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,包含了组成地球的所有凝聚态物质。布里渊球在一个点上接触地球,从原点开始并穿过该点的径向线称为奇异径向线。大约60年来,人们一直在担心外部引力势是否存在球面谐波(SH)扩展,V,将汇聚在布里渊球下。最近,证明了这种收敛的概率为零。这些证明之一提供了渐近关系,叫做Costin\的公式,对于上限,EN,在预测误差的绝对值上,eN,SH系列模型,VN(θ,λ,r),最大程度地截断,N=nmax。当SH系列被限制为(或投影到)特定径向线时,它简化为1/r中的泰勒级数(TS)。Costin的公式是*BN-b(R/r)N,其中Ri是布里渊球的半径。这个公式取决于两个正参数:b,它控制误差幅度的衰减,作为Nwhenris固定的函数,和比例因子B。我们在这里证明Costin公式来自上界的类似渐近关系,根据TS系数的绝对值,an,对于相同的径向线。这个公式,AnKn-k,取决于程度,n,和两个正参数,kandK,类似于bandB。我们使用合成行星,我们可以计算出潜力,V,以及重力加速度的径向分量,gr=13CV/13Cr,到数百个有效数字,来验证这两个渐近公式。让superscriptV引用与引力系数和预测误差相关的渐近参数,以及与gr相关的系数和预测误差。对于密度均匀的多面体行星,我们表明bV=kV=7/2和bg=kg=5/2几乎无处不在。我们证明了TS系数的振荡频率(约为零)和序列预测误差,对于给定的径向线,由地心角控制,α,在该径向线和奇异径向线之间。我们还推导出连接KV的有用身份,BV,Kg,和Bg.这些身份以各种比例因子的商表示。这些身份中唯一涉及的其他数量是α和R。“级数发散”和预测误差(当R,从非常低的值增加,当达到某个特定或最优值时,上误差边界会缩小,直到达到其最小(最佳)值,Nopt.当N>Nopt,预测误差随着N的不断增加而增加。最终,当NNopt,预测误差随N的增加呈指数增加。如果我们确定值的值,并允许R/r变化,然后我们发现布里渊球下自由空间的预测误差随深度呈指数增加,D,布里渊球下.因为bg=bV-1无处不在,发散驱动的预测误差比V更快地加剧,无论是对Nandd的依赖。如果我们把Nandd都修好,并关注预测误差的“横向”变化,我们观察到,随着我们接近高振幅地形,发散和预测误差趋于增加(正如B那样)。
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.