Non-Thermal Radiative Processes in Astrophysics Leptonic & Hadronic Elementary Processes

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Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes

Non-thermal radiative processes in Astrophysics Leptonic & hadronic elementary processes

Alexandre Marcowith 1 [email protected]

1Laboratoire Univers et Particules de Montpellier Université de Montpellier-2, IN2P3/CNRS

27 avril 2013

1/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Preliminaries-I

Non-thermal processes in Astrophysics : A great variety of objects (see the different lectures) 1 Compact sources : Sources associated with residues of massive stars Stellar size compact sources : black holes and X-ray binaries, pulsars, gamma-ray binaries, gamma-ray bursts. Galactic size compact sources : Central black-hole (Sagitarus A∗), Active galactic nuclei. 2 Diffuse or extended sources : Sources linked with a compact object but spread over larger scales Stellar size extended objects : Pulsar nebula, supernova remnants, massive star clusters. Interstellar medium. Galactic size extended objects : Galaxy starburst, Clusters of galaxy. 3 Special effects associated with relativity (special or general) All share a common property : to emit a large fraction of their bolometric luminosity into non-thermal radiation.

2/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Preliminaries-II

Non-thermal processes in Astrophysics : A great variety of processes

1 Processes related to leptons (electrons, electron-positron pairs) Interaction with matter : Coulomb or ionization losses and Bremsstrahlung radiation, pair creation/annihilation. Interaction with magnetic ﬁelds : Cyclo-synchrotron radiation, curvature radiation. Interaction with radiation : Inverse Compton, Comptonization.

2 Processes related to hadrons (mostly protons and helium) Interaction with matter : Coulomb or ionization losses and Bremsstrahlung radiation - Pion production - neutrinos. Interaction with radiation : Pion production - neutrinos. Interaction with magnetic ﬁeld : cyclo-Synchrotron radiation.

3 Impact of losses over particle distribution 4 Radiative transfer and radiation in a plasma

3/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Preliminaries -III

The three different approaches to the problem of radiation.

1 Classical radiation theory (CRT), or classical electrodynamics (CED). Based on a classical treatment of fields/particles. Consider the interaction between particle and fields using interaction kinematics. Derive the radiation intensity solving a radiative transfer equation. 2 Quantum radiation theory (QRT), or quantum electrodynamics (QED)/ quantum chromodynamics (QCD) depending we are considering leptons or hadrons. Based on a quantum treatment of fields and particle-field interaction theory. Mandatory to derive the cross sections. 3 Quantum Plasma Dynamics (QPD). A theory that describes plasma kinetics using a semi-classical approach. Largely developed by D.B. Melrose. I will not discuss it (see Melrose D.B. Springer Verlag books).

4/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Bibliography

Books Bekeﬁ G., 1966, Radiation processes in plasmas, Wiley. (B66) Ginzburg, V.L., 1979, Theoretical Physics and Astrophysics (Oxford :Pergamon)(G64) Jackson, J.D., 1962, Classical Electrodynamics, Wiley. (J62) Landau, L. & Lifschitz E., 1971, Field theory, Mir. (LL71) Longair, M.S., 1981, High energy Astrophysics, vol 1. and 2., Cambridge university press. (L81) Rybicki G. B. & Lightmann A.P. , 1979, Radiative processes in Astrophysics, Wiley (RL79) Schlickeiser R., 2002, Cosmic Ray Astrophysics, Springer. (S02) Articles Blumenthal G.R. & Gould R.J., 1970, Rev. of Mod. Phys., 42, 237 (BG70) Strong, A.W. & Moskalenko, I.V., 1998, ApJ, 509, 212 (SM98)

5/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Some recurrent notations

See NRL plasma formulary These lectures are given in CGS units. For a particle we deﬁne β = v/c and its Lorentz factor γ = (1 − β2)−1/2. Electromagnetic processes : The classical electron radius 2 2 −13 re = e /mec = 2.8 × 10 cm, Thomson cross section : 2 −25 2 σT = 8π/3re = 6.65 × 10 cm . Hadronic processes : The cross section are in units of milli barns=10−27 cm2. Energy units 1erg =1/1.60 TeV

6/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

4 Non-thermal leptonic processes Interaction with matter Interaction with magnetic ﬁelds Interaction with radiation 5 Non-thermal hadronic processes Hadron-matter interaction Hadron-radiation interaction Hadron-magnetic ﬁeld interaction

7/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Radiative transfer

The radiative transfer theory is the macroscopic theory of propagation of light through a medium. This theory involves the description of systems with size L λ. It involves the concept of rays. Rays are described in the eikonal approximation : curves whose tangent at each point lie along the direction of the propagation of the wave. If the wave function W(~x, t) = A(~x, t) exp(iφ(~x, t)) → A is a slowly varying function over a wavelength, φ varies rapidly over the wavelength. The radiative transfer and ray approximation are intrinsically limited by the uncertainty principle of quantum mechanics. Associated to rays different quantities characterize the radiation ﬁeld.

8/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Speciﬁc intensity and its moments

1 Specific intensity or brightness : Iν (~r, Ω, ν) = energy per unit of time frequency and solid angle traversing a surface element [erg/s cm2 St Hz], a quantity conserved along one ray (RL79 §1) dW I = ν dtdSdΩdν R 2 2 Mean intensity : Jν = (1/4π) Iν dΩ[erg/s cm Hz]. R 2 3 Energy flux (1st moment) : Fν = Iν cos(θ)dΩ[erg/s cm Hz]. 4 Radiative pressure (2nd moment, flux of momentum) : R 2 2 Pν = (1/c) Iν cos(θ) dΩ[dynes/cm Hz]. θ = (~n,~d) ~n is the normal to dS and ~d is the ray direction.

9/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Radiative transfer equation

In vacuo dIν /ds = 0, s is the ray path. However in a medium Iν varies through the combined effect of :

1 Source term or spontaneous emission coefﬁcient jν = dW/dVdtdν [erg/cm3s St Hz] (synchrotron, bremsstrahlung ...) −1 2 Absorption coefﬁcient αν [cm ] (synchrotron, bremsstrahlung ...) The radiative transfer Eq. reads :

dI ν = −α I + j . (1) ds ν ν ν

Also useful is the source function Sν = jν /αν = a quantity towards which Iν tends to relax. −1 3 Diffusion coefﬁcient σν [cm ] (Comptonization (lect.2) ...). For isotropic scattering jν = σν Jν .

10/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Thermal processes

In case of thermal (6= Blackbody Iν = Bν ) radiation :

3 2 Sν = Bν (T) = 2hν /c /(exp(hν/kBT) − 1) , hνmax = 2.82kBT . (2)

The transfer Eq. can be written as the Kirchoff law for thermal emission

jν = αν Bν (T) . (3)

The brightness temperature is the temperature of a blackbody having the same speciﬁc intensity at ν : Iν = Bν (Tb). In the Rayleigh-Jeans c2 regime hν kBT and Tb = 2 Iν , Iν = Sν /∆Ω. Important diagnostic 2ν kB in compact sources with resolved solid angle ∆Ω.

