radiative processes in high energy · pdf filejuri poutanen (tuorla observatory, university of...

RADIATIVE PROCESSES in

HIGH ENERGY ASTROPHYSICS

Juri Poutanen (Tuorla Observatory, University of Turku, Finland)

Zelenogorsk, Russia 2014 1

1.  Introduction 2.  Kinetic equations 3.  Synchrotron radiation 4.  Compton scattering 5.  Pair production



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RADIATIVE PROCESSES: Literature

1.  General Aharonian F.A., 2004, Very high energy cosmic gamma-ray radiation Blumenthal G.B., & Gould R.J., 1970, Rev. Mod. Phys., 42, 237 Rybicki G.B., Lightman A.P., 1979, Radiative processes in astrophysics Vurm I., Poutanen J., 2009, ApJ, 698, 293 Vurm I., 2010, PhD thesis 2. Synchrotron radiation, absorption, thermalization Ginzburg V.L., Syrovatskii S.I., 1965, ARA&A, 3, 297 Ghisellini G., Guilbert P.W., Svensson R., 1988, ApJ, 334, L5 Ghisellini G. & Svensson R., 1991, MNRAS, 252, 313 3. Compton scattering Pozdnyakov L.A., Sobol’ I.M., Sunyaev R.A., 1983, Astr. Sp. Phys., 2, 189 Nagirner D.I., Poutanen J., 1994, Astr. Sp. Phys., 9, 1 Poutanen J., Svensson R., 1996, ApJ, 470, 249 4. Photon-photon pair production Gould R.J., Shreder G.P., 1967, Phys. Rev., 155, 1404 Svensson R., 1982, 258, 321 Zdziarski A.A., 1988, ApJ, 335, 786

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RADIATIVE PROCESSES Physical Units

Use Gauss system of units (CGS):

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RADIATIVE PROCESSES Astrophysical Units

5

8.64 × 104 s

INTRODUCTION

Most of the information about the Universe comes to us via electro-magnetic (EM) radiation.

Exceptions: 1. measurements in situ at the surfaces of a few planets

and their satellites (e.g. Earth and Moon, Mars, Venus, Saturn's rings and moons Enceladus and Titan)

2. measurements of the particles in the interplanetary space

3. cosmic rays 4. neutrinos 5. gravitational waves - hopefully in the near future 6

INTRODUCTION EM radiation

EM radiation: 1.  Optical - was used for thousands of years. 2.  Radio - radars from 1940s 3.  IR - observations already in 19th century, but IR telescopes in 1969 4.  UV - observations of the Sun in 1946 from rockets 5.  X-rays from the Sun in 1949, first X-ray source in 1962. 6.  Gamma-rays, cosmic background discovered in 1961

Riccardo Giacconi (1931- ) Nobel prize in 2002

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INTRODUCTION Multiwavelength Universe

Many sources emit EM radiation over many different wavelengths.

Crab nebula emits radiation over many decades in energy.

Blazars - active galaxies with relativistic jets - shine from radio up to TeV energies.

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INTRODUCTION Main continuum radiative processes

Understanding the microphysics of the radiative processes giving rise to the observed radiation allows to relate the observed properties to the physical parameters in a source using physical models. 9


1.  Bremsstrahlung - breaking radiation - free-free radiation : involves interactions between charged particles, e.g. electron and proton.

Among the astrophysical sources where

bremsstrahlung is important are HII regions, clusters of galaxies, accreting white dwarfs

Coma cluster in the optical (left) and X-rays (right) Intermediate polars (upper) as sources of the Galactic ridge emission (lower)

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2. Synchrotron radiation: emission of charged particles spiraling in the magnetic field.

Astrophysical application started after realizing that polarized radiation from the Crab nebula is produced by synchrotron process.

Theory was developed mostly in the 2nd half of the 20th century (Ginzburg, Syrovatskii), while some important papers have already appeared in the 1910s (Shott 1912).

Crab nebula Jets from active galaxies also emit synchrotron radiation .

