<p>The gravitational potential and its first-order derivatives induced by finite mass distributions are evaluated numerically and analytically. Three asteroid shape models have been used for the implementation, namely Eros which is the most irregular one, Didymos which is nearly spherical and Dimorphos which is a perfect ellipsoid. For the numerical approach, the spherical harmonic series up to maximum expansion degree 100 were computed. For the analytical approach on the other hand, the line integral algorithm of general polyhedra was applied. The two methods are compared in terms of numerical convergence between them with respect to maximum expansion degree of the corresponding harmonic series and relative position between computation point and modeled body. Additionally, emphasis is given on the different geometric characteristics of the applied shapes and their influence on the induced gravity signal evaluation. The results are separated for points located inside and outside the Brillouin sphere. Inside Brillouin sphere, better agreement between methods is provided for the case of Didymos due to its spherical-like shape. Outside Brillouin sphere, Dimorphos secured the highest convergence between analytical and numerical methods, due to its smooth exterior boundary, with the maximum difference being <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10712_2025_9892_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$6\text{E}{-10}\text{ m}^2/\text{s}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>6</mn> <mtext>E</mtext> <mrow> <mo>-</mo> <mn>10</mn> </mrow> <mspace width="0.333333em" /> <msup> <mtext>m</mtext> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mtext>s</mtext> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for the gravitational potential and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10712_2025_9892_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$7\text{E}{-10}\text{ m}/\text{s}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>7</mn> <mtext>E</mtext> <mrow> <mo>-</mo> <mn>10</mn> </mrow> <mspace width="0.333333em" /> <mtext>m</mtext> <mo stretchy="false">/</mo> <msup> <mtext>s</mtext> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for its first-order derivatives. For gravitational potential the highest differences are observed for Eros (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10712_2025_9892_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$2\text{E}{-}6\text{ m}^2/\text{s}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mtext>E</mtext> <mo>-</mo> <mn>6</mn> <mspace width="0.333333em" /> <msup> <mtext>m</mtext> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mtext>s</mtext> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>). For the first-order derivatives, both Eros and Didymos provided differences of the same magnitude, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10712_2025_9892_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$1\text{E}{-}8\text{ m}^2/\text{s}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mtext>E</mtext> <mo>-</mo> <mn>8</mn> <mspace width="0.333333em" /> <msup> <mtext>m</mtext> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mtext>s</mtext> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10712_2025_9892_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$2\text{E}{-}8\text{ m}^2/\text{s}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mtext>E</mtext> <mo>-</mo> <mn>8</mn> <mspace width="0.333333em" /> <msup> <mtext>m</mtext> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <msup> <mtext>s</mtext> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. Finally, regarding the maximum expansion degree, the convergence between the two methods at degree 100 for Eros and Didymos are provided by Dimorphos at degrees 27 and 35, respectively.</p>

Degree Wise Validation of Gravity Spherical Harmonics for Polyhedral Sources

期刊论文
发表日期：

2025年6月30日

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Dimitrios Tsoulis,
Georgia Gavriilidou,
Mohammad Poursina,
Margrethe Wold

摘要

The gravitational potential and its first-order derivatives induced by finite mass distributions are evaluated numerically and analytically. Three asteroid shape models have been used for the implementation, namely Eros which is the most irregular one, Didymos which is nearly spherical and Dimorphos which is a perfect ellipsoid. For the numerical approach, the spherical harmonic series up to maximum expansion degree 100 were computed. For the analytical approach on the other hand, the line integral algorithm of general polyhedra was applied. The two methods are compared in terms of numerical convergence between them with respect to maximum expansion degree of the corresponding harmonic series and relative position between computation point and modeled body. Additionally, emphasis is given on the different geometric characteristics of the applied shapes and their influence on the induced gravity signal evaluation. The results are separated for points located inside and outside the Brillouin sphere. Inside Brillouin sphere, better agreement between methods is provided for the case of Didymos due to its spherical-like shape. Outside Brillouin sphere, Dimorphos secured the highest convergence between analytical and numerical methods, due to its smooth exterior boundary, with the maximum difference being $6\text{E}{-10}\text{ m}^2/\text{s}^2$ $6 E - 10 m^{2} / s^{2}$ for the gravitational potential and $7\text{E}{-10}\text{ m}/\text{s}^2$ $7 E - 10 m / s^{2}$ for its first-order derivatives. For gravitational potential the highest differences are observed for Eros ( $2\text{E}{-}6\text{ m}^2/\text{s}^2$ $2 E - 6 m^{2} / s^{2}$ ). For the first-order derivatives, both Eros and Didymos provided differences of the same magnitude, $1\text{E}{-}8\text{ m}^2/\text{s}^2$ $1 E - 8 m^{2} / s^{2}$ and $2\text{E}{-}8\text{ m}^2/\text{s}^2$ $2 E - 8 m^{2} / s^{2}$ . Finally, regarding the maximum expansion degree, the convergence between the two methods at degree 100 for Eros and Didymos are provided by Dimorphos at degrees 27 and 35, respectively.