Non Thermal Radiative Processes in Astrophysics

Loss & EscapesSpecial relativistic effects in radiative transfer

Radiative processes in a plasmaConclusions

Non-thermal radiative processes in AstrophysicsLoss / relativistic / plasma effects

Alexandre Marcowith 1

[email protected]

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

28 avril 2013

1/40 Radiative professes in Astrophysics.



Bibliography

1 BooksKrall N.A. Trivelpiece A.W., 1973, Principles of plasma physics,McGraw-Hill (KT73)Melrose D.B., 1980, Plasma Astrophysics. (M80)Rybicki G. B. & Lightmann A.P. , 1979, Radiative processes inAstrophysics, Wiley.Schlickeiser R., 2002, Cosmic Ray Astrophysics, Springer.

2 ArticlesBlumenthal G.R. & Gould R.J., 1970, Review of Modern Physics, 42, 237




Outlines

1 Loss & EscapesLoss-Escape equation

2 Special relativistic effects in radiative transferLorentz transformationsSome applications to relativistic sources

3 Radiative processes in a plasmaGeneralitiesSome practical applicationsThe particular case of hot plasmas

4 Conclusions




Loss-Escape equation

Outlines




4 Conclusions





Introduction

The aim of this section is to describe the time spectral impact of radiativelosses over (non-thermal) particle distribution. We will restrict the analysisto [E,t] space and evaluate the time evolution of the particle energy densityN(E, t).

This supposes that efficient mechanisms produce a particleisotropization in the momentum (energy) space (see A. Bykovs, D.Ellisons and H.Spruits lectures)But no explicit acceleration mechanisms will be treated (see again nextlectures).The solution will be space-averaged ; N(E, t) = (1/V)

dVN(E,~x, t),

where V is the volume of the region under consideration = one-zonemodel.





The diffusion-loss equation

This a Fokker-Planck Eq. May be derived from conservation arguments orfrom more fundamental kinetic Eq. (Vlasov).

Nt

= (DN) + E

[a(E)N] + Q(E, x, t) . (1)

a(E) 1/t(E) < 0Catastrophic losses maybe added (nuclei fragmentation) N/C,lossThe diffusive term producing spatial escape is simplified as N/esc(E)in the one-zone model we consider. Hence :

Nt

= Nesc

+

E[a(E)N] + Q(E, x, t) . (2)





Lepton loss timescales in the interstellar medium

Electron loss timescales in theISM with a target density of1 cm3. Different loss regimes areidentified t(E) E, a(E) E0(Coulomb/ionization,E 1 GeV).





Hadron loss timescales in the interstellar medium

Proton loss rate. Implyt(E) E, a(E) E0(Coulomb/ionization,E1GeV).





Some solutions : only with losses

1 Case of stationary solution with Q(E) = Q0Ep

N(E) = Q0 E(p1)

(p 1)a(E) . (3)

Hence for electrons the emerging spectrum is harder(Coulomb/ionization), identical (Bremsstrahlung), softer (Synchrotron,IC-Thomson).

2 Case of a time-dependent solution with N(E, t = 0) = N0Ep but notime injection see 1

N(E, t) = N0(1 aEt)p2EpH(1 aEt) . (4)H(x) Heaviside function.

1. BG70, Kardashev I.N., 1962, 6, 3179/40 Radiative professes in Astrophysics.




Relaxation by losses

see L84.




Lorentz transformationsSome applications to relativistic sources

Outlines




4 Conclusions





Lorentz transformations of the radiation field

Based on the fact that the phase space density f = dN/d3xd3p is an invariantquantity. The following transformation rules can be applied :

Specific intensity I : as

hf (x, p)p2dpd = Udd ,U = I/c ,

we haveI3

= INV . (5)

S behaves similarly.Absorption coefficient :

= INV . (6)

Emission coefficient : j = S :

j2

= INV . (7)





Outlines




4 Conclusions





Luminosity of relativistic sources

Between the comoving plasma frameR and the observer frameR severalimportant relations exist. The plasma is moving with a Lorentz factor andan angle wrt to the observer frame.

Observed frequency (Doppler factor D = (1 cos())1)

= D

If D > 1 : blue shifting, if D < 1 red shifting.Differential luminosity :

L = D3L .Total luminosity :

L = D4L .





Superluminal motionsA source moving with an angle wrt to the line of sight at a velocityV < c emits two photon flashes ata time interval of t = tT tS.Hence the transversal velocityV = V sin() and the apparentdisplacement tV. The effectivetime difference between the twoflashes is tapp = x/c =(1 V/c cos())t.

