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<p>The oblate spheroidal harmonic algorithms for forward modelling for the external gravitational field of polyhedral bodies with constant density including the gravitational potential and its derivatives up to third order are presented. The volume integral forms of the oblate spheroidal harmonic potential coefficients are converted into the surface integrals over the polygonal faces of the polyhedron using the Gauss divergence theorem and then into the line integrals along the edges of the polyhedron using the Stokes theorem. The discontinuities of the oblate spheroidal coordinates on the equatorial circular disc and the half-plane with the longitude being zero or perigon (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="190_2025_1954_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(2\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>) as well as the vertical axis which may lead to the discontinuous vector field and its derivatives, are taken into account in the integral conversions. The normalized forms of the algorithms of the oblate spheroidal harmonic coefficients are given for numerical implementations and can handle the oblate bodies with eccentricity of any size and high- and ultra-high-degree harmonic coefficients. The numerical experiments with two tested polyhedral bodies including the asteroids Atlas and 4 Vesta whose circular discs intersect with or are enclosed by the surfaces of the bodies on the equatorial planes show the good convergences and numerical accuracies with the truncated degree/order (d/o) of the oblate spheroidal harmonic expansions up to d/o 300 and the stabilities for computing the harmonic coefficients up to d/o 300 using the line integral algorithms. Compared with the spherical harmonic expansions of the external gravitational field, the oblate spheroidal harmonic expansions converge faster and also have better achievable precisions of the numerical results.</p>

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Spheroidal harmonic expansions for the gravitational field of homogeneous polyhedral bodies I: using oblate spheroidal harmonics

  • 期刊论文

  • 发表日期：

    2025年5月27日

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  • Cheng Chen,
  • Shaofeng Bian

摘要

The oblate spheroidal harmonic algorithms for forward modelling for the external gravitational field of polyhedral bodies with constant density including the gravitational potential and its derivatives up to third order are presented. The volume integral forms of the oblate spheroidal harmonic potential coefficients are converted into the surface integrals over the polygonal faces of the polyhedron using the Gauss divergence theorem and then into the line integrals along the edges of the polyhedron using the Stokes theorem. The discontinuities of the oblate spheroidal coordinates on the equatorial circular disc and the half-plane with the longitude being zero or perigon ( \(2\pi \) 2 π ) as well as the vertical axis which may lead to the discontinuous vector field and its derivatives, are taken into account in the integral conversions. The normalized forms of the algorithms of the oblate spheroidal harmonic coefficients are given for numerical implementations and can handle the oblate bodies with eccentricity of any size and high- and ultra-high-degree harmonic coefficients. The numerical experiments with two tested polyhedral bodies including the asteroids Atlas and 4 Vesta whose circular discs intersect with or are enclosed by the surfaces of the bodies on the equatorial planes show the good convergences and numerical accuracies with the truncated degree/order (d/o) of the oblate spheroidal harmonic expansions up to d/o 300 and the stabilities for computing the harmonic coefficients up to d/o 300 using the line integral algorithms. Compared with the spherical harmonic expansions of the external gravitational field, the oblate spheroidal harmonic expansions converge faster and also have better achievable precisions of the numerical results.