@let@token High Energy Radiation Processes: I ...€¦ · 3 Source morphology 3 Time variability 3 Polarization of EMR The conventional astronomy features tools that allow studying

High Energy Radiation Processes:I. Electromagnetic Processes

DMITRY KHANGULYANRikkyo University (Tokyo)

DIAS Summer School in High-EnergyAstrophysics 2018

D.Khangulyan High Energy Electromagnetic Radiation Processes

Introduction

“High Energy Electromagnetic Radiation Processes”


Introduction

“High Energy Electromagnetic Radiation Processes”The talk is about production ofelectromagnetic radiation


Introduction

“High Energy Electromagnetic Radiation Processes”The talk is about production ofelectromagnetic radiation

Four Tools to Study EMR3 Study of spectra3 Source morphology3 Time variability3 Polarization of EMR

The conventional astronomy features tools that allow studying EM radia-tion across entire electromagnetic spectrum, fine angular and time resolu-tion to reveal the source morphology and time-dependent processes andfurther details with polarimetry observations.


Introduction

“High Energy Electromagnetic Radiation Processes”The talk is limited to the processesenabled by particle’s electric charge:

LorentzTransformation

for EM field

Ex = E ′xEy = ΓE ′y

Ez = ΓE ′zBx = 0

By = −βΓE ′zBz = βΓE ′y

Poynting Flux

S =c

4π~E × ~B


Introduction

“High Energy Electromagnetic Radiation Processes”The talk is limited to the processesenabled by particle’s electric charge:

LorentzTransformation

for EM field

Ex = E ′xEy = ΓE ′y

Ez = ΓE ′zBx = 0

By = −βΓE ′zBz = βΓE ′y

Poynting Flux

S =c

4π~E × ~B

If a charged particle gets accelerated, then the configuration ofthe electromagnetic field is such that the Poynting flux is non-zero. Magnetic field provides ideal conditions for radiation. Ac-celeration in electric field saturates quickly, so one needs timevariable E-field (i.e., a wave) or strong spatial dependence (e.g.,collisions of charged particles).


Introduction

“High Energy Electromagnetic Radiation Processes”“High Energy” stays for “gamma ray”?Yes, but what is more important: thisis about the emission of high-energyparticles. This may seem to be a trivialstep: high energy emission can onlybe produced by high energy particles,however there are two important points:

Emission peak energy dependson the radiation mechanism, thusa high energy particle may emitphotons with very different energy

High Energy particles are essen-tially non-thermal, i.e., there isno fundamental/general way todescribe their distribution

10 14 10 12 10 10 10 8 10 6 10 4 10 2 100 102Energy (TeV)

1025

1026

1027

1028

1029

1030

1031

L,W

H.E.S.S.

Chandra

XMM

UVW1

1.0 TeV < Ee < 3.0 TeV 3.0 TeV < Ee < 10.0 TeV 10.0 TeV < Ee < 30.0 TeV


Introduction

“High Energy Electromagnetic Radiation Processes”“High Energy” stays for “gamma ray”?Yes, but what is more important: thisis about the emission of high-energyparticles. This may seem to be a trivialstep: high energy emission can onlybe produced by high energy particles,however there are two important points:

Emission peak energy dependson the radiation mechanism, thusa high energy particle may emitphotons with very different energy

High Energy particles are essen-tially non-thermal, i.e., there isno fundamental/general way todescribe their distribution

10 14 10 12 10 10 10 8 10 6 10 4 10 2 100 102Energy (TeV)

1025

1026

1027

1028

1029

1030

1031

L,W

H.E.S.S.

Chandra

XMM

UVW1

1.0 TeV < Ee < 3.0 TeV 3.0 TeV < Ee < 10.0 TeV 10.0 TeV < Ee < 30.0 TeV

Four tools for studying non-thermal sources:

Depends onparticle distribution radiation process

Spectrum 3 3Morphology 3 3Time Variability 3 3Polarization 3 3

where 3 means a strong dependence

For interpretation of non-thermal emission an accurate de-scription of the particle spectrum is as important as accuratecalculation of the emission. When particles lose energy byradiating this influences the particle distribution thus the de-scription of the particle spectrum should include the impact ofradiation. However, non-radiative processes may also have asubstantial influence on the particle spectrum.