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3. Compton scattering: involves interactions between photons and charged particles.

Thomson scattering is important in

atmospheres of hot stars. Among the astrophysical sources where

(inverse) Compton scattering is important are compact sources: such as accreting black holes, neutron stars, jets from active galaxies and gamma-ray bursts as well as in the Universe and cluster of galaxies (Sunyaev-Zeldovich effect).

Sunyaev-Zeldovich effect (deviation of CMB from a black body) in clusters of galaxies due to Compton scattering by the hot gas.

Galactic black hole Cyg X-1. Photons above few keV are produced by Compton scattering.

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4. Photon-photon pair production : involves interactions between high-energy photons and other photons.

Among the astrophysical sources where it is

important are compact sources: pulsars, jets from active galaxies and gamma-ray bursts.

Observed TeV spectrum of a blazar (lower red points) and the reconstructed intrinsic spectrum (upper blue points). The difference is caused by absorption of TeV photons on extragalactic IR light.

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1.  Introduction 2. Kinetic equations 3.  Synchrotron radiation 4.  Compton scattering 5.  Pair production 14



2. Kinetic equations Plan: a.  Definitions, description of photons and electrons b.  General form of relativistic kinetic equations c.  Radiative transfer equation (phenomenological derivation) d.  Electron kinetic equation (phenomenological derivation)

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Definitions (photons) •  Description of photons:

•  Photon distribution is described by the photon occupation number or intensity :

four-momentum x = x, x{ }= x 1,ω{ }, where ω is the unit vector in the photon propagation direction

Metric g=diag{1,-1,-1,-1} x2 = x2 − x2ω ⋅ω = 0

!nν or !nλIν or Iλ

Iν =2hν 3

c2!nν

Thermal radiation: Planck function Bν =2hν 3

c21

ehν /kT −1, !nν =

1ehν /kT −1

photon energy ε = hνdimensionless energy x = hν /mec

2

nph = 2λC

3 d 2∫ ω x2 dx ∫ !nph (x)

λC = h /mec−Compton wavelength

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Definitions (electrons) •  Description of electrons:

•  Electron distribution function: normalized as Electron number density (isotropic) distribution: Electron occupation number:

four-momentum p = γ, p{ }= γ, pΩ{ }= γ 1,βΩ{ }, where Ω is the unit vector in the electron propagation direction

γ and p = γ 2 −1 are electron Lorentz factor and its momentum in units mec;β = p /γ is electron velocity in units of speed of light c

p2 = γ 2 − p2Ω ⋅Ω =1

fe(p)

dp ∫ ne(p) = nene(p) = 4π p2 fe(p)

d 2Ω∫ p2 dp∫ fe(p) = ne

dγ ∫ ne(γ ) = nene(γ ) = 4π pγ fe(p)

ne = 2λC

3 d 2∫ Ω p2 dp ∫ !ne(p) 17

Relativistic kinetic equation •  General form of the RKE

Full equation contains collision integrals for many processes, which should be summed up. In the frame where distribution is isotropic we can write:

∇ = ∂ / c∂t,−∇{ }

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Relativistic kinetic equation

Form of equations useful for calculations

Processes

Numerical solution on a grid: Vurm & Poutanen 2009, ApJ Large Particle Monte Carlo method: Stern et al. 1995, MNRAS

In physical situation, particles and photons depend on each other and one needs to solve these equations simultaneously.

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Relativistic kinetic equation •  RKE for Compton scattering

The rhs can be reduced to the standard form of the radiative transfer equation

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From integral to differential eqs. •  Very often these complicated integro-

differential kinetic equations can be reduced to much simpler, Fokker-Planck differential equations.

•  In most of our examples during these lectures we are going to use simple forms.

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Eλ

= Iλdλ dA dΩ dt

Consider light within the wavelength interval dλ passing through a surface area dA in a narrow cone of solid angle dΩ in time dt. Intensity relates to energy via

The (specific) intensity Iλ is then a measure of brightness with units of erg/(s cm2 str Å). Iλ is proportional to energy per solid angle, and so is independent of distance.

