Divergence beneath the Brillouin sphere and the phenomenology of prediction error in spherical harmonic series approximations of the gravitational field

Abstract

Abstract 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, E N , on the absolute value of the prediction error, e N , of a SH series model, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , truncated at some maximum degree, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mo movablelimits="true">max</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math> . When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math> . Costin’s formula is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>≃</mml:mo><mml:mi>B</mml:mi><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> , where R is the radius of the Brillouin sphere. This formula depends on two positive parameters: b , which controls the decay of error amplitude as a function of N when r is fixed, and a scale factor B . We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound, A n on the absolute value of the TS coefficients, a n , for the same radial line. This formula, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≃</mml:mo><mml:mi>K</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> , depends on degree, n , and two positive parameters, k and K , that are analogous to b and B . We use synthetic planets, for which we can compute the potential, V , and also the radial component of gravitational acceleration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>∂</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>∂</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> , to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscript V refer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscript g to the coefficients and predictions errors associated with g r . For polyhedral planets of uniform density we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>7</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> almost 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 connecting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>K</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>B</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mi>g</mml:mi></mml:msup></mml:mrow></mml:math> , and B g . These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities are α and R . The phenomenology of ‘series divergence’ and prediction error (when r &lt; R ) can be described as a function of the truncation degree, N , or the depth, d , beneath the Brillouin sphere. For a fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>r</mml:mi><mml:mtext>⩽</mml:mtext><mml:mi>R</mml:mi></mml:mrow></mml:math> , as N increases from very low values, the upper error bound E N shrinks until it reaches its minimum (best) value when N reaches some particular or optimum value, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> . When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> , prediction error grows as N continues to increase. Eventually, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>≫</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> , prediction errors increase exponentially with rising N . If we fix the value of N and allow <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math> to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth, d , beneath the Brillouin sphere. Because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> everywhere, divergence driven prediction error intensifies more rapidly for g r than for V , both in terms of its dependence on N and d . If we fix both N and d , and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as does B ) as we approach high-amplitude topography.

Topics & Concepts