New proof of the convergence of the Laplacian series at physical surfaces of the planets (Q1081054)

It is proven that the external gravitational potential of a planet may be written as a series in spherical functions \[ V(r,\phi,\lambda)=f\sum^{\infty}_{n=0}\frac{1}{r^{n+1}}\int \mu (r',\phi',\lambda')P_ n(\cos \gamma)r^{'n+2} dr'd \sin \phi'd\lambda', \] uniformly converging everywhere on and outside the physical surface of the planet, with a normalized nonlinear dispersion of the relief height conforming to the condition \[ \bar D_ J<\frac{1+\hat H/R}{\beta J\sqrt{2J+1}}(\frac{R-\Delta}{R+\hat H})^ J, \] where \(\Delta >0\) is the maximum depth of the relief with respect to the mean sphere radius R; \(\hat H\) is the height; the density of the surface masses (lying above the sphere passing through the observation point r, \(\phi\), \(\lambda)\) is an analytic function of the coordinates.