Divergence beneath the Brillouin sphere and the phenomenology of prediction error in spherical harmonic series approximations of the gravitational field
Authors: Bevis M , Ogle C , Costin O , Jekeli C , Costin R D , Guo J , Fowler J , Dunne G V , Shum C K , Snow K 
Published: June 2024

The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential,V, will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin’s formula, for the upper bound,EN, on the absolute value of the prediction error,eN, of a SH series model,VN(θ,λ,r), truncated at some maximum degree,N=nmax. When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in1/r. Costin’s formula isEN≃BN−b(R/r)N, whereRis the radius of the Brillouin sphere. This formula depends on two positive parameters:b, which controls the decay of error amplitude as a function ofNwhenris fixed, and a scale factorB. We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound,Anon the absolute value of the TS coefficients,an, for the same radial line. This formula,An≃Kn−k, depends on degree,n, and two positive parameters,kandK, that are analogous tobandB. We use synthetic planets, for which we can compute the potential,V, and also the radial component of gravitational acceleration,gr=∂V/∂r, to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscriptVrefer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscriptgto the coefficients and predictions errors associated withgr. For polyhedral planets of uniform density we show thatbV=kV=7/2andbg=kg=5/2almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle,α, between that radial line and the singular radial line. We also derive useful identities connectingKV,BV,Kg, andBg. These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities areαandR. The phenomenology of ‘series divergence’ and prediction error (whenr < R) can be described as a function of the truncation degree,N, or the depth,d, beneath the Brillouin sphere. For a fixedr⩽R, asNincreases from very low values, the upper error boundENshrinks until it reaches its minimum (best) value whenNreaches some particular or optimum value,Nopt. WhenN>Nopt, prediction error grows asNcontinues to increase. Eventually, whenN≫Nopt, prediction errors increase exponentially with risingN. If we fix the value ofNand allowR/rto vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth,d, beneath the Brillouin sphere. Becausebg=bV−1everywhere, divergence driven prediction error intensifies more rapidly forgrthan forV, both in terms of its dependence onNandd. If we fix bothNandd, and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as doesB) as we approach high-amplitude topography.