ) scattered photon n i n f incident photon e - recoil Energy and momentum conservation: Pi + Q i = P f + Q f ) Q 2 f =( P i + Q i Pf ) 2 = P 2 i + Q 2 i + P 2 f +2 P i Qi 2 P i Pf 2 Q i Pf 9

10 With Q 2 = Q Q = 2 m e c 2 2 m e v 2 = m e c 2 and P 2 =0: ) P i P f = Q i ( P i Pf ) or: With ~n i ~n f =cos : h i h f c 2 (1 ~n i ~n f )=m e (h i h f ) h i f (1 cos ) =m e c 2 ( i f ) So that f = i 1+ h i m e c 2 (1 cos ) or in terms of wavelength with Compton wavelength = c/ : or f = i 1+ i m e c 2 (1 cos ) f i = h m e c (1 cos ) =: c (1 cos ) 0 c := h/m e c = cm. Note: f > i for all angles ( 6= 0),photonsalwayslosesenergy. 10

11 Fractional energy change: averaging over,withtaylorexpansionforh i m e c 2 (=Thomson regime): := h h i = f i i ' 1 i apple i 1 h i m e c2[1 cos ] i = h i m e c 2 [1 < cos >] = h i m e c 2 1 Compton scattering: Scattering of photon o an electron accompanied by energy transfer (decrease in photon energy). Thomson scattering: in regime h i m e c 2,transfersmall,scatteringalmost elastic ( i = h i = h f = f ), Thomson cross-section applies (initial and final wavelength quasi identical). Klein-Nishina scattering: in regime h i >m e c 2 transfer large, scattering deeply inelastic, needtousecross-sectionderivedfromqed. 11

12 6 Klein-Nishina Formula General formula for di erential cross-section derived by Klein & Nishina 1929 based on QED: d d = 3 2 f i 16 T + f sin 2 i f i with = h and f = i /(1 + i [1 cos ]) (kinematics). d /d measures mc 2 probability that photon gets scattered into angle. If i ' f (for i m e c 2 ) Thomson regime: d d! 3 16 T 1+[1 sin 2 ] = 3 16 T 1+cos 2 = d d T homson,unpolarized Note: Principal e ects is to reduce cross-section from classical value increases. T as energy Increased forward-scattering with increasing energy. At small energies, crosssection is forward-backward (, + ) symmetric. 12

13 d /d [m 2 /sr] Thomson limit: Wiki

14 7 Total Compton Cross-section Integration over solid angle gives total cross-section: = 2 =... = 3 4 where Z T 0 d sin d d apple 1+x 2x(1 + x) x 3 1+2x ln(1 + 2x) + 1 2x ln(1 + 2x) 1+3x (1 + 2x) 2 x h i m e c 2 Limits: (x) ' T (1 2x +...) for x 1 (Thomson) (x) ' T ln 2x + 1 for x 1 (extreme KN) x 2 14

15 1 asymptotics! T! KN /! T E/m e c 2 Figure 1: Total Compton cross-section as a function of normalized photon energy x = h i /m e c 2 along with asymptotics to high energies. 15

16 8 Inverse Compton Scattering If electron moves with velocity v (lab. frame K), energy can be transferred from electron to photon = Inverse Compton Scattering (ICS). In electron rest frame K 0,previousresultholds(writtenintermsofrest-frame variables = primed), e.g.: 0 0 apple i f = i m e c 2 (1 cos 0 ) with 0 scattering angle in K 0, i 0, 0 f propagation directions: ' 0 i cos 0 =cos 0 i cos 0 f +sin 0 i sin 0 f cos( 0 i 0 i m e c 2(1 cos 0 ) polar angles between electron and photon 0 0 where i, f azimuthal angles of incident and scattered photon in electron rest frame [ERF], noting that cos 0 = ~n 0 f ~n 0 i and in spherical coordinates ~n 0 i = (cos i 0, sin 0 i cos 0 i, sin i 0 sin 0 i )withcos(a b) =cosa cos b +sinasin b etc. Last relation on rhs in energy eq. valid in Thomson regime. 0 f) 16

17 Photon energies s in K 0 and K are related by Doppler formula i = D 0 i $ 0 i = i (1 cos i ) (1) f = 0 f (1 cos f ) = 0 f (1+ cos 0 f) (2) where D =1/( [1 cos ]), = v/c, =1/ p 1 v 2 /c 2. Thomson regime for 0 i m ec 2,i.e., i m e c 2 / Limits: (1) for i =0(photonapproachesfrombehind): 0 i = i (1 )! i /[2 ]. (2) for i = (head-on collision): 0 i = i (1 + )! 2 i ) Maximum energy in Thomson regime (using equation (2) above): f,max =2 0 f =2 0 i =4 2 i. ) in Klein-Nishina regime: f,max < m e c 2 + i m e c 2 (energy conservation). 17

18 Alternatively, usingaberration formula (lecture 4): cos 0 i,f = cos i,f 1 cos i,f Isotropic distribution in lab. frame K: halfthephotonshave i between (headon) and /2. ) in electron rest frame K 0, cos 0 i = for i = /2. ) For relativistic electrons ( ' 1), most photons are close to head-on in ERF. In Thomson regime: for 0 i m ec 2, 0 i = 0 f f = 2 i (1 cos i )(1 + cos 0 f)= 2 i (1 cos i ) = 2 i (1 cos i ) (1 cos f ) (1 2 )= i (1 cos i ) (1 cos f ) with eqs. (1),(2) before: 1+ cos f 1 cos f For head-on scattering i = and f =0(photonsturnsaroundafterscattering), f,max i = (1 + ) (1 ) = 2 (1 + ) 2 '

