Models

Work In Progr

Self-Absorption

To derive analytic expressions for the self-absorption frequency, I follow the method in VDH09 which starts with the formula for the absorption coefficient from (Ginzburg & Syrovatskii 1965):

$$\alpha_\nu = -\frac{1}{8 \pi m_e \nu^2} \int P_\nu(\gamma_e) \gamma_e^2 \frac{d}{d\gamma_e} \left( \frac{N(\gamma_e)}{\gamma_e^2} \right) d\gamma_e$$

Slow Cooling Case:

For slow cooling, the self-absorption coefficient becomes:

Where $\Gamma$ is the incomplete gamma function.

For $x << 1$ for $\Gamma(a, x)$:

$$\Gamma(a, x) \approx \Gamma(a) - \frac{x}{a}^{a}$$

For $x >> 1$ for $\Gamma(a, x)$:

$$\Gamma(a, x) \approx x^{a -1} e^{-x} \approx 0$$

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For the case of $\nu_a << \nu_m << \nu_c$ :

Using $\nu_c >> \nu_m$:

$$\frac{p - 1}{2} \left[ 1 - \frac{1}{p} \left( \frac{\nu_c}{\nu_m} \right)^{-\frac{p - 1}{2}} \right]^{-1} \approx \frac{p - 1}{2}$$

and the $p + 3$ term goes to zero. Then we are left with:

$$=3 \left( \frac{3}{2} \right)^{\frac{1}{3}} \frac{n_e q_e^3 B \Gamma_2}{8 \pi m_e^2 c^2 \gamma_m \nu^2} \cdot \frac{p - 1}{2} \cdot \left\{ (p + 2) \left( \frac{\nu_a}{\nu_m} \right)^{-\frac{p}{2}} \left[ \Gamma\left( \frac{p}{2} + \frac{1}{3}, \frac{\nu_a}{\nu_c} \right) - \Gamma\left( \frac{p}{2} + \frac{1}{3}, \frac{\nu_a}{\nu_m} \right) \right] \right. \\$$

Using the above incomplete gamma properties above:

$$=3 \left( \frac{3}{2} \right)^{\frac{1}{3}} \frac{n_e q_e^3 B \Gamma_2}{8 \pi m_e^2 c^2 \gamma_m \nu^2} \cdot \frac{p - 1}{2} \cdot (p + 2) \left( \frac{\nu_a}{\nu_m} \right)^{-\frac{p}{2}} \left[ \Gamma\left( \frac{p}{2} + \frac{1}{3}\right) - \Gamma\left( \frac{p}{2} + \frac{1}{3} \right) + \frac{1}{\frac{p}{2} + \frac{1}{3}}\left( \frac{\nu_a}{\nu_m} \right)^{p/2 + 1/3}\right] \\$$

Simplifying:

$$=\left( \frac{3}{2} \right)^{\frac{4}{3}} \frac{n_e q_e^3 B \Gamma_2}{8 \pi m_e^2 c^2 \gamma_m \nu^2} \cdot (p - 1) \cdot (p + 2) \left( \frac{\nu_a}{\nu_m} \right)^{-\frac{p}{2}} \left[\frac{2}{p + \frac{2}{3}}\left( \frac{\nu_a}{\nu_m} \right)^{p/2 + 1/3}\right] \\$$

$$\alpha_{\nu_a}=2\left( \frac{3}{2} \right)^{\frac{4}{3}} \frac{n_e q_e^3 B \Gamma_2}{8 \pi m_e^2 c^2 \gamma_m} \cdot \frac{(p - 1) (p + 2)}{p + \frac{2}{3}} \nu_m^{-1/3}\nu_a^{-5/3} . \\$$

To solve for $\nu_a$, set $\alpha_{\nu_a} = \Delta R$, where $\Delta R$ id thickness of the shell that emits synchrotron radiation.

$$\nu_a=\left[2\left( \frac{3}{2} \right)^{\frac{4}{3}} \frac{n_e q_e^3 B \Gamma_2}{8 \pi m_e^2 c^2 \gamma_m} \cdot \frac{(p - 1) (p + 2)}{p + \frac{2}{3}} \nu_m^{-1/3}\right]^{3/5}\\$$