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## Emission spectrum from given particle spectrum Edit

As shown in Synchrotron radiation the power emitted by a relativistic electron through synchrotron radiation is $P \propto \gamma^2$. For a distribution of electrons, this is equivalent to $P \propto \nu$, where we have assumed that electrons of different energies dominate the emission at their respective synchrotron frequencies (this is equivalent to approximating the contribution of each electron as a delta function emission function around its corresponding synchrotron frequency). In particular it implies that $\frac{dP}{d\nu}=const$. Assuming a power law distribution of electron energies: $\frac{dN}{d\gamma} \propto \gamma^{-p}$, we obtain the well known result:

$F_{\nu}=\int_{\gamma(\nu)}^\infty d\gamma \frac{dN}{d\gamma} P(\nu)\propto \nu^{-(p-1)/2}$

## Effect of synchrotron cooling on the electron distribution Edit

A characteristic frequency in the synchrotron spectrum, is the so called "cooling frequency", $\nu_c=\nu_{syn}(\gamma_c)$. This is the typical frequency of radiation emitted by electrons that are cooling on a given time-scale $t$ (this is typically the dynamical time of a system or its life-time). Electrons with $\gamma>\gamma_c$ are cooled significantly by time $t$, while lower energy electrons are not. We can estimate $\gamma_c$ by equating the total energy losses through synchrotron over a time $t$ with the initial energy in the electron:

$P t = E \rightarrow \frac{4}{3}\sigma_T c \gamma_e^2 \frac{B^2}{8 \pi} t =\gamma_c m_e c^2 \rightarrow \gamma_c=\frac{6 \pi m_e c}{\sigma_T B^2 t}$.

The distribution of electron energies at $\gamma>\gamma_c$ will be affected by cooling. This can be seen from the continuity equation for electrons in the energy space:

$\frac{\partial}{\partial t}\bigg(\frac{dN}{d\gamma}\bigg)+\frac{\partial}{\partial \gamma}\bigg(\dot{\gamma}\frac{dN}{d\gamma}\bigg)=S(\gamma)$

where $\dot{\gamma}\propto \gamma^2B^2$ is the cooling rate and $S(\gamma)$ is the rate at which electrons with Lorentz factor $\gamma$ are injected into the system. A steady state solution ($\frac{\partial}{\partial t}=0$) of this equation (assuming a time dependent magnetic field) is $\frac{dN}{d\gamma} \propto \gamma^{-2}$ for $\gamma_c <\gamma <\gamma_m$, where $\gamma_m$ is the minimum Lorentz factor of injected electrons (i.e. $S(\gamma)=0$ for $\gamma <\gamma_m$). The synchrotron spectrum corresponding to this energy range can be calculated as above (with $p=2$) and gives: $F_{\nu}\propto \nu^{-1/2}$. For $\gamma>\gamma_m>\gamma_c$ we plug $S(\gamma)\propto \gamma^{-p}$ in the continuity equation, to get: $\frac{dN}{d\gamma} \propto \gamma^{-(p+1)}$, leading to $F_{\nu}\propto \nu^{-p/2}$. These results are known as the fast cooling synchrotron spectrum.