19 Summary: Scattered photon energy in lab frame: f ' ( 2 i, i m e c 2 / Thomson regime m e c 2, i m e c 2 / Klein Nishina limit 19

20 Figure 2: 2 -energy boost in Thomson regime as a result of relativistic beaming. Top left: Electron moving with velocity v in lab frame, incoming photons are isotropically distributed. Top right: Incoming photons as seen in the rest frame of the electron. They are now highly anisotropic, electron sees them as nearly head-on, their typical energies are boosted by a factor. Bottom left: Photons after scattering in electron rest frame. They are approximately isotropic and have roughly the same energy (Thomson regime) they had before being scattered. Bottom right: Scattered photons as seen in the lab-frame. They are now again highly collimated, with their typical energies boosted by a further factor, so that in lab-frame the overall energy is boosted by a factor 2 [Credits: N. Kaiser].

21 2 Single Particle Power radiated in Inverse Compton Scattering Total power emitted by an electron exposed to some photon field n ph ( ): Thomson regime ( h i m e c 2, 0 = 0 f )-scattered power in e -rest frame K0 : de 0 Z = dt 0 T curad 0 = T c n 0 ph( 0 ) 0 f( 0, 0 ) d 0 with n 0 ph ( 0 )d 0 =numberdensityofincidentphotonswithenergyind 0. Recall 1: de 0 /dt 0 = de/dt = Lorentz invariant (lecture 4). Recall 2: Phase space distribution f(x, p) := dn d 3 ~xd 3 ~p = f 0 (x 0, p 0 )=invariant. Let n ph ( )d := f ph (x, p)d 3 p,usethatd 3 p transforms same as energy (lecture 4): ) n ph( )d = n0 ph ( 0 )d 0 0 =Lorentzinvariant. Thus, with Doppler shift 0 = (1 cos ) andthomson 0 f = 0 : U 0 rad := Z 0 n 0 ph( 0 )d 0 = Z 02n0 ph ( 0 )d 0 0 = Z 02n ph( )d = Z 2 2 (1 cos ) 2n ph( )d 3

22 Assuming isotropic incident radiation field, angle average gives Z 1 1 cos ) 2 1(1 2 d cos = 1 Z 1 2 cos + 2 1(1 2 cos 2 ) d cos = so Urad 0 = /3 U rad Angle-averaged Compton scattered power (lab. frame) [erg/s]: de dt = de0 = dt 0 T curad 0 = T c U rad 3 Energy lost by electron = IC power =scatteredpowerminus incoming power: de e dt = P IC = T c(urad 0 U rad )= T c U rad = with 2 1= 2 2. Notes: (1) Total P IC in Thomson regime is independent of input photon spectrum. (2) Number of photons scattered per unit time dn s /dt = T cu rad /(h i ) ) Average energy increase (Thomson regime) via P IC =<h f >dn s /dt: T cu rad = P IC =<h f > T c U rad h i! = T cu rad ) < f >= i 4

23 Notes: If energy transfer in K 0 is not neglected (general KN case), Blumenthal and Gould (1970) showed that for incident isotropic photon distribution de e dt = P IC = T cu rad apple 1 63 < 2 i > 10 m e c 2 < i > with < i >= R i n ph ( i )d i /( R n ph ( i )d i )etcaveragesoverincidentphoton spectrum (spectrum now matters). ) Energy can be given or taken from electron. For power-law electron number distribution N e ( )=C p between min apple apple max,totalthomson IC power [erg/s] for 1: Z dee P tot = dt ( ) N e( )d = 4 C 3 T 3 p 3 p U rad max min 3 p For thermal distribution of non-relativistic electrons (< 2 >=3kT/m e c 2, ' 1): 4kT P tot = c m e c 2 T N e U rad [erg s 1 ] 5

24 3 Inverse Compton Cooling Timescale Electron cooling timescale due to ICS for isotropic incident radiation field in Thomson regime: t cool = E e de e /dt = m ec 2 P IC ' U rad [cgs] sec. Electron cooling timescale in general case: useknformulaabove. ) analytical approximation for isotropic electrons and photons: Moderski et al. (2005): Energy Loss Rate: with: where de e dt F KN := 1 U rad f KN (x) ' = 4 3 c T Z i,max i,min 2 U rad F KN f KN (x) i n ph ( i )d i 1 (1 + x) 1.5 for x := 4 i m e c 2 <

25 and U rad = R i,max i,min i n ph ( i )d i =totalenergydensityinincidentfield. ) in KN regime the cooling timescale increases as some function of! log F KN (x) log x Figure 1: Compton weighting function F KN (x) formono-energeticphotonfield. Atlow(x 1) energies, Thomson regime applies (F KN ' 1). In KN regime (x 1), radiated power is much reduced due to reduction in cross-section. Note: In principle, energy losses in the KN regime are not continuous, but catastrophic (electron loses significant fraction of its energy in a single interaction), yet noted approximation still turns out to be useful for many cases. 7

Radiation processes and mechanisms in astrophysics I. R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 2009

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