Introduction

“High Energy Electromagnetic Radiation Processes”There are many excellent

text books covering radiationprocess: Landau&Lifshitz,Longair, Rybicki&Lightman

There are advance numer-ical tools that allow quickcalculations of spectra, e.g.naima by Victor Zabalza:


Introduction




naima allows not only computing synchrotron, IC, bremsstrahlung,and pion decay emission, but also advance MCMC fitting radiativemodels to observed spectra


Introduction




This talk aims to discuss aspects that are beyondthe standard textbooks and that are still useful formodeling of high-energy sources.

This

talk


Description of Non-Thermal ParticlesDistribution of high energy parti-cles depends on some parame-ter(s):

- Energy: dN = fdE- Momentum: dN = fd3p- Coordinate: dN = fdEdx- Coordinates: dN = fdEd3r- Phase-space coordinates:

dN = fd3rd3pHere f is distribution function,and its definition may vary de-pendingly on the context. Foreach problem one needs to se-lect an adequate distributionfunction that allows accountingfor all relevant processes.


Description of Non-Thermal ParticlesDistribution of high energy parti-cles depends on some parame-ter(s):

- Energy: dN = fdE- Momentum: dN = fd3p- Coordinate: dN = fdEdx- Coordinates: dN = fdEd3r- Phase-space coordinates:

dN = fd3rd3pHere f is distribution function,and its definition may vary de-pendingly on the context. Foreach problem one needs to se-lect an adequate distributionfunction that allows accountingfor all relevant processes.

Aharonian&Atoyan 1998

One-zone model

MHD model for particle

transport and cooling


Description of Non-Thermal ParticlesIf one ignores the particle spin –which still might be important insome astrophysical conditions,e.g., in pulsar magnetosphere–the phase-space distribution func-tion provides the most completedescription: dN = fd3rd3p.

There is a quite simple equa-tion for the distribution function,Boltzmann Equation:

∂f∂t

+ v∂f∂r

+ F∂f∂p

= 0∂∂t

∫X0

f dx = −∫∂X0F(f ) dSx∫

X0

(∂f∂t + divF(f )

)dx = 0

∂fa∂t +

∂(ṙfa)∂r +

∂(ṗfa)∂p = 0


Description of Non-Thermal ParticlesIf one ignores the particle spin –which still might be important insome astrophysical conditions,e.g., in pulsar magnetosphere–the phase-space distribution func-tion provides the most completedescription: dN = fd3rd3p.

What about particle collisions?

There is a quite simple equa-tion for the distribution function,Boltzmann Equation:

∂f∂t

+ v∂f∂r

+ F∂f∂p

=

[∂f∂t

]col

∂∂t

∫X0

f dx = −∫∂X0F(f ) dSx∫

X0

(∂f∂t + divF(f )

)dx = 0

∂fa∂t +

∂(ṙfa)∂r +

∂(ṗfa)∂p = 0


Boltzmann Equation

∂f∂t

+ v∂f∂r

+ F∂f∂p

=

[∂f∂t

]col

The collision integral[∂f∂t

]col

accounts formany processes: particle injection, ac-celeration, scattering, energy losses, etc– i.e., ALL plasma physics (lecture byM.Malkov illustrated the complexity of thetopic)In the simplest case, Boltzmann collisionintegral is[

∂fa∂t

]st

=∑

b

∫dp1vreldσ

(fa(x ′)fb(x ′1)− fa(x)fb(x1)

)D.Khangulyan High Energy Electromagnetic Radiation Processes

Boltzmann Collision Integral in Astrophysics[∂fa∂t

]st

=∑

b

∫dp1vreldσ

(fa(x ′)fb(x ′1)− fa(x)fb(x1)

)

Boltzmann collision integral is widely usedin kinetics of neutral gases, e.g., to de-scribe an admixture propagationIn astrophysics the collision integral in thisform is used to describe, e.g., the electro-magnetic cascading:

∂fe∂t

+ v∂fe∂r

+ F∂fe∂p

=

∫dpγc dσ̄γγfγ(pγ)− cσ̄icfe

∂fγ∂t

+ c∂fγ∂r

=

∫dpec dσ̄icfe(pe)− cσ̄γγfγ


Boltzmann Equation

∂f∂t

+ v∂f∂r

+ F∂f∂p

=

[∂f∂t

]col

Equation with the collision integral[∂f∂t

]col

cannot besolved for astrophysical applicationsIt is possible to divide the physics included in the col-lision integral in two parts: complex (acceleration –talks by Bykov, Malkov, Blasi, Lemoine) and simple(cooling, which can be treated under the continuous-loss approximation )Also in the most cases particles are isotropic in somesystem, thus particle energy is a good parameter


Significant simplification in the case of energy losses

Cont. Loss Approx.

