Emission mechanisms. I Emission mechanisms. I Giorgio Giorgio Matt - - PowerPoint PPT Presentation

Emission mechanisms. I Emission mechanisms. I

Giorgio Giorgio Matt Matt

(Dipartimento di Fisica, Universit (Dipartimento di Fisica, Università à Roma Tre, Roma Tre, Italy Italy) )

Reference: Reference: Rybicki Rybicki & & Lightman Lightman, , “ “Radiative Radiative processes in astrophysics processes in astrophysics” ”, Wiley , Wiley

Outline of the lecture Outline of the lecture

  Basics (emission, absorption, Basics (emission, absorption, radiative radiative transfer) transfer)   Bremsstrahlung Bremsstrahlung   Synchrotron emission Synchrotron emission   Compton Scattering (Inverse Compton) Compton Scattering (Inverse Compton)

Any charged particle in accelerated motion emits e.m. radiation. The intensity of the radiation is governed by Larmor’s formulae: where q is the electric charge, v the particle velocity, _ the angle between the acceleration vector and the direction of emission ( Averaged over emission angles)  The power is in general inversely proportional to the square of the mass of the emitting particle !! (dv/dt = F/m) Electrons emit much more than protons !

2 2 2 3 2 2 2 3

3 2 sin 4 1       ∝       = = Θ       = Ω m F dt dv q c dt dW P dt dv q c dtd dW π

The previous formulae are valid in the non relativistic case. If the velocity of the emitting particle is relativistic, then the formula for the angle-averaged emission is: where _ is the Lorenzt factor ( _=(1-v2/c2)-1/2 ) of the emitting particle, and the acceleration vector is decomposed in the components parallel and perpendicular to the velocity. Of course, for _~1 the non relativistic formula is recovered.

              +       =

parallel perp

dt dv dt dv q c P

2 2 . 2 4 2 3

3 2 γ γ

( )

t coefficien absorption ds dI I ds j dI emissivity d dVdtd dE j Flux d I F Intensity d dAdtd dE I

v ν ν ν ν ν ν ν ν

α ν ϑ ν 1 cos − = = Ω ≡ Ω = Ω ≡

∫

ν ν ν ν ν ν ν ν ν ν ν ν ν ν

α τ τ α τ α j S S I d dI depth

• ptical

ds d with

• r

j I ds dI ≡ + − = ≡ = + − = ) ( ,

The equation of radiative transfer is: If matter is in local thermodynamic equilibrium, S_ is a universal function of temperature: S_ = B_(T) (Kirchoff’s law). B_(T) is the Planck function:

        − = 1 2 ) (

2 3 kT h

e c h T B

ν ν

ν

Polarization Polarization

) ( 2 2 sin ) ( 2 cos ) (

2 2 2 2 2 2 2 2 2

V U Q I AB V B A U B A Q B A I + + = ± = − = − = + = θ θ

The polarization vector (which is a pseudovector, i.e. modulus π) rotates forming an ellipse. Polarization is described by the Stokes parameters:

If V=0, radiation is linearly polarized If Q=U=0, radiation is circularly polarized

Polarization Polarization

2 2 2 T T T T

V U Q I + + ≥

Summing up the contributions

• f all photons, I increases while

this is not necessarily so for the

• ther Stokes parameters.

Therefore:

T T T T T T

Q U I V U Q arctan 2 1

2 2 2

= + + = Π χ

The net polarization degree and angle are given by:

Black Body emission Black Body emission

kT h Wien e c h I kT h Jeans Rayleigh c kT I kT hv T F B I

kT h v v

82 . 2 2 2 ) (

max 2 3 2 2 4

= = >> − = << = =

−

ν ν ν ν σ

ν ν ν

If perfect thermal equilibrium between radiation and matter is reached throughout the material, I_ is independent of _. In this case the matter emits as a Black Body:

Black body emission occurs when __∞, so in practice there are always deviations from a pure Black Body spectrum due to finite opacities and surface layers effects. The only perfect Black Body is the Cosmic Microwave Background radiation.

