Non-thermal radiative processes in Astrophysics - ISSI · Preliminaries Elements of radiative...

PreliminariesElements of radiative transfer

Elements of classical ElectrodynamicsNon-thermal leptonic processes

Non-thermal hadronic processes

Non-thermal radiative processes in AstrophysicsLeptonic & hadronic elementary processes

Alexandre Marcowith 1

[email protected]

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

27 avril 2013

1/97 Radiative processes in Astrophysics.




Preliminaries-I

Non-thermal processes in Astrophysics : A great variety of objects (see thedifferent lectures)

1 Compact sources : Sources associated with residues of massive starsStellar 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∗), Activegalactic nuclei.

2 Diffuse or extended sources : Sources linked with a compact object butspread over larger scales

Stellar size extended objects : Pulsar nebula, supernova remnants, massivestar 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 bolometricluminosity into non-thermal radiation.





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 andBremsstrahlung radiation, pair creation/annihilation.Interaction with magnetic fields : Cyclo-synchrotron radiation, curvatureradiation.Interaction with radiation : Inverse Compton, Comptonization.

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

3 Impact of losses over particle distribution4 Radiative transfer and radiation in a plasma





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 theinteraction 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 consideringleptons or hadrons. Based on a quantum treatment of fields andparticle-field interaction theory. Mandatory to derive the cross sections.

3 Quantum Plasma Dynamics (QPD). A theory that describes plasmakinetics using a semi-classical approach. Largely developed by D.B.Melrose. I will not discuss it (see Melrose D.B. Springer Verlag books).





Bibliography

BooksBekefi 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., Cambridgeuniversity press. (L81)Rybicki G. B. & Lightmann A.P. , 1979, Radiative processes inAstrophysics, Wiley (RL79)Schlickeiser R., 2002, Cosmic Ray Astrophysics, Springer. (S02)

ArticlesBlumenthal G.R. & Gould R.J., 1970, Rev. of Mod. Phys., 42, 237 (BG70)Strong, A.W. & Moskalenko, I.V., 1998, ApJ, 509, 212 (SM98)





Some recurrent notations

See NRL plasma formularyThese lectures are given in CGS units.For a particle we define β = v/c and its Lorentz factorγ = (1− β2)−1/2.Electromagnetic processes : The classical electron radiusre = e2/mec2 = 2.8× 10−13 cm, Thomson cross section :σT = 8π/3r2

e = 6.65× 10−25 cm2.Hadronic processes : The cross section are in units of milli barns=10−27

cm2.Energy units 1erg =1/1.60 TeV





Outlines

1 Preliminaries

2 Elements of radiative transfer

3 Elements of classical ElectrodynamicsDefinitionsThe dipolar approximation and the Larmor formula

4 Non-thermal leptonic processesInteraction with matterInteraction with magnetic fieldsInteraction with radiation

5 Non-thermal hadronic processesHadron-matter interactionHadron-radiation interactionHadron-magnetic field interaction





Radiative transfer

The radiative transfer theory is the macroscopic theory of propagationof light through a medium. This theory involves the description ofsystems with size L λ. It involves the concept of rays.

Rays are described in the eikonal approximation : curves whose tangent ateach point lie along the direction of the propagation of the wave. If thewave function W(~x, t) = A(~x, t) exp(iφ(~x, t))→ A is a slowly varyingfunction over a wavelength, φ varies rapidly over the wavelength.

The radiative transfer and ray approximation are intrinsically limited bythe uncertainty principle of quantum mechanics.Associated to rays different quantities characterize the radiation field.





Specific intensity and its moments

1 Specific intensity or brightness : Iν(~r,Ω, ν) = energy per unit of timefrequency and solid angle traversing a surface element[erg/s cm2 St Hz], a quantity conserved along one ray (RL79 §1)

Iν =dW

dtdSdΩdν2 Mean intensity : Jν = (1/4π)

∫IνdΩ [erg/s cm2Hz].

3 Energy flux (1st moment) : Fν =∫

Iν cos(θ)dΩ [erg/s cm2Hz].4 Radiative pressure (2nd moment, flux of momentum) :

Pν = (1/c)∫

Iν cos(θ)2dΩ [dynes/cm2Hz].

θ = (~n,~d) ~n is the normal to dS and ~d is the ray direction.





Radiative transfer equation

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

1 Source term or spontaneous emission coefficient jν = dW/dVdtdν[erg/cm3s St Hz] (synchrotron, bremsstrahlung ...)

2 Absorption coefficient αν [cm−1] (synchrotron, bremsstrahlung ...)The radiative transfer Eq. reads :

dIνds

= −ανIν + jν . (1)

Also useful is the source function Sν = jν/αν = a quantity towardswhich Iν tends to relax.

3 Diffusion coefficient σν [cm−1] (Comptonization (lect.2) ...). Forisotropic scattering jν = σνJν .





Thermal processes

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

Sν = Bν(T) = 2hν3/c2/(exp(hν/kBT)− 1) , hνmax = 2.82kBT . (2)

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

jν = ανBν(T) . (3)

The brightness temperature is the temperature of a blackbody havingthe same specific intensity at ν : Iν = Bν(Tb). In the Rayleigh-Jeansregime hν kBT and Tb = c2

2ν2kBIν , Iν = Sν/∆Ω. Important diagnostic

in compact sources with resolved solid angle ∆Ω.The blackbody Wien regime (hν kBT) leads toIν = (2hν3/c2)exp(−hν/kBT).

All these macroscopic quantities can be obtained from a microscopicanalysis.





Einstein coefficients

• Kirchoff law relies on a detailed balance on a microscopic level(reversibility) between emission and absorption processes. Consider twodiscrete energy levels (E, with a statistical weight g1, population n1) and(E + hν0,g2,n2).

