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We present two novel additions to the semi-analytic solution of Lyman$\alpha$ (Ly$\alpha$) radiative transfer in spherical geometry: (1)implementation of the correct boundary condition for a steady source, and (2)solution of the time-dependent problem for an impulsive source. For thesteady-state problem, the solution can be represented as a sum of two terms: apreviously-known analytic solution of the equation with mean intensity $J=0$ atthe surface, and a novel, semi-analytic solution which enforces the correctboundary condition of zero-ingoing intensity at the surface. This solution iscompared to that of the Monte Carlo method, which is valid at arbitrary opticaldepth. It is shown that the size of the correction is of order unity when thespectral peaks approach the Doppler core and decreases slowly with line centeroptical depth, specifically as $(a \tau_0)^{-1/3}$, which may explaindiscrepancies seen in previous studies. For the impulsive problem, the time,spatial, and frequency dependence of the solution are expressed using aneigenfunction expansion in order to characterize the escape time distributionand emergent spectra of photons. It is shown that the lowest-ordereigenfrequency agrees well with the decay rate found in the Monte Carlo escapetime distribution at sufficiently large line-center optical depths. Thecharacterization of the escape-time distribution highlights the potential for aMonte Carlo acceleration method, which would sample photon escape propertiesfrom distributions rather than calculating every photon scattering, therebyreducing computational demand.

A Novel Solution for Resonant Scattering Using Self-Consistent Boundary Conditions