Thomson scattering Thomson scattering

θ σ

2 4 2 4

sin c m e d d

T =

Ω

2 25 4 2 2

10 65 . 6 3 8 cm c m e

T −

× = = π σ

It is the interaction between a photon and an electron (at rest), with h_«mc2. It is an elastic

• process. The cross section is:

The differential cross section is:

θ θ

2 2

cos 1 cos 1 + − = P

The scattered radiation is polarized. E.g., the polarization degree of a parallel beam of unpolarized radiation is:

Pair production and annihilation Pair production and annihilation

137 1 ≈ ≈ → ≈ →

− + − +

α ασ σ γ σ σ γγ

γ γγ T p T

p e e p e e

A e+-e- pair may annihilate producing two _-rays (to conserve momentum). If the electrons are not relativistic, the two photons have E=511 keV. Conversely, two _-rays (or a _-ray with the help of a nucleus) may produce a e+-e- pair The pair production cross sections are: Pair production is of course a threshold process. E_1 E_2 > 2m2c4 _-rays interacts with IR photons. The maximum distance at which an extragalactic _-ray source is observed provides an estimate of the (poorly known) cosmic IR background

Bremsstrahlung Bremsstrahlung ( (“ “braking radiation braking radiation” ”) ) a.k.a. free-free emission a.k.a. free-free emission

It is produced by the deflection

• f a charged particle (usually an

electron in astrophysical situations) in the Coulombian field of another charged particle (usually an atomic nucleus). Also referred to as free-free emission because the electron is free both before and after the deflection.

The interaction occurs on a timescale _t ≈ 2b/v A Fourier analysis leads to the emitted energy per unit frequency in a single collision, which is inversely proportional to the square of: the mass, velocity of the deflected particle (electron) and impact parameter.

( ) ( )

ω ν ω ν / / 3 16

2 2 2 3 6 2

v b d dW v b b v m c e Z d dW >> ≈ << ≈

b is called the impact parameter

Integrating over the impact parameter, we obtain: bmax and bmin must be evaluated taking into account quantum

• mechanics. They are

calculated numerically. gff is the so called Gaunt factor. It is of

• rder unity for large

intervals of the parameters. To get the final emissivity, we have to integrate over the velocity distribution of the electrons.

        = =

min max 2 3 6 2 2

ln 3 3 3 32 b b g where g n n v m c e Z dtdV d dW

ff ff i e

π π ν

Thermal Thermal Bremsstrahlung Bremsstrahlung

If electrons are in thermal equilibrium, their velocity distribution is Maxwellian. The bremsstrahlung emission thus becomes:

 Integrated over frequencies

The above formulae are valid in the

• ptically thin case. If _ >> 1, we of

course have the Black Body emission

∫

= = =

− −

dV n n T f dt dW g n n T m k m hc e Z dtdV dW g e n n T km m c e Z dtdV d dW

i e ff i e ff kT h i e

) ( 3 2 3 64 3 2 3 64

2 1 2 3 6 2 2 2 1 2 3 6 2 2

π π π π ν

ν

Emission measure

Free-free absorption Free-free absorption

ff kT h i e ff ff ff

g e n n T km m hc e Z T B dtdV d dW j ) 1 ( 3 2 3 8 ) ( 4

3 2 1 2 3 6 2 ν ν ν ν ν

ν π π α α ν π

− − −

− = = =

A photon can be absorbed by a free electron in the Coulombian field of an atom: it is the free-free absorption, which is the aborption mechanism corresponding to bremsstrahlung. Thus, for thermal electrons: At low frequencies matter in thermal equilibrium is optically thick to free-free aborption, becoming thin at high frequencies.

