# Maurizio Paolillo » 4.Emission processes – Part I

### Contents

The main processes responsible for the production of high energy radiation are:

• Bremsstrahlung.
• Radiative recombination of heavy ions.
• Inverse Compton scattering.
• e-/e+ annihilation.
• Synchrotron emission.

Free-Free radiation is produced by the interaction of charged particles. The emission is due to the energy radiated by the acceleration experienced by one of the particles moving in the electric field produced by the other one. For such reason it is also called Bremsstrahlung Radiation (which means ‘braking’ in German).

In Astrophysics the usual treatment considers the problem of unbound interacting charged particles of very different mass, since this is the most common case where an electron interacts with a heavy ion.

Schematic view of Bremsstrahlung emission.

### Free-Free or Bremsstrahlung Radiation (cont’ed)

In the frame of the heavier particle the light particle is moving at velocity v with an impact parameter b The electric field of the heavier particle accelerates the lighter particle and, as we know (see lecture 3) accelerating charges radiate (Larmor radiation). We suppose that the particles are unbound throughout the problem, hence the name free-free. The lighter particle, usually an electron, is gradually decelerated because it is losing energy to radiation.

Since the initial and final states of the accelerated particles are unbound states, the result is a continuum emission.

Scheme of the trajectory of a free electron of speed v, passing by a heavy charged ion +Ze with impact parameter b.

### Free-Free or Bremsstrahlung Radiation (cont’ed)

In an ionised plasma free-free radiation comes from electrons encountering ions. Encounters between two electrons cannot produce radiation by electric or magnetic dipole processes; they will emit quadrupole radiation for which the power is lower by ~(v/c)2.

An ion of course is also accelerated by the field of a passing electron but its mass mi is much greater than me and so it is the acceleration: thus its radiated power is negligible in comparison to the electrons.

We assume that the electrons trajectory is a straight line which will be true of in a typical astrophysical gas. We neglect the effect of radiation reaction on the orbit, generally a good approximation.

### Emission from single encounter

To derive the Bremsstrahlung spectal energy distribution, we start considering the simpler case where the electron is moving non-relativistically. The acceleration is: $\vec{a}=\frac{Z e^2}{m_e d^2}\hat{d}=\frac{Z e^2}{m_e (b^2+v^2t^2)}\hat{d}$

where t=0 is taken to be the instant of closest approach and d is the distance from the electron to the ion. The dipole electric field at distance r from the electron is $|E|=\frac{ea}{c^2r}\sin\theta=\frac{Z e^3\sin\theta}{m_e c^2 r(b^2+v^2t^2)}$

To derive the spectrum of the emitted radiation, we take the Fourier transform of this electric field:

$E^*_{\nu}=\int_{-\infty}^{+\infty}E(t) e^{2\pi i \nu t} dt=\frac{Z e^3\sin\theta}{m_e c^2 r}\int_{-\infty}^{+\infty} \frac{e^{2\pi i \nu t}}{(b^2+v^2t^2)}dt=\frac{Z e^3\sin\theta}{c^2 r m_e}\frac{\pi}{bv} e^{-2\pi\nu b/v}$

.

The spectrum radiated by a single electron is thus:

$\frac{dW}{d\nu}=\frac{c}{2\pi}\int_{4\pi} |E^*_{\nu}|^2 r^2 d\Omega=\frac{4\pi^2}{3} \frac{Z^2 e^6}{ m_e^2 c^3 v^2 b^2}e^{-4\pi\nu b/v}$

### Emission from single encounter (cont’ed)

This spectrum is flat for v<< ν/b and has a high-energy exponential cutoff for v>> ν/b.

This result is due essentially to the fact that the electron cannot exchange more than its total energy, and thus the radiated spectum must fall to zero rapidly around that energy.

In practice electrons will interact with ions over a broad range of parameters of impact b. Thus the total emission is obtained integrating over all possible impact parameters.

Spectrum produced in the Bremsstrahlung process by a single ion-electron encounter. The spectrum is flat up to a cutoff frequency ωcut=ν/2πb and falls off exponentially at higher frequencies.

### Emission from single encounter (cont’ed)

There are a few important facts about bremsstrahlung to keep in mind:

1. The process is not an oscillatory one so the spectrum covers a broad band of frequencies.
2. The total power will be dominated by the closest approach where the acceleration is largest.
3. The emissivity vs frequency is nearly constant up to an exponential cutoff at high frequencies.

Note that absorption in a plasma can alter the spectrum at low frequencies where the material becomes optically thick (see Lecture 2).