The blackbody Wien regime (hν kBT) leads to 3 2 Iν = (2hν /c )exp(−hν/kBT). All these macroscopic quantities can be obtained from a microscopic analysis.

11/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Einstein coefﬁcients

• Kirchoff law relies on a detailed balance on a microscopic level (reversibility) between emission and absorption processes. Consider two discrete energy levels (E, with a statistical weight g1, population n1) and (E + hν0,g2,n2). 3 coefﬁcients with probabilities/s : spontaneous (incoherent) emission A2→1, absorption B1→2J, stimulated emission B2→1J. R ∞ J = 0 Jν φ(ν)dν , φ(ν) is the line proﬁle centered on ν0 At thermodynamic equilibrium (but the relations are valid out of thermodynamical eq.) the detailed balance gives :

n1B12J = n2A21 + n2B21J (4)

Also Jν = Bν which provides relations between the coefﬁcients

g1B12 = g2B21 (5) 2hν3 A = B (6) 21 c2 21

12/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Relation to macroscopic coefﬁcients

Spontaneous emission hν j = n A φ(ν) ν 4π 2 21 Absorption+stimulated emission hν α = × (n B − n B ) φ(ν) ν 4π 1 12 2 21

Hence at the TE n1/n2 = g1/g2exp(hν/kT) gives Sν = Bν (T), the Kirchoff law.

13/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Polarization and Stokes parameters In the case of a monochromatic wave elliptically polarized. We transform the electric ﬁeld ~E → ~E0 where (x,y) is the frame of the observer (instrument) and (x’,y’) is the frame of the polarization ellipse. We apply a rotation of an angle χ E1, E2, φ1, φ2 → E0, β, χ (see RL section 2.4). β describes ~E in (x’,y’). The 4 Stokes parameters express this link :

2 2 2 I = E1 + E2 = E0 > 0 , ≡ Intensity 2 2 2 Q = E1 − E2 = E0 cos(2β) cos(2χ) , 2 U = 2E1E2 cos(φ1 − φ2) = E0 cos(2β) sin(2χ) , 2 V = 2E1E2 sin(φ1 − φ2) = E0 sin(2β) , ≡ degree of circularity .

14/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Polarization and Stokes parameters for an ensemble of waves

1 Monochromatic wave case : Pure elliptical polarization I2 = Q2 + U2 + V2. V = 0 is the condition for the linear polarization. Q = U = 0 is the condition for circular polarization.

2 In fact light ≡ sum of wave packets hence the Stokes parameters are averaged h.iT over the time T of the signal (i.e. Iensemble = ΣkhImono,kiT ). p A wave is the sum of an unpolarized I − Q2 + U2 + V2 and a polarized p Q2 + U2 + V2 part. The latter can be cast into two oppositely polarized p 2 2 2 parts I± = 1/2(I ± Q + U + V ). Degree of polarization of an ensemble of waves

p I Q2 + U2 + V2 (I − I ) Π = pol = = + − . (7) I I I

15/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Elements of classical Electrodynamics Non-thermal leptonic processes Non-thermal hadronic processes Useful spectral quantities : power-law radiation

Photon differential spectrum (or also per unit of energy keV,...,TeV) dn ∝ ν−s [nb/cm2 s Hz] . dSdtdν Flux or energy differential spectrum 1 dW Z νdn 4πJ = = I dΩ ≡ ∝ ν1−s [erg/cm2s Hz] . ν dSdtdν ν dSdtdν Energy spectrum per unit log of frequency (or energy) dW νF = =∝ ν2−s [erg/cm2s] . ν dSdtd log(ν) Differential luminosity Z dW L = dS [erg/s Hz] . ν dSdtdν 2 The luminosity reported at Earth is an observed ﬂux Fν = Lν /4πR . 1. In radio the Jansky unit = 10−23 erg/cm2s Hz = 10−26 W/m2Hz. 16/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

17/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Deﬁnition : Power spectrum

The radiation spectrum is given by the time variation of the electric ﬁeld of the electromagnetic waves. The energy in the spectrum per unit of time and per unit of surface can be obtained from the Poynting vector Π~ = c~E ∧ (~B/4πµ). dW c = |E(t)|2 . dSdt 4π • The spectrum is expressed using the Fourier transform of the electric ﬁeld : dW = c|E(ω)|2 . (8) dSdω

18/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Retarded potentials : geometry

If~r = ~r0(t) is the position of the particle at time t, the solution of Maxwell Eq. using the retarded potential method gives the electric ﬁeld produced by a ~ charge with a velocity ~v = β(tret)c and with an acceleration ~a = ~v˙(tret) (see J62 §14, LL71 §8, RL79 §3). 0 0 The retarded time is tret = t − R(tret)/c. R(t ) = |~r0(t) −~r0(t )|, ~n = ~R/R.

19/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Retarded potentials : electric ﬁeld solution

• The electric field has two components : generalization of a Coulomb field and radiation field.

(~n − β~) q ~n ~ ~E(~r, t) = q + ∧ (~n − β~) ∧ β˙| , (9) γ2d3R2 c d3R tret ~ ~ ~ ~ = ECoul,v + Erad , B(~r, t) = ~n ∧ E|tret

It differs to the static case by two effects (i) retarded times (ii) beaming effects : d = 1 − ~n.β~ accounts for angular effects. If the particle is non relativistic tret = t(O(β)) and d ∼ 1. If the particle is relativistic one can have d 1.

20/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Spectrum radiated by one particle

Using Eq. 8 one gets the energy radiated per unit of solid angle and frequency 2 Z 2 dW q = × Re(E(t)) exp(iωt)dt . (10) dωdΩ 4π2c • Including the radiation part (second one) of Eq. 9 (still evaluated at retarded times t0 = t − R(t0)/c) gives :

2 Z 2 dW q ~n ~ ~˙ 0 0 0 = × ∧ (~n − β) ∧ β exp(iω(t − ~n.~r0(t )/c))dt . dωdΩ 4π2c d2 (11)

21/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

22/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes The electric dipolar approximation

We consider the sources of the electromagnetic ﬁeld as an ensemble of charges displayed over a region of size a. We consider the radiation produced at an observer place by this ensemble of charges at large distances r a.

In this configuration we only retain a part of the solution of the electromagnetic field produced by one charge, we discard the Coulomb contribution which scales as 1/r2 and has a Poynting flux vanishing at infinity (see Jackson, Rybicki & Lightman). The electric dipole approximation consists in writing |~r −~r0| ∼ |~r|. It can be translated into a condition over the wavelength of the radiation

a λ . (12)

23/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Total power radiated by an ensemble of particle : The Larmor formula

The total power emitted by an ensemble non-relativistic particles is obtained by integrating the Poynting ﬂux over a sphere of radius r using solutions 9 for one particle and summing over the ensemble contribution. One gets the Larmor formula (with D~ = Σiqi~ri)

2 D¨ 2 P = . (13) em 3 c3 ¨ For one charge e with an acceleration ~A = D~ /e one gets :

2 e2A2 P = . (14) em 3 c3 2 • The power radiated per unit of solid angle P(θ) = 3/8πPem sin (θ) is not a relativistic invariant (θ = (~A,~n)).

