Celestial Mechanics and Dynamical Astronomy

, Volume 122, Issue 4, pp 391–403 | Cite as

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

  • Konstantin V. Kholshevnikov
  • Vakhit Sh. Shaidulin
Original Article


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 \(\sigma =3\).


Gravitational potential Laplace series Convergence rate Conic bodies 



We are greatful to the reviewers for valuable remarks. This work is supported by Saint Petersburg State University, research Grant 6.37.341.2015.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Konstantin V. Kholshevnikov
    • 1
    • 2
  • Vakhit Sh. Shaidulin
    • 1
    • 3
  1. 1.St. Petersburg State UniversitySt. Petersburg, Stary PeterhofRussia
  2. 2.Institute of Applied Astronomy of Russian Academy of SciencesSt. PetersburgRussia
  3. 3.Main (Pulkovo) Astronomical Observatory of Russian Academy of SciencesSt. PetersburgRussia

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