The apparent transverse velocity is 2

Vapp, =tVtapp

=V sin()

(1 V/c cos()) . (8)

It peaks at cos() = ; hence Vapp,,max = .2. Rees M.J., 1966, Nature, 211, 468





Apparent velocity





The brightness temperature problem

Relativistic motions also introduce a difference between the rest-frame andthe observer source size.

From signal variation over a time one can estimate that the sourcesize must be (from causality arguments) : R = c .Relativistic effects implies that the intrinsic source size is in reality :R = R/D.Combining the source effect with the Lorentz transformtion of S thebrightness temperature in the source frame scales as :

Tb = Tb/D3 .

Hence this effect has been argued to explain the brightness temperatureexceeding 1012 K in compact radio sources. But this was not sufficientfor some sources 3.

3. see Wagner S.J. & Witzel A., 1995, ARAA, 33, 163 for a review on alternative scenarii17/40 Radiative professes in Astrophysics.



GeneralitiesSome practical applicationsThe particular case of hot plasmas

Outlines




4 Conclusions





The special plasma effects

... wrt to vacuo results.

The section is a bit odd because plasmas are everywhere in Astrophysics.But up to now we have supposed the propagation medium to be a vacuum(synchrotron, Inverse Compton, hadronic processes). Considering themedium as a plasma permits to include all these processes into a generalelegant formalism.

Plasma kinetic theory provides identical results (hopefully) as the one usedin lesson I. It has been developed by several author (Kaplan, Tsytovich ...),but D.B. Melrose did likely provided the most comprehensive description(see bibliographical notes). I briefly discuss it here.





Plasma theories

Treats dynamical evolution of plasma waves (radiation) coupled withparticles solving

1 Maxwell equations (~E, ~B) in terms of (,~j) coupled with1 Fluid equations in case one neglects thermal motions : Cold plasma

approximation.2 Kinetic equation (eg. Vlasov) in case we account for momentum particle

effect : kinetic plasma theory - see particle acceleration lectures(analytical, numerical)-

This system permits to derive the wave dispersion relation at differentapproximation level.





Characteristic quantities

1 Plasma frequency : pe = (4pine2/me)1/2 = 5.4 104ne,cc [s1] or afrequency fp = (8.98 kHz)

ne,cc : Involves processes at lowfrequencies (usually radio).

2 Electron gyro-frequency : ce = 1.76 107BGauss [s1] orfe = (2.80 Mhz)BGauss : Involves processes at high frequencies inextreme cases (pulsars, magnetars)





Dispersion relation and normal modes of a plasma :un-magnetized medium

In the cold approximation in the case of an un-magnetized medium(KT73), two normal modes are obtained : Longitudinal (~E ~k) modesor electrostatic modes or Langmuir waves, transversal modes (~E ~k).For transverse waves the dispersion relation is :

2 = 2pe + k2c2 , (9)

and the refractive index n2 = (kc/)2 = (1 2pe/2) < 1. Transversewaves with < pe do have an imaginary wave number k and aredamped.For Langmuir waves :

2 = 2pe + 3k2V2e . (10)

The index of refraction can be > 1.22/40 Radiative professes in Astrophysics.




Dispersion relation and normal modes of a plasma :magnetized medium

In the cold approximation in the case of magnetized medium (KT73)the dispersion relation becomes more complicated. Different frequencyregimes are relevant :

> ce, pe : magnetoionic modes < ce cos() < pe : whistler modes < cp : magneto-hydrodynamic modes (Alfvn, magnetosonic).





General formalism

A plasma has different responses to electromagnetic perturbations ~E, ~B. Thefirst kind is a linear response corresponding to the propagation of normalmodes and is associated with an induced current ~Jind,l(~k, ) such that :

Jind,li (~k, ) = ij(~k, )Ej(~k, ) (11)

and

ij(~k, ) =k2c2

2(ki/kkj/k ij) + (ij + 4pi

ij(~k, )) = 0

is the dispersion relation.But a plasma can also have external currents corresponding to otherresponses (non-linear, external sources) ...hence the above Eq. is :

ij(~k, )Ej(~k, ) = 4piJexti (~k, ) (12)





Power radiated in a plasma

In plasma physics the power radiated by a source is calculated with ~Jext.~Einstead of calculating the Poynting flux through a sphere as in a vacuo (slide18 lecture 1). The method is the following :

Express Jexti (~k, ) in terms of Ej(~k, ) using Eq. 12

Account from the zeros of ij(~k, ) corresponding to normal modes = n(~k).Integrate the work ~Jext.~E over .

