Radial evolution of the strahl pitch-angle width

Suprathermal electron distributions observed in the solar wind are highly anisotropic. The field-aligned component, called the strahl, is typically characterized by its angular width—the "Strahl Pitch-Angle Width", or SPAW. The radial evolution of the SPAW provides valuable information about the scattering mechanisms acting on the electron population for energies roughly between $100$ and $1000$ eV. We theoretically examine how the SPAW evolves in the interplanetary medium, considering competing effects such as magnetic focusing in the Parker spiral and scattering from various sources. Coulomb collisions are studied as a specific case. The electron dynamics are described by a Fokker-Planck equation. We employed the stochastic differential equation formalism to derive an analytical expression for the SPAW—defined as the second moment of the pitch angle distribution—as a function of radial distance. These analytical formulas were compared with numerical solutions of the Fokker-Planck equation. We find that relatively simple formulas can be used to obtain robust estimates of the electrons' pitch-angle diffusion coefficients (or scattering mean-free paths) from SPAW data, especially when the scattering mean-free path is assumed to vary with distance as a power law. Additionally, the interplay between different scattering mechanisms can be tracked through the radial evolution of the SPAW. Notably, we show that the distance at which the SPAW is minimal, as observed by Parker Solar Probe, results from the competition between Coulomb collisions—which are dominant close to the Sun—and a turbulent scattering mechanism that prevails farther out.