3 coefficients with probabilities/s : spontaneous (incoherent) emissionA2→1, absorption B1→2J, stimulated emission B2→1J.J =

∫∞0 Jνφ(ν)dν , φ(ν) is the line profile centered on ν0

At thermodynamic equilibrium (but the relations are valid out ofthermodynamical eq.) the detailed balance gives :

n1B12J = n2A21 + n2B21J (4)

Also Jν = Bν which provides relations between the coefficients

g1B12 = g2B21 (5)

A21 =2hν3

c2 B21 (6)





Relation to macroscopic coefficients

Spontaneous emission

jν =hν4π

n2A21φ(ν)

Absorption+stimulated emission

αν =hν4π× (n1B12 − n2B21)φ(ν)

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





Polarization and Stokes parametersIn the case of a monochromatic waveelliptically polarized. We transformthe electric field ~E → ~E′ where (x,y)is the frame of the observer(instrument) and (x’,y’) is the frameof the polarization ellipse. We apply arotation of an angle χE1,E2, φ1, φ2 → E0, β, χ (see RLsection 2.4). β describes ~E in (x’,y’).

The 4 Stokes parameters express this link :

I = E21 + E2

2 = E20 > 0 ,≡ Intensity

Q = E21 − E2

2 = E20 cos(2β) cos(2χ) ,

U = 2E1E2 cos(φ1 − φ2) = E20 cos(2β) sin(2χ) ,

V = 2E1E2 sin(φ1 − φ2) = E20 sin(2β) ,≡ degree of circularity .





Polarization and Stokes parameters for an ensemble ofwaves

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 areaveraged 〈.〉T over the time T of the signal (i.e. Iensemble = Σk〈Imono,k〉T ).

A wave is the sum of an unpolarized I −√

Q2 + U2 + V2 and a polarized√Q2 + U2 + V2 part. The latter can be cast into two oppositely polarized

parts I± = 1/2(I ±√

Q2 + U2 + V2).Degree of polarization of an ensemble of waves

Π =Ipol

I=

√Q2 + U2 + V2

I=

(I+ − I−)

I. (7)





Useful spectral quantities : power-law radiation

Photon differential spectrum (or also per unit of energy keV,...,TeV)dn

dSdtdν∝ ν−s [nb/cm2 s Hz] .

Flux or energy differential spectrum 1

4πJν =dW

dSdtdν=

∫IνdΩ ≡ νdn

dSdtdν∝ ν1−s [erg/cm2s Hz] .

Energy spectrum per unit log of frequency (or energy)

νFν =dW

dSdtd log(ν)=∝ ν2−s [erg/cm2s] .

Differential luminosity

Lν =

∫dS

dWdSdtdν

[erg/s Hz] .

The luminosity reported at Earth is an observed flux Fν = Lν/4πR2.1. In radio the Jansky unit = 10−23 erg/cm2s Hz = 10−26 W/m2Hz.





DefinitionsThe dipolar approximation and the Larmor formula

Outlines

1 Preliminaries










Definition : Power spectrum

The radiation spectrum is given by the time variation of the electric field ofthe electromagnetic waves.The energy in the spectrum per unit of time and per unit of surface can beobtained from the Poynting vector ~Π = c~E ∧ (~B/4πµ).

dWdSdt

=c

4π|E(t)|2 .

• The spectrum is expressed using the Fourier transform of the electric field :

dWdSdω

= c|E(ω)|2 . (8)






Retarded potentials : geometry

If~r = ~r0(t) is the position of the particle at time t, the solution of MaxwellEq. using the retarded potential method gives the electric field produced by acharge with a velocity~v = ~β(tret)c and with an acceleration ~a = ~v(tret) (seeJ62 §14, LL71 §8, RL79 §3).The retarded time is tret = t − R(tret)/c. R(t′) = |~r0(t)−~r0(t′)|, ~n = ~R/R.






Retarded potentials : electric field solution

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

~E(~r, t) = q(~n− ~β)

γ2d3R2 +qc~n

d3R∧ (~n− ~β) ∧ ~β|tret , (9)

= ~ECoul,v + ~Erad , ~B(~r, t) = ~n ∧ ~E|tret

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






Spectrum radiated by one particle

Using Eq. 8 one gets the energy radiated per unit of solid angle andfrequency

dWdωdΩ

=q2

4π2c×∣∣∣∣∫ Re(E(t)) exp(iωt)dt

∣∣∣∣2 . (10)

• Including the radiation part (second one) of Eq. 9 (still evaluated atretarded times t′ = t − R(t′)/c) gives :

dWdωdΩ

=q2

4π2c×∣∣∣∣∫ ~n

d2 ∧(

(~n− ~β) ∧ ~β)

exp(iω(t′ −~n.~r0(t′)/c))dt′∣∣∣∣2 .

(11)






Outlines

1 Preliminaries










The electric dipolar approximation

We consider the sources of the electromagnetic field as an ensemble ofcharges displayed over a region of size a. We consider the radiation producedat 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 theelectromagnetic field produced by one charge, we discard the Coulombcontribution which scales as 1/r2 and has a Poynting flux vanishing atinfinity (see Jackson, Rybicki & Lightman).The electric dipole approximation consists in writing |~r −~r′| ∼ |~r|. It can betranslated into a condition over the wavelength of the radiation

a λ . (12)






Total power radiated by an ensemble of particle : TheLarmor formula

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

Pem =23

D2

c3 . (13)

For one charge e with an acceleration ~A = ~D/e one gets :

Pem =23

e2A2

c3 . (14)

• The power radiated per unit of solid angle P(θ) = 3/8πPem sin2(θ) is not arelativistic invariant (θ = (~A,~n)).