24/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Total power radiated by relativistic particles : The relativistic Larmor formula

We apply the Larmor formula in the instantaneous rest frame of the radiating particle and hence proceed with a Lorentz transformation to get the emitted power in the observer frame. In fact the total power emitted Pem = ∆E/∆t is a scalar and a relativistic invariant. We can express the total power emitted as Pem ∝ A.A ; in terms of a scalar product of the quadri-acceleration A. In the observer frame :

2 −e2 . 2 e2γ4 P = A A = × A2 + γ2A2 . em 3 3 k ⊥ (15) 3 4π0c 3 c

2 −1/2 With β = v/c, γ = (1 − β ) . Ak,⊥ are the components of the acceleration vector parallel and perpendicular to the boost ~v in the observer frame.

25/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Angular power

We use the Lorentz transformation of the energy-momentum 4 vector from the particle rest frame (R’) to the observer frame (R). The energy emitted dW in a direction dΩ = d cos(θ)dφ is

dW dW0 = (γ(1 − βcosθ))−3 × (16) dΩ dΩ0 We distinguish among the emitted power in frame R : dW −4 −3 dW0 Pe = dtdΩ = γ (1 − βcosθ) × dt0dΩ0 and the received power by a ﬁx −4 dW0 observer in frame R : Pr = (γ(1 − βcosθ)) × dt0dΩ0 . The latter includes a retardation effect due to the motion of the source. The term dW0/dt0dΩ0 is deduced from Eq. 14. • The Doppler factor D = (γ(1 − βcosθ))−1 is 1 if γ 1 and cos θ ∼ β. The power emitted is highly ampliﬁed in the particle direction of motion in a cone of size ∼ 1/γ.

26/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Deﬁnitions Elements of classical Electrodynamics The dipolar approximation and the Larmor formula Non-thermal leptonic processes Non-thermal hadronic processes Reaction force

• As the particle is radiating a part of its own energy hence there must be a force acting on the particle due to its radiation (J62, LL71, RL74). Braking force due to radiation (Abraham-Lorentz force) 2e2 ~F = ~u¨ + (~F ) . (17) 3c3 ext

~Fext is an external force ; e.g. Lorentz force by an EM ﬁeld. The reaction force is a perturbation if the condition λ re is fulﬁlled, λ is the wavelength of the radiation in the particle rest-frame (see LL71). In the case of relativistic particles a 4-vector formulation is used : 2e2 d2uµ P gµ = − rad uµ . (18) 3c ds2 c2

And Prad is given by Eq. 15 and U = (γ, γ~v/c). In fact as γ 1 the second term is dominant (LL71) and the force can be seen as a friction.

27/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

28/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Lepton-matter interaction : Generalities

The relevant processes here are of two kinds depending if the lepton do emit or not a photon during the interaction. Process with no photon : Coulomb interaction in a fully ionized plasma - Ionization losses in a partially ionized plasma. Process with photon production : Bremsstrahlung (active in fully/partially ionized plasma). Bremsstrahlung process of a non-relativistic electron in the Coulomb ﬁeld of a charged ion using a classical approach- Thermal Bremsstrahlung. Bremsstrahlung process of a relativistic electron in interaction with atoms. This involves at least some connection with QED as the energy of the 2 emitted photon may be ∼ mec . NB : Thermal Bremsstrahlung can be treated using the methods developed in QED.

29/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Coulomb and ionization losses

Loss rates (see G64, L81, SM98) : 2 1 Coulomb losses in a cold ionized plasma (ne is the background electron density)

dE 1 πEm c2 3 − = πcr2m c2 n e − [ / ] . 2 e e e ln 2 2 erg s (19) dt β reh c ne 4

2 Ionization in a neutral plasma (IH = 13.6eV, IHe = 24.6eV are the ionization potentials) :

dE 1 (γ − 1)β2E2 1 − = πcr2m c2 ×Σ Z n × + [ / ] . 2 e e s=H,He s s ln 2 erg s dt β 2Is 8 (20) Both processes dominate at energies < GeV in the interstellar medium (S02). 2 2. The case of hot plasma kBTe ∼ mec is treated in Moskalenko I. & Jourdain E., 1997, A&A, 325, 401 and the relativistic case in L81 30/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung

At higher energies (but still non-relativistic for the moment) the interaction with the Coulomb ﬁeld of a charge induces an acceleration of the electron and the emission of a photon. The acceleration is perpendicular to the velocity. The power radiated by the particle is given by Eq. 8 expressed in terms of the Fourier transform of the electric ﬁeld with the FT of D¨ : dW 8πω4 = D(ω)2 (21) dSdω 3c3

31/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung radiation by one non-relativistic particle

The timescale of the interaction is τ = b/v.

Interaction of an electron of charge e with an ion of charge Ze at an impact parameter b The dipole moment D¨ = −ev˙ leads to : |D(ω)| = e∆v(b)/2πω2 for ωτ 1. The velocity variation along the trajectory is due to the centripete acceleration produced by the normal electric Coulomb ﬁeld of the ion (see RL79 Eq.4.70b (CGS), L81 Eq.3.35 (SI)). The ﬂux of incident electron and the density of targets leads to the total power radiated per unit time and volume : dW Z bmax dW(b) 16πe6 b = (n v)×n × πb db = n n Z2ln( max ) e i 2 2 4 e i (22) dωdtdV bmin dbdω 3mec β bmin

32/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes

The Gaunt factor : bmax and bmin

We have followed a classical treatment but include quantum corrections for bmin.

bmax is obtained for ωτ = ωb/v = 1 thus bmax = v/ω.

bmin is obtained either in the limit ∆v = v or in the quantum limit where the classical trajectories cease to be valid. 2 2 Condition 1 gives : bmin = 4Ze /πmev . It dominates in the low energy 2 2 2 4 2 limit Ek = mv /2 Z Ry, Ry = 4π mee /2h . Condition 2 gives : applying the uncertainty principle : bmin = h/mv. It dominates in the high energy limit T = mv2/2 Z2Ry. We can deﬁne the Gaunt factor √ 3 bmax gff (v, ω) = ln( ) (23) π bmin

33/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung spectrum by a population of thermal particles

We consider the case of an isotropic non-relativistic Maxwellian distribution of electrons with a temperature T

2 2 2 F(v)4πv dv = 4πv × exp(−mev /2kBT)dv

6 2 4 3 2 We get (KTh = 32πe /3(mec ) = 32π/3 × re mec ):

2 1/2 dW 2 2πmec 3 = KTh×ne×ni×Z exp(−hν/kBT)g¯ff (T, ν)[erg/s Hz cm ] . dtdνdV 3kBT (24) The power radiated is almost independent (g¯ff is not strongly dependent on ν) of the frequency except if hν kBT where it is cut off exponentially. The factor g¯ff is in the range (1-5). It can be found in RL79 in ﬁgure 5.2 and in references therein.

