Spherical harmonic expansion of the photon angular distribution in the laboratory frame for photon emission by a rapidly moving source

Analysis of photon emission by a rapidly moving source requires making a relativistic transformation between the moving frame of reference and the laboratory frame of reference in which the photon detectors are located. Use of the exact relativistic transformation requires numerical integration over finite-sized detector geometries, a procedure which is inconvenient as well as not being conducive to making rotational transformations between different orientations of the coordinate systems. A spherical harmonic expansion of the photon angular distribution in the laboratory frame facilitates rotational transformations between coordinate systems during the data analysis as well as integration over the photon detector geometry. In this paper a relativistic transformation is derived that directly relates the spherical harmonic expansion of the photon distribution in the moving frame to the spherical harmonic expansion in the laboratory frame. The transformation involves an expansion to second order in ββ which is accurate to ≤0.1%≤0.1% when β≤0.1β≤0.1 and ≤1%≤1% when β≤0.2.β≤0.2. The usefulness of this spherical harmonic expansion becomes marginal for β≈0.3β≈0.3 when the second order expansion errors become several percent.