Total power radiated by relativistic particles : Therelativistic Larmor formula

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

Pem =23−e2A.A4πε0c3 =

23

e2γ4

c3 ×(

A2‖ + γ2A2

⊥

). (15)

With β = v/c, γ = (1− β2)−1/2. A‖,⊥ are the components of theacceleration vector parallel and perpendicular to the boost~v in the observerframe.






Angular power

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

dWdΩ

= (γ(1− βcosθ))−3 × dW ′

dΩ′(16)

We distinguish among the emitted power in frame R :Pe = dW

dtdΩ = γ−4(1− βcosθ)−3 × dW′dt′dΩ′ and the received power by a fix

observer in frame R : Pr = (γ(1− βcosθ))−4 × dW′dt′dΩ′ . The latter includes a

retardation effect due to the motion of the source. The term dW ′/dt′dΩ′ isdeduced from Eq. 14.• The Doppler factor D = (γ(1− βcosθ))−1 is 1 if γ 1 andcos θ ∼ β. The power emitted is highly amplified in the particle direction ofmotion in a cone of size ∼ 1/γ.






Reaction force

• As the particle is radiating a part of its own energy hence there must be aforce acting on the particle due to its radiation (J62, LL71, RL74).

Braking force due to radiation (Abraham-Lorentz force)

~F =2e2

3c3~u + (~Fext) . (17)

~Fext is an external force ; e.g. Lorentz force by an EM field. Thereaction force is a perturbation if the condition λ re is fulfilled, λ isthe wavelength of the radiation in the particle rest-frame (see LL71).In the case of relativistic particles a 4-vector formulation is used :

gµ =2e2

3cd2uµ

ds2 −Prad

c2 uµ . (18)

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





Interaction with matterInteraction with magnetic fieldsInteraction with radiation

Outlines

1 Preliminaries










Lepton-matter interaction : Generalities

The relevant processes here are of two kinds depending if the lepton do emitor 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 infully/partially ionized plasma).

Bremsstrahlung process of a non-relativistic electron in the Coulomb fieldof 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 theemitted photon may be ∼ mec2. NB : Thermal Bremsstrahlung can betreated using the methods developed in QED.






Coulomb and ionization losses

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

electron density)

−dEdt

= 2πcr2e mec2 1

βne ln

(πEmec2

reh2c2ne− 3

4

)[erg/s] . (19)

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

−dEdt

= 2πcr2e mec2 1

β×Σs=H,HeZsns×ln

((γ − 1)β2E2

2I2s

+18

)[erg/s] .

(20)Both processes dominate at energies < GeV in the interstellar medium (S02).

2. The case of hot plasma kBTe ∼ mec2 is treated in Moskalenko I. & Jourdain E., 1997,A&A, 325, 401 and the relativistic case in L81






Bremsstrahlung

At higher energies (but stillnon-relativistic for the moment) theinteraction with the Coulomb field of acharge induces an acceleration of theelectron and the emission of a photon.The acceleration is perpendicular to thevelocity.The power radiated by the particle isgiven by Eq. 8 expressed in terms of theFourier transform of the electric fieldwith the FT of D :

dWdSdω

=8πω4

3c3 D(ω)2 (21)






Bremsstrahlung radiation by one non-relativistic particle

Interaction of an electron of charge e with an ion of charge Ze at an impact parameter b

The timescale of theinteraction is τ = b/v.

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 centripeteacceleration produced by the normal electric Coulomb field of the ion (seeRL79 Eq.4.70b (CGS), L81 Eq.3.35 (SI)). The flux of incident electron andthe density of targets leads to the total power radiated per unit time andvolume :

dWdωdtdV

= (nev)×ni×∫ bmax

bmin

2πbdW(b)

dbdωdb =

16πe6

3m2ec4β

neniZ2ln(bmax

bmin) (22)






The Gaunt factor : bmax and bmin

We have followed a classical treatment but include quantum corrections forbmin.

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 wherethe classical trajectories cease to be valid.

Condition 1 gives : bmin = 4Ze2/πmev2. It dominates in the low energylimit Ek = mv2/2 Z2Ry, Ry = 4π2mee4/2h2.Condition 2 gives : applying the uncertainty principle : bmin = h/mv. Itdominates in the high energy limit T = mv2/2 Z2Ry.

We can define the Gaunt factor

gff (v, ω) =

√3π

ln(bmax

bmin) (23)






Bremsstrahlung spectrum by a population of thermalparticles

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

F(v)4πv2dv = 4πv2 × exp(−mev2/2kBT)dv

We get (KTh = 32πe6/3(m2ec4) = 32π/3× r3

e mec2) :

dWdtdνdV

= KTh×ne×ni×Z2(

2πmec2

3kBT

)1/2

exp(−hν/kBT)gff (T, ν) [erg/s Hz cm3] .

(24)The power radiated is almost independent (gff is not strongly dependent onν) of the frequency except if hν kBT where it is cut off exponentially. Thefactor gff is in the range (1-5). It can be found in RL79 in figure 5.2 and inreferences therein.






Bremsstrahlung self-absorption

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

αν =dW

dtdνdV× (4πBν(T))−1 [cm−1] ,Bν(T) =

2hν3/c2

exp(hν/kBT)− 1.

The effect of absorption is strong at hν kBT if the opacity ανR 1 (R isthe size of the emitting region).

Typical thermal free-freespectrum as produced in HIIregions (in radio wavebands).The flux is ∝ ν2 at lowfrequencies (Rayleigh-Jeansspectrum) and exponentiallycut (Wien spectrum) off athigh frequencies.






Bremsstrahlung in the relativistic regime

In that regime hν may be not negligible wrt mec2.Three approaches are possible :

Derive an equivalent expression as Eq.22 in the relativistic case fromthe radiated spectrum by an accelerated particle (L81).Uses the method of virtual quanta (Weiszäcker & Williams method, seeBG70 section 3.2).Use QED calculations to derive the differential Bremsstrahlung crosssection (BG70 section 3.3, 3.5). This calculation give more accurateresults.