34/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung self-absorption

An electron can either absorb a photon in the ﬁeld of an ion with an absorption coefﬁcient αν (Kirchoff’s law)

3 2 dW −1 −1 2hν /c αν = × (4πBν (T)) [cm ] , Bν (T) = . dtdνdV exp(hν/kBT) − 1

The effect of absorption is strong at hν kBT if the opacity αν R 1 (R is the size of the emitting region). Typical thermal free-free spectrum as produced in HII regions (in radio wavebands). The ﬂux is ∝ ν2 at low frequencies (Rayleigh-Jeans spectrum) and exponentially cut (Wien spectrum) off at high frequencies.

35/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung in the relativistic regime

2 In that regime hν may be not negligible wrt mec . Three approaches are possible : Derive an equivalent expression as Eq.22 in the relativistic case from the radiated spectrum by an accelerated particle (L81). Uses the method of virtual quanta (Weiszäcker & Williams method, see BG70 section 3.2). Use QED calculations to derive the differential Bremsstrahlung cross section (BG70 section 3.3, 3.5). This calculation give more accurate results. We have the following chain to get the loss timescale dN dσ −dE Z dN = cΣ n → = dνhν → t = E/|dE/dt| . dtdν i i dν dt dtdν loss

2 Hereafter we will note E¯ = E/mec and the Bremsstrahlung photon energy 2 = hν/mec .

36/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung cross section

Naked ion of charge Ze (see BG70, eq.3.1 and references therein) :

d 2 2E¯(E¯ − ) 1 dσ = 4Z2αr2 E¯2 + (E¯ − )2 − E¯(E¯ − ) ln − . e E¯2 3 2 (25) An atom : d 2 dσ = αr2 (E¯2 + (E¯ − )2)φ − E¯(E¯ − )φ (26) e E¯2 1 3 2

φ1/2,s are the screening functions given in BL70 (Fig9, table 2). At low 2 ¯ ¯ energies φ1/2,s → 4(Z + Z)(ln(2E(E − )/) − 1/2), (weak shielding case ≡ ion + Z electrons). Low energy means ∆ = 1/(2α)/(E¯(E¯ − )) > 1.

37/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung cross section

dσ/d on atomic hydrogen for different electron energies 3.

3. Sacher W. & Schoenfelder V., 1984, ApJ, 879, 217 38/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung loss timescale

1 Weak shielding case (ionized medium or an atom with low energy electrons) : (i=ions of charge Z+ Z electrons) dE¯ − = 4αr2cΣ n Z(Z + 1) ln(2E¯ − 1/3)E¯ [s−1]! (27) dt e i i

2 Strong shielding case : (s=species : bound electrons, ions and high-energy electrons) dE¯ 4 1 − = αr2cΣ n φ − φ . (28) dt e s s 3 1,s 3 2,s The typical loss timescale is (ionized case) in a plasma composed of hydrogen + 10% of helium.

8 −1 ¯ −1 tloss ' 1.45 × 10 × nH,cm−3 [ln(2E − 1/3)] [years] (29)

39/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Bremsstrahlung spectrum by a population of non-thermal relativistic particles

We consider the case of an isotropic relativistic power-law distribution of electrons with total density ne.

Z Emax −s N(E)dE = KeE¯ dE , Emin < E < Emax , ne = N(E)dE Emin The differential ﬂux is (weak shielding) : (R dσ/d(E)N(E)dE) dW = 4αr2cK m c2Σn (Z2 + Z) × I(, E¯ , s) , E¯ = max(E¯ , ) (30) dtddV e e e i L L min ! 4 E¯−(s−1) E¯−s 3 2E¯−s−1 I(, E¯ , s) = × L − L + L × φ ∝ −s+1 (E¯ = ) L 3 (s − 1) s 4 (s + 1) L

φLE = ln(2E¯min(E¯min − )/ − 1/2) , < E¯min

φHE = ln(4 − 1/2) , > E¯min

40/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Summary

2 1 Thermal Bremsstrahlung. Self-absorption produces F ∝ ν at high opacity and low frequencies and F ∝ exp(−hν/kBT) at high optically thin frequencies. 2 Loss timescale is large in the interstellar medium, but Bremsstrahlung usually dominates loss processes at energies ∼ GeV (see lesson II). Important loss in denser media (HII regions, supernova remnant in interaction with molecular clouds ...) −s 3 Non-thermal Bremsstrahlung. A distribution E do produce an energy (photon) spectrum in −s+1 (−s).

41/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Electron-positron pair production

Reaction : γ + γ → e− + e+ ~ ~ Pair creation threshold : 12(1 − cosχ) ≥ 2, χ = (k1, k2) : angle of interaction.

42/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Electron-positron pair production cross section

CM 3σT 2 4 1 + βCM 2 σγγ = (1 − βCM) (3 − βCM) ln( ) − 2βCM(2 − βCM) 16 1 − βCM s 2 βCM = 1 − . (31) 12(1 − cos χ)

See 4. One uses it to derive the photon-photon pair production opacity.

Z Z d cos χ dφ τ ( , Ω ) = dZ d 1 1 n ( , Ω )σCM( , , χ)(1−cos χ(χ )) . γγ 2 2 1 4π ph,1 1 1 γγ 1 2 1/2 (32) Integral over Z the length along the line of sight, 1, Ω1 the energy and solid angle of the low energy photons ... 5 integrals needed to get τγγ (2) !

4. Gould R.J. & Schréder G.P. 1967 Phys Rev 155 1408 43/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Electron-positron pair annihilation

In-ﬂight reaction : e− + e+ → γ + γ : This produces an annihilation line that peaks at 511keV if the particles annihilate at rest. Other processes : positronium formation. 2 ground states.

1 Para-positronium p-Ps (fraction 1/4) : Anti-parallel spins produce a singlet 1 state S0. Lifetime = 125ps in vacuum. Decay into 2 gamma-rays at 511keV. 2 Ortho-positronium o-Ps (fraction 3/4) : Parallel spins produce a multiplet 3 state S1. Lifetime = 142.05 ± 0.02ns in vacuum. Decay into 3 gamma-rays through a continuum spectrum.

44/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Contribution to the galactic annihilation line

Fit of the diffuse annihilatetion as observed by SPI instrument on board INTEGRAL satellite 5. See Prantzos, N. et al (AM) 2011, Rev.Mod.Phys. 83, 1001 for a review about diffuse galactic annihilation line. 5. Jean P. et al (AM) 2006, A&A, 445, 579 45/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes (integrated) Annihilation cross section

CM 3σT 2 1 1 + βCM 1 σan = 2 2 (2 + 2 − 4 ) ln( ) − 2βCM(1 + 2 ) 32βCMγCM γCM γCM 1 − βCM γCM r 1 + γ γ (1 − β β µ) γ = + − + − = (1 − β2 )−1/2 . (33) CM 2 CM

~ ~ 6 With µ = (β+, β−). The electron-positron production rate can be deduced .

Z 2 ~ ~ ~ ~ CM γCM −3 −1 R± = 2 dβ+dβ−f (β+)f (β−)βCMcσan × [cm s ]. (34) γ+γ−

~ ± R ~ ~ 7 f (β±) : e distribution normalized wrt to the densities n± = f (β±)dβ± .

