<p>The algorithm for estimating the uncertainties of the gravitational potential induced by polyhedral shape variations is evaluated for several asteroid shapes. The method provides variations in spherical harmonic coefficients by evaluating partial derivatives of their expressions with respect to shapes coordinates. Four stochastic models that describe the statistical behavior of the evaluated uncertainties are investigated, namely a Gaussian, an exponential, an inverse quadratic and an inverse multiquadric. The results are compared with the uncertainty range determined by differences in the gravity signal from various stochastic shape considerations using the line integral analytical approach. Our aim is to quantify the level of convergence between the derived uncertainties and the analytical gravity signal, as well as to investigate the interplay between the polyhedron's geometry, the stochastic parameters and the series maximum degree of expansion to the obtained uncertainty results. The numerical implementation is applied to three asteroids, namely Eros, Didymos and Dimorphos. Outside the uncertainty region defined by the Brillouin sphere, the differences between the computed normalized uncertainties of the gravitational potential and the spectrum limits derived from the analytical approach, range up to 0.1% for Eros, 0.75% for Didymos and 0.1% for Dimorphos. This agreement improves as the selected stochastic parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> decreases and as distance <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$l$</EquationSource> <EquationSource Format="MATHML"><math> <mi>l</mi> </math></EquationSource> </InlineEquation> increases. Inside the Brillouin sphere, the geometry of the considered asteroids strongly influences the uncertainty results, e.g., for the irregular shape of Eros the differences between the solutions with increasing degree of expansion range up to nearly 4% up to degree 20, while at the distance of 30 km these differences vanish from degree 2 on. Between the four examined stochastic models, the uncertainties derived from the adopted inverse multiquadric model achieve the highest convergence rate, approaching the analytical method down to 0.2%. Finally, the exponential model diverges from the other selections up to 2.2% in terms of a direct comparison between them.</p>

Implementation of different stochastic models in the frame of a dynamic polyhedral gravitational approach

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发表日期：

2025年8月28日

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Georgia Gavriilidou,
Dimitrios Tsoulis

摘要

The algorithm for estimating the uncertainties of the gravitational potential induced by polyhedral shape variations is evaluated for several asteroid shapes. The method provides variations in spherical harmonic coefficients by evaluating partial derivatives of their expressions with respect to shapes coordinates. Four stochastic models that describe the statistical behavior of the evaluated uncertainties are investigated, namely a Gaussian, an exponential, an inverse quadratic and an inverse multiquadric. The results are compared with the uncertainty range determined by differences in the gravity signal from various stochastic shape considerations using the line integral analytical approach. Our aim is to quantify the level of convergence between the derived uncertainties and the analytical gravity signal, as well as to investigate the interplay between the polyhedron's geometry, the stochastic parameters and the series maximum degree of expansion to the obtained uncertainty results. The numerical implementation is applied to three asteroids, namely Eros, Didymos and Dimorphos. Outside the uncertainty region defined by the Brillouin sphere, the differences between the computed normalized uncertainties of the gravitational potential and the spectrum limits derived from the analytical approach, range up to 0.1% for Eros, 0.75% for Didymos and 0.1% for Dimorphos. This agreement improves as the selected stochastic parameter $\sigma $ $σ$ decreases and as distance $l$ $l$ increases. Inside the Brillouin sphere, the geometry of the considered asteroids strongly influences the uncertainty results, e.g., for the irregular shape of Eros the differences between the solutions with increasing degree of expansion range up to nearly 4% up to degree 20, while at the distance of 30 km these differences vanish from degree 2 on. Between the four examined stochastic models, the uncertainties derived from the adopted inverse multiquadric model achieve the highest convergence rate, approaching the analytical method down to 0.2%. Finally, the exponential model diverges from the other selections up to 2.2% in terms of a direct comparison between them.