We have the following chain to get the loss timescale

dNdtdν

= cΣinidσdν→ −dE

dt=

∫dνhν

dNdtdν

→ tloss = E/|dE/dt| .

Hereafter we will note E = E/mec2 and the Bremsstrahlung photon energyε = hν/mec2.






Bremsstrahlung cross section

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

dσ = 4Z2αr2e

dεεE2

(E2 + (E − ε)2 − 2

3E(E − ε)

)(ln(

2E(E − ε)ε

)− 1

2

).

(25)An atom :

dσ = αr2e

dεεE2

((E2 + (E − ε)2)φ1 −

23

E(E − ε)φ2

)(26)

φ1/2,s are the screening functions given in BL70 (Fig9, table 2). At lowenergies φ1/2,s → 4(Z2 + Z)(ln(2E(E − ε)/ε)− 1/2), (weak shieldingcase ≡ ion + Z electrons).Low energy means ∆ = 1/(2α)ε/(E(E − ε)) > 1.






Bremsstrahlung cross section

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

3. Sacher W. & Schoenfelder V., 1984, ApJ, 879, 21738/97 Radiative processes in Astrophysics.





Bremsstrahlung loss timescale

1 Weak shielding case (ionized medium or an atom with low energyelectrons) : (i=ions of charge Z+ Z electrons)

−dEdt

= 4αr2e cΣiniZ(Z + 1) ln(2E − 1/3)E [s−1]! (27)

2 Strong shielding case : (s=species : bound electrons, ions andhigh-energy electrons)

−dEdt

= αr2e cΣsns

(43φ1,s −

13φ2,s

). (28)

The typical loss timescale is (ionized case) in a plasma composed ofhydrogen + 10% of helium.

tloss ' 1.45× 108 × n−1H,cm−3 [ln(2E − 1/3)]−1 [years] (29)






Bremsstrahlung spectrum by a population of non-thermalrelativistic particles

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

N(E)dE = KeE−sdE ,Emin < E < Emax , ne =

∫ Emax

Emin

N(E)dE

The differential flux is (weak shielding) : (∫

dσ/dε(E)N(E)dE)

dWdtdεdV

= 4αr2e cKemec2Σni(Z2 + Z)× I(ε, EL, s) , EL = max(Emin, ε) (30)

I(ε, EL, s) =43×

(E−(s−1)

L

(s− 1)− εE−s

L

s+

34ε2E−s−1

L

(s + 1)

)× φ ∝ ε−s+1 (EL = ε)

φLE = ln(2Emin(Emin − ε)/ε− 1/2) , ε < Emin

φHE = ln(4ε− 1/2) , ε > Emin






Summary

1 Thermal Bremsstrahlung. Self-absorption produces F ∝ ν2 at highopacity and low frequencies and F ∝ exp(−hν/kBT) at high opticallythin frequencies.

2 Loss timescale is large in the interstellar medium, but Bremsstrahlungusually dominates loss processes at energies ∼ GeV (see lesson II).Important loss in denser media (HII regions, supernova remnant ininteraction with molecular clouds ...)

3 Non-thermal Bremsstrahlung. A distribution E−s do produce an energy(photon) spectrum in ε−s+1 (ε−s).






Electron-positron pair production

Reaction : γ + γ → e− + e+

Pair creation threshold : ε1ε2(1− cosχ) ≥ 2, χ = (~k1,~k2) : angle ofinteraction.






Electron-positron pair production cross section

σCMγγ =

3σT

16(1− β2

CM)

[(3− β4

CM) ln(1 + βCM

1− βCM)− 2βCM(2− β2

CM)

]βCM =

√1− 2

ε1ε2(1− cosχ). (31)

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

τγγ(ε2,Ω2) =

∫dZ∫

dε1d cosχ1dφ1

4πnph,1(ε1,Ω1)σCM

γγ (ε1, ε2, χ)(1−cosχ(χ1/2)) .

(32)Integral over Z the length along the line of sight, ε1,Ω1 the energy and solidangle of the low energy photons ... 5 integrals needed to get τγγ(ε2) !

4. Gould R.J. & Schréder G.P. 1967 Phys Rev 155 140843/97 Radiative processes in Astrophysics.





Electron-positron pair annihilation

In-flight reaction : e− + e+ → γ + γ : This produces an annihilationline 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 singletstate 1S0. Lifetime = 125ps in vacuum. Decay into 2 gamma-rays at511keV .

2 Ortho-positronium o-Ps (fraction 3/4) : Parallel spins produce a multipletstate 3S1. Lifetime = 142.05± 0.02ns in vacuum. Decay into 3gamma-rays through a continuum spectrum.






Contribution to the galactic annihilation line

Fit of the diffuse annihilatetion as observed by SPI instrument on boardINTEGRAL 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, 57945/97 Radiative processes in Astrophysics.





(integrated) Annihilation cross section

σCMan =

3σT

32β2CMγ

2CM

[(2 +

2γ2

CM− 1γ4

CM) ln(

1 + βCM

1− βCM)− 2βCM(1 +

1γ2

CM)

]γCM =

√1 + γ+γ−(1− β+β−µ)

2= (1− β2

CM)−1/2 . (33)

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

R± = 2∫

d~β+d~β−f (~β+)f (~β−)βCMcσCMan ×

(γ2

CM

γ+γ−

)[cm−3s−1]. (34)

f (~β±) : e± distribution normalized wrt to the densities n± =∫

f (~β±)d~β± 7.

6. Svensson R., 1982, ApJ, 258, 321 and references therein7. To derive a spectrum on must turn back to the differential cross section






Outlines

1 Preliminaries










Lepton-magnetic field interaction : Generalities

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

Hadrons do also radiate in a magnetic field but at lower rate and at differentfrequencies. 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 thencollective charge effects associated with a plasma can be discarded (seelecture II for the case of plasma effects). Hence the results in these sectionsare those obtained in vacuo.






