We investigate the conditions under which parallel-propagating Alfvén/ion-cyclotron waves are driven unstable by an isotropic (T ⊥α = T ∥α) population of alpha particles drifting parallel to the magnetic field at an average speed U α with respect to the protons. We derive an approximate analytic condition for the minimum value of U α needed to excite this instability and refine this result using numerical solutions to the hot-plasma dispersion relation. When the alpha-particle number density is 5% of the proton number density and the two species have similar thermal speeds, the instability requires that βp 1, where βp is the ratio of the proton pressure to the magnetic pressure. For 1 βp 12, the minimum U α needed to excite this instability ranges from 0.7v A to 0.9v A, where v A is the Alfvén speed. This threshold is smaller than the threshold of 1.2v A for the parallel magnetosonic instability, which was previously thought to have the lowest threshold of the alpha-particle beam instabilities at βp 0.5. We discuss the role of the parallel Alfvénic drift instability for the evolution of the alpha-particle drift speed in the solar wind. We also analyze measurements from the Wind spacecraft's Faraday cups and show that the U α values measured in solar-wind streams with T ⊥α ≈ T ∥α are approximately bounded from above by the threshold of the parallel Alfvénic instability.
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