Ilana HARRUS (USRA/NASA/GSFC)

1. ContinuumProcesses (in High-Energy Astrophysics)3rd X-ray Astronomy schoolWallops Island May 12-16 May Ilana HARRUS (USRA/NASA/GSFC)

2. Bremsstrahlung What is Bremsstrahlung? (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 .

3. Bremsstrahlung Important because for relativistic particles, this can be the dominant mode of energy loss. Whenever there is hot ionized gas in the Universe, there will be Bremsstrahlung emission. Provide information on both the medium and the particles (electrons) doing the radiation. The complete treatment should be based on QED ===> in every reference book, the computations are made “classically” and modified (“Gaunt” factors) to take into account quantum effects.

4. Bremsstrahlung (non relativistic) Use the dipole approximation (fine for electron/nucleus bremsstrahlung)--- v -e b R Ze Electron moves mainly in straight line-- Dv = Ze2/me (b2+v2t2)-3/2 bdt = 2Ze2/mbv Electric field: E(t) =Ze3sinq/mec2R(b2+v2t2)

5. Bremsstrahlung (for one NR electron) (using Fourier transform) E(w) =Ze3sinq/mec2R p/bv e-bw/v Energy per unit area and frequency is: dW/dAdw = c|E(w)|2 So that integrated on all solid angles: dW(b)/dw = 8/3p (Z2e6/me2c3)(1/bv)2 e-2bw/v

6. Bremsstrahlung For a distribution of electrons in a medium with ion density ni. Electron density ne and same velocity v. Emission per unit time, volume, frequency: dW/dVdtdw = neni2pv bmin dW(b)/dw bdb Approximation: contributions up to bmax This implies (after integration on b) dW/dVdtdw = (16e6/3me2vc3) neniZ2 ln(bmax /bmin) with bmin ~ h/mv and bmax~ v/w

7. Bremsstrahlung If QED used, the result is: dW/dVdtdw = (16pe6/33/2me2vc3) neniZ2 gff(v,w) Karzas & Latter, 1961, ApJS, 6, 167

8. Now for electrons with a Maxwell-Boltzmann velocity distribution. The probability dP that a particle has a velocity within d3v is: dP e-E/kT d3v v2e(-mv2/2kT)dv Integration limits: 1/2 mv2 >> hn (Photon discreteness effect) and using dw=2pdn dW/dVdtdn = (32pe6/3mec3) (2p/3kTme)1/2 neniZ2 e(-hn/kT) <gff> == 6.8 10-38 T-1/2neniZ2 e(-hn/kT) <gff> erg s-1 cm-3 Hz-1 <gff> is the velocity average Gaunt factor Bremsstrahlung

9. Bremsstrahlung Approximate analytic formulae for <gff> From Rybicki & Lightman Fig 5.2 (corrected) -- originally from Novikov and Thorne (1973)

10. Bremsstrahlung u =hn/kT ; g2 = Ry Z2/kT =1.58x105 Z2/T Numerical values of <gff>. From Rybicki & Lightman Fig 5.3 -- originally from Karzas & Latter (1961) When integrated over frequency: dW/dVdt = (32pe6/3hmec3) (2pkT/3me)1/2 neniZ2 <gB> == 1.4x10-27 T1/2neniZ2 <gB> erg s-1 cm-3 (1+4.4x10-10T)

11. Cyclotron/Synchrotron Radiation Radiation emitted by charge moving in a magnetic field. First discussed by Schott (1912). Revived after 1945 in connection with problems on radiation from electron accelerators, … Very important in astrophysics: Galactic radio emission (radiation from the halo and the disk), radio emission from the shell of supernova remnants, X-ray synchrotron from PWN in SNRs…

12. Cyclotron/Synchrotron Radiation As with Bremsstrahlung, complete (rigorous) derivation is quite tricky. First for a non-relativist electron: frequency of gyration in the magnetic field is wL =eB/mc = 2.8 B1G MHz (Larmor) Frequency of radiation == wL

13. Angular distribution of radiation (acceleration  velocity). Rybicki & Lightman Synchrotron Radiation Because of relativistic effects: beaming and Dq~1/g Gyration frequency wB = wL/g Observer sees radiations for duration Dt<<< T =2p/wB This means that the spectrum includes higher harmonics of wB. Maximum is at a characteristic frequency which is: wc ~ 1/Dt ~ g2eB /mc

14. Total emitted radiation is: P= 2e4B2/3m2c3 b2g2 = 2/3 r02 c g2 B2when g>>1 Or P  g2 c sTUBsin2q (UB is the magnetic energy density) P~1.6x10-15g2 B2 sin2qerg s-1 Synchrotron Radiation Life time of particle of energy t  E/P ~ 20/gB2 yr Example of Crab-- Life time of X-ray producing electron is about 20 years. P  1/m2 : synchrotron is negligible for massive particles.