Particle trajectory in a magnetic field

We consider an uniformmagnetic field (MF) ofstrength B0. The electronmotion is helical with apulsation Ω0 = qB0/γmec(Ωb = γΩ0) and with apitch-angle α = (~B0,~v) andωb = qB0/mec. The Larmorradius is defined as :rL = p sin(α)/qB0. We havenoted θ = (~R, ~B0) : thedirection between the observerand the magnetic field.






Equation of motion

The equation of motion (relativistic case) :

d(γme~v)

dt=

ec~v ∧ ~B0

d(γmec2)

dt=

ec~v.~E = 0 (35)

The second Eq. gives γ = cat and hence |v| = cst.This leads to :

γmed~v⊥dt

=ec~v⊥ ∧ ~B0

d~v‖dt

= 0 , (36)

thus |v‖| = 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 toan acceleration perpendicular to~v of constant intensity. In this circularmotion the pulsations at m× Ω0 are contributing to the radiation.






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 frequencyinterval (Eq.11)Develop the Fourier exponential term in terms of Bessel function : thisproduces the development over harmonics.The emitted power is obtained by dividing W(Ω, ω) by∫ +∞−∞ exp(−iyt) = 2πδ(y) the time of the radiation has been produced.

Where y = mΩ0 − ω(1− β‖ cos(θ)) implies that radiation is producedat harmonics mΩ0 modified by the Doppler term of the motion theelectron along the MF.

The cyclotron emissivity is (in [erg/s Hz st]) : (x = ω/Ω0β⊥ sin(θ))

j(ω, β, θ) =q2ω2

2πc× Σ∞m=1

[(cos(θ)− β‖

sin(θ)

)2

J2m(x) + β2

⊥J′m2(x)

]δ(y) ,

(37)51/97 Radiative processes in Astrophysics.





Emission cyclotron lines

Cyclotron line emissivityas function of β = v/c forthe first 10 harmonics.Dotted lines : synchrotronapproximation(Marcowith A. & MalzacJ., 2003, A&A., 409, 9. !use ν instead of ω andνb = Ωb/2π).

The ratio of the totalemissivity of twosuccessive harmonicsis ∝ β−2 and theinterval between twoharmonics is Ω0.






Radiation by a relativistic particle : The synchrotron limit

Once v→ c the emission is beamed along~v⊥ and focused within a cone ofangle ∼ 1/γ. In the relativistic (non-relativistic) limit the electric field E(t)is a pulse over a time T < 2π/Ω0 (periodical 2π/Ω0) hence |E(ω)|2 is spredover ∆ω = γ3sinα Ω0 > Ω0 (∆ω < Ω0).






Differential synchrotron emissivity

Detailed calculations are provided in RL79 section 6.4 and can be derivedfrom the Schott formula in the limit β → 1 as well (see B66).

dWdωdt

=

√3q3B sin(α)

4πmec2 × F(x) , (38)

with

F(x) = x∫ ∞

xK5/3(u)du , x =

ω

ωc, ωc =

3qB sin(α)γ2

2mec.

Function F(x) wrt to x. Itpeaks at 0.29x and can beapproximated by x0.3 exp(−x).






The synchrotron radiation is polarized

In the relativistic limit, two terms remain in the Schott formulacorresponding to two electric polarizations 1/ one parallel to ~B0 2/ oneperpendicular to ~B0. The total emissivity can be decomposed into twocomponents either. The power emitted Eq.38 can be re-written as (see B66section 6.3 or RL section 6.4) :

dWdtdω

=dW‖dtdω

+dW⊥dtdω

, (39)

P‖/⊥ = Pmin/max =dW‖/⊥dtdω

=

√3q3B sin(α)

4πmec2 × (F(x)∓ G(x)) ,

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






Synchrotron loss timescale

From the total power radiated by a relativistic particle Eq.15 withA‖ = 0 for an isotropically distributed particle population so averagedover the pitch-angle 〈sin2(α)〉Ω = 2/3 we get :

tloss =

[1E|dE

dt|]−1

=3(mec2)2

4σTcEUB[s] . (40)

The magnetic energy density is UB = B2/8π.Interstellar medium estimates :

tloss = 5× 108 E−1GeV B−2

5µG [years] . (41)






Spectrum produced by a population of particles

The spectrum emitted by a population of particles with a densityn =

∫N(p, α)dp is given :

dWdνdVdt

=

∫dW

2πdωdt(p, α)N(p, α)dpdΩ [erg/Hz cm3s] . (42)

We consider an isotropic power-law distribution of electrons :N(p, α) = Kep−s with mec pmin < p < pmax. The spectrum hence scalesas ν−(s−1)/2 in the limit νc(pmin) ν νc(pmax).

4πjν =dW

dνdVdt= (4πKereqB)×

(3νb

2ν

)(s−1)/2

× E(s) . (43)

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






Cyclotron-synchrotron self-absorption

It is possible to define an absorption cross section of a photon of frequency νby a particle of momentum p from an analysis based on Einsteincoefficients 8 : (j(ν, p) is the particle emissivity given by Eq.38.)

σ(ν, p) =1

2meν2 ×1γp∂γ(γpj(ν, p)) . (44)

In the non-relativistic regime integrating j(ν, p) around νb givesσ(ν = νb) ∼ (σT/α)Bcr/B as γ → 1 (Bcr = 4× 1013 Gauss).

Angle averagedcyclo-synchrotron absorptioncross section wrt to ν fordifferent 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.





