Electromagnetic Radiation in Astrophysics

Abstract

In the present chapter we first discuss all electromagnetic processes and interactions that are relevant for electrons, positrons and photons in high energy astrophysics. More comprehensive reviews on electromagnetic processes can be found in standard textbooks, e.g. Refs. [230, 528], and also in Ref. [529]. In the second part we will apply this to the main scenarios and models of $\gamma -$ray sources, separated into Galactic and extragalactic sources.

Appendices

Appendix 8

1.1 The Kramers–Kronig Relation

Imagine a time-dependent quantity A(t) that in response to a perturbation Bf(t) changes from its unperturbed evolution $A_0(t)$. To linear order one can then write

$$\begin{aligned} A(t)-A_0(t)=\int _{-\infty }^{+\infty }dt^\prime \chi _{AB}(t-t^\prime )f(t^\prime ), \end{aligned}$$

(8.88)

where s the dynamic susceptibility $\chi _{AB}(t)=0$ for $t<0$ if the perturbation is causal. One can show that it is given by the retarded Green’s function for the operators A and B corresponding to these quantities,

$$\begin{aligned} \chi _{AB}(t-t^\prime )=i\varTheta (t-t^\prime )\left\langle [A(t),B(t^\prime )]_\mp \right\rangle = \int d^3\mathbf{r}\,G^r_{AB}(t,\mathbf{r},t^\prime ,\mathbf{r}), \end{aligned}$$

(8.89)

where $[.,.]_\mp$ stands for the commutator if at least one of the operators is bosonic and for the anticommutator otherwise. The last form of Eq. (8.89) refers to Eqs. (4.225) and (4.226) from Sect. 4.7.2 where we have discussed Green’s functions in general situations out of thermodynamic equilibrium in the context of baryo - and leptogenesis . Causality thus implies that the Fourier transform defined for $z\in \mathbb {C}$

$$\begin{aligned} \chi _{AB}(z)\equiv \int _{-\infty }^{+\infty }dt\,e^{izt}\chi _{AB}(t) \end{aligned}$$

(8.90)

is analytic in the upper half plane. With this one can write the response to a slowly switched on perturbation, $\epsilon \rightarrow 0+$,

$$\begin{aligned} V(t)=B f e^{-i\omega t}e^{\epsilon t} \end{aligned}$$

(8.91)

as

$$\begin{aligned} A(t)-A_0(t)=\chi _{AB}(\omega )fe^{-i\omega t}. \end{aligned}$$

(8.92)

The Cauchy theorem implies

$$\begin{aligned} \chi _{AB}(z)=\frac{1}{2\pi i}\oint _C dz^\prime \,\frac{\chi _{AB}(z^\prime )}{z^\prime -z}, \end{aligned}$$

(8.93)

where C is an arbitrary closed curve in the area where $\chi _{AB}(z)$ is analytic which is to be followed counter-clockwise. For example, one can compose C of an integration along the real axis from $-L$ to $+L$ and a semi-circle in the upper half plane with center at the origin and radius L. If $\chi _{AB}(t)$ grows at most as a power of t for large t, then the half-circle does not contribute in the limit $L\rightarrow \infty$ and one gets

$$\begin{aligned} \chi _{AB}(z)=\frac{1}{2\pi i}\int _{-\infty }^{+\infty } dx^\prime \,\frac{\chi _{AB}(x^\prime )}{x^\prime -z}. \end{aligned}$$

(8.94)

For $z=x\in \mathbb {R}$ real this implies

$$\begin{aligned} \chi _{AB}(x)=\lim _{\epsilon \rightarrow 0+}\chi _{AB}(x+i\epsilon )= & {} \lim _{\epsilon \rightarrow 0+}\int _{-\infty }^{+\infty }\frac{dx^\prime }{2\pi i}\, \frac{\chi _{AB}(x^\prime )}{x^\prime -x-i\epsilon }=\\= & {} \int _{-\infty }^{+\infty }\frac{dx^\prime }{2\pi i}\,\left[ \mathcal{P}\,\frac{1}{x^\prime -x} +i\pi \delta (x^\prime -x)\right] \chi _{AB}(x^\prime ),\nonumber \end{aligned}$$