6. Svensson R., 1982, ApJ, 258, 321 and references therein 7. To derive a spectrum on must turn back to the differential cross section 46/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

47/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Lepton-magnetic ﬁeld interaction : Generalities

The relevant process name here depends on the energy of the radiating particle. We speak about cyclotron radiation for a non-relativistic particle (v < 0.5c). We speak about gyromagnetic radiation at mildly relativistic speeds. The synchrotron radiation is produced by relativistic particles.

Hadrons do also radiate in a magnetic ﬁeld but at lower rate and at different frequencies. The calculations will be here displayed only for leptons. Protons are rapidly discussed at the end of lecture I. Here we consider free electrons. When the plasma is tenuous enough then collective charge effects associated with a plasma can be discarded (see lecture II for the case of plasma effects). Hence the results in these sections are those obtained in vacuo.

48/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Particle trajectory in a magnetic ﬁeld

We consider an uniform magnetic field (MF) of strength B0. The electron motion is helical with a pulsation Ω0 = qB0/γmec (Ωb = γΩ0) and with a pitch-angle α = (~B0,~v) and ωb = qB0/mec. The Larmor radius is defined as : rL = p sin(α)/qB0. We have noted θ = (~R, ~B0) : the direction between the observer and the magnetic field.

49/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Equation of motion

The equation of motion (relativistic case) : d(γm ~v) e e = ~v ∧ ~B dt c 0 d(γm c2) e e = ~v.~E = 0 (35) dt c The second Eq. gives γ = cat and hence |v| = cst. This leads to : d~v e γm ⊥ = ~v ∧ ~B e dt c ⊥ 0 d~v k = 0 , (36) dt

thus |vk| = cst hence an uniform motion along the MF and also |v⊥| = cst (no work from the MF) hence the helical motion. The particle is subject to an acceleration perpendicular to ~v of constant intensity. In this circular motion the pulsations at m × Ω0 are contributing to the radiation. 50/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Cyclotron emissivity : The Schott formula

The procedure is as follows (see B66 chapter 6, ! in MKSA units) Use W(Ω, ω) the energy emitted per unit solid angle and frequency interval (Eq.11) Develop the Fourier exponential term in terms of Bessel function : this produces the development over harmonics. The emitted power is obtained by dividing W(Ω, ω) by R +∞ −∞ exp(−iyt) = 2πδ(y) the time of the radiation has been produced. Where y = mΩ0 − ω(1 − βk cos(θ)) implies that radiation is produced at harmonics mΩ0 modiﬁed by the Doppler term of the motion the electron along the MF.

The cyclotron emissivity is (in [erg/s Hz st]) : (x = ω/Ω0β⊥ sin(θ)) " # q2ω2 cos(θ) − β 2 j(ω, β, θ) = × Σ∞ k J2 (x) + β2 J0 2(x) δ(y) , 2πc m=1 sin(θ) m ⊥ m (37)

51/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Emission cyclotron lines

Cyclotron line emissivity as function of β = v/c for the ﬁrst 10 harmonics. Dotted lines : synchrotron approximation (Marcowith A. & Malzac J., 2003, A&A., 409, 9. ! use ν instead of ω and νb = Ωb/2π). The ratio of the total emissivity of two successive harmonics is ∝ β−2 and the interval between two harmonics is Ω0.

52/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Radiation by a relativistic particle : The synchrotron limit

Once v → c the emission is beamed along ~v⊥ and focused within a cone of angle ∼ 1/γ. In the relativistic (non-relativistic) limit the electric ﬁeld E(t) 2 is a pulse over a time T < 2π/Ω0 (periodical 2π/Ω0) hence |E(ω)| is spred 3 over ∆ω = γ sinα Ω0 > Ω0 (∆ω < Ω0).

53/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Differential synchrotron emissivity

Detailed calculations are provided in RL79 section 6.4 and can be derived from the Schott formula in the limit β → 1 as well (see B66). √ dW 3q3B sin(α) = × F(x) , 2 (38) dωdt 4πmec with Z ∞ ω 3qB sin(α)γ2 F(x) = x K5/3(u)du , x = , ωc = . x ωc 2mec

Function F(x) wrt to x. It peaks at 0.29x and can be approximated by x0.3 exp(−x).

54/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The synchrotron radiation is polarized

In the relativistic limit, two terms remain in the Schott formula corresponding to two electric polarizations 1/ one parallel to ~B0 2/ one perpendicular to ~B0. The total emissivity can be decomposed into two components either. The power emitted Eq.38 can be re-written as (see B66 section 6.3 or RL section 6.4) :

dW dW dW = k + ⊥ , (39) dtdω dtdω dtdω √ dW 3q3B sin(α) P = P = k/⊥ = × (F(x) ∓ G(x)) , k/⊥ min/max 2 dtdω 4πmec

with G(x) = xK2/3(x). The degree of polarization is (see Eq. 7) : Π(ω) = (Pmax − Pmin)/Ptot = G(x)/F(x).

55/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Synchrotron loss timescale

From the total power radiated by a relativistic particle Eq.15 with Ak = 0 for an isotropically distributed particle population so averaged 2 over the pitch-angle hsin (α)iΩ = 2/3 we get :

−1 2 2 1 dE 3(mec ) tloss = | | = [s] . (40) E dt 4σT cEUB

2 The magnetic energy density is UB = B /8π. Interstellar medium estimates :

8 −1 −2 tloss = 5 × 10 EGeV B5µG [years] . (41)

56/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Spectrum produced by a population of particles

The spectrum emitted by a population of particles with a density n = R N(p, α)dp is given :

dW Z dW = (p, α)N(p, α)dpdΩ[erg/Hz cm3s] . (42) dνdVdt 2πdωdt We consider an isotropic power-law distribution of electrons : −s N(p, α) = Kep with mec pmin < p < pmax. The spectrum hence scales −(s−1)/2 as ν in the limit νc(pmin) ν νc(pmax).

dW 3ν (s−1)/2 4πj = = (4πK r qB) × b × E(s) . (43) ν dνdVdt e e 2ν

The function E(s) is given by Eq.4.60 in BG70. The degree of polarization is : Π(s) = (s + 1)/(s + 7/3) .

57/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Cyclotron-synchrotron self-absorption

It is possible to deﬁne an absorption cross section of a photon of frequency ν by a particle of momentum p from an analysis based on Einstein coefﬁcients 8 :(j(ν, p) is the particle emissivity given by Eq.38.) 1 1 σ(ν, p) = × ∂ (γpj(ν, p)) . 2 γ (44) 2meν γp

In the non-relativistic regime integrating j(ν, p) around νb gives 13 σ(ν = νb) ∼ (σT /α)Bcr/B as γ → 1 (Bcr = 4 × 10 Gauss).

Angle averaged cyclo-synchrotron absorption cross section wrt to ν for different particle momenta (see 8 for details)

8. Ghisellini G. & Svensson R., 1991, MNRAS, 252, 313 ; section 2.1 and RL74 section 6.8. 58/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Case of power-law distribution of relativistic particles

The absorption coefﬁcient is : Z −1 αν = dpN(p)σ(ν, p)[cm ] (45)

−s One gets for N(p) = Kep and mec pmin < p < pmax.

