Synchrotron Emission and Absorption

Abstract

This chapter introduces the synchrotron process, a classic radiation process of high energy particles. It is treated in many text-books, therefore the emphasis is once again on explaining the phenomena from a physically intuitive point of view. The Appendix instead presents exact results and some useful approximations for the Γ-functions entering the exact formulae. The adopted steps are: (i) find out the total synchrotron losses; (ii) find out the frequency at which the single particle emits most of the power; (iii) discuss the passage from cyclotron to synchrotron emission; (iv) discuss the emissivity produced by an ensemble of electrons; (v) introduce the absorption process from two points of view: the point of view of the photon, that has to calculate its probability of surviving, and the point of view of an electron, that has to calculate its energy gain when absorbing synchrotron photons. In the end we will discuss how to find out the physical parameters of a synchrotron emitting source on the basis of the observed properties of the received flux.

Appendix: Useful Formulae

In this section we collect several useful formulae concerning the synchrotron emission. When possible, we give also simplified analytical expressions. We will often consider that the emitting electrons have a distribution in energy which is a power law between some limits γ ₁ and γ ₂. Electrons are assumed to be isotropically distributed in the comoving frame of the emitting source. Their density is

$$ N(\gamma) = K\gamma^{-p}; \quad \gamma_1<\gamma< \gamma_2 $$

(4.40)

The Larmor frequency is defined as:

$$ \nu_L \equiv {e B \over 2\pi m_{\rm e} c} $$

(4.41)

4.1.1 A.1 Emissivity

The synchrotron emissivity $j_{\rm s}(\nu,\theta)$ [erg cm⁻³ s⁻¹ sterad⁻¹] is

$$ j_{\rm s}(\nu,\theta) \equiv {1\over 4\pi} \int _{\gamma_1}^{\gamma_2} N(\gamma) P_{\rm s}(\nu,\gamma, \theta) d\gamma $$

(4.42)

where $P_{\rm s}(\nu, \gamma,\theta)$ is the power emitted at the frequency ν (integrated over all directions) by the single electron of energy $\gamma m_{\rm e}c^{2}$ and pitch angle θ. For electrons making the same pitch angle θ with the magnetic field, the emissivity is

$$ j_{\rm s}(\nu,\theta) = {3\sigma_{\rm T} c K U_B \over 8 \pi^2 \nu_L} \biggl( {\nu\over \nu_L} \biggr)^{-{p-1\over 2}} (\sin\theta)^{p+1\over 2} 3^{p\over 2} { \varGamma ( {3p-1 \over 12} ) \varGamma ( {3p+19\over 12} ) \over p+1} $$

(4.43)

between $\nu_{1} \gg \gamma_{1}^{2}\nu_{L}$ and $\nu_{2}\ll \gamma_{2}^{2}\nu_{L}$. If the distribution of pitch angles is isotropic, we must average the $(\sin\theta)^{p+1\over 2}$ term, obtaining

$$ \bigl\langle(\sin\theta)^{p+1\over 2}\bigr\rangle = \int_0^{\pi\over 2} (\sin\theta)^{p+1\over 2} \sin\theta d\theta = {\sqrt{\pi} \over 2} {\varGamma ( {p+5 \over 4} ) \over \varGamma ( {p+7 \over 4} ) } $$

(4.44)

Therefore the pitch angle averaged synchrotron emissivity is

$$ j_{\rm s}(\nu) = {3\sigma_{\rm T} c K U_B \over 16 \pi \sqrt{\pi} \nu_L} \biggl( {\nu\over \nu_L} \biggr)^{-{p-1\over 2}} f_j(p) $$

(4.45)

The function f _j (p) includes all the products of the Γ-functions:

(4.46)

where the simplified fitting function is accurate at the per cent level.

4.1.2 A.2 Absorption Coefficient

The absorption coefficient α _ν (θ) [cm⁻¹] is defined as:

$$ \alpha_\nu (\theta) \equiv {1\over 8\pi m_{\rm e}\nu^2} \int _{\gamma_1}^{\gamma_2} {N(\gamma)\over \gamma^2} {d\over d\gamma} \bigl[ \gamma^2 P(\nu, \theta) \bigr] d\gamma $$

(4.47)

Written in this way, the above formula is valid even when the electron distribution is truncated. For our power law electron distribution α _ν (θ) becomes:

$$ \alpha_\nu (\theta) \equiv {1\over 8\pi m_{\rm e}\nu^2} \int _{\gamma_1}^{\gamma_2} {N(\gamma)\over \gamma^2} {d\over d\gamma} \bigl[ \gamma^2 P(\nu, \theta) \bigr] d\gamma $$

