Exact analytical expression for the synchrotron radiation spectrum in the Gaussian turbulent magnetic field

Abstract

We demonstrate that the exact solution for the spectrum of synchrotron radiation from an isotropic population of mono-energetic electrons in turbulent magnetic field with Gaussian distribution of local field strengths can be expressed in the simple analytic form: ${\left(\frac{\mathrm{d}\dot{N}}{\mathrm{d}\omega }\right)}_t=\frac{\alpha }{3}\frac{1}{{\gamma }^2}\left(1+\frac{1}{x^{2/3}}\right)\exp \left(-2x^{2/3}\right)$, where $x=\frac{\omega }{{\omega }_0};{\omega }_0=\frac{4}{3}{\gamma }^2\frac{e{B}_0}{m_{e}c}.$ We use this expression to find approximate synchrotron spectra for power-law electron distributions with $\propto \exp \left(-{\left[\gamma /{\gamma }_0\right]}^{\beta }\right)$ type high-energy cut-off; the resulting synchrotron spectrum has the exponential cut-off factor with frequency raised to $2\beta /(3\beta +4)$ power in the exponent. For the power-law electron distribution without high-energy cut-off, we find the coefficient $a_m$ as a function of the power-law index, which results in exact expression for the synchrotron spectrum when using monochromatic (i.e., each electron radiates at frequency ${\omega }_m=a_m{\gamma }^2\frac{e{B}_0}{m_{e}c}$) approximation.

final version published in ApJ. Several changes to the inroduction, minor changes in notation

References in corpus (2)

• Analytical solutions for energy spectra of electrons accelerated by nonrelativistic shock-waves in shell type supernova remnants
• Reacceleration of electrons in supernova remnants

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