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: $\left( \frac{ { \rm d} \dot{ N}}{ { \rm d} \omega} \right)_t = \frac{ \alpha}{ 3} \frac{ 1}{ \gamma^2} \left( 1 + \frac{ 1}{ x^{ 2/3}} \right) \exp \left( - 2 x^{ 2/3} \right)$, where $x = \frac{ \omega}{ \omega_0}\, ; \omega_0 = \frac{ 4}{ 3} \gamma^2 \frac{ eB_0}{ m_e c}\, .$ We use this expression to find approximate synchrotron spectra for power-law electron distributions with $\propto \exp\left( -\left[ \gamma/\gamma_0 \right]^\beta\right)$ 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.
Categories: astro-ph.HE