Galactic Synchrotron and free-free emission model
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>