# Effect of Cooling Time on Synchrotron Spectra

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\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \subsection*{Effective Source Sizes} Imagine that we had a continuous, power law source of highly relativistic electrons that are subsequently carried away by a non-relativistic magnetized flow at speed v. An example of this situation would be the shock of a supernova remnant. Typical speeds are of order v $\sim$ 1250 $km/s$. We want to see if the finite cooling time for synchrotron radiation will effect what we observe.

The distance such an electron will travel will before cooling will be simply: $$\begin{aligned}vt_{cool}&\sim.02 pc\left({1\,Gauss\over B}\right)^2\left(\inv{\gamma}\right)\\ &\sim 10^{7} pc\left({40\,\mu G\over B}\right)^2\left(\inv{\gamma}\right)\\ \end{aligned}$$

where 40 $\mu$G is a more appropriate magnetic field for the post shock environment in a supernova remnant. We can convert the electron energy, $\gamma$ to a photon energy using the critical frequency found in past lectures: $$\begin{aligned}\nu_c = \frac{3\gamma^{2}eB}{2 m_e c}\\ \Rightarrow \gamma = \sqrt{\frac{2\nu_c m_e c}{3 e B}}\end{aligned}$$ so $$\begin{aligned} vt_{cool} \sim .9 pc\left({40\,\mu G\over B}\right)^{3/2}\left({1 keV \over h\nu}\right)^{1/2}\end{aligned}$$

Which is smaller than the typical supernova remnant radius around keV energies. As a result, when viewing supernova remnants at higher energies ($\sim$ keV and above), we see synchrotron emission concentrated around very thin nonthermal filaments that should decrease in size with energy. However, electrons also have the ability to diffuse through media, which is characterized by a diffusion coefficient, $\kappa$ with dimensions of [length][velocity] and can be written as : $$\begin{aligned} \kappa = \lambda_{mfp} v. \end{aligned}$$ For very relativistic electrons, $v\sim c$ and $\lambda_{mfp}$ is taken as some constant multiple of the gyroradius $\left(\frac{\gamma m_e c^2}{e B}\right)$, so that $$\begin{aligned} \kappa \sim \frac{\gamma m_e c^3}{e B} \end{aligned}$$ Quick dimensional analysis tells us the the characteristic length scale of diffusion is

$$\begin{aligned}

\sqrt{\kappa t_{cool}} &\sim 5 \times 10^{-8} pc\left(\frac{1 Gauss}{B}\right)^{3/2} \\ &\sim 30 pc \left(\frac{40 \mu G}{B}\right)^{3/2} \end{aligned}$$

which is independent of the emitting frequency and comparable to $vt_{cool}$ at keV energies. At higher energies, then, diffusion will be the limiting factor for effective source size while for lower energies convection will be the limiting factor. Since both results have depend on the magnetic field as $B^{-3/2}$, we can use effective source sizes to infer local values of the magnetic field, a parameter that is otherwise hard to get at.

\subsection*{Spectrum} The cooling time also has an impact on the shape of the spectra. To calculate the total $F_\nu$ from our source, we would integrate the specific intensity produced by these electrons over the solid angle subtended by the source: $$\begin{aligned} F_\nu &= \int I_{\nu} d \Omega

\\ \end{aligned}$$

Where D is the distance to the source. The integration over A is in principle of the area of the entire source, but as we have shown above this is limited to a region of width $vt_{cool}$ or $\sqrt{\kappa t_{cool}}$ before the radiation will effectively shut off. For the case of a spherical shock and constant specific intensity, then, the integral becomes

$$\begin{aligned}

F_\nu &= \int I_\nu \frac{dA}{D^2} \\ &\sim \inv{D^2} \int \limits ^{vt_{cool}}_{0} \int \limits ^{2\pi}_{0} I_\nu rdrd\phi\\ &\sim \frac{I_\nu}{D^2}(vt_{cool})^2 \\ &\propto I_\nu \nu^{-1}B^{-3}\end{aligned}$$

for convection-limited electrons and

$$\begin{aligned}

F_\nu &= \int I_\nu \frac{dA}{D^2} \\ &\sim \inv{D^2} \int \limits ^{\sqrt{\kappa t_{cool}}}_{0} \int \limits ^{2\pi}_{0} I_\nu rdrd\phi\\ &\sim \frac{I_\nu}{D^2}(\sqrt{\kappa t_{cool}})^2 \\ &\propto I_\nu B^{-3}\end{aligned}$$ for diffusion-limited electrons.

Thus, the two transport mechanism have qualitatively different effects on the shape of the spectrum. In one case there is an extra steeping by a power of $\nu^{-1}$ while for the other the spectrum remains unchanged. We can use these differences in many situations to infer properties of diffusion and convection in the plasma. This effect is also important in other astronomical objects, such as relativistic jets and pulsar wind nebulae, but these will have different geometries which will change the degree of steepening when integrating over the area of the source.

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