Initial matter profiles used in the deleptonization problem, taken from Liebendörfer et al. (2005). Plotted vs. radius are mass density (upper left panel), temperature (upper right panel), and electron fraction (lower left panel) for various postbounce times: 50 (solid), 100 (dashed), 150 (dotted), and 250 (dashed–dotted) ms. The neutrinosphere radii, defined in Equation (76), for the respective profiles—for electron neutrinos (black) and electron antineutrinos (gray)—are also plotted vs. neutrino energy (lower right panel).