变密度球体重力位的推导及其球谐展开

The gravity potential of varied-density sphere and its spherical harmonic expansion

天体内部密度的变化,往往引起天体表面重力位的非均匀分布.在计算变密度的重力位时,普通数值积分耗时较长,效率低下,不利于大规模的重力正反演.针对该问题,提出基于广义傅里叶变换的球谐算法,发现重力位球谐展开系数能直接由密度展开系数求出,简化了变密度问题重力位繁琐的积分过程.将算法应用于具体事例,发现球谐算法估算的重力位分布与普通数值积分方法所得一致,而计算时间得到了极大的降低.将本文算法应用于月壳变密度问题,发现相应的重力位分布与密度分布规律一致,验证了算法的合理性和实用性.

The varied-density in the interior of a celestial body,generally results in an anisotropic gravity potential outside of the celestial.The general numerical integration method needs too much time in the potential determination to be efficiently applied in a large-scaled gravity inversion as well as the forward case.To deal with this problem,this paper proposes an updated model of spherical harmonic expansion,which is based on the generalized Fourier Transformation.Results indicate that the spherical harmonic expanded coefficients of the gravity potential can be directly estimated according to the expanded coefficients of density.Our method simplifies the potential calculation in the case of density varied in a celestial body.By applying our algorithm to the actual case,the potential from our approach shows a similar distribution with the general method,but our method spents less time in the computation than its counterpart.To the issue of lunar crustal density varied,the estimated gravity potential from our method shows the same behavior with the varied-density.Our method is validated and can be applied in the gravity inversion as well as its forward problem.