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<p>Divergence of the spherical harmonic gravity field is a well-known issue. This divergence is caused by the divergence of the generating function for the Legendre polynomial inside the Brillouin (circumscribing) sphere of the body. Interior to this sphere, a complementary expansion in terms of spherical Bessel harmonics provides a rigorous model of the field. Our past article derived an algorithm for uniquely calculating the Bessel harmonic coefficients directly from the surface integral of an arbitrary shape with homogeneous density. In another article, we solved for these harmonics for the simplest body: a homogeneous sphere. We discovered that this special case has an analytical expression. Coincidentally, we realize that this is the same problem Newton solved in his Principia in order to derive the Shell Theorem. The equivalency is corroborated by some special infinite series, involving the Riemann zeta function, Dirichlet series, and Bernoulli polynomials, to name a few. In this article, we provide the full derivation and elaborate on the intricate properties of the interior spherical Bessel harmonics.</p>

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On the connection between Newton’s shell theorem, interior spherical Bessel gravity field, and special infinite series

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Yu Takahashi,
  • Justin R. Mansell,
  • Yohsuke Matsuzawa

摘要

Divergence of the spherical harmonic gravity field is a well-known issue. This divergence is caused by the divergence of the generating function for the Legendre polynomial inside the Brillouin (circumscribing) sphere of the body. Interior to this sphere, a complementary expansion in terms of spherical Bessel harmonics provides a rigorous model of the field. Our past article derived an algorithm for uniquely calculating the Bessel harmonic coefficients directly from the surface integral of an arbitrary shape with homogeneous density. In another article, we solved for these harmonics for the simplest body: a homogeneous sphere. We discovered that this special case has an analytical expression. Coincidentally, we realize that this is the same problem Newton solved in his Principia in order to derive the Shell Theorem. The equivalency is corroborated by some special infinite series, involving the Riemann zeta function, Dirichlet series, and Bernoulli polynomials, to name a few. In this article, we provide the full derivation and elaborate on the intricate properties of the interior spherical Bessel harmonics.