Case of power-law distribution of relativistic particles

The absorption coefficient is :

αν =

∫dpN(p)σ(ν, p) [cm−1] (45)

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

αν =√

3× recKe

νB×(

2ν3νB

)−(s+4)/2

× A(s) , (46)

A(s) = 2π × 2(s−2)/2Γ(3s + 2

12)Γ(

3s + 2212

)×∫ π

0sin(α)(s+4)/2dα .

• The self-absorbed spectrum is using Eq.43 : Sν = jν/αν ∝ ν5/2.






The curvature radiation processThe curvature radiation process is similar to thesynchrotron radiation process except rL → rc : themagnetic field curvature radius.• Synchrotron emission comes from a cone offset by theparticle pitch-angle (down) / curvature radiation theradiation is emitted along the magnetic field lines (up).Similar spectrum emitted per particle a

P(ω) =

√3

2πe2

rcγ × F(x) ;ωc =

32

crcγ3 ; x =

ω

ωc.

The power radiated by one particle is (see Eq 40) :

P =23

q2cr2

c× β3γ4 . (47)

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






Outlines

1 Preliminaries










Lepton-radiation interaction : Generalities

The relevant process here is the Compton process : the diffusion of a photonby an electron (or a positron). We speak about Inverse Compton radiationwhen the electron is relativistic. We speak about Comptonization whenInverse 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).






The Thomson scattering

If the scattering electron is at "rest", v c and if hν mec2, thephoton 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 electricfield of the incoming wave (~d = e~r). The flux of the wave beingF = cE2/8π the differential cross section is

dσT

dΩ=

dPdΩ× 1

F=

r2e

2(1 + sin(Θ)2) , (48)

with Θ = (~d,~k1) and ~k1 is propagation direction of the out-comingphoton.Integrated over Θ this gives σT .






The Compton scattering

In case photons have a momenta hν ∼ mec2. The electron has a recoiland the process involves QED. Using conservation of momentum andenergy the incoming ε = hν/mec2 and out-coming ε1 = hν1/mec2

photon energies are linked by :

ε1

ε=

11 + ε(1− cos(θ))

(49)

Or alternatively : λ1 − λ = λc × (1− cosθ), λc = hmec . Hence if λ λc

or hν mec2 the diffusion is closely elastic λ1 ∼ λ.






The Compton process cross section

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

dσdΩ

=r2

e

2ε2

1

ε2 ×(ε

ε1+ε1

ε− sin2(θ)

)(50)

Non-relativistic regime (Thomson limit) : ε 1, ε1 = ε and σ = σT .Ultra-relativistic regime : ε 1 and the angle-integrated cross sectiontends to

σ =38σT ×

1ε

(ln(2ε) +

12

) σT . (51)






The Inverse Compton process

If now the electron is relativistic v ∼ c hence even a small recoil canprovide 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 aspeed v wrt to the observer frame (R). Applying a double Lorentztransformation one gets in the Thomson regime (ε′ < 1) the maximumenergy of the diffused photon (See RL section 7.1) :

ε1 ' γ2ε× (1− β cos(θ′1))(1− β cos(θ))ε1,Max = 4γ2ε , (52)

θ (θ′1) is the angle between the incident (diffused) photon and theelectron in the R (R’) frame.






Results in the Klein-Nishina regime

The efficiency of the scattering process depends on the energy of theincoming photon in the electron rest-frame, i.e. if ε′ ∼ 1 henceKlein-Nishina effects become important and the process efficiency isreduced : Scattering process (Thomson regime : ε′ 1)→Catastrophic loss process (Klein-Nishina regime : ε′ ≥ 1).In the Klein-Nishina regime (ε′(1− cos(θ′)) > 1), ε′1 becomesindependent of ε :

ε1 ' γ(1 + cos(θ′))(1− β cos(θ′1)) (53)






Spectrum produced by one relativistic electron : case studyin 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 photondensity dn/ε where dn(ε, x = cos(θ)) = n(ε, x)dεdx

2 From this the calculation is performed in the R’ frame we have :dσ/dΩ′1dε′1 = (r2

e/2)(1 + cos2(θ′1))δ(ε′1 − ε′)

From 1/ one can deduce the total emitted power in R’

dε′1dt′

= cσT

∫ε′1dn(ε′) = INV = − 1

mec2

dEe

dt.

From 1/ and 2/ one can deduce the photon spectrum produced by onerelativistic electron.

dNdt′dε′dΩ′1dε′1

= dn′cdσ

dΩ′1dε′1[Nb/s st ε′ε′1] .






Inverse Compton loss timescale

We find :Loss rate (or radiated power) in the case of an isotropic incident photondistribution in R :

− dEdt

= σTc(γ2(1 +

13β2)− 1

)Uph ,

Uph =

∫εdn(ε) [erg/cm3] (54)

Loss timescale (β = 1)

tloss =E|E|

=3(mec2)2

4σTcEUph[s] , (55)

tloss ' [3× 108]E−1GeVU−1

ph,eV/cm3 [years] .






Inverse Compton photon spectrum produced by one particle

Here I use the general Klein-Nishina cross section in step 2 of slide 68. Thediffused photon spectrum is (BG70 Eq.2.48) :with Γe = 4εγ, q = E1/Γe(1− E1), E1 = ε1/γ.

dNdtdε1

=2πr2

e cγ2

n(ε)dεε× F(Γe, q) (56)

F(Γe, q) = 2q ln(q) + (1 + 2q)(1− q) +12

(1− q)(Γeq)2

(1 + Γeq).

F(E1,Γe) as function ofx = E1 = E1/(Γe(Γe + 1)−1) fordifferent values of Γe =0 (dotted), 1,10, 50, 100 (long dashed).Γe 1 (Γe 1) corresponds to theThomson regime (the Klein-Nishinaregime).