Polarization Polarization

Bremsstrahlung photons are polarized with the electric vector perpendicular to the plane of interaction. In most astrophysical situations, and certainly in case of thermal bremsstrahlung, the planes of interaction are randomly distributed, resulting in null net polarization. For an anisotropic distribution of electrons, however, bremsstrahlung emission can be polarized.

Cooling time Cooling time

dt dE E tcool / = For any emission mechanism, the cooling time is defined as:

yr T g Z n x t

ff e cool 2 1 2 3

10 6 ≈

where E is the energy of the emitting particle and dE/dt the energy lost by

• radiation. For thermal bremsstrahlung:

The cooling time is of order one thousand years for a HII regions, and of a few times 1010 years (i.e. more than the age of the Universe) for a Cluster of galaxies

 Radio image of the Orion Nebula X-ray emission of the Coma Cluster 

Synchrotron emission Synchrotron emission

mc qB r v qB mcv r

B B B

γ α ω α γ = = = sin sin

It is produced by the acceleration of a moving charged particle in a magnetic field due to the Lorentz force: The force is always perpendicular to the particle velocity, so it does not do work. Therefore, the particle moves in a helical path with constant |v| (if energy losses by radiation are neglected). The radius

• f gyration and the frequency of the orbit

are (_ is the angle between v and B):

) ( B v c q F r r r × =

π γ β σ 8 3 4

2 2 2

B U U c P

B B T

≡ =

Let us assume the charged particle is an electron. Using the relativistic Larmor formulae, and averaging over _, the power emitted by an electron is (__v/c): The synchrotron spectrum from a single electron is peaked at:

mc eB π α γ ν 4 sin 3

2 0 =

To get the total spectrum from a population of electrons, we must know their energy distribution. A particularly relavant case is that of a power law distribution, N(E)=KE-p. The total spectrum is also a power law, F(_)=C_-_, with _=(p-1)/2 F(_)=A(p)KB(p+1)/2_-_

Synchrotron self-absorption Synchrotron self-absorption

2 4 2 2

) (

+ − +

=

p p syn

K B p G ν αν

If the energy distribution of the electrons is non-thermal, e.g. a power law, N(E)=KE-p, the absorption coefficient cannot be derived from the Kirchhoff’s law. The direct calculation using Einstein’s coefficient yields: In the optically thick region, the spectrum is independent of p The transition frequency is related to the magnetic field and can be used to determine it.

Polarization Polarization

The radiation is polarized perpendicularly to the projection of B

• n the plane of the

sky For a power law distribution of emitting particles, the degree of polarization is _=(p+1)/(p+7/3). This is actually un upper limit, because the magnetic field is never perfectly ordered.

Cooling time Cooling time

The cooling time is:

s B cU mc t

B T cool

γ γ β γ σ γ

2 8 2 2 2

10 75 . 7 ) 1 ( 3 4 ) 1 ( × ≈ >> ≈ − ≈ For the interstellar matter (B ~ a few _G, _~104): _~108 yr For a radio galaxy (B ~ 103 G, _~104): _~0.1 s  continuous acceleration

(Electron’s rest mass is irreducible)

Equipartition Equipartition

The energy in the magnetic field is proportional to B2. Given a synchrotron luminosity, the energy in particles is proportional to B-3/2. If it is assumed that the system is in the minimum of total energy, the magnetic field can be estimated. The minimum occurs when WB~Wp (“equipartition”). Beq _ L2/7

(Inverse) Compton scattering (Inverse) Compton scattering

( ) ( ) ( ) ( )

2 2 3

2 1 3 1 2 2 1 ln 2 1 ln 2 1 1 2 1 4 3 mc E x x x x x x x x x x x

T KN

≡       + + − + +       + − + + + = σ σ

In the electron rest frame, the photon changes its energy as:

E E

( )

ϑ cos 1 1

2

− + = mc E E E

The cross section is the Klein-Nishina:

_KN _ _T for x _ 0

In the laboratory rest frame E ≈ E0_2 (the calculation is done in the electron rest frame, where the photon energy is E ≈E0_, the other _ arising in the transformation back to the lab frame). For __1, the classic Compton scattering is recovered, while for ultrarelativistic electrons E ≈_mc2 The power per single scattering is, assuming E « mc2 in the electron rest frame (Urad energy density of the radiation field): (Note that this formula is equal to the Synchrotron one, but with Urad instead of UB) Assuming a thermal distribution for the electrons, the mean percentage energy gain of the photons is:

rad TcU

P σ β γ

2 2

3 4 =

2

4 mc E kT E E − = Δ

4kT > E0 _ Energy transferred from electrons to photons 4kT < E0 _ Energy transferred from photons to electrons

Cooling Cooling

The formula for the energy losses by a single electron is identical to the synchrotron one, once Urad replaces UB. Therefore PIC/Psyn=Urad/UB IC losses dominate when the energy density of the radiation field is larger than that of the magnetic field (“Compton catastrophe”)

2 2 2

3 4 ) 1 ( β γ σ γ

rad T cool

cU mc t − ≈

which may be very short for relativistic electrons in a strong radiation field The cooling time is:

Comptonization Comptonization

Let us define the Comptonization parameter as: y = _E/E0 x Nscatt Assuming non relativistic electrons, the mean energy gain of the photons is: E = E0 ey To derive the spectral shape, one has to solve the diffusione equation, also known as the Kompaneets equation:

In general , it should be solved numerically. In case of unsaturated Comptonization (i.e. not very opt. thick matter):

1 5 . 1 1 ; 1 ( 4 4 9 2 3

3

≈ ≈ << − >> + + ± − = ∝

− +

y for m y for y for y m e I

kT h m ν ν

ν

Synchrotron Self-Compton (SSC) Synchrotron Self-Compton (SSC)

Electrons in a magnetic field can work twice: first producing Synchrotron radiation, and then Comptonizing it (SSC). The ratio between SSC and Synchrotron emission is given by (spectral shape is the same):

min max

: , ln ν ν τ = Λ Λ ≈ where j j

c Syn SSC

Mkn 501

SSC emission may be relevant in Blazars, where two peaks are actually

• bserved. The first peak is

due to Synchrotron, the second to IC (either SSC or external IC)

Polarization Polarization

In Blazars, the radiation field may be either the synchrotron emission (SSC)

• r the thermal emission from

the accretion disc (external IC). The polarization properties are different in the two cases: e.g. while in the SSC the pol. angle of IC and S are the same, in the external IC the two are no longer directly related.

SSC

Compton scattering radiation is polarized (but less than Thomson

• scattering. Polarization degree decreases with h_/mc2 in the

reference frame of the electron). The degree of polarization depends on the geometry of the system.

Sunayev Sunayev-

• Zeldovich

Zeldovich effect effect

CMB photons are Comptonized by the IGM in Clusters of

• Galaxies. As a result, the CMB

spectrum in the direction of a CoG is shifted _I_/I_ ≈ -2y (in the R-J regime)

4 3 2 2 3 2

10 10 10 5 4 10 10

− − − − −

− ≈ × ≈ ≈ Δ − ≈ ≈ y mc kT R ne

T

ν ν σ τ

Sunayev Sunayev-

• Zeldovich

Zeldovich effect effect

The S-Z effect is potentially a very efficient tool to search for Clusters and, when combined with X-ray

• bservations, can be used to

estimate the baryonic mass fraction and even the Hubble constant.

4 3 2 2 3 2

10 10 10 5 4 10 10

− − − − −

− ≈ × ≈ ≈ Δ − ≈ ≈ y mc kT R ne

T

ν ν σ τ

Cherenkov Cherenkov radiation radiation

It occurs when a charged particle passes through a medium at a speed greater than the speed of light in that medium. It is used to detect high energy (~TeV) _-

• rays. Projects like

Veritas, HESS and Magic are indeed providing outstanding results.