−(s+4)/2 √ recKe 2ν αν = 3 × × × A(s) , (46) νB 3νB

3s + 2 3s + 22 Z π A(s) = 2π × 2(s−2)/2Γ( )Γ( ) × sin(α)(s+4)/2dα . 12 12 0 5/2 • The self-absorbed spectrum is using Eq.43 : Sν = jν /αν ∝ ν .

59/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic fields Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The curvature radiation process The curvature radiation process is similar to the synchrotron radiation process except rL → rc : the magnetic field curvature radius. • Synchrotron emission comes from a cone offset by the particle pitch-angle (down) / curvature radiation the radiation is emitted along the magnetic field lines (up). Similar spectrum emitted per particle a √ 2 3 e 3 c 3 ω P(ω) = γ × F(x); ωc = γ ; x = . 2π rc 2 rc ωc The power radiated by one particle is (see Eq 40) :

2 q2c P = × β3γ4 . 2 (47) 3 rc

a. Ochelkov Y.P., Usov V.V., 1980, Ap&ss 69, 4390

60/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

61/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Lepton-radiation interaction : Generalities

The relevant process here is the Compton process : the diffusion of a photon by an electron (or a positron). We speak about Inverse Compton radiation when the electron is relativistic. We speak about Comptonization when Inverse Compton radiation proceeds in a hot plasma (like the coronal gas (see lesson II) found in stellar corona or the gas in an accretion disk).

62/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The Thomson scattering

2 If the scattering electron is at "rest", v c and if hν mec , the photon energy is conserved. The power emitted per unit solid angle has been obtained in slide 24. This is the radiation produced by an electron accelerated by the electric ﬁeld of the incoming wave (~d = e~r). The ﬂux of the wave being F = cE2/8π the differential cross section is

dσ dP 1 r2 T = × = e (1 + sin(Θ)2) , (48) dΩ dΩ F 2 ~ ~ ~ with Θ = (d, k1) and k1 is propagation direction of the out-coming photon. Integrated over Θ this gives σT .

63/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The Compton scattering

2 In case photons have a momenta hν ∼ mec . The electron has a recoil and the process involves QED. Using conservation of momentum and 2 2 energy the incoming = hν/mec and out-coming 1 = hν1/mec photon energies are linked by : 1 1 = (49) 1 + (1 − cos(θ))

h Or alternatively : λ − λ = λc × (1 − cosθ), λc = . Hence if λ λc 1 mec 2 or hν mec the diffusion is closely elastic λ1 ∼ λ.

64/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The Compton process cross section

The cross section is given by the Klein-Nishina formula (see RL79 chapter 7 and references therein).

dσ r2 2 = e 1 × + 1 − 2(θ) 2 sin (50) dΩ 2 1

Non-relativistic regime (Thomson limit) : 1, 1 = and σ = σT . Ultra-relativistic regime : 1 and the angle-integrated cross section tends to 3 1 1 σ = σ × ln(2) + σ . (51) 8 T 2 T

65/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The Inverse Compton process

If now the electron is relativistic v ∼ c hence even a small recoil can provide the diffused photon with a lot of energy in contrast to Eq.49 : Inverse Compton process. Formula 49 is now applied in the electron rest-frame (R’) moving at a speed v wrt to the observer frame (R). Applying a double Lorentz transformation one gets in the Thomson regime (0 < 1) the maximum energy of the diffused photon (See RL section 7.1) :

2 0 1 ' γ × (1 − β cos(θ1))(1 − β cos(θ)) 2 1,Max = 4γ , (52)

0 θ (θ1) is the angle between the incident (diffused) photon and the electron in the R (R’) frame.

66/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Results in the Klein-Nishina regime

The efﬁciency of the scattering process depends on the energy of the incoming photon in the electron rest-frame, i.e. if 0 ∼ 1 hence Klein-Nishina effects become important and the process efﬁciency is reduced : Scattering process (Thomson regime : 0 1)→ Catastrophic loss process (Klein-Nishina regime : 0 ≥ 1). 0 0 0 In the Klein-Nishina regime ( (1 − cos(θ )) > 1), 1 becomes independent of :

0 0 1 ' γ(1 + cos(θ ))(1 − β cos(θ1)) (53)

67/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Spectrum produced by one relativistic electron : case study in the Thomson regime

The calculation is done in BG70 section 2.6, RL79 section 7.2 is a bit (too) lengthy. 1 The calculation is based on the invariance of the differential photon density dn/ where dn(, x = cos(θ)) = n(, x)ddx 2 From this the calculation is performed in the R’ frame we have : 0 0 2 2 0 0 0 dσ/dΩ1d1 = (re /2)(1 + cos (θ1))δ(1 − ) From 1/ one can deduce the total emitted power in R’ d0 Z 1 dE 1 = cσ 0 dn(0) = = − e . 0 T 1 INV 2 dt mec dt From 1/ and 2/ one can deduce the photon spectrum produced by one relativistic electron.

dN 0 dσ 0 0 0 0 0 0 = dn c 0 0 [Nb/s st 1] . dt d dΩ1d1 dΩ1d1 68/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Inverse Compton loss timescale

We ﬁnd : Loss rate (or radiated power) in the case of an isotropic incident photon distribution in R : dE 1 − = σ c γ2(1 + β2) − 1 U , dt T 3 ph Z 3 Uph = dn()[erg/cm ] (54)

Loss timescale (β = 1)

2 2 E 3(mec ) tloss = = [s] , (55) |E˙ | 4σT cEUph

8 −1 −1 tloss ' [3 × 10 ]EGeV Uph,eV/cm3 [years] .

69/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Inverse Compton photon spectrum produced by one particle

Here I use the general Klein-Nishina cross section in step 2 of slide 68. The diffused photon spectrum is (BG70 Eq.2.48) : with Γe = 4γ, q = E1/Γe(1 − E1), E1 = 1/γ. dN 2πr2c n()d = e × F(Γ , q) 2 e (56) dtd1 γ 2 1 (Γeq) F(Γe, q) = 2q ln(q) + (1 + 2q)(1 − q) + (1 − q) . 2 (1 + Γeq)

F(E¯1, Γe) as function of −1 x = E¯1 = E1/(Γe(Γe + 1) ) for different values of Γe =0 (dotted), 1, 10, 50, 100 (long dashed).

Γe 1 (Γe 1) corresponds to the Thomson regime (the Klein-Nishina regime).

70/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Spectrum produced by a population of particles

We consider the case of an isotropic power-law distribution of electrons : −s R Ne(γ) = Keγ , γmin < γ < γmax (with Ne(γ)dγ = ne). Integrating Eq. 56 over Ne(γ) yields to a complicated expression (BL 70 Eq. 2.75). Thomson limit : 1 dW dN = 1 = πr2cK (s, n ) × −(s−1)/2 × F(s) 2 e e e 1 (57) mec dtd1dV dtd1dV 2s+3(s2 + 4s + 11) Z F(s) = (s−1)/2n()d . (s + 3)2(s + 1)(s + 5) Klein-Nishina limit (C(s) is given by Fig6. BL70) :

1dN 2 −(s−1) = πre cKe(s, ne) × 1 × F(s, 1) (58) dtd1dV Z n() F(s, ) = (ln( ) + C(s))d . 1 1 The Klein-Nishina spectrum is steeper than the Thomson spectrum.