Spectrum produced by a population of particles

We consider the case of an isotropic power-law distribution of electrons :Ne(γ) = Keγ

−s, γmin < γ < γmax (with∫

Ne(γ)dγ = ne). Integrating Eq. 56over Ne(γ) yields to a complicated expression (BL 70 Eq. 2.75).

Thomson limit :1

mec2

dWdtdε1dV

=ε1dN

dtdε1dV= πr2

e cKe(s, ne)× ε−(s−1)/21 × F(s) (57)

F(s) =2s+3(s2 + 4s + 11)

(s + 3)2(s + 1)(s + 5)

∫ε(s−1)/2n(ε)dε .

Klein-Nishina limit (C(s) is given by Fig6. BL70) :

ε1dNdtdε1dV

= πr2e cKe(s, ne)× ε−(s−1)

1 × F(s, ε1) (58)

F(s, ε1) =

∫n(ε)

ε(ln(εε1) + C(s))dε .

The Klein-Nishina spectrum is steeper than the Thomson spectrum.71/97 Radiative processes in Astrophysics.





Some spectral examples

Inverse Compton spectrum produced by a power-law electron distributionover a black-body photon source in the Thomson regime (blue) andKlein-Nishina regime (orange)






Synchrotron self-Compton (SSC) process

The soft photons are synchrotron radiation produced by the same (notalways the case) electron population. In the Thomson regime : if thesynchrotron spectrum extend over [εs,min, εs,max] hence the SSC spectrumextends over γ2 × [εs,min, εs,max] for each particle, hence SSC is moreextended than IC emission produced from a black-body.

In the Thomson regime the ratio of the power radiated by IC tosynchrotron is (see Eqs. 38, 54) :

PIC

PS=

Uph

UB. (59)

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

P2IC

P1IC

=P1

IC

PS=

Uph

UB(60)






Spectral energy distribution : example of blazars

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






The synchro-Compton loss catastrophy

•We note η the ratios in Eq. 60. In the case of homogeneous synchrotronself-absorbed source it is possible to calculate this ratio in terms of a thebrightness temperature Tb (see slide 11) 9.

The brightness temperature such that η = 1 :

Tb,η=1 = 1012K ×( ν

1Ghz

)−1/5. (61)

Hence at 1 Ghz the brightness temperature should not exceed 1012 Kotherwise it induces a loss catastrophy producing huge amount of Xand gamma-rays as the power in secondary generations exceeds thepower 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, L7175/97 Radiative processes in Astrophysics.





Induced Compton scattering

• In this process photons scatter off free electrons at rate enhanced withrespect to Compton scattering by the photon intensity (occupation number).

The condition for Induced Compton scattering (ICS) to be importantis : f

(kBTB/mec2

)τT ≥ 1 and τT = neσTR. f ∼ Ω0/4π accounts for the

anisotropy of the incident photon beam 10.The transfer Eq. becomes non-linear. In the case of isotropic photondistribution and uniform particle distribution confined in the sameregion, noting y(ν) = νkBTB/mec2 we have (with t = neσTct) :

∂y∂ t

= 2y∂yν, (62)

The effect of ICS is to pump photons from high to low frequencies as ymoves along the characteristic (ν, t) with a speed −2y. Hence thebrightness temperature at low energies increases.

10. Kuncic Z. et al 1998 ApJ 495 L35, Coppi P.S. et al 1993 MNRAS 262 60376/97 Radiative processes in Astrophysics.





Isotropic homogeneous Induced Compton scattering

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






Polarization

The degree of polarization depends on the incoming photon polarization. Incase of an isotropic electron distribution with Ne(E) ∝ E−sone may considertwo 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 andfrequency) of photon in the Thomson regime

ΠIC '3 + 4s + s2

11 + 4s + s2 , (63)

In the case of synchrotron photon spectrum Nω ∝ ω−α scattered by thesame 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 therein78/97 Radiative processes in Astrophysics.





• (upper panel) IC polarization degree(s=1.01,1.5,2,3,4 black to magenta) incase of a power-law electron and abeam of photons at an angle of 85o

with l.o.s.Klein-Nishina case :ΠKN ∼ 0.5/(1 + ε′).• (lower panel) SSC polarizationdegree in case of a photon distributionNω ∝ ω−(α+1) (α = 1, p = 3 red andα = 0.5, p = 2 black) with an anglebetween MF and l.o.s. of 85o.In both figures the properenergy/frequency limits have to beconsidered.





Hadron-matter interactionHadron-radiation interactionHadron-magnetic field interaction

Outlines

1 Preliminaries










Ionization and Coulomb losses

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

dEdt

= −4πcr2e nemec2Z2 ln Λ× x2

m + β2

β2 , (65)

xm = 0.0286(Te/2× 106K)1/2 and ln Λ = [30− 40] in the ISMconditions 12.

2 Ionization losses :

dEdt

= −2πcr2e mec2neZ2 × 1

β× Σs=H,Hens × Ks(Is, β,E) . (66)

The complicated function Ks is described in S02 and SM98 andreferences therein.

12. Dermer C.D. 1985, ApJ, 285, 2881/97 Radiative processes in Astrophysics.





Inverse Bremsstrahlung

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

Particle distribution at shocks and Bremsstrahlung/Inverse Bremsstrahlungspectra (Baring et al 2000).FIB/FB ∝ (meTp/mpTe)

(s−1)/2, s : non-thermal particle distribution index.

13. see introduction section of Baring M.G. et al 2000, ApJ, 528, 77682/97 Radiative processes in Astrophysics.





High energy hadron-matter interaction processes

see S02Proton pion production (n1, n2=multiplicities).

p + p → p + p + n1(π+ + π−) + n2π0 ,

π± → µ± + νµ/νµ ,

π0 → 2γ ,µ± → e± + νe/νe + νµ/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 inparticular sources one may be concerned with heavier particles.

