Existence of a class of irregular bodies with a higher convergence rate of Laplace series for the gravitational potential

Abstract

The main form of the representation of a gravitational potential $V$ for a celestial body $T$ in the outer space is the Laplace series in solid spherical harmonics $(R/r)^{n+1}Y_n(R,\theta ,\lambda )$ with $R$ being the radius of enveloping $T$ sphere. It is well known that $Y_n$ satisfy the inequality

$$\begin{aligned} \langle Y_n\rangle <Cn^{-\sigma }. \end{aligned}$$

The angular brackets mark the maximum of a function’s modulus over a unit sphere. For bodies of irregular structure $\sigma =5/2$, and this value cannot be increased in general case. At the same time modern models of the geopotential show more rapid rate of decreasing of $Y_n$. We have found a class $\mathcal {T}$ of irregular bodies for which $\sigma =3$. The Earth and (at least a part of) other terrestrial planets, satellites, and asteroids most likely belong to this class. In this paper we describe $\mathcal {T}$ proving the above inequality for $\sigma =3$.