Authors: Nooshin Davis(University of New Hampshire), Benjamin Chandran(University of New Hampshire)

In-situ observations of the solar wind indicate that velocity distribution functions (VDFs) of protons show distinctly non-thermal kinetic features, such as the presence of a more dense core and a field-aligned beam with a drift velocity higher than the core. The proton-beam instabilities has been suggested to play an important role in decelerating proton beams. In this study, we investigate the parallel F/M instability in a plasma containing the proton core, proton beam, and electron components. We derive analytic expressions for this instability threshold, which show the effects of the proton temperature anisotropy $T_{\perp p}/T_{\parallel p}$ and parallel proton beta $\beta_\parallel$ on the real frequency and growth rate of the proton-beam instability. We develop a general dispersion relation solver in Python, which provides numerical solutions to the full hot-plasma dispersion relation. We compare the analytical threshold expressions with numerical results and discuss the relevance of our results to solar wind observations.