Distribution function & Injection

dN = f (E , t)dE dN = q(E , t)dEdt

F (E , t) =

∞∫E

f (E ′, t)dE ′

F (E + Ėdt , t + dt) =

F (E , t)+dt

∞∫E

q(E ′, t)dE ′

Fokker-Planck Equation

∂f∂t

+∂(Ė f )∂E

= q(E , t)


Significant simplification in the case of energy losses

F (E + Ėdt , t + dt) = F (E , t) + dt

∞∫E

q(E ′, t)dE ′

F (E , t) +∂F∂E

Ėdt +∂F∂t

dt = F (E , t) + dt

∞∫E

q(E ′, t)dE ′

∂

∂E=⇒

accounting for ∂F∂E = −f∂∂E

∞∫E

q(E ′, t)dE ′ = −q

∂f∂t

+∂(Ė f )∂E

= q(E , t)


Particle Energy Distribution

Fokker-Planck Equation Solution

f (E , t) =1Ė

Eeff∫E

dE ′ q(E ′), where t =

Eeff∫E

dE ′

Ė ′

Ė = Ėsyn + Ėic + Ėad + etc/Ėsyn + Ėpp + Ėpγ + etc

Fast Cooling (Saturation)Eeff →∞

f (E) =1Ė

∞∫E

dE ′ q(E ′)

Slow Cooling

t � E/Ė

f (E , t) = q(E) · t


Spectral Breaks: Particle Distribution

Solution of the Fokker-Planck Equation:

f (E, t) =1

Ė

Eeff∫E

dE ′ q(E ′), where t =Eeff∫E

dE ′

Ė ′

Let us consider the simplest case:

q(E, t) = θ(E−EMIN)θ(EMAX−E)E−α, where Ė ∝ Eβ

Cooling energy is Ec = Ė(Ec)t , where t is the source age

Ec > EMIN then break at Ec and rangeof energy is EMIN < EMAX:

f (E) ∝{

(E−α+1 − E−α+1MAX )E−βE−α

Ec < EMIN then break at EMIN andrange of energy is Ec < EMAX

f (E) ∝{

(E−α+1 − E−α+1MAX )E−βE−β


Spectral Breaks: Particle Distribution

Solution of the Fokker-Planck Equation:

f (E, t) =1

Ė

Eeff∫E

dE ′ q(E ′), where t =Eeff∫E

dE ′

Ė ′

Let us consider the simplest case:

q(E, t) = θ(E−EMIN)θ(EMAX−E)E−α, where Ė ∝ Eβ

Cooling energy is Ec = Ė(Ec)t , where t is the source age

Ec > EMIN then break at Ec and rangeof energy is EMIN < EMAX:

f (E) ∝{

(E−α+1 − E−α+1MAX )E−βE−α

Ec < EMIN then break at EMIN andrange of energy is Ec < EMAX

f (E) ∝{

(E−α+1 − E−α+1MAX )E−βE−β

In case with pure power-law injection andcooling the particle distribution may haveone break and three different slops α+β+1,α, and β are allowed (here α and β are thepower-law indexes of the acceleration spec-trum and the cooling rate). For example, theparticle distribution cannot have two breaks.More complicated particle distributions areallowed if (i) the injection spectrum is nota power-law (e.g., each acceleration spec-trum has a high-energy cutoff); (ii) the lossrate has a non-power-law dependence onenergy.