71/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Some spectral examples

Inverse Compton spectrum produced by a power-law electron distribution over a black-body photon source in the Thomson regime (blue) and Klein-Nishina regime (orange)

72/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Synchrotron self-Compton (SSC) process

The soft photons are synchrotron radiation produced by the same (not always the case) electron population. In the Thomson regime : if the synchrotron spectrum extend over [s,min, s,max] hence the SSC spectrum 2 extends over γ × [s,min, s,max] for each particle, hence SSC is more extended than IC emission produced from a black-body. In the Thomson regime the ratio of the power radiated by IC to synchrotron is (see Eqs. 38, 54) : P U IC = ph . (59) PS UB

In the case of SSC Uph is produced by the synchrotron radiation and hence if we have several IC generations :

2 1 PIC PIC Uph 1 = = (60) PIC PS UB

73/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Spectral energy distribution : example of blazars

Spectral energy distribution(radio loud quasar) : Synchrotron emission (Sy) in red, synchrotron self-compton (SSC) emission in blue, External Comptonization of Direct disk radiation (ECD) in green, and External Comptonization of radiation from the clouds (ECC) in yellow

74/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes The synchro-Compton loss catastrophy

• We note η the ratios in Eq. 60. In the case of homogeneous synchrotron self-absorbed source it is possible to calculate this ratio in terms of a the 9 brightness temperature Tb (see slide 11) . The brightness temperature such that η = 1 :

ν −1/5 T = 1012K × . (61) b,η=1 1Ghz Hence at 1 Ghz the brightness temperature should not exceed 1012 K otherwise it induces a loss catastrophy producing huge amount of X and gamma-rays as the power in secondary generations exceeds the power in primary produced photons.

The point is that several radio sources do have Tb > Tb,η=1.

9. Kellerman I.N. & Pauliny-Toth Pauliny-Toth, I. I. K. 1969, ApJ, 155, L71 75/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Induced Compton scattering

• In this process photons scatter off free electrons at rate enhanced with respect to Compton scattering by the photon intensity (occupation number). The condition for Induced Compton scattering (ICS) to be important 2 is : f kBTB/mec τT ≥ 1 and τT = neσT R. f ∼ Ω0/4π accounts for the anisotropy of the incident photon beam 10. The transfer Eq. becomes non-linear. In the case of isotropic photon distribution and uniform particle distribution conﬁned in the same 2 region, noting y(ν) = νkBTB/mec we have (with ¯t = neσT ct): ∂y ∂y = 2y , (62) ∂¯t ν The effect of ICS is to pump photons from high to low frequencies as y moves along the characteristic (ν,¯t) with a speed −2y. Hence the brightness temperature at low energies increases. 10. Kuncic Z. et al 1998 ApJ 495 L35, Coppi P.S. et al 1993 MNRAS 262 603 76/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Isotropic homogeneous Induced Compton scattering

Time evolution of an isotropic photon ﬁeld under the effect of ICS by an homogeneous electron distribution. y(ν, 0) ∝ ν1.5(1 − exp(−ν3)). A series of shocks developed that in nature is controlled by the electron thermal velocity, the shock thickness is Ve/c = δν/ν (see Coppi et al (1993)).

77/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes Polarization

The degree of polarization depends on the incoming photon polarization. In −s case of an isotropic electron distribution with Ne(E) ∝ E one may consider two cases : (i) Incoming photons are not strongly polarized (i.e. (most of) accretion disk, background radiation) (ii) Incoming photons are polarized (i.e. Synchrotron radiation). The results are 11 : In the case of an incoming beam (in direction and frequency) of photon in the Thomson regime

3 + 4s + s2 Π ' , (63) IC 11 + 4s + s2

−α In the case of synchrotron photon spectrum Nω ∝ ω scattered by the same electrons :

ΠSSC = η(α, p) × Πsync × ΠIC , Πsync = (3s + 3)/(3s + 7) . (64)

11. See recent work by Krawczynski H. 2012 ApJ, 744, 30 and references therein 78/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Interaction with matter Elements of classical Electrodynamics Interaction with magnetic ﬁelds Non-thermal leptonic processes Interaction with radiation Non-thermal hadronic processes

• (upper panel) IC polarization degree (s=1.01,1.5,2,3,4 black to magenta) in case of a power-law electron and a beam of photons at an angle of 85o with l.o.s. Klein-Nishina case : 0 ΠKN ∼ 0.5/(1 + ). • (lower panel) SSC polarization degree in case of a photon distribution −(α+1) Nω ∝ ω (α = 1, p = 3 red and α = 0.5, p = 2 black) with an angle between MF and l.o.s. of 85o. In both ﬁgures the proper energy/frequency limits have to be considered.

79/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

80/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Ionization and Coulomb losses

1 Coulomb losses (S02, SM98 and ref. therein) : Case of an ion of charge Z and velocity β = v/c and energy E. The losses are dominated by scattering off thermal electrons :

dE x2 + β2 = −4πcr2n m c2Z2 ln Λ × m , (65) dt e e e β2

6 1/2 xm = 0.0286(Te/2 × 10 K) and ln Λ = [30 − 40] in the ISM conditions 12. 2 Ionization losses : dE 1 = −2πcr2m c2n Z2 × × Σ n × K (I , β, E) . (66) dt e e e β s=H,He s s s

The complicated function Ks is described in S02 and SM98 and references therein. 12. Dermer C.D. 1985, ApJ, 285, 28 81/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Inverse Bremsstrahlung

Bremsstrahlung produced by relativistic hadrons over a charged plasma. It has been invoked in the contexts of cosmic X-ray background, diffuse galactic X-ray emission and at shocks 13.

Particle distribution at shocks and Bremsstrahlung/Inverse Bremsstrahlung spectra (Baring et al 2000). (s−1)/2 FIB/FB ∝ (meTp/mpTe) , s : non-thermal particle distribution index. 13. see introduction section of Baring M.G. et al 2000, ApJ, 528, 776 82/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes High energy hadron-matter interaction processes

see S02 Proton pion production (n1, n2=multiplicities). + − 0 p + p → p + p + n1(π + π ) + n2π , ± ± π → µ + νµ/ν¯µ , π0 → 2γ , ± ± µ → e + νe/ν¯e +ν ¯µ/νµ . (67) Other channels for protons : n, 2n, p¯, D, K0, K±, Σ0, Σ±, Λ0, Λ±... Other hadrons : essentially in the interstellar medium α particles but in particular sources one may be concerned with heavier particles. All these processes involve inelastic interactions : a1 + a2 → Σiai with a total mass M = Σimi. The threshold condition for the interaction (prime is expressed in the center of mass frame) :

0 2 4 2 4 2 1/2 2 Eth = m1c + m2c + 2E1E2 − 2p1p2c cos(θ1,2) ≥ Mc . (68)

83/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Neutron production

Neutrons have a short lifetime (γ × 103 s) but have no charge so are insensitive to electromagnetic ﬁelds and can transport a substantial fraction of the energy from the sources 14. Fraction of protons converted into neutrons : 1/4 in pp collisions and 1/2 in pγ collisions

14. Derishev, E.V. et al 1999, ApJ, 521, 640, Begelman M. et al 1990, ApJ, 362, 38 84/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes (neutral) Pion production cross section

Here is shown a parametrization of the inelastic cross section 15

L = ln(EP/1TeV) and from Eq. 68 the threshold energy is : 2 Eth = 2mπ + mp + mπ/2mp = 1.23 GeV (mπ ' 135 MeV, mp ' 0.938 GeV).

