The wave energy density and growth rate for the resonant instability in relativistic plasmas

ABSTRACT The wave instability acts in astrophysical plasmas to redistribute energy and momentum the absence of frequent collisions. There are many different types waves, it is important quantify density growth rate for understanding what instabilities possible plasma regimes. situations throughout our Universe where contain a significant fraction relativistic particles. Theoretical estimates constrained either field-aligned propagation angles or non-relativistic considerations. Based on linear theory, we derive analytic expressions an arbitrary resonant with angle plasmas. For this derivation, calculate Hermitian anti-Hermitian parts relativistic-plasma dielectric tensor. We demonstrate that expression presents explicit increase waves wavenumber range positive (i.e. driven). demonstration, numerically analyse loss-cone driven instability, as specific example, which whistler-mode scatter electrons into loss cone radiation belt. Our results further develop basis theory better understand have potential combine quasi-linear allows study time evolution not only particle distribution function but also properties through instability.