Used simplifications

Phenomenological treatment of the accel-eration process

- High energy cut-off may depend on theloss rate

- Some acceleration processes cannot betreated as a power-law injection, e.g.converter mechanism by Derishev

Particles lose energy by small fractions,which is not true for some processes, e.g.IC in the Klein-Nishina regimeThe Fokker-Planck equation describes aone-zone model


Continuous Loss Approximation for Klein-Nishinaregime

Klein-Nishina and Continues Loss approximation

Particles lose energy bysmall fractions, which is nottrue for some processes,e.g. IC in the Klein-Nishinaregime

Solid line: cσicf (γ) = q(γ) + c∞∫γ

dγ′f (γ′) dσdEγ (γ′, γ′ − γ)

Dash-dotted line: f (γ) = 1Ėic

∞∫γ

dγ′q(γ′)


Transport Equation

Transport Equation with Diffusion and Escape

∂f∂t

+∂(Ė f )∂E

+∇ (D∇f ) + fτ

= q(E , t)

also can be solved analytically, see e.g. Ginzburg’s "Astro-physics of Cosmic Rays"


Radiation Production

Emission of a Particle (two channels)

Single particle spectra:dNidEγ

= Ki (Eγ ,E0)

Total luminosity:

L = Ė1 + Ė2

Luminosity per channel:

Li =Ėi

Ė1 + Ė2L

Ratio of the humps:

L1L2

=Ė1Ė2


νFν peak gives the luminosity

νFν peaking distribution

νFν = E2γdNγdEγ

Lγ =∫

dEγ EγdNγdEγ

=

E0∫Emin

dEγ E1−αγ +

Emax∫E0

dEγ E1−βγ

Emin � E0 � Emax

Lγ = L0

(1

2− α+

1β − 2

)For α = 1.5 and β = 2.5

Lγ = 4L0


Synchrotron Radiation

Single Particle Spectrum:

dIsyndω

=

√3

2πe3Bmc2

F(ω

ωc

)

where ωc = 3eBγ2

2mc and F (x) = x∞∫x

K5/3(x ′)dx ′

Energy Losses: Ėsyn = −43UBcγ2

Spectrum transformation: α⇒ Γ = α+12

Acceleration of non-thermal parti-

cle proceeds in magnetized me-

dia therefore accelerated particles

unavoidable interact with magnetic

field generating non-therm emis-

sion – synchrotron radiation


Synchrotron Radiation

Approximation for the peak

x � 1

F (x) =(π

2

)1/2x1/2e−x

x � 1

F (x) =4π√

3Γ(1/3)x1/3

Useful approximation (but not an asymptotic):

F (x) = x

∞∫x

K5/3(x ′)dx ′ = 1.76 x0.29e−x


Synchrotron Test for MHD Regime

MHD regime implies that

E = B/η η > 1

Acceleration Time: E/eE

tacc = ηrgc

= 0.1ηETeVB−1G s

Synchrotron Cooling Time

tsyn ≈ 400E−1TeVB−2G s

Synchrotron Self Cutoff

� = 1.15~ωc ≈ 300η−1MeV




E = B/η η > 1


tacc = ηrgc

= 0.1ηETeVB−1G s




� = 1.15~ωc ≈ 300η−1MeV

RX J1713.7 Aharonian+

Shock acceleration: η = 2π(

cvsh

)2

Ebr ∼ 1 keV, i.e. η ∼ 105 (vsh ∼ 103 km/s)




E = B/η η > 1


tacc = ηrgc

= 0.1ηETeVB−1G s




� = 1.15~ωc ≈ 300η−1MeV

RX J1713.7 Aharonian+

Shock acceleration: η = 2π(

cvsh

)2

Ebr ∼ 1 keV, i.e. η ∼ 105 (vsh ∼ 103 km/s)

Aharonian&Atoyan

PWN: η → 1(?)

Ebr ∼ 1021−22Hz, i.e. η ∼ 2π (vsh → c ?)


Particle Acceleration in Astrophysics

Matsumoto+ 2015

Gyro freq. is Ωge = eBmc , so

ΩgeT = γT/tacc

dγd(ΩgeT )

= η−1

Different acceleration regimesDSA (v = 0.1c)→ η = 103

Crab Nebula→ η = 10Can it be even more efficient?η → 1



Matsumoto+ 2015

η = 103

Gyro freq. is Ωge = eBmc , so

ΩgeT = γT/tacc

dγd(ΩgeT )

= η−1





Matsumoto+ 2015

η = 10Gyro freq. is Ωge = eBmc , so

ΩgeT = γT/tacc

dγd(ΩgeT )

= η−1




Crab Flares: Even More Efficient Acceleration?

Buehler+2012

Ebr > 300 MeV




E = B/η η > 1


tacc = ηrgc

= 0.1ηETeVB−1G s




� = 1.15~ωc ≈ 300η−1MeV



Matsumoto+ 2015

η = 1Gyro freq. is Ωge = eBmc , so

ΩgeT = γT/tacc

dγd(ΩgeT )

= η−1




Synchrotron Model

ωMAXωC

Synchrotron Emission component

If synchrotron losses are thedominant

ωMAX = 300η−1MeV

If the break is cooling

ωc = 60(

t400s

)−2B−3G keV

If the break is cooling

∆Γ =12

Hardest cooled spectrum

Γ = 1.5


Proton Synchrotron

ElectronsCooling: tcool =400E−1TeVB

−2G s

Photon Energy: ~ω =20E2TeVBG keVHighest Energy Cooling:tcool = 6η1/2B

−3/2G s

Highest Energy Photons:~ω = 100η−1 MeV

ProtonsCooling: tcool =

400E−1TeVB−2G

(mpme

)4s

Photon Energy:~ω =

20E2TeVBG(

memp

)3keV

Highest Energy Cooling:

tcool = 6η1/2B−3/2G

(mpme

)2s

Highest Energy Photons:~ω = 100η−1

(mpme

)MeV


Inverse Compton Scattering


dIicdω

=r2oπm3e c4κT 2

3~3E×[

(ω/E)2

2(1− ω/E)+ 1]

Energy Losses: Ėsyn = −43Uphcγ2

Spectrum transformation: α⇒ Γ = α+12

Background photons shouldpresent in any source, in manycases IC scattering appears tobe comparable to the synchrotronradiation.


Synchrotron-IC Model

Very similar processesEnergy Losses

ĖIC = −43

Uphcγ2 Ėsyn = −43

UBcγ2

Mean Energy

~ω =(

43~ω0

)γ2 ~ω =

(eB~

2πmec

)γ2

LsynLic

=UBUph

~ωsyn~ωic

=

(eB~

2πmec

)(4

3~ω0)


Synchrotron-IC Model

Two identical humps?

Ratio of the peaks

LsynLic

=UBUph

Ratio of the energies

~ωsyn~ωic

=

(eB~

2πmec

)(4

3~ω0)


Kein-Nishina Cut-off

Photon Energy < Electron Energy

~ω =(

43~ω0

)γ2 < γmec2

γ <

(mec243~ω0

)γ < 5× 105ω−10, eV

~ω =(

eB~2πmec

)γ2 < γmec2

γ <

(2πm2e c3

eB~

)γ < 3× 1014B−1G


Inverse Compton Radiation

Interact with photon field

Maximum Energy of Gamma Rays Eγ <γ(

1− 14γ2

)1+ 14γ�


dσγdEγ

=πr2e �2Eγγ3(γ − Eγ)

[ln

Eγγ�(γ − Eγ)

+(

4�γ2 − 4γ�Eγ − Eγ)

(2�γ2 − 2γ�Eγ + �E2γ + Eγ

)4Eγγ�(γ − Eγ)

]


Treatment of IC losses and scattering

IC scattering on BB:

dNani/isodω dt

=T 3m3e c

3κ

π2~3

∞∫�ani/iso/T

dνani/isodω dNph dt

x2dxex − 1

.

Approximate treatment as in Khangulyan+2014

dNani/isodω dt

=2r2om

3e c

4κT 2

π~3E2×[

z2

2(1− z)F1 (x0) + F2 (x0)

]where x0 = z2(1−z) EeT (1−cos θ)tθ , z = Eγ/Ee, and F1,2 aresimple functions of 1 argument


