Free-free absorption in hot relativistic flows: application to fast radio bursts

Magnetic flares create hot relativistic shocks outside the light cylinder radius of a magnetised star. Radio emission produced in such a shock or at a radius smaller than the shock undergoes free-free absorption while passing through the shocked medium. In this work, we demonstrate that this free-free absorption can lead to a negative drift in the frequency-time spectra. Whether it is related to the downward drift pattern observed in fast radio bursts (FRBs) is unclear. However, if the FRB down drifting is due to this mechanism then it will be pronounced in those shocks that have isotropic kinetic energies $\gtrsim10^{44}$ erg. In this model, for an internal shock with a Lorentz factor $\sim100$, the normalised drift rate $|{\rm DR_{\rm obs}}|/\nu_{\rm mean}$ is $\sim10^{-2}$ per ms, where $\nu_{\rm mean}$ is the central frequency of the radio pulses. The corresponding radius of the shocked shell is, therefore, in the range of $10^{10}$ cm and $10^{11}$ cm. This implies that, for an outflow consisting of hydrogen ion, the upper limit on the mass of the relativistic shocks is a few $\times~10^{-10}~M_\odot$, which is considerably low compared to that ejected from SGR 1806-20 during the 2004 outburst.