SCIENTIA SINICA Physica, Mechanica & Astronomica, Volume 49 , Issue 8 : 084507(2019) https://doi.org/10.1360/SSPMA-2019-0011

## Large-scale modeling of parametric asteroid surfaces using polynomial series

• AcceptedFeb 19, 2019
• PublishedJun 5, 2019
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### Abstract

In the last twenty years, numerous deep space probes have been sent to space to detect asteroids, comets and planetary satellites, indicating that small Solar System bodies have become one of the main targets for deep space exploration. Many of asteroids are remnants of the materials from which the solar system accreted. The asteroid contains information about the origins and development of the solar system. The size and shape of the asteroid contains their thermal, collision, and kinetic history, and is fundamental to the analysis of gravitational field and dynamics of the asteroid. Studies on the asteroid geological structure shows that most asteroids have asymmetric and irregular mass distributions. It posed a challenge in accurately modelling of such asteroids’ surfaces. And for the solar system small bodies, the most widely-applied method is to discretize their surfaces is the polyhedron method. However, the shape model of the asteroid obtained by polyhedron method is triangular mesh model. In order to study the dynamics of the asteroid’s surface, we need to build a continuous, analytical model of asteroid surfaces. For these questions, the following research is done in this paper. Firstly, the CALD algorithm which controls both length distortion and area distortion is used to map the sample point data of the asteroid to the unit sphere. After obtaining the bijection relationship between the asteroid surface and the unit sphere, the polynomial series is used as the basis function to characterize the bijection relationship. Then, the sample point coordinates and the corresponding coordinates on the unit sphere are substituted into the polynomial model established to obtain a set of large-scale linear equations. Since the fundamental function of polynomial series is non-orthogonal, the system is a pathological equation. In this paper, the method of solving large-scale ill equations is adopted to control the error of solving equations within acceptable range. Finally, after the establishment of the polynomial model, this paper discusses in detail the modified parameters in the polynomial model that affect the model accuracy.

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• 图 1

CALD算法实现球面参数化. (a) (216) kleopatra; (b) (1620) geographos; (c) (1998) ky26; (d) (433) Eros; (e) (2063) bacchus; (f) (6489) golevka

• 图 2

球坐标的符号约定

• 图 3

(1620) geographos小行星表面及其多项式重构. (a) (1620) geographos; (b) 一阶重构; (c) 五阶重构; (d) 十二阶重构

• 图 4

小天体的重构模型. (a) (216) kleopatra; (b) (1620) geographos; (c) (1998) ky26; (d) (433) Eros; (e) (2063) bacchus; (f) (6489) golevka

• 图 5

修正参数对模型重构精度的影响

• Table 1   The area distortion cost of spherical parameterization
 小天体 初始参数化($Ca$) 最终的球面参数化($Ca$) (216) kleopatra 3362.608701 1.380304 (1620) geographos 1.770708 1.322214 (1998) ky26 2.539985 1.251203 (433) Eros 1.504304 1.216207 (2063) bacchus 1.606943 1.206612 (6489) golevka 2.196672 1.564225
• Table 2   Model error
 小天体 $esum$ $emax$ (216) kleopatra 0.009201 3.883887 (1620) geographos 0.007357 0.054517 (1998) ky26 0.009439 0.000566 (433) Eros 0.005609 0.377891 (2063) bacchus 0.007792 0.01279 (6489) golevka 0.02113 0.022253
• Table 3   The influence of modified parameters on the precision of model reconstruction
 修正参数 $A$的条件数 $esum$ $emax$ $1/k!$ $2.480×1040$ 0.023342 6.866587 1 $1.080×1021$ 0.009201 3.883887 $ek$ $9.788×1021$ 0.008976 3.871908 $6k$ $1.091×1024$ 0.009825 3686919. $10k$ $3.523×1025$ 0.014497 4.325611 $k!$ $2.649×1027$ 0.028531 10.829981

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