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, 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) in 1/r . Costin’s formula is EN≃BN−b(R/r)N , 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, An≃Kn−k , 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, gr=∂V/∂r , 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 bV=kV=7/2 and bg=kg=5/2 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 KV,BV,Kg , 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 < R) can be described as a function of the truncation degree, N, or the depth, d, beneath the Brillouin sphere. For a fixed r⩽R , 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, Nopt . When N>Nopt , prediction error grows as N continues to increase. Eventually, when N≫Nopt , prediction errors increase exponentially with rising N. If we fix the value of N and allow R/r 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 bg=bV−1 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.

中文翻译：

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布里渊球下的散度和引力场球谐级数近似中的预测误差现象学


布里渊球被定义为最小的球体，以地心坐标系的原点为中心，包含构成行星的所有凝聚态物质。布里渊球在一点与地球接触，从原点开始并经过该点的径向线称为奇异径向线。大约 60 年来，人们一直担心外部引力势是否会发生球谐函数 (SH) 膨胀， V ，将在布里渊球下方汇聚。最近，事实证明这种收敛的概率为零。其中一个证明为上限提供了一种渐近关系，称为 Costin 公式， _CN ，关于预测误差的绝对值， _eN ，SH系列型号， V氮（ θ , λ , r ） ，在某个最大程度处被截断，氮= n最大限度。 当 SH 级数限制在（或投影到）特定的径向线上时，它会简化为泰勒级数 (TS) 1 / r 。科斯汀的公式是乙氮≃乙氮-乙（右/ r ）氮， 在哪里右是布里渊球的半径。该公式取决于两个正参数：乙，它控制误差幅度的衰减作为函数氮什么时候r是固定的，并且比例因子乙。我们在此证明 Costin 公式源自类似的上限渐近关系，_一个关于 TS 系数的绝对值，_一个，对于同一条径向线。 这个公式，一个n ≃ K n - k ，取决于程度， n ，和两个正参数， k和K ，类似于乙和乙。我们使用合成行星，我们可以计算其潜力， V ，以及重力加速度的径向分量，克r = ∂ V / ∂ r ，到数百位有效数字，以验证这两个渐近公式。让上标V指与引力势系数和预测误差相关的渐近参数，上标克与相关的系数和预测误差_克。 对于密度均匀的多面体行星，我们证明 乙V = k V = 7 / 2和 乙克= k克= 5 / 2几乎无处不在。我们表明，对于给定的径向线，TS 系数的振荡频率（在零附近​​）和序列预测误差是由地心角控制的， α ，在该径向线和奇异径向线之间。我们还得出有用的身份连接 K V ,乙V , K克， 和^血红蛋白。这些恒等式以各种比例因子的商来表示。这些恒等式中涉及的唯一其他数量是α和右。 “级数分歧”和预测误差的现象学（当r <右) 可以描述为截断度的函数，氮，或深度， d ，位于布里渊球下方。对于固定的r ⩽右， 作为氮从非常低的值开始增加，误差上限_CN收缩直到达到其最小值（最佳）值氮达到某个特定或最佳值， 氮选择。什么时候氮>氮选择，预测误差增长为氮持续增加。最终，当氮≫氮选择，预测误差随着增加而呈指数增加氮。 如果我们固定值氮并允许右/ r变化，然后我们发现布里渊球下方自由空间的预测误差随着深度呈指数增加， d ，位于布里渊球下方。因为 乙克=乙V - 1在任何地方，分歧驱动的预测误差都会更快地加剧_克比V ，无论是在其依赖性方面氮和d 。如果我们修复两者氮和d ，并关注预测误差的“横向”变化，我们观察到分歧和预测误差趋于增加（正如乙）当我们接近高振幅地形时。