All these processes involve inelastic interactions : a1 + a2 → Σiai with atotal mass M = Σimi. The threshold condition for the interaction (prime isexpressed in the center of mass frame) :

E′th =(m2

1c4 + m22c4 + 2E1E2 − 2p1p2c2cos(θ1,2)

)1/2 ≥ Mc2 . (68)






Neutron production

Neutrons have a short lifetime (γ × 103 s) but have no charge so areinsensitive to electromagnetic fields and can transport a substantialfraction of the energy from the sources 14.Fraction of protons converted into neutrons : 1/4 in pp collisions and1/2 in pγ collisions

14. Derishev, E.V. et al 1999, ApJ, 521, 640, Begelman M. et al 1990, ApJ, 362, 3884/97 Radiative processes in Astrophysics.





(neutral) Pion production cross section

Here is shown a parametrization of the inelastic cross section 15

L = ln(EP/1TeV) and from Eq. 68 thethreshold energy is :Eth = 2mπ + mp + m2

π/2mp = 1.23GeV (mπ ' 135 MeV, mp ' 0.938GeV).

15. see Kelner S.R. et al 2006, PhysRevD, 74, 034018 ; Kamae T. et al 2005, ApJ, 620, 244and 2006, ApJ, 647, 692 for details






Gamma-ray emissivity by one particle

Each pion produces a differential number of gamma-ray photons :

dNγdEγ

(Eγ) = 2×∫ ∞

Eγ+4mπc2/Eγ

dNπdEπ

(Eπ)dEπ√

E2π − m2

πc4. (69)

Now each proton produces pions at a differential rate 16 :

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

Typical pion energy : 〈Eπ〉 ∼ 1/6Tp (Ep ≤ 10 TeV, Tp is the kineticenergy) and Ep(Eπ) = mpc2 + 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 moredetailed calculations






Proton loss timescale

tloss = 3× Ep ×[∫

R(Eπ)dEπ

]−1

∼ 5× 107 n−1H,cm−3 [years] . (71)

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






pp interaction process is anisotropic

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

Gamma-ray spectra at different angles wrt the beam for Np ∝ E−2p .

17. Karlsson N. & Kamae T., 2008, ApJ, 674, 27888/97 Radiative processes in Astrophysics.





Gamma-ray emissivity by a population of particles

We define the pion rate integrated over the proton distribution

qπ(Eπ) =

∫R(Eπ)np(EP)dEπ =

nHcKπ

σpp(Ep(Eπ))np(Ep(Eπ)) [nb/cm3s Eπ] .

(72)again, Ep(Eπ) = mpc2 + Eπ/Kπ and then use Eq.69. In case of a power lawgamma-ray and proton indices are identical.

EF(E) (TeV/cm−3 s) vs E(TeV)gamma-ray spectrum from pion decayfor different power-law protondistribution n(Ep) ∝ E−p : p = 2 (3dotted-dashed), p = 2.5dotted-dashed), other lines correspondto fits of the diffuse galacticgamma-ray emissivity (see Aharonian& Atoyan 2000).






Electron-positron and neutrino production

Here are displayed the results from Kelner et al 2006 using parametrizationof 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 protonsnp ∝ E−αp × exp(−(Ep/E0)β).Left panel α = 2, β = 1,E0 = 1 PeV. Left panel α = 1.5, β = 1,E0 = 1PeV. In dashed line : delta approximation Eq.72.• One must account for the radiation (synchrotron, Bremsstrahlung, InverseCompton) radiation from secondary leptons.






Outlines

1 Preliminaries










Hadron-radiation interaction processes

Photo-pair production

p + γ → p + e− + e− .

Photon threshold E′γ = ∆mc2(1 + ∆m/mp)= 1 MeV (proton restframe)Photo-pion production

p + γ → p + π0 ,

p + γ → p + π+ + π− .

Photon threshold E′γ = ∆mc2(1 + ∆m/mp)= 145 MeV (protonrestframe)Photo-disintegration

A + γ → (A− 1) + n/p .

Important process in the context of ultra high-energy cosmic raypropagation in the intergalactic medium 18

18. see e.g. Khan E. et al 2005, Aph, 23 191 for further details92/97 Radiative processes in Astrophysics.





Photo processes cross sections

Photo-pion cross section asfunction of the incomingphoton energy in the protonrestframe. Peakσ ∼ 5× 10−28cm2 atε′ ∼ 2ε′th.

See 19. From this we can deduce the collision rate in the observer frame

R =cγp

∫n(ε′,Ω′)σπ(ε′)dε′dΩ′ [nb/s] , (73)

with∫

n(ε′,Ω′)dε′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






Loss timescales

The loss timescale is derived using Eq.73 but accounting for the inelasticityparameter K(ε′)→ 0.5 (at high energy, ε′ 1) 20

tloss = [cγp

∫n(ε′,Ω′)K(ε′)σπ(ε′)dε′dΩ′]−1 . (74)

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

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

where P(s) are tabulated in20 for both photo-pion and photo-pair processesalso in case of black body radiation.Typically for s = 1, tloss ∼ 108U−1

rad,erg/cm3 × E−1p,GeV years.

20. Begelman M. et al 1990, ApJ, 362, 38 appendix94/97 Radiative processes in Astrophysics.





Loss timescales over the cosmic microwave background

We have UCMB ' 4× 10−13 erg/cm3.

Inverse of loss timescale with respectto the cosmic ray energy.

Typically for ∼ 1020 eV protons tloss ∼ 108 years 21.

21. Kelner et al 200895/97 Radiative processes in Astrophysics.





Outlines

1 Preliminaries










Synchrotron radiation

Loss timescale

tsyn,p = [43

(me

mp

)3

× cσTUB

mec2 Ep,GeV ]−1 = 18363 × tsyn,e . (76)

Peak of particle radiation

νc,p =me

mp× νc,e =

11836

νc,e . (77)


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