15. see Kelner S.R. et al 2006, PhysRevD, 74, 034018 ; Kamae T. et al 2005, ApJ, 620, 244 and 2006, ApJ, 647, 692 for details 85/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Gamma-ray emissivity by one particle

Each pion produces a differential number of gamma-ray photons : Z ∞ dNγ dNπ dEπ (Eγ ) = 2 × (Eπ)p . (69) dE 2 dE 2 2 4 γ Eγ +4mπ c /Eγ π Eπ − mπc

Now each proton produces pions at a differential rate 16 :

R(Eπ) ' cnHσpp(Ep(Eπ)) × δ(Eπ − hEπi)H(Ep − Eth) , (70)

Typical pion energy : hEπi ∼ 1/6Tp (Ep ≤ 10 TeV, Tp is the kinetic 2 energy) and Ep(Eπ) = mpc + 6Eπ, Kπ = 1/6 is the inelasticity factor

16. Aharonian F.A. & Atoyan A., 2000, A&A 362, 937, Mannheim K. & Schlickeiser R., 1994, A&A, 286, 983 gave approximate expressions or see Kelner et al 2006 Eq.58 for more detailed calculations 86/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Proton loss timescale

Z −1 7 −1 tloss = 3 × Ep × R(Eπ)dEπ ∼ 5 × 10 nH,cm−3 [years] . (71)

I have used nH as the total hydrogen target density nH = nHI + nHII + 2nH2 .

87/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes pp interaction process is anisotropic

• Kinematical effect : the neutral pion and gamma-rays are mostly produced in the direction of the incident proton especially at high energies 17. This calculation requires to retain the angles of secondary particles in the cross section.

−2 Gamma-ray spectra at different angles wrt the beam for Np ∝ Ep . 17. Karlsson N. & Kamae T., 2008, ApJ, 674, 278 88/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Gamma-ray emissivity by a population of particles

We deﬁne the pion rate integrated over the proton distribution Z nHc 3 qπ(Eπ) = R(Eπ)np(EP)dEπ = σpp(Ep(Eπ))np(Ep(Eπ)) [nb/cm s Eπ] . Kπ (72) 2 again, Ep(Eπ) = mpc + Eπ/Kπ and then use Eq.69. In case of a power law gamma-ray and proton indices are identical. EF(E) (TeV/cm−3 s) vs E(TeV) gamma-ray spectrum from pion decay for different power-law proton −p distribution n(Ep) ∝ E : p = 2 (3 dotted-dashed), p = 2.5 dotted-dashed), other lines correspond to ﬁts of the diffuse galactic gamma-ray emissivity (see Aharonian & Atoyan 2000).

89/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Electron-positron and neutrino production

Here are displayed the results from Kelner et al 2006 using parametrization of the gamma-ray, secondary lepton and neutrino production rate.

EF(E)(TeV/cm−3 s) vs E(TeV) gamma-ray, lepton(electron and positron), muonic neutrinos emissivities for different distribution of protons −α β np ∝ Ep × exp(−(Ep/E0) ). Left panel α = 2, β = 1, E0 = 1 PeV. Left panel α = 1.5, β = 1, E0 = 1 PeV. In dashed line : delta approximation Eq.72. • One must account for the radiation (synchrotron, Bremsstrahlung, Inverse Compton) radiation from secondary leptons.

90/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

91/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Hadron-radiation interaction processes

Photo-pair production p + γ → p + e− + e− . 0 2 Photon threshold Eγ = ∆mc (1 + ∆m/mp)= 1 MeV (proton restframe) Photo-pion production p + γ → p + π0 , p + γ → p + π+ + π− . 0 2 Photon threshold Eγ = ∆mc (1 + ∆m/mp)= 145 MeV (proton restframe) Photo-disintegration A + γ → (A − 1) + n/p . Important process in the context of ultra high-energy cosmic ray propagation in the intergalactic medium 18 18. see e.g. Khan E. et al 2005, Aph, 23 191 for further details 92/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Photo processes cross sections

Photo-pion cross section as function of the incoming photon energy in the proton restframe. Peak σ ∼ 5 × 10−28cm2 at 0 0 ∼ 2th.

See 19. From this we can deduce the collision rate in the observer frame Z c 0 0 0 0 0 R = n( , Ω )σπ( )d dΩ [nb/s] , (73) γp

R 0 0 0 0 with n( , Ω )d dΩ = nph. 19. Kelner S.R. & Aharonian, F.A., 2008 Phys Rev D, 78, 034013, Muecke A. et al, 2000, Comp Phys Com, 124, 290 93/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Loss timescales

The loss timescale is derived using Eq.73 but accounting for the inelasticity parameter K(0) → 0.5 (at high energy, 0 1) 20 Z c 0 0 0 0 0 0 −1 tloss = [ n( , Ω )K( )σπ( )d dΩ ] . (74) γp

−s In case of a power-law energy density of soft photons Uν ∝ ν one gets :

s −1 tloss = [Urad,erg/cm3 × Ep,GeV × P(s)] , (75)

where P(s) are tabulated in20 for both photo-pion and photo-pair processes also in case of black body radiation. 8 −1 −1 Typically for s = 1, tloss ∼ 10 Urad,erg/cm3 × Ep,GeV years.

20. Begelman M. et al 1990, ApJ, 362, 38 appendix 94/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Loss timescales over the cosmic microwave background

−13 3 We have UCMB ' 4 × 10 erg/cm .

Inverse of loss timescale with respect to the cosmic ray energy.

20 8 21 Typically for ∼ 10 eV protons tloss ∼ 10 years .

21. Kelner et al 2008 95/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical Electrodynamics Deﬁnitions The dipolar approximation and the Larmor formula

96/97 Radiative processes in Astrophysics. Preliminaries Elements of radiative transfer Hadron-matter interaction Elements of classical Electrodynamics Hadron-radiation interaction Non-thermal leptonic processes Hadron-magnetic ﬁeld interaction Non-thermal hadronic processes Synchrotron radiation

Loss timescale

4 m 3 cσ U t = [ e × T B E ]−1 = 3 × t . syn,p 2 p,GeV 1836 syn,e (76) 3 mp mec

Peak of particle radiation

me 1 νc,p = × νc,e = νc,e . (77) mp 1836

97/97 Radiative processes in Astrophysics.