Recent Advances in Modeling Gravity Field of Small Bodies

Abstract

Small celestial bodies in the Solar system，such as asteroids and comets，have become the key targets in the field of deep space exploration. The exploration of small celestial bodies is of great significance for studying the formation and evolution of the solar system，the origin of life，planetary defense，and the exploitation of space resources. With the continuous development of aerospace technology，the way of small-body exploration has gradually shifted from flyby and orbiting to close-proximity detection methods，such as landing，sampling，and touring. The construction of a high-accuracy gravitational field model in the vicinity of the given small body is crucial to the design and implementation of such close-proximity exploration missions and to the study of dynamics near the small bodies. Thus，this paper first reviews the two-hundred-year history of development of the modeling of gravitational field，and elucidates the basic principles and drawbacks in the different methods. The relationship between the gravitational field near the binary asteroid system and the dynamics of such system is second reviewed. This paper also reviews the techniques for modelling the gravitational interactions in the study of the dynamics of the binary asteroid system. These techniques provide fundamental tools for the binary system mission designs and the study of the formation and evolution of the binaries. Finally，the future research trends are discussed.

Key words： small body; gravity field model; binary asteroid system; gravitational interactions

References

[1] CARRY B. Density of asteroids[J]. Planetary and Space Science,2012,73(1):98-118
[2] YEOMANS D. Small bodies of the solar system[J]. Nature,2000,404(6780):829-832
[3] IRVINE W M,LESCHINE S B,SCHLOERB F P. Thermal history,chemical composition and relationship of comets to the origin of life[J]. Nature,1980,283(5749):748-749
[4] LI M,HUANG Y K,GONG S P. Assessing the risk of potentially hazardous asteroids through mean motion resonances analyses[J]. Astrophysics and Space Science,2019,364(5):1-12
[5] HARRIS A W,D’ABRAMO G. The population of near-Earth asteroids[J]. Icarus,2015(257):302-312
[6] SONTER M. The technical and economic feasibility of mining the near-Earth asteroids[J]. Acta Astronautica,1997,41(4):637-647
[7] INGEBRETSEN M. Mining asteroids[J]. IEEE Spectrum,2001,38(8):34-39
[8] JOHNSON T V，YEATES C M，YOUNG R. Space science reviews volume on Galileo mission overview[M]. Dordrecht：Springer Netherlands，1992.
[9] RAYMAN M D,VARGHESE P. The deep space 1 extended mission[J]. Acta Astronautica,2001,48(5):693-705
[10] RAYMAN M D,FRASCHETTI T C,RAYMOND C A,et al. Dawn:a mission in development for exploration of main belt asteroids vesta and ceres[J]. Acta Astronautica,2006,58(11):605-616
[11] ZOU X,LI C,LIU J,et al. The preliminary analysis of the 4179 toutatis snapshots of the Chang’E-2 flyby[J]. Icarus,2014,229:348-354
[12] 黄江川,王晓磊,孟林智. 嫦娥二号卫星飞越 4179 小行星工程参数分析[J]. 中国科学:技术科学,2013,43(6):596-601
[13] 乔栋,黄江川,崔平远. 嫦娥二号卫星飞越探测小行星的目标选择[J]. 中国科学:技术科学,2013,43(6):602-608
[14] FUJIWARA A,KAWAGUCHI J,YEOMANS D K,et al. The rubble-pile asteroid itokawa as observed by hayabusa[J]. Science,2006,312(5778):1330-1334
[15] WATANABE S I,TSUDA Y,YOSHIKAWA M,et al. Hayabusa 2 mission overview[J]. Space Science Reviews,2017,208(1):3-16
[16] TAYLOR M G G T,ALTOBELLI N,BURATTI B J,et al. The Rosetta mission orbiter science overview:the comet phase[J]. Philosophical Transactions of the Royal Society A:Mathematical,Physical and Engineering Sciences,2017,375(2097):1-16
[17] YEOMANS D K,ANTREASIAN P G,BARRIOT J P,et al. Radio science results during the NEAR-Shoemaker spacecraft rendezvous with eros[J]. Science,2000,289(5487):2085-2088
[18] LAURETTA D S,BALRAM-KNUTSON S S,BESHORE E,et al. OSIRIS-REx:sample return from asteroid (101955) Bennu[J]. Space Science Reviews,2017,212(1):925-984
[19] YOSHIMITSU T，KUBOTA T，NAKATANI I. Operation of Minerva rover in Hayabusa asteroid mission[C]//AIAA 57th International Astronautical Congress. Valencia，Spain：IAC，2006.
[20] CHENG A F,ATCHISON J,KANTSIPER B,et al. Asteroid impact and deflection assessment mission[J]. Acta Astronautica,2015(115):262-269
[21] THOMSON W，TAIT P G. Cambridge library collection- mathematics：volume 1 treatise on natural philosophy[M]. 2nd ed. ，Cambridge：Cambridge University Press，2009.
[22] HOBSON E W. The theory of spherical and ellipsoidal harmonics[M]. New York：Chelsea Publishing Company，1931.
[23] WALTER H G. Association of spherical and ellipsoidal gravity coefficients of the Earth’s potential[J]. Celestial Mechanics,1970,2(3):389-397
[24] DECHAMBRE D,SCHEERES D J. Transformation of spherical harmonic coefficients to ellipsoidal harmonic coefficients[J]. Astronomy & Astrophysics,2002,387(3):1114-1122
[25] HU X,JEKELI C. A numerical comparison of spherical,spheroidal and ellipsoidal harmonic gravita tional field models for small non-spherical bodies:examples for the martian moons[J]. Journal of Geodesy,2015,89(2):159-177
[26] HU X. The exact transformation from spherical harmonic to ellipsoidal harmonic coefficients for gravitational field modeling[J]. Celestial Mechanics and Dynamical Astronomy,2016,125(2):195-222
[27] 梁磊,于锦海,万晓云. 椭球谐和球谐系数之间一个简单的转换关系[J]. 测绘学报,2019,48(2):185-190
LIANG L,YU J H,WAN X Y. A simple transformation between ellipsoidal harmonic cofficients and spherical harmonic cofficients[J]. Acta Gedaetica et Cartographica Sinica,2019,48(2):185-190
[28] GARMIER R,BARRIOT J P. Ellipsoidal harmonic expansions of the gravitational potential:theory and application[J]. Celestial Mechanics and Dynamical Astronomy,2001,79(4):235-275
[29] TAKAHASHI Y,SCHEERES D J. Surface gravity fields for asteroids and comets[J]. Journal of Guidance,Control,and Dynamics,2013,36(2):362-374
[30] HERRERA-SUCARRAT E,PALMER P L,ROBERTS R M. Modeling the gravitational potential of a nonspherical asteroid[J]. Journal of Guidance,Control,and Dynamics,2013,36(3):790-798
[31] TAKAHASHI Y,SCHEERES D J. Small body surface gravity fields via spherical harmonic expansions[J]. Celestial Mechanics and Dynamical Astronomy,2014,119(2):169-206
[32] RICHARDSON D,MICHEL P,WALSH K,et al. Numerical simulations of asteroids modelled as gravitational aggregates with cohesion[J]. Planetary and Space Science,2009,57(2):183-192
[33] MICHEL P,RICHARDSON D C. Collision and gravitational reaccumulation:possible formation mechanism of the asteroid Itokawa[J]. Astronomy & Astrophysics,2013,554:1-4
[34] GEISSLER P,PETIT J M,DURDA D D,et al. Erosion and ejecta reaccretion on 243 Ida and its moon[J]. Icarus,1996,120(1):140-157
[35] SHANG H B,WU X Y,REN Y,et al. An efficient algorithm for global periodic orbits generation near irregular-shaped asteroids[J]. Communications in Nonlinear Science and Numerical Simulation,2017,48:550-568
[36] PARK R S,WERNER R A,BHASKARAN S. Estimating small-body gravity field from shape model and navigation data[J]. Journal of Guidance,Control,and Dynamics,2010,33(1):212-221
[37] WERNER R A,SCHEERES D J. Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia[J]. Celestial Mechanics and Dynamical Astronomy,1996,65(3):313-344
[38] TSOULIS D,PETROVIC? S. On the singularities of the gravity field of a homogeneous polyhedral body[J]. Geophysics,2001,66(2):535-539
[39] TAKAHASHI Y,SCHEERES D. Morphology driven density distribution estimation for small bodies[J]. Icarus,2014,233:179-193
[40] WEI B W,SHANG H B. Global gravity field modeling of small bodies with heterogeneous mass distributions[J]. Journal of Guidance,Control,and Dynamics,2022,45(2):248-261
[41] JUNKINS J L. Investigation of finite-element representations of the geopotential[J]. AIAA Journal,1976,14(6):803-808
[42] ENGELS R C,JUNKINS J L. Local representation of the geopotential by weighted orthonormal poly nomials[J]. Journal of Guidance,Control,and Dynamics,1980,3(1):55-61
[43] BEYLKIN G,CRAMER R. Toward multiresolution estimation and efficient representation of gravitational fields[J]. Journal of Guidance,Control,and Dynamics,2002,84(1):87-104
[44] JONES B A,BORN G H,BEYLKIN G. Comparisons of the cubed-sphere gravity model with the spherical harmonics[J]. Journal of Guidance,Control,and Dynamics,2010,33(2):415-425
[45] ARORA N,RUSSELL R P. Efficient interpolation of high-fidelity geopotentials[J]. Journal of Guidance,Control,and Dynamics,2015,39(1):128-143
[46] COLOMBI A,HIRANI A H,VILLAC B F. Adaptive gravitational force representation for fast trajectory propagation near small bodies[J]. Journal of Guidance,Control,and Dynamics,2008,31(4):1041-1051
[47] WEI B W,SHANG H B,QIAO D. Hybrid model of gravitational fields around small bodies for efficient trajectory propagations[J]. Journal of Guidance,Control,and Dynamics,2020,43(2):232-249
[48] GAO A,LIAO W T. Efficient gravity field modeling method for small bodies based on Gaussian process regression[J]. Acta Astronautica,2019,157:73-91
[49] YU S，CHENG L，GONG S P. Fast estimation of gravitational field of irregular asteroids based on deep neural network and its application[C]//29th AAS/AIAA Space Flight Mechanics Meeting. Hawaii：AIAA，2019.
[50] CHENG L,WANG Z B,SONG Y,et al. Real-time optimal control for irregular asteroid landings using deep neural networks[J]. Acta Astronautica,2020,170:66-79
[51] FURFARO R,LINARES R,REDDY V,et al. Modeling irregular small bodies gravity field via extreme learning machines and Bayesian optimization[J]. Advances in Space Research,2021,67(1):617-638
[52] MACIEJEWSKI A J. Reduction,relative equilibria and potential in the two rigid bodies problem[J]. Celestial Mechanics and Dynamical Astronomy,1995,63(1):1-28
[53] SCHEERES D J. Relative equilibria for general gravity fields in the sphere-restricted full 2-body problem[J]. Celestial Mechanics and Dynamical Astronomy,2006,94(3):317-349
[54] SCHEERES D J. Stability of relative equilibria in the full two-body problem[J]. Annals of the New York Academy of Sciences,2006,1017(1):81-94
[55] BELLEROSE J,SCHEERES D J. Energy and stability in the full two body problem[J]. Celestial Mechanics and Dynamical Astronomy,2008,100(1):63-91
[56] SCHEERES D J. Rotational fission of contact binary asteroids[J]. Icarus,2007,189(2):370-385
[57] BORDERIES N. Mutual gravitational potential of N solid bodies[J]. Celestial Mechanics and Dynamical Astronomy,1978:295-307
[58] COMPÈRE A,LEMAITRE A. The two-body interaction potential in the STF tensor formalism:an application to binary asteroids[J]. Celestial Mechanics and Dynamical Astronomy,2014,119(3):313-330
[59] TRICARICO P. Figure–figure interaction between bodies having arbitrary shapes and mass distributions:a power series expansion approach[J]. Celestial Mechanics and Dynamical Astronomy,2008,100(4):319-330
[60] GIACAGLIA G E O,BURA M. Transformations of spherical harmonics and applications to geodesy and satellite theory[J]. Studia Geophysica et Geodaetica,1980,24(1):1-11
[61] SCHUTZ B E. The mutual potential and gravitational torques of two bodies to fourth order[J]. Celestial Mechanics and Dynamical Astronomy,1981,24(2):173-181
[62] PAUL M K. An expansion in power series of mutual potential for gravitating bodies with finite sizes[J]. Celestial Mechanics and Dynamical Astronomy,1988,44(1):49-59
[63] ASHENBERG J. Mutual gravitational potential and torque of solid bodies via inertia integrals[J]. Celestial Mechanics and Dynamical Astronomy,2007,99(2):149-159
[64] WERNER R A,SCHEERES D J. Mutual potential of homogeneous polyhedra[J]. Celestial Mechanics and Dynamical Astronomy,2005,91(3):337-349
[65] FAHNESTOCK E G,SCHEERES D J. Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach[J]. Celestial Mechanics and Dynamical Astronomy,2006,96(3-4):317-339
[66] FAHNESTOCK E G,SCHEERES D J. Simulation and analysis of the dynamics of binary near-Earth Asteroid (66391) 1999 KW4[J]. Icarus,2007,194(2):410-435
[67] HOU X,SCHEERES D J,XIN X. Mutual potential between two rigid bodies with arbitrary shapes and mass distributions[J]. Celestial Mechanics and Dynamical Astronomy,2017,127(3):369-395
[68] SHI Y,WANG Y,XU S. Mutual gravitational potential,force,and torque of a homogeneous polyhedron and an extended body:an application to binary asteroids[J]. Celestial Mechanics and Dynamical Astronomy,2017,129(3):307-320
[69] YU Y,CHENG B,HAYABAYASHI M,et al. A finite element method for computational full two-body problem:I. the mutual potential and derivatives over bilinear tetrahedron elements[J]. Celestial Mechanics and Dynamical Astronomy,2019,131(11):51
[70] GAO Y,YU Y,CHENG B,et al. Accelerating the finite element method for calculating the full 2-body problem with CUDA[J]. Advances in Space Research,2022,69(5):2305-2318