�

Eλ

= Iλcosθ dλ dA dΩ dt

Now consider the energy passing through a surface area dA at an angle θ with respect to the normal of this surface area, the effective area is reduced by cos θ

dA

dA

dΩ

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dA’

Radiation terms: specific intensity

dE = Iλ1

dλ1 cosθ

1dA

1 dΩ

1 dt

=Iλ 2

dλ2 cosθ

2dA

2 dΩ

2 dt

dΩ1 =cosθ

2dA

2/ r

2

dΩ2 =cosθ

1dA

1/ r

2

dλ1 = dλ

2

Iλ1

= Iλ 2

Iλ = const

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Constancy of specific intensity in free space

One can also define intensity in frequency units such that Note that Iλ ≠ Iν ! Integrated (bolometric) intensity is The mean intensity Jλ is the directional average of the specific intensity (over 4π steradians) The flux, Fλ, is the projection of the specific intensity in the radial direction (integrated over all solid angles).

Jλ=

1

4πIλ dΩ∫

Iλdλ = I

νdν

�

Fλ = Iλ cosθ dΩ∫

I = Iλdλ

0

∞

∫ = Iνdν

0

∞

∫

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Mean intensity and flux

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•  Consider radiation shining through a layer of material containing particles with number density n [cm-3] and cross-section σλ [cm2].

•  In the column of length ds and cross-section 1 cm2 the number of particles is n×1×ds and the fraction of area they cover is nσλ ds

•  Thus the relative change of radiation intensity due to absorption (or scattering) is dIλ/Iλ = – nσλds = – αλds=-dτλ

•  Here αλ is the absorption coefficient [cm-1]. It can also be represented as αλ=ρκλ, where ρ is the density, and κλ is the so-called mass absorption coefficient (or opacity) [cm2 g-1].

•  The photon mean free path is inversely proportional to αλ (note that subscript λ just means that absorption is photon-wavelength dependent).

ds

Iλ I

λ+ dI

λ

Radiation terms: absorption coefficient

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•  Two physical processes contribute to the opacity: •  (i) true absorption where the photon is destroyed

and the energy thermalized; •  (ii) scattering where the photon is shifted in

direction and removed from the solid angle under consideration.

•  The radiation sees a combination of αλ and ds over some path length, L given by a dimensionless quantity, the `optical depth’

�

τλ = αλds0

L

∫

Radiation terms: optical depth

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We can write the change in specific intensity over a path length as

using the definition of optical depth. This may be directly integrated to give the usual extinction law:

An optical depth of τ=0 corresponds to no reduction in

intensity. An optical depth of τ= 1 corresponds to a reduction in

intensity by a factor of e=2.7. If the optical depth is large (τ >>1) negligible intensity

reaches the observer.

�

dIλ

= −Iλdτ

λ

λτ

λλλτ

−= eII )0()(

Importance of optical depth

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We can also treat emission processes in the same way as absorption via a (volume) emission coefficient jλ erg/s/cm3/str/Å

Physical processes contributing to jλ, are (i)  real emission – the creation of photons; (ii) scattering of photons into a given direction

from other directions.

dIλ = jλds

Radiation terms: emission coefficient

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The radiative transfer equation describes how the physical properties of the material are coupled to the spectrum of radiation that is passing through that material.

Energy can be removed from (true absorption or scattered), or added to (true emission or scattered) a ray of radiation:

The rate of change of (specific) intensity is: where α is the absorption and j is the emission coefficients. We can re-write this equation in terms of the optical depth τ and ratio of emission to absorption coefficients,

the source function

�

dτλ

= αλds

dIλ = −αλIλds+ jλds

λλλdIIdsI +

Sλ = jλ /αλ

Radiative transfer equation

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i.e. •  This is the (parallel-ray) equation of radiative transfer. •  If Sλ<Iλ, the intensity will decrease with increasing τλ, and

increase if Sλ>Iλ. When τλ è∞, Iλ è Sλ •  One can formally solve the transfer equation with an

integrating factor exp(τλ):

�

dIλ

dτλ

= −Iλ

+ Sλ

�

Iλ (τλ ) = Iλ 0e−τ λ + e

−(τ λ −τ λ′)Sλ (τλ

′)

0

τ λ

∫ dτλ′

This is the formal solution of the RTE which assumes that the source function is known. The first term on the RHS describes the amount of radiation left over from the intensity entering the box, after it has passed through an optical depth τλ, the second term gives the contribution to the intensity from the radiation emitted along the path.