15. The computation of the spectral distribution of the total radiation from one UR electron: (computation done in both polarization directions -- parallel and perpendicular to the direction of the magnetic field). P(w) = (3e3/4pmc2) B sinq [F(x)+G(x)] ; x=w/wc P(w) =(3e3/4pmc2) B sinq [F(x)-G(x)] Where F(x)==x∫∞x K 5/3(y)dy ; G(x)==xK 5/3(x) and K modified Bessel function and wc = 3/2g2 wLsinq Total emitted power per frequency: P(w)=(3e3/2pmc2) B sinq F(w/ wc) Synchrotron Radiation

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17. Hypothesis: Energy spectrum of the electrons between energy E1 and E2 can be approximated by a power-law -- N(E)=KE-r dE (isotropic, homegeneous). Number of e- per unit volume, between E and E+dE (in arbitrary direction of motion) Synchrotron Radiation Intensity of radiation in a homogeneous magnetic field: I(n,k)=(3/r+1) G(3r-1/12) G(3r+19/12) e3/mc2 (3e/2pm3c5)(r-1)/2 K [B sinq](r+1)/2 n-(r-1)/2 Average on all directions of magnetic field (for astrophysical applications). L is the dimension of the radiating region I(n)= a(r) e3/mc2(3e/4pm3c5)(r-1)/2 B(r+1)/2 K L n-(r-1)/2erg cm-2 s-1 ster-1 Hz-1 where a(r) = 2(r-1)/2 (3/p)G(3r-1/12) G(3r+19/12) G(r+5/4)/[8(r+1) G(r+7/4)]

18. Synchrotron Radiation If the energy distribution of the electrons is a power distribution

19. Estimating the two boundaries energies E1 and E2 of electrons radiating between n1 andn2. Synchrotron Radiation E1(n) ≤ mc2 [4pmcn1/3eBy1(r)] 1/2 = 250 [n1/By1(r)] 1/2 eV E2(n) ≤ mc2 [4pmcn2/3eBy2(r)] 1/2 = 250 [n2/By2(r)] 1/2 eV y1(r) and y2(r) are tabulated (or you can compute them yourself..). If interval n2/ n1 << y1(r)/y2(r) or if r<1.5 this is only rough estimate

20. Synchrotron Radiation Expected polarization: (P(w) - P(w))/(P(w) + P(w)) == (r+1)/(r+ 7/3) can be very high (more than 70%).

21. 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-1/2v5/2 independent of the spectral index. ====> break frequency

22. For low energy photons (hn << mc2), scattering is classical Thomson scattering (Ei=Es; sT = 8p/3 r02) Compton/Inv Compton Scattering Es=hns q Ei=hni Pe, E More general: Es=Ei (1+Ei(1-cosq)/mc2)-1 or ls-li=lc(1-cosq) (lc =h/mc) This means that Es is always smaller than Ei Even more general: (Klein-Nishina) ds/d = 1/2 r02 y2(y+1/y -sin2q) with y=Es/Ei

23. Compton/Inv Compton Scattering If electron kinetic energy is large enough, energy transferred from electron to the photon: Inverse Compton One can use previous formula (valid in the rest frame of the electron) and then Lorentz transform. So : Eifoe=Eilabg(1-bcosq) then Eifoe becomes Esfoe and Eslab=Esfoeg(1+bcosq') This means that Eslab  Eilab g2 The boost can be enormous!

24. Inverse Compton Scattering Inverse Compton Scattering

25. Compton/Inv Compton Scattering The total power emitted: Pcompt = 4/3 sTcg2b2Uph[1-f(g,Eilab)] ~ 4/3 sTcg2b2Uph And Uph is the initial photon energy density We had Psync  g2 c sTUB In fact : Psync/Pcompt = UB /Uph Synchrotron == inverse Compton off virtual photons in the magnetic field. Both synchrotron and IC are very powerful tools ===> direct access to magnetic and photon energy density.

26. Not covered • Thermal bremsstrahlung absorption (energy absorbed by free moving electrons) • Black body radiation • Transition radiation (often not mentioned)

27. Books and references • Rybicki & Lightman "Radiative processes in Astrophysics" • Longair "High Energy Astrophysics" • Shu "Physics of Astrophysics" • Tucker "Radiation processes in Astrophysics" • Jackson "Classical Electrodynamics" • Pacholczyk "Radio Astrophysics" • Ginzburg & Syrovatskii "Cosmic Magnetobremmstrahlung" 1965 Ann. Rev. Astr. Ap. 3, 297 • Ginzburg & Tsytovitch "Transition radiation"