Galactic Synchrotron and free-free emission model

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Symbols

p = Energy spectral index of cosmic ray electrons

In Rayleigh-Jeans limit the surface brightness temperature can be written as,

<math>T_b(\nu) = \frac{c^2}{2k_B\nu^2} I(\nu)</math>

The emission coefficient is the energy emitted per unit solid angle per unit volume. It can also be written in terms of temperature,

<math>j_b(\nu)=\frac{c^2}{2k_B\nu^2}I(\nu)</math>

In that case,

<math>T_b(\nu) = \int j_b(\nu)ds</math>

Synchrotron emission

Emission coefficient for total and polarized synchrotron emission is,

<math>j_b^{I,PIsyn} = C_{I,PIsyn} \left(\frac{2\pi m_e c}{3e}\right)^{-\frac{p-1}{2}} n_{CR} B_\perp^{\frac{p+1}{2}} \nu^{-\frac{p+3}{2}}</math>

where the constants are different for total and intrinsically polarized intensities,

<math>\begin{align}

C_{Isyn} = \frac{\sqrt{3}e^3}{8\pi m_e k_B(p+1)} \Gamma \left(\frac{p}{4}-\frac{1}{12}\right) \Gamma \left(\frac{p}{4}+\frac{19}{12} \right) \\ C_{PIsyn} = \frac{\sqrt{3}e^3}{32\pi m_e k_B} \Gamma \left(\frac{p}{4}-\frac{1}{12}\right) \Gamma \left(\frac{p}{4}+\frac{7}{12} \right) \end{align}</math> Than the intrinsic degree of polarization is,

<math>\Pi = \frac{p+1}{p+7/3}</math>

The emission coefficients for Stokes Q and U (that determine polarized emission) are,

<math>j_b^Q = j_b^{PIsyn} \cos2\phi \text{, } j_b^U = j_b^{PIsyn} \sin2\phi</math>

We can integrate the emission coefficients expressed in terms of temperature along some LOS to get the actual brightness temperature and then the observed polarized emission is given by,

<math>T_b^{PI} = \sqrt{(T_b^Q)^2+(T_b^U)^2}</math>

And the polarization angle is given by,

<math>\Phi_{obs}= \frac{1}{2} \tan^{-1}\frac{T_b^U}{T_b^Q}</math>

Free-free emission

The optical depth of warm ionized gas in the intra-Galactic medium at a given frequency is,

<math>\tau_\nu^{ff} = 3.01\times 10^{-8} g_{ff} \left(\frac{T_e}{[K]}\right)^{-3/2} \left(\frac{\nu}{[MHz]}\right)^{-2} \frac{EM}{[cm^{-6} pc]}</math>

where the Gaunt factor is given by,

<math>g_{ff} = \ln \left[4.95\times 10^{-5} \left(\frac{\nu}{[MHz]}\right)^{-1}\right] + 1.5\ln \left(\frac{T_e}{[K]}\right)</math>

The Emission measure is defined as,

<math>\frac{EM}{[cm^{-6} pc]} = \int n_e^2 ds</math>

The Galactic free-free emission is,

<math>j_b^{ff} = T_e(1-e^{-\tau^{ff}})</math>

Of course for optically thin gas it is just,

<math>j_b^{ff} = T_e \tau^{ff}</math>

Faraday rotation

Polarization angle of the wave after a rotation (<math>\Delta\Phi=\Phi-\Phi_0</math>) is given by,

<math>\Phi = \Phi_0 + \frac{e^3}{2\pi m_e^2 c^2} \nu^{-2} \int n_e B_\parallel ds = \Phi_0 + RM \lambda^2</math>

And Faraday depth (which is a kind of distance measure in Faraday space) is just,

<math>\frac{\phi}{[rad\ m^{-2}]} = 0.81 \int \frac{n_e}{[cm^{-3}]} \frac{B_\parallel}{[\mu G]} \frac{ds}{[pc]}</math>