Spectropolarimetry of synchrotron radiation from relativistic electrons with anisotropic pitch-angle and various energy distributions

For synchrotron radiation from relativistic electrons having a power-law energy distribution with a power-law index $p$ in an optically thin medium with a locally uniform magnetic field, it is generally adopted that the spectral index $\alpha =(p-1)\mathrm{/}2$ and the degree of linear polarisation ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}=(p+1)\mathrm{/}(p+7\mathrm{/}3)$, and hence that ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}=(\alpha +1)\mathrm{/}(\alpha +5\mathrm{/}3)$. These ($\alpha$, ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}$; $p$) relations are derived assuming that the electrons have an isotropic momentum distribution and a power-law energy distribution, and they have been used, almost universally, in the interpretation of polarimetry observations. In this study, we assess the validity of the ($\alpha$, ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}$; $p$) relations in different scenarios, such as anisotropic electron momenta and non-power-law distributions of electron energy. We calculate the synchrotron radiation and polarisation spectra for power-law, kappa and log-parabola electron-energy distributions, with isotropic and two anisotropic (beamed and loss cone) electron-momentum distributions. Our calculations show that when the electron momenta are isotropic, the usual ($\alpha$, ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}$; $p$) relations are generally applicable. However, if the electrons are anisotropic, the usual ($\alpha$, ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}$; $p$) relations could break down. Applying the usual ($\alpha$, ${\mathrm{\Pi }}_{\mathrm{L},\mathrm{p}\mathrm{l}}$; $p$) relations indiscriminately without caution on anisotropy in the electron momentum distribution, would lead to incorrect interpretations of polarimetry data.