Journal Article

Estimate for the decay rate of the general term of the laplace series for the geopotential

2011

Overview

The exact estimates with respect to the uniform (Chebyshev) norm of the general term Vn of the Laplace series in spherical harmonics for the gravitational potential of a planet are known to be functions of the differential properties of distribution of masses. However, it is difficult to use them in practice because the norm for large n is difficult to calculate. The mean square (Euclidean) norm, however, is used almost entirely in applied research, and so the translation of the estimates from one norm to the other can be effected only in particular cases. In the present paper, a power law estimate of the general term of the Laplace series with respect to the mean square norm is obtained, and its parameters are evaluated numerically. For the EGM2008 geopotential model (Pavlis et al., 2008), the exponent ranges between 2.7 and 3.4. Preliminary calculations are made for a model body for which the harmonic factors are known exactly. A cautious conclusion is made: the geopotential model put forward is capable of adequately describing spherical harmonics for n ≤ 1000; for 1000 ≤ n ≤ 2000 the model gives a correct description, at least qualitatively; for larger n the spherical harmonics of the model do not correspond to reality.

• Publisher: Springer Nature B.V
• Subject: Chebyshev approximation / Decay rate / Estimates / Geophysics / Geopotential / Gravity / Mathematical models / Partial differential equations / Spherical harmonics