On the exactness of estimates for irregularly structured bodies of the general term of Laplace series

Abstract

The main form of the representation of a gravitational potential V for a celestial body T in outer space is the Laplace series in solid spherical harmonics \((R/r)^{n+1}Y_n(\theta ,\lambda )\) with R being the radius of the enveloping T sphere. The surface harmonic \(Y_n\) satisfies 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 with an irregular structure \(\sigma = 5/2\), and this value cannot be increased generally. However, a class of irregular bodies (smooth bodies with peaked mountains) has been found recently in which \(\sigma = 3\). In this paper, we will prove the exactness of this estimate, showing that a body belonging to the above class does exist and $$\begin{aligned} 0<\varlimsup n^3\langle Y_n\rangle <\infty \end{aligned}$$ for it.