Evolution of Tau-Neutrino Lepton Number in Protoneutron Stars due to Active-Sterile Neutrino Mixing

We present an approximate treatment of the mixing between ${\nu }_{\tau }$ (${\bar{\nu }}_{\tau }$) and a sterile species ${\nu }_s$ (${\bar{\nu }}_s$) with a vacuum mass-squared difference of $\sim$ 10${}^2$-10${}^3$ keV${}^2$ in protoneutron stars created in core-collapse supernovae. Including production of sterile neutrinos through both resonant flavor conversion and collisions, we track the evolution of the ${\nu }_{\tau }$ lepton number due to both escape of sterile neutrinos and diffusion. Our approach provides a reasonable treatment of the pertinent processes discussed in previous studies and serves a pedagogical purpose to elucidate the relevant physics. We also discuss refinements needed to study more accurately how flavor mixing with sterile neutrinos affects protoneutron star evolution.