# Presentation: 572390

## Emission mechanisms. I 

***                      ******Giorgio****** ******Matt***
***(Dipartimento di Fisica, Università Roma Tre, Italy)***
***Reference: Rybicki & Lightman, “Radiative processes in astrophysics”, Wiley***

## Outline of the lecture 

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

## Any charged particle in accelerated motion emits e.m. radiation.

**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,**** ****Θ**** 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 !**

## The previous formulae are valid in the non relativistic case.

**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 ****γ**** is the Lorenzt factor**** ( ****γ****=(1-v****2****/c****2****)****-1/2**** ****) of the emitting particle,  and the acceleration vector is decomposed in the components parallel and perpendicular to the velocity. **
**Of course, for ****γ****~1 ****the non relativistic formula is recovered.**

## The equation of radiative transfer is:

***The equation of radiative transfer is:***

**If matter is in local thermodynamic equilibrium, *****S******ν***** is a universal function of temperature: *****S******ν****** = B******ν******(T)***** ****(Kirchoff’s law). **
***B******ν******(T)***** is the *****Planck function:***

## Polarization

**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

**Summing up the contributions of all photons, ****I ****increases while this is not necessarily so for the ****other Stokes parameters****. Therefore:**

**The ****net polarization degree and angle**** are**
**given by:**

## Black Body emission

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

## Black body emission occurs when τ→∞, so in practice there 

**Black body emission occurs when ****τ→∞****, 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

** ****It is the interaction between a photon and an electron (at rest), **
**with *****h******ν******«mc******2*****. **** It is an  elastic **
**process. The cross section is:**

**The differential **
**cross section is:**

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

## Pair production and annihilation

** ****A ****e****+****-e****-**** pair**** may annihilate producing two **
**γ****-rays**** (to conserve momentum). If the electrons are not relativistic, **
**the two photons have ****E=511 keV. **
**Conversely, two ****γ****-rays**** (or a ****γ****-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****γ****1**** E****γ****2**** > 2m****2****c****4**

**γ****-rays**** interacts with ****IR**** photons. The maximum distance at which **
**an extragalactic ****γ****-ray source is observed provides an estimate of the**
**(poorly known) ****cosmic IR background**** **

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

** ****It is produced by the**** deflection ****of 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

**The interaction occurs on a timescale**
***Δ******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. **

***b is called the impact parameter***

## Integrating over the impact parameter, we obtain: 

**Integrating over the impact parameter, we obtain: **

**b****max**** ****and ****b****min**** ****must be evaluated taking into account quantum mechanics.  They are calculated numerically.**
**g****ff**** ****is the so called**** *****Gaunt factor*****. ****It is of order unity for large intervals of the parameters.**

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

## Thermal 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 ****optically thin case****.  If ****τ**** >> 1****, we of course have the Black Body emission**

***Emission measure***

## Free-free absorption

** ****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

**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

**For any emission mechanism, the cooling time is defined as:**

**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 10****10**** 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

** ****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 of gyration and the frequency of the orbit are**
**(******** is the angle between v and B):**

##  Let us assume the charged particle is an electron. 

** ****Let us assume the charged particle is an electron. **
**Using the relativistic Larmor formulae, and averaging over , the power emitted by an electron is (****β≡****v/c****)****:**

** ****The synchrotron spectrum from a single electron is**
**peaked at:**

## 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. 

**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(****ν****)=C****ν****-****, with ****=(p-1)/2**

** ****F(****ν****)=A(p)KB****(p+1)/2****ν****-**

## Synchrotron self-absorption

**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

**The radiation is polarized perpendicularly to the projection of B  on the plane of the sky**

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

## Cooling time

**The cooling time is:**

**For the ****interstellar matter**** (B ~ a few ****μ****G, ****γ****~10****4****): ****τ****~10****8**** yr**

**For a ****radio galaxy****  (B ~ 10****3**** G, ****γ****~10****4)****: ****τ****~0.1 s**
****** ****continuous acceleration**

**(Electron’s rest mass **
**is irreducible)**

## Equipartition

**The energy in the magnetic field is proportional to**** B****2****. 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**
**W****B****~W****p**** (“equipartition”).**
**B****eq****   L****2/7**

## (Inverse) Compton scattering

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

***E******0***

***E***

**The cross section is the ****Klein-Nishina:**

***σ******KN****** → ******σ******T****** ***
***for x → 0***

## In the laboratory rest frame E ≈ E0γ2  (the calculation is done in the electron rest frame, where the photon energy is E ≈E0γ, the other γ arising in the transformation back to the lab frame). 

**In the laboratory rest frame *****E ≈ E******0******γ******2 ****** *****(the calculation is done in the electron rest frame, where the photon energy is *****E ≈E******0******γ*****, the other ****γ**** arising in the transformation back to the lab frame). **
**For *****γ→******1*****, the classic Compton scattering is recovered, **
**while for ultrarelativistic electrons *****E ≈******γ******mc******2***
**The power per single scattering is, assuming *****E « mc******2***** **** in the **
**electron rest frame (****U****rad**** energy density of the radiation field): **

***(Note that this formula is equal to the Synchrotron one,***
***but with U******rad******  instead of U******B******)***

**Assuming a thermal distribution for the electrons, **
**the mean percentage energy gain of the photons is:**

***4kT > E******0****** → Energy transferred from electrons to photons***** **
***4kT < E******0****** → Energy transferred from photons to electrons ***

## Cooling 

**The formula for the energy losses by a single electron is identical to the synchrotron one, once***** U******rad******  *****replaces***** U******B*****.**
**Therefore  *****P******IC******/P******syn******=U******rad******/U******B ***
**IC losses dominate when the energy density of the **
**radiation field is larger than that of the magnetic field**
***(“Compton catastrophe”)***

**which may be very short for relativistic electrons in a **
**strong radiation field**

**The cooling time is:**

## Comptonization

**Let us define the Comptonization parameter as: *****y = ******Δ******E/E******0  ******x N******scatt***** **
**Assuming non relativistic electrons, the mean energy gain of the photons is:      *****E = E******0 ******e******y****** ***
**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):***

## 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): **

***Mkn 501***

**SSC emission may be relevant in *****Blazars,***** where two peaks are actually observed. The first peak is due to Synchrotron, the second to IC (either ****SSC ****or ****external IC****)**

## Polarization 

**In *****Blazars*****, the radiation field may be either the synchrotron emission ****(SSC) ****or 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******ν******/mc******2 ******in the reference frame of the electron*****). The degree of polarization depends on the geometry of the system. **

## Sunayev-Zeldovich 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**
***Δ******I******ν******/I******ν******  ******≈******  -2y***
**(in the R-J regime)**

## Sunayev-Zeldovich effect

**The S-Z effect is potentially a very efficient tool to search for Clusters and, when combined with X-ray observations, can be used to estimate**
**the ****baryonic mass fraction**** and even the ****Hubble constant****.**

## Cherenkov 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) ******γ******-rays*****. Projects like **
***Veritas, HESS***** and *****Magic***** are indeed**
**providing outstanding results. **