The energy radiated into a particular mode n is (see M80 eq.3.18)

Wn(~k) = 4piRnE(~k)|~en,(~k).~Jext(~k, n(~k))|2 , (13)

where~en, is the cc of the polarization vector and RE(~k) is a function of(~k, n), the fraction of the energy imparted into the electric field (seeEq.2.70, 2.85 in M80). The power radiated is Pn(~k) = limtWn(~k)/t.





Outlines




4 Conclusions





We are considering several relevant effects on radiation occurring in aplasma.

1 Cerenkov radiation2 Razin-Tsytovich effect3 Faraday rotation4 Processes in hot electron/positron plasmas

ComptonizationPair production/annihilation in hot plasmas





Cerenkov radiation

The Cerenkov radiation is the emission produced by a uniformly movingcharge in a medium with n > 1. One derives ~Jext associated with such acharge moving as~r(t) = ~r0 +~v0t :

~J(~k, ) = 2piq~v0 exp(i~k.~r0)( ~k.~v) . (14)The condition :

= ~k.~v , (15)

is the Cerenkov resonance the particle sees a static periodic EM field inits rest-frame.

Cerenkov radiation is forbidden in vacuo as /k = c > v.Only modes with n > 1 are emitted. The total power radiated is givenby Eq. 13 and (3.58 in M80) in Langmuir modes is :

P =q22pv ln(v/Vthe) .





Magnetized media

We extend the discussion to the case of a medium imbedded in a magneticfield (M80, sect. 4)

~Jext is given by the spiraling around B0 characterized by a Larmorradius rL = v sin/0, 0 = qB0/mec, is the particle pitch-angle.The emissivity gives an analog to the Schott formula (Eq.37 lecture 1)in vacuo. But in a plasma the Synchrotron resonancen(~k) m0 kv = 0, n(~k) includes the refractive index of themedium 6= 1.In the case of transverse waves the critical frequency, the peak of thesynchrotron emission (see slide 54) is transformed as :

c =32

02 sin

(1 + 22p/2)3/2. (16)

hence c 3/202 sin (vacuo solution) at high frequencies.29/40 Radiative professes in Astrophysics.




Razin-Tsytovich effect

The Razin-Tsytovich effect is the suppression of low frequency transversewaves in the range p < < p ( is the particle Lorentz factor), hence itis particularly relevant at low radio frequencies ; a process in competitionwith synchrotron self-absorption 4.

Considering c the synchrotron radiation is exponentially damped(see slide 54). Using Eq. 16 one can derive a limit value ofR = 2p/30 sin where Rp = R are suppressed.For R one cannot find any frequency range where the emissivity isof the order of 1.

4. see C. Fransson & C.-I. Bjornsson, 1998, ApJ, 509, 86130/40 Radiative professes in Astrophysics.




Faraday Rotation

Let a circularly polarized electromagnetic wave (~E, ~B) propagate :~E = E exp(it)(~e1 ~e2) : superposition of a left(+) and right(-)handed wave, ~B0 = B0~e3.The wave propagation in a plasma is supported by the motion of freeelectrons. The fluid velocity is obtained solving the Lorentz Eq.

~v(t) =ie

m( ce)~E(t)

The response of the plasma proceeds a different velocity wrt to thewave polarization. The plan of polarization rotates as the wavepropagate in the medium : Faraday rotation

The plan of polarization rotates by an angle for a medium of width d anddepends on the component of B along the line of sight.

=2pie3

m2ec22

d0neBds . (17)





Outlines




4 Conclusions





Processes in hot plasmas

Hot means that thermal temperature e = kBTe/mec2 may be close to 1. This kind of medium is relevant in accretion discs corona around compactobjects, in jets (Active galactic nuclei, gamma-ray bursts) or in pulsarmagnetospheres. There opacity to pair production can be high. The transferof radiation has to include pair production and annihilation as well as thepair distribution evolution.

This investigation is eased considering a system of coupled kinetic equations(i) fe+/e (ii) f . Soft photons will be subject to (Inverse) Compton diffusionby the particles. Upscattered photons can produce new pairs. The pairs cancool and radiate new photons ... etc ... this questions the problem of pairequilibrium distribution 5.

