  In this work, we analyze the kinetic stability of a solar wind electron
distribution composed of core and strahl subpopulations. The core is modeled by
a drifting Maxwellian distribution, while the strahl is modeled by an analytic
function recently derived in (Horaites et al. 2018) from the collisional
kinetic equation. We perform a numerical linear stability analysis using the
LEOPARD solver (Astfalk & Jenko 2017), which allows for arbitrary gyrotropic
distribution functions in a magnetized plasma. We find that for typical solar
wind conditions, the core-strahl distribution is unstable to the kinetic
Alfv\'en and magnetosonic modes. The maximum growth rates for these
instabilities occur at wavenumbers $k d_i \lesssim 1$, at moderately oblique
angles of propagation, thus providing a potential source of kinetic-scale
turbulence. In contrast with previous reports, we however do not find evidence
for a whistler instability directly associated with the electron strahl. This
may be related to the more realistic shape of the electron strahl distribution
function adopted in our work. We therefore suggest that the whistler modes
often invoked to explain anomalous scattering of strahl particles could appear
as a result of nonlinear mode coupling and turbulent cascade originating at
scales $k d_i \lesssim 1$.
