Wave dispersion in pulsar plasma. Part 1. Plasma rest frame

Abstract

Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle$\unicode[STIX]{x1D703}=0$) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation$|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$, where$z=\unicode[STIX]{x1D714}/k_{\Vert }c$is the phase speed and$\unicode[STIX]{x1D6FD}_{\text{A}}c$is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function,$z^{2}W(z)$, which is negative for$|z|<z_{0}$and has a sharp maximum at$|z|=z_{\text{m}}$, with$1-z_{\text{m}}<1-z_{0}\ll 1$. We give numerical estimates for the maximum of$z^{2}W(z)$and for$z_{\text{m}}$and$z_{0}$for a one-dimensional Jüttner distribution. The L and A modes reconnect, for$z_{\text{A}}>z_{0}$, to form the O and Alfvén modes for oblique propagation ($\unicode[STIX]{x1D703}\neq 0$). For$z_{\text{A}}<z_{0}$the Alfvén and O mode curves reconnect forming a new mode that exists only for$\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$. The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around$\unicode[STIX]{x1D703}\approx 0$. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.