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If the source function is constant along the ray: Consider simple limiting cases: A simple case of thermal emission:

(i) τ λ >>1 : Iλ (τ λ ) = Sλ = jλ /αλ

(ii) τ λ <<1 and I0 = 0 : Iλ (τ λ ) = Sλτ λ =jλαλ

αλR = jλR

�

Iλ(τ

λ) = I

λ 0e−τ

λ + Sλ(1− e

−τλ )

Sλ = Bλ −Planck function

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Consider optically thick sphere of radius R. Flux at the surface Luminosity: Consider now optically thin sphere of radius R. Luminosity: The observed flux is

Lλ ≈4π3R3 4π jλ( )

Lλ ≈ 4πR2 πSλ( )

Fλ =Lλ4πD2

Fλ,surf ≈ πSλ

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INJECTION REGION

size R

acceleration sites

Injection is homogeneous. The space inbetween acceleration sites is filled with accelerated electrons trying to escape the injection region on a time-scale

tesc=(R/c) x diffusion factor Before escaping they may cool on a

time-scale tcool

1. tesc <tcool (Leaky box)

γ γmax

γ-Γ

Q(γ) cm-3 s-1 Injection rate

γmax γ

γ-Γ

Steady-state particle distribution n(γ)

When cooling is inefficient compared to escape, then the steady-state

density distribution is n(γ) = tesc Q(γ), i.e. has the same shape.

Simple form of kinetic equation for electrons

2. tesc >tcool particles cool before escaping

Steady-state particle distribution

)()]([)(

γγγγ

γQn

t

n=

∂

∂+

∂

∂

energy loss rate= velocity in γ-space

γ γ+dγ

γ

)(γγ nflux

n(γ ) =

Q(γ ')dγ 'γ

γ max

∫

− γ (γ )

For synchrotron (and Compton) cooling γ ∝γ 2 thus n(γ )∝1/γ 2 (for Γ <1 )

n(γ ) ∝1/γ Γ+1 (for Γ >1 )

Q(γ ) = γ −Γif Γ <1 ⇒n(γ )∝

1

γ (γ )

if Γ >1 ⇒n(γ )∝1

γ Γ−1γ (γ )

�

Q(γ) = δ(γ - γmax

) ⇒ n(γ)∝1

˙ γ (γ)

Examples: i)  monoenergetic injection

ii)  power-law injection

Kinetic equation for electrons

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35

γ

γmax

γ2-Γ

γ

γ1-Γ

Shock acceleration è Γ>2 è most power at low energies α>1

γ γmax

particle injection γ2Q(γ)

è cool

density distribution γ2ne(γ)

1) MONOENERGETIC INJECTION Q(γ)~δ(γ-γmax)

1

0

2

γγ

γ∝

è scatter

xs x

photon injection

p=2 α=1/2 Define spectral index

x1/2

2) POWER-LAW INJECTION WITH COOLING

γ2Q(γ)

Γ>2

most power injected here

è cool

γ2ne (γ)

è scatter

γmax

x

x1-Γ/2

x2 n(x)= x Fx

p = Γ+ 1

α=(p-1)/2=Γ/2

γ

most power emitted here

most power emitted here

Kinetic equation for electrons

x2 n(x)= x Fx

ne(γ )∝γ− p

Fx ∝ x−α,α = (p−1) / 2

radiative processes in high energy · pdf filejuri poutanen (tuorla observatory, university of...

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