(8.95)

where for a real valued function f(x) which has a singularity at $x=x_0$ the principal value is defined as

$$\begin{aligned} \mathcal{P}\,\int _a^b dx\,f(x)\equiv \lim _{\epsilon \rightarrow 0+}\int _a^{x_0-\epsilon }dx f(x)+\int _{x_0+\epsilon }^b dx f(x) \end{aligned}$$

(8.96)

for $a<x_0<b$, provided the limit exists. From Eq. (8.95) one then obtains

$$\begin{aligned} \chi _{AB}(x)= \frac{1}{\pi i}\,\mathcal{P}\int _{-\infty }^{+\infty } dx^\prime \,\frac{\chi _{AB}(x^\prime )}{x^\prime -x}. \end{aligned}$$

(8.97)

From this follows the dispersion relation , also known as Kramers–Kronig relation  [231]

$$\begin{aligned} \mathfrak {R}\left[ \chi _{AB}(\omega )\right]= & {} \frac{1}{\pi }\,\mathcal{P}\int _{-\infty }^{+\infty } d\omega ^\prime \,\frac{\mathfrak {I}\left[ \chi _{AB}(\omega ^\prime )\right] }{\omega ^\prime -\omega },\nonumber \\ \mathfrak {I}\left[ \chi _{AB}(\omega )\right]= & {} -\frac{1}{\pi }\,\mathcal{P}\int _{-\infty }^{+\infty } d\omega ^\prime \,\frac{\mathfrak {R}\left[ \chi _{AB}(\omega ^\prime )\right] }{\omega ^\prime -\omega }. \end{aligned}$$

(8.98)

We can now apply this to a plane wave photon field propagating along the $z-$direction in a medium with refractive index $n(\omega )$. The photon field amplitude at $z=z_0$ can be written as

$$A(t)=\exp [-i\omega t+i\omega n(\omega )z_0]\simeq i\omega [n(\omega )-1]z_0\exp [-i\omega (t-z_0)],$$

for $n(\omega )-1\ll 1$. Comparing this with Eq. (8.92) shows that $\chi (\omega )=i\omega [n(\omega )-1]z_0$ is a dynamic susceptibility to which the Kramers–Kronig relation Eq. (8.98) apply. We note that the absorption rate of $|A(t)|^2$ with $z_0$ is

$$\begin{aligned} R(\omega )=2\omega \mathfrak {I}[n(\omega )] \end{aligned}$$

(8.99)

and satisfies $R(-\omega )=R(\omega )$ since both waves with positive and negative frequency propagate in the positive $z-$direction. Then the second equation in Eq. (8.98) implies the Kramers–Kronig relation Eq. (8.74) for the refractive index .

Problems

8.1

The Power Emitted by a Non-Relativistic Accelerated Charge

(a) Using the electric and magnetic fields of the radiation emitted by an accelerating non-relativistic electric charge, Eq. (8.1), compute the Poynting flux $\mathbf{S}=\mathbf{E}\times \mathbf{B}/(4\pi )$ , Eq. (2.204), explicitly as a function of distance to and orientation relative to the second derivative $\ddot{\varvec{\mu }}_e$ of the electric dipole moment .

(b) Compute the total power emitted by integrating the Poynting flux over all directions and verify that it is given by Eq. (8.3).

8.2

Energy Transfer in Compton Scattering

(a) Derive the relation Eq. (8.27) between the incoming and outgoing photon energies in Compton scattering with an electron at rest by using energy-momentum conservation.

(b) Show that to first order in $\hbar \omega /m_e$ and $v_e^2$ the energy change of a photon of energy $\hbar \omega$ undergoing Compton scattering with an electron of velocity $v_e$ is given by Eq. (8.56),

$$\frac{\varDelta \epsilon }{\epsilon }=\frac{\omega ^\prime -\omega }{\omega }\simeq v_e\left( \cos \theta _*^\prime -\cos \theta \right) +v_e^2+\frac{\hbar \omega }{m_e}\left( \cos \theta \cos \theta _*^\prime -1\right) . $$

Hints: Use Eq. (8.55) and $v_*\simeq v_e+\hbar \omega \cos \theta /m_e$ to this order which follows from $v_*=P/E$.

(c) Show Eq. (8.57) for the scattering angle in the CM frame by performing suitable Lorentz transformations.