Treatment of IC losses and scattering


Klein-Nishina Effect



dNedE

=1Ė

∞∫E

dE ′Q(E) Ė = Ėsyn + Ėic



X-ray: harden-ingγ-rays: noKlein-Nishinacutoff


Compton Scattering Spectrum

dNγdEγ

=

∫dEec(1− cos θ)nph

dNedEe

dσdEγ

Aharonian&Atoyan, 1981


dNγdEγ

=

∫dEec(1− cos θ)nph

dNedEe

dσdEγ

Anisotropic inverse Compton


Klein-Nishina + Anisotropic IC


Radiation in Turbulent Fieldscredit Medvedev2000

Reville&Kirk2010

Study of emission in turbulent mediahas a long history (e.g., Gatmant-sev1973, Tsytovich&Kaplan1973,Chiuderi&Veltri1974)

There are two basic approaches: per-turbation theory and full descriptionof trajectories (via kinetic equation ornumerically)


Radiation in Turbulent Field (Jitter Case)

dEnωdω dΩ =

e24π2c3

∣∣∣∣∫R

n×[(n−β(t))×a(t)](1−nβ(t))2 e

iω(t−nr(t)/c) dt∣∣∣∣2

eBλmc2 � 1

Pω(t) =e4〈B2〉

6π2m2c4

∞∫1

u(ξ)(

ωξ

2cγ2

)2Ψ

(ωξ

2cγ2

)dξξ

K̃ρσ(q) =12

(δρσ −

qρqσq2

)Ψ(|q|)〈B2〉Kρσ ≡ 〈Bρ(r1, t1)Bσ(r2, t2)〉

Kelner+2014: Emission spectrum produced in small-scale turbulence is deter-

mined by the spectrum of turbulence only (both in isotropic and anisotropic regimes)


Synchrotron v.s. Jitter Emission Spectra

low energy partx4/3 → x

maximumωc → mc

2ωceBλ

high energy partxe−x → x1−α,where Ψ ∝ q−2−α for q � 1


Applicability of Jitter Radiation

Jitter Regime is realized if mc2

eBλ � 1, i.e.,

λ� 1.7× 102(m/me)(B/1 G)−1 cm

For conditions typical for SNR, blazar jet, and GRB thatgives 10(m/me) km, 1(m/me) m, and 1(m/me) mm, respec-tively. That conditions might be hard to realize for electronsHowever, for protons the requirements are significantly re-laxed

However, for relativistic particles there is an intermediateradiation regime (given γ � 1):

mc2eB � λ�

mc2γeB


Radiation in Turbulent Field (Large-Scale Case)credit Medvedev2000

the emission spectrum is determined by thetrajectory curvature

if λ � mc2

eB , the momentary spectrum issynchrotron-type

the time-averaged spectrum can be obtainedby averaging the distribution of magneticfield: w(B)


Radiation in Turbulent Field (Large-Scale Case)

the emission spectrum is determined by thetrajectory curvature

if λ � mc2

eB , the momentary spectrum issynchrotron-type

the spectrum from an ensemble of particlescan be obtained by averaging the distributionof magnetic field

w(B)dB is probability to interacts with field in therange (B,B + dB)

w(B)dB = δ(B/B0 − 1)dB/B0w(B)dB = 3

√6√π

(B/B0)2e−3(B/B0)2/2dB/B0

w(B)dB = 32 (B/B0)2

π(1+(B/B0)2)4dB/B0


Radiation in Turbulent Field (Large-Scale Case)Teraki&Takahara2011

for a power-law distributions: w(B →∞) ∝ B−σ

the photon spectrum is also power-law:νFν ∝ ω2−σ


High-Energy radiative models typically treat plasma physicsa simplified way: phenomenological treatment of the accel-eration and continues-loss approximationNevertheless, obtaining the particle distribution is the mostimportant part of the non-thermal emission modelingParticle spectra cannot feature arbitrary breaks, each breakneeds to be justified by cooling, injection or dominant radi-ation channel changeOne-zone models very useful and can address many physi-cal features, but much more sophisticated models are widelyused for more than 20 yearsIn the high energy regime IC mechanism may proceed inthe Klein-Nishina or anisotropic regime which has a strongimpact on particle distribution, synchrotron, and IC spectraSynchrotron radiation may proceed differently in the case ofa turbulent magnetic field which has a strong impact on thesynchrotron spectrum, (particle distribution and IC spectraremain not affected)


Documents

@let@token High Energy Radiation Processes: I ...€¦ · 3 Source morphology 3 Time variability 3 Polarization of EMR The conventional astronomy features tools that allow studying