The Liquid Droplet Radiator is a proposed lightweight radiator for the dissipation of waste heat generated by power plants in space. The hot working fluid is sprayed into space as coherent streams of tiny, discrete droplets, which then cool by transient radiative heat transfer, and are later collected for recirculation. An improved, hybrid trial function is presented for the solution by Galerkin's method of the equation of radiative transfer, which governs the emission, scattering, and absorption within the layer formed by the droplet streams. The trial function compensates for the effects of the particular solution corresponding to the inhomogeneous source term. The method of analysis is demonstrated for a source function having a polynomial profile. The improved-profile Galerkin solution is then applied to a non-dimensional analysis of gray, isotropically, scattering droplet layers which are asymmetrically heated by normal, isotropic, external radiation. The analysis identifies a critical value for the magnitude of external radiation. Droplet layers exposed to external radiation greater than this critical value will heat initially, and then either recover and cool, or for magnitudes sufficiently stronger than the critical value will achieve a state of equilibrium.