8.3

Power Law Solutions from Comptonization

Show that the the version Eq. (8.65) of the Kompaneets equation has power law solutions $n\propto x^m$ where the index is given by Eq. (8.66),

$$ m=-\frac{3}{2}\pm \left( \frac{9}{4}+\frac{1}{y_\mathrm{esc}}\right) ^{1/2}. $$

8.4

The Sunyaev-Zeldovich (SZ) effect

(a) Show that for small Compton optical depth, $y\ll 1$, after traversing an electron plasma at temperature $T_e$ the distortion of a thermal photon distribution initially in thermal equilibrium at temperature $T_0$ is given by Eq. (8.67),

$$ \frac{\varDelta I(\epsilon )}{I(\epsilon )}=\frac{\varDelta n(\epsilon )}{n(\epsilon )}\simeq y(x_e-1)\frac{xe^{x_0}}{e^{x_0}-1}\left( x_0\frac{e^{x_0}+1}{e^{x_0}-1}-4\right) , $$

where $x_0\equiv \epsilon /(k_\mathrm{B}T_0)$, $x\equiv \epsilon /(k_\mathrm{B}T_e)$, and $x_e\equiv T_e/T_0$.

(b) Show that in the limit $x_0\ll 1$ this expression turns into Eq. (8.68), $\varDelta I(\epsilon )/I(\epsilon )\simeq -2y$.

8.5

Pair Production Cross Section from Dimensional Analysis

Estimate the cross section for pair production by photons, $\gamma \gamma \rightarrow e^+e^-$ close to the threshold, $s\sim m_e^2$ with $m_e$ the electron mass, i.e. in the Thomson regime , and for $s\gg m_e^2$ (Klein–Nishina regime ) . You can use dimensional analysis for these estimates.

8.6

Synchrotron Energy Loss of Galactic Electrons

(a) The power of synchrotron emission by an electron of energy E in a magnetic field B is given by Eq. (8.13). Assuming the average Galactic magnetic field strength is ${\simeq }5\,\mu$G, estimate the energy loss length of an electron in the Galaxy as a function of its energy.

(b) How far does the electron diffuse before it looses most of its energy assuming that the diffusion coefficient is given by the Bohm limit Eq. (6.9), i.e. by the gyro-radius? For comparison, verify the corresponding length scale Eq. (8.25) for the effective diffusion coefficient from CR abundance fits.

8.7

Synchrotron Radiation

Assume that electrons with an energy distribution $\varPhi (E)\propto E^{-\alpha }$ are injected in an environment with a magnetic field of a given strength B. Derive the shape of the resulting synchrotron spectrum for the following two cases:

(a): The electron energy loss length is much larger than the size of the magnetized region.

(b) The electrons loose most of their energy before leaving the magnetized region.

Hints: For simplicity assume that the injection spectrum extend to arbitrarily high energies. For case (b) first derive the equilibrium electron spectrum by using Eq. (7.9).

8.8

Triplet Pair Production

By which power of the fine structure constant $\alpha _\mathrm{em}\equiv e^2/(4\pi )$ is the cross section of triplet pair production, $e+\gamma \rightarrow e e^+e^-$ suppressed compared to inverse Compton scattering, $e+\gamma \rightarrow e+\gamma$ , and pair production , $\gamma +\gamma \rightarrow e^+e^-$? Hint: Count the number of vertices in the corresponding Feynman diagrams .

8.9

Pion Production and Pair Production by Protons

At very high energies the cross sections for pair production by protons, $p+\gamma \rightarrow p+e^+e^-$, and for triplet pair production , $e+\gamma \rightarrow e e^+e^-$ , become very similar and roughly are equal to $\alpha _\mathrm{em}^n\sigma _T$, with $\sigma _T\simeq 0.6\,$barn the Thomson cross section Eq. (8.30) and n the power determined in Problem 8.7. Compare this with the cross section $\sigma _{p\gamma }\simeq 100\,\mu$barn for pion production, $p+\gamma \rightarrow N+\pi$. How do the energy loss rates by pair production compare to the ones by pion production if the proton looses a fraction ${\sim }m/m_p$ of its energy with m the mass of the produced particle, i.e. the pion mass and the electron mass, respectively?