Hereafter we will consider a region of size R and luminosity L (e.g.accretion disk luminosity), composed of an isotropic and homogeneousdensity of electron/pairs ne.

5. See for instance Svensson R. 1982 ApJ 258 36133/40 Radiative professes in Astrophysics.




The Comptonization process

Photons will be submitted to multiple interactions with electrons in a hotplasma= Comptonization process. No pair creation is considered yet. Weconsider a thermal electron distribution of temperature e and photons withenergies mec2/ (RL74).

Compton parameter defines the total relative energy change of a photontraversing the medium = [average fractional energy change perscattering] [mean number of scatterings].For thermal non-relativistic (fixed) temperatures :

yTh = 4e ( + 2) , = neTR . (18)





Comptonized photon spectrum

The Eq. of photon transfer is a Fokker-Planck Eq. in the regime ofnon-relativistic leptons as the relative energy transfer per scattering is/ 1 6. Power-law spectrum regimes exist (diffusion dominance,y 1) :

Comptonized photonintensity (Log(I)) wrtto Log(h/kBTe) fory=1. The spectrumcut-off exponentiallyat x e.

6. A.S. Kompaneets, 1957, Sov Phys JETP, 4, 73035/40 Radiative professes in Astrophysics.




Electron-positron in hot plasmas

e pairs are produced if ` = LT4piRmec3 > 1 : compactness as` = nphTR . Pairs modified the continuum spectrum : (i)up-scattering soft photons (Comptonization) (ii) pairs with energies mec2 may produce an annihilation line (iii) relativistic particlesinduce photon/particle cascades and reprocess gamma-rays into X-rays(iv) pair creation/annihilation induces variability.The time-dependent transfer problem in hot plasmas is quite complex 7

Source geometry poorly known (escape, relativistic motion effects)Energy supply not strongly constrained.Many processes involved : Comptonization of thermal/synchrotronradiation, radiative cooling (Coulomb, Bremsstrahlung, synchrotron, IC),pair creation/annihilation and cascades ...

Numerical approaches considerably simplify it : (i) Monte-Carlomethods particle per particle propagation :-( bad statistics at highenergies (ii) finite element methods :-( integro-differential Eqs. to solve.

7. see discussion in Coppi P.S., 1992, MNRAS, 258, 65736/40 Radiative professes in Astrophysics.




An illustration : dynamics of photons and particles in disccorona

Time dependent numerical solutions of a system of coupled Fokker-PlanckEq. (photons, leptons ; Belmont et al 2008) or (photons, leptons, plasmawaves ; Marcowith et al 2013, Belmont et al 2013). Example with largeparticle compactness (`e = T Einj/mec3R) 8. A high compactness inducesintense pair production and annihilation line

Stationary solutionsfor different numericalcalculations and fordifferent particlecompactnessle = le+ + le .

8. cont. Belmont, Malzac, AM, 2008, A&A, 491, 617, dashed : Coppi P.S., 1992, dottedStern B. et al, 1995, MNRAS, 272, 291




All subjects that I skipped

Hope that you can find it in my colleagues lectures...Losses processes

I mentioned to rapidly Coulomb/ionization losses : important especially atlow energies and in the energy balance of the interstellar medium.Adiabatic losses : loss process produced by the expansion of the fluidcontaining the particles.Nuclei photo-disintegration is important especially for high energycosmic rays.

Several processes important in high magnetic field environments :magnetic pair creation, photon-splitting ....Discussion about the maximum energy of the synchrotron cut-off.The jitter radiation : Radiation produced by particles in a turbulentmagnetic field.

... and much more ...my apologies for this ...




Conclusions

If you would have a few points to retain from this course :

Most common radiative processes in astrophysics : synchrotron,bremsstrahlung, inverse Compton, pion decay.Most common loss processes in astrophysics : synchrotron,bremsstrahlung, inverse Compton, pion decay and at low energiesCoulomb/ ionization.Special relativistic effects expected in : jets, gamma-ray bursts(amplification , Doppler shift, superluminal motions).Plasma effects are expected at radio frequencies (Razin-Tsytovich) orcan help to constraint magnetic fields (Faraday rotation).Hot plasmas involve Comptonization and pair production/annihilation.




Thank you for your attention


Loss & EscapesLoss-Escape equation

Special relativistic effects in radiative transferLorentz transformationsSome applications to relativistic sources

Radiative processes in a plasmaGeneralitiesSome practical applicationsThe particular case of hot plasmas

Conclusions

Documents

Non Thermal Radiative Processes in Astrophysics