(4.48)

Above $\nu = \gamma_{1}^{2}\nu_{L}$, we have:

$$ \alpha_\nu (\theta) = {e^2 K \over 4 m_{\rm e} c^2} ( \nu_L \sin\theta)^{p+2\over 2} \nu^{-{p+4 \over 2}} 3^{p+1\over 2} \varGamma \biggl( {3p+22 \over 12} \biggr) \varGamma \biggl( {3p+2 \over 12} \biggr) $$

(4.49)

Averaging over the pitch angles we have:

$$ \bigl\langle(\sin\theta)^{p+2\over 2}\bigr\rangle = \int_0^{\pi\over 2} (\sin\theta)^{p+2\over 2} \sin\theta d\theta = {\sqrt{\pi} \over 2} {\varGamma ( {p+6 \over 4} ) \over \varGamma ( {p+8 \over 4} ) } $$

(4.50)

resulting in a pitch angle averaged absorption coefficient:

$$ \alpha_\nu = {\sqrt{\pi} e^2 K \over 8 m_{\rm e} c} \nu_L^{p+2\over 2} \nu^{-{p+4 \over 2}} f_\alpha(p) $$

(4.51)

where the function f _α (p) is:

(4.52)

The simple fitting function is accurate at the per cent level.

4.1.3 A.3 Specific Intensity

Simple radiative transfer allows to calculate the specific intensity:

$$ I(\nu) = {j_{\rm s}(\nu) \over \alpha_\nu } \bigl( 1-e^{-\tau_\nu} \bigr) $$

(4.53)

where the absorption optical depth τ _ν ≡α _ν R and R is the size of the emitting region. When τ _ν ≫1, the exponential term vanishes, and the intensity is simply the ratio between the emissivity and the absorption coefficient. This is the self-absorbed, or thick, regime. In this case, since both $j_{\rm s}(\nu)$ and α _ν depends linearly upon K, the resulting self-absorbed intensity does not depend on the normalization of the particle density K:

$$ I(\nu) = {2m_{\rm e} \over \sqrt{3} \nu_L^{1/2}} f_I(p) \bigl( 1-e^{-\tau_\nu} \bigr) $$

(4.54)

we can thus see that the slope of the self-absorbed intensity does not depend on p. Its normalization, however, does (albeit weakly) depend on p through the function f _I (p), which in the case of averaging over an isotropic pitch angle distribution is given by:

(4.55)

where again the simple fitting function is accurate at the level of 1 per cent.

4.1.4 A.4 Self-absorption Frequency

The self-absorption frequency $\nu_{\rm t}$ is defined by $\tau_{\nu_{\rm t}}=1$:

$$ \nu_{\rm t} = \nu_L \biggl[ {\sqrt{\pi} e^2 R K \over 8 m_{\rm e} c\nu_L} f_\alpha(p) \biggr]^{4\over p+4} = \nu_L \biggl[ {\pi \sqrt{\pi}\over 4} {e R K \over B} f_\alpha(p) \biggr]^{2\over p+4} $$

(4.56)

Note that the term in parenthesis is dimensionless, and since RK has units of the inverse of a surface, then e/B has the dimension of a surface. In fact we have already discussed that this is the synchrotron absorption cross section of a relativistic electron of energy $\gamma m_{\rm e} c^{2}$ absorbing photons at the fundamental frequency ν _L /γ.

The random Lorentz factor $\gamma_{\rm t}$ of the electrons absorbing (and emitting) photons with frequency $\nu_{\rm t}$ is $\gamma_{\rm t} \sim [3\nu_{\rm t}/ (4\nu_{L})]^{1/2}$.

4.1.5 A.5 Synchrotron Peak

In an F(ν) plot, the synchrotron spectrum peaks close to $\nu_{\rm t}$, at a frequency ν _s,p given by solving

$$ {d I(\nu) \over d\nu} = 0 \to {d \over d\nu} \bigl[ \nu^{5/2} \bigl( 1-e^{-\tau_\nu} \bigr) \bigr] = 0 $$

(4.57)

which is equivalent to solve the equation:

$$ \exp ( \tau_{\nu_{s,p}} ) -{ p+4 \over 5} \tau_{\nu_{s,p}} -1 = 0 $$

(4.58)

whose solution can be approximated by

$$ \tau_{\nu_{s,p}} \sim {2\over 5} p^{1/3}\ln{p} $$

(4.59)