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

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: $\left( \frac{{\rm d} \dot{N}}{{\rm d} ω} \right)_t = \fracα{3} \frac{1}{γ^2} \left( 1 + \frac{1}{x^{2/3}} \right) \exp \left( - 2 x^{2/3} \right)$, where $x = \fracω{ω_0}\, ; ω_0 = \frac{4}{3} γ^2 \frac{eB_0}{m_e c}\, .$ We use this expression to find approximate synchrotron spectra for power-law electron distributions with $\propto \exp\left( -\left[ γ/γ_0 \right]^β\right)$ type high-energy cut-off; the resulting synchrotron spectrum has the exponential cut-off factor with frequency raised to $2β/(3β+4)$ 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 $ω_m = a_m γ^2 \, \frac{e B_0}{m_e c}$) approximation.