### Emission from multiple encounters

Lets assume that electrons all move at speed v, then we can calculate the number of electrons passing through an annulus of thickness db around a single ion per unit time: ne=2πvb db. To get the power radiated per unit volume we need to multiply by the density ni of ions:

$\frac{dW}{d\nu~dt~dV}=n_e n_i v\int^{\infty}_{b_{min}}\frac{dW}{d\nu}2\pi b~db \simeq \frac{8\pi^3}{3}\frac{Z^2e^6}{m_e^2 c^3 v}\int^{b_{max}}_{b_{min}}\frac{db}{b}$

where dW/dν is the single encounter spectrum discussed above.

Since this spectrum drops exponentially for ν>v/b we may approximate the exponential by a step function which goes from 0 to 1 at ν~v/b; this is equivalent to choosing a maximum impact parameter for the integration bmax~v/ν and ignoring the exponential factor. An alternative path to approximate the value of the maximum impact parameter can be found in Longair (vol.1).

We put all of our ignorance (both the approximations made thus far and additional quantum mechanical factors into the ‘Gaunt factor’ gff(v,ν)~1 in most regimes of interest. See Rybicki-Lightman (p.159-161) a more accurate discussion of this factor.

### Emission from multiple encounters (cont’ed)

The minimum impact parameter bmin will be set by one of the two following considerations:

• The uncertainty principle prohibits the electron from getting any closer than bmin~h/p~h/me v.
• The small angle approximation will begin to break down if b<bmin where bmin is set by Z e2/bmin ~me v2.

The classical criterion will be important when v≤Ze2/ħ (v/c≤αZ where α=e2/ħc~1/137); this will be true for Bremsstrahlung from a typical HII region (T~104 K).
At the higher velocities found in the intergalactic gas in galaxy clusters (T~108 K), the uncertainty principle determines bmin.

### Thermal Bremsstrahlung

To make further progress we need to know the velocity distribution of the particles in the gas. If dP(v) is the probability of a velocity in the range (v,v+dv), to find the emission coeffcient from a thermal plasma we need to average over the velocity distribution:

$\epsilon_\nu=\int_{v_{min}}^\infty \frac{dW}{d\nu~dt~dV}dP(v)$

The minimum allowable velocity vmin arises because the electron must have a kinetic energy of at least (1/2) me v2=hν in order to emit a photon of frequency ν.

For a Maxwell-Boltzmann distribution $dP\propto \exp(-E/kT)4\pi v^2 dv$where we have assumed that the velocities are isotropic. Thus in a plasma at temperature T:

$P(v)=4\pi n_e\left ( \frac{m_e}{2\pi kT} \right )^{3/2} v^2 e^{-m_e v^2/2kT}$

The result of the integration gives the monocromatic emission due to a thermal plasma:

$\epsilon_\nu=\frac{32\pi}{3}\left ( \frac{2\pi}{3} \right )^{1/2} \frac{Z^2 e^6}{m_e^2 c^3} \left (\frac{m_e}{kT} \right )^{1/2} n_e n_i e^{-h\nu/kT} \bar{g}_{ff} \approx 6.8\times 10^{-38} Z^2 n_e n_i T^{-1/2} e^{-h\nu/kT} \bar{g}_{ff}~\mbox{erg s}^{-1} \mbox{cm}^{-3} \mbox{Hz}^{-1}$
where $\bar{g}_{ff}$ is the Gaunt factor averaged over velocities.

### Thermal Bremsstrahlung (cont’ed)

The exponential cutoff is due to the photon energy limit and the exponentially small number of very high energy electrons in a Maxwellian distribution. The T-1/2 arises because the power emitted by a single electron scales as v-1 i.e. T-1/2. Furthermore, since the emission is produced by an ion-electron encounter, the result is proportional to ne ni .

The total power loss to bremsstrahlung can be found by integrating over all frequencies:

$\frac{dW}{dVdt}\int_0^{\infty} \epsilon_\nu d\nu=\frac{32\pi}{3} \left ( \frac{2\pi}{3} \right )^{1/2}\frac{e^6}{hm_e c^3} \left (\frac{kT}{m_e} \right )^{1/2} Z^2 n_e n_i \bar{g} \approx 1.4\times 10^{-27} Z^2 n_e n_i T^{1/2} \bar{g}$

Where g is obtained averaging over all frequencies the velocity-averaged Gaunt factor. This factor will be in the range 1.1-1.5 and a good approximation is 1.2 . Note that the total power lost to the electron is now proportional to $T^{1/2}\propto v\propto E^{1/2}$, where E is the energy of the electron.

This result holds for non-relativistic Bremsstrahlung, since in the relativistic regime the power becomes proportional to E (see e.g. Longair, vol.1).

### Thermal Bremsstrahlung spectrum

Example of the Bremsstrahlung spectrum produced by the thermal plasma of a galaxy cluster at different temperatures. As the temperature drops the emission lines due to radiative recombination (see next lecture) becomes apparent.

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