On a spectral method for $\beta$-particle bound excitation collisions in kilonovae

The interaction of $\beta$-particles with the weakly ionized plasma background is an important mechanism for powering the kilonova transient signal from neutron star mergers. For this purpose, we present an implementation of the approximate fast-particle collision kernel, described by Inokuti (1971) following the seminal formulation of Bethe (1930), in a spectral solver of the Vlasov-Maxwell-Boltzmann equations. In particular, we expand the fast-particle plane-wave atomic excitation kernel into coefficients of the Hermite basis, and derive the relevant discrete spectral system. In this fast-particle limit, the approach permits the direct use of atomic data, including optical oscillator strengths, normally applied to photon-matter interaction. The resulting spectral matrix is implemented in the MASS-APP spectral solver framework, in a way that avoids full matrix storage per spatial zone. We numerically verify aspects of the matrix construction, and present a proof-of-principle 3D simulation of a 2D axisymmetric kilonova ejecta snapshot. Our preliminary numerical results indicate that a reasonable choice of Hermite basis parameters for $\beta$-particles in the kilonova are a bulk velocity parameter $\vec{u}=0$, a thermal velocity parameter $\vec{\alpha}=0.5c$, and a 9x9x9 mode velocity basis set (Hermite orders 0 to 8 in each dimension). For ejecta-interior sample zones, we estimate the ratio of thermalization from large-angle ($\gtrsim2.5^{\circ}$) bound excitation scattering to total thermalization is $\sim$0.002-0.003.