Volume and surface area of a spherical harmonic surface approximation to a NIF implosion core defined by HGXI/GXD images from the equator and pole

A solid object, such as a simplified approximation to an implosion core defined by the 17% intensity contour, can be described by a sum of spherical harmonics, following the notation of Butkov (Mathematical Physics, ISBN 0-201-00727-4, 1968; there are other notations so care is required), with Pl(x) being the usual (apparently standard) Legendre polynomial. For the present purposes, finding the volume and surface area of an implosion core defined by P0, P2, P4, M0, and M4, I will restrict the problem to consider only A{sub 00}, A{sub 20}, A{sub 40}, and A{sub 44}, with the phase angle set to eliminate the sin(m{phi}) term. Once the volume and surface area are determined, I will explore how these coefficients relate to measured quantities A0, A2/A0, A4/A0, M0, and M4/M0