Instabilities Driven by the Drift and Temperature Anisotropy of Alpha Particles in the Solar Wind

We investigate the conditions under which parallel-propagating Alfv\'en/ion-cyclotron (A/IC) waves and fast-magnetosonic/whistler (FM/W) waves are driven unstable by the differential flow and temperature anisotropy of alpha particles in the solar wind. We focus on the limit in which $w_{\parallel \alpha} \gtrsim 0.25 v_{\mathrm A}$, where $w_{\parallel \alpha} $ is the parallel alpha-particle thermal speed and $v_{\mathrm A}$ is the Alfv\'en speed. We derive analytic expressions for the instability thresholds of these waves, which show, e.g., how the minimum unstable alpha-particle beam speed depends upon $w_{\parallel \alpha}/v_{\mathrm A}$, the degree of alpha-particle temperature anisotropy, and the alpha-to-proton temperature ratio. We validate our analytical results using numerical solutions to the full hot-plasma dispersion relation. Consistent with previous work, we find that temperature anisotropy allows A/IC waves and FM/W waves to become unstable at significantly lower values of the alpha-particle beam speed $U_\alpha$ than in the isotropic-temperature case. Likewise, differential flow lowers the minimum temperature anisotropy needed to excite A/IC or FM/W waves relative to the case in which $U_\alpha =0$. We discuss the relevance of our results to alpha particles in the solar wind near 1 AU.