Hostname: page-component-5d59c44645-k78ct Total loading time: 0 Render date: 2024-03-02T20:58:06.168Z Has data issue: false hasContentIssue false

Orbit propagation around small bodies using spherical harmonic coefficients obtained from polyhedron shape models

Published online by Cambridge University Press:  30 May 2022

P. Peñarroya
Deimos Space S.L.U., Ronda de Poniente, 19, 28760 Tres Cantos (Madrid), Spain email:
R. Paoli
Department of Mathematics, Universitatea Alexandru Ioan Cuza, Bd. Carol I, nr. 11, 700506, Iasi Romania email:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Missions to asteroids have been the trend in space exploration for the last years. They provide information about the formation and evolution of the Solar System, contribute to direct planetary defense tasks, and could be potentially exploited for resource mining. Be their purpose as it may, the factor that all these mission types have in common is the challenging dynamical environment they have to deal with. The gravitational environment of a certain asteroid is most of the times not accurately known until very late mission phases when the spacecraft has already orbited the body for some time.

Shape models help to estimate the gravitational potential with a density distribution assumption (usually constant value) and some optical measurements of the body. These measurements, unlike the ones needed for harmonic coefficient estimation, can be taken from well before arriving at the asteroid’s sphere of influence, which allows to obtain a better approximation of the gravitational dynamics much sooner. The disadvantage they pose is that obtaining acceleration values from these models implies a heavy computational burden on the on-board processing unit, which is very often too time-consuming for the mission profile.

In this paper, the technique developed on [1] is used to create a validated Python-based tool that obtains spherical harmonic coefficients from the shape model of an asteroid or comet, given a certain density for the body. This validated software suite, called AstroHarm, is used to analyse the accuracy of the models obtained and the improvements in computational efficiency in a simulated spacecraft orbiting a small body.

The results obtained are shown offering a qualitative comparison between different order spherical harmonic models and the original shape model. Finally, the creation of a catalogue for harmonics is proposed together with some thoughts on complex modelling using this tool.

Research Article
© The Author(s), 2022. Published by Cambridge University Press on behalf of International Astronomical Union


Werner, R. A. Spherical harmonic coefficients for the potential of a constant-density polyhedron. Computers & Geosciences 23(10), 10711077 December (1997).CrossRefGoogle Scholar
Cheng, A. F., Santo, A. G., Heeres, K. J., Landshof, J. A., Farquhar, R. W., Gold, R. E., and Lee, S. C. Near-Earth Asteroid Rendezvous: Mission overview. Journal of Geophysical Research: Planets 102(E10), 2369523708 (1997).CrossRefGoogle Scholar
Schwehm, G. H. and Schulz, R. The International Rosetta Mission. In Laboratory Astrophysics and Space Research, Ehrenfreund, P., Krafft, C., Kochan, H. , and Pirronello, V., editors, volume 236, 537546. Springer Netherlands, Dordrecht (1999).CrossRefGoogle Scholar
Glassmeier, K.-H., Boehnhardt, H., Koschny, D., Kührt, E., and Richter, I. The Rosetta Mission: Flying Towards the Origin of the Solar System. Space Sci Rev 128(1), 121 February (2007).CrossRefGoogle Scholar
Kawaguchi, J., Fujiwara, A., and Uesugi, T. Hayabusa–Its technology and science accomplishment summary and Hayabusa-2. Acta Astronautica 62(10-11), 639647 May (2008).CrossRefGoogle Scholar
Lauretta, D. S., Balram-Knutson, S. S., Beshore, E., Boynton, W. V., Drouet d’Aubigny, C., DellaGiustina, D. N., Enos, H. L., Golish, D. R., Hergenrother, C. W., Howell, E. S., Bennett, C. A., Morton, E. T., Nolan, M. C., Rizk, B., Roper, H. L., Bartels, A. E., Bos, B. J., Dworkin, J. P., Highsmith, D. E., Lorenz, D. A., Lim, L. F., Mink, R., Moreau, M. C., Nuth, J. A., Reuter, D. C., Simon, A. A., Bierhaus, E. B., Bryan, B. H., Ballouz, R., Barnouin, O. S., Binzel, R. P., Bottke, W. F., Hamilton, V. E., Walsh, K. J., Chesley, S. R., Christensen, P. R., Clark, B. E., Connolly, H. C., Crombie, M. K., Daly, M. G., Emery, J. P., McCoy, T. J., McMahon, J. W., Scheeres, D. J., Messenger, S., Nakamura-Messenger, K., Righter, K., and Sandford, S. A. OSIRIS-REx: Sample Return from Asteroid (101955) Bennu. Space Sci Rev 212(1), 925984 October (2017).CrossRefGoogle Scholar
Watanabe, S.-i., Tsuda, Y., Yoshikawa, M., Tanaka, S., Saiki, T., and Nakazawa, S. Hayabusa2 Mission Overview. Space Sci Rev 208(1), 316 July (2017).CrossRefGoogle Scholar
Rivkin, A. S., Chabot, N. L., Stickle, A. M., Thomas, C. A., Richardson, D. C., Barnouin, O., Fahnestock, E. G., Ernst, C. M., Cheng, A. F., Chesley, S., Naidu, S., Statler, T. S., Barbee, B., Agrusa, H., Moskovitz, N., Daly, R. T., Pravec, P., Scheirich, P., Dotto, E., Corte, V. D., Michel, P., Küppers, M., Atchison, J., and Hirabayashi, M. The Double Asteroid Redirection Test (DART): Planetary Defense Investigations and Requirements. Planet. Sci. J. 2(5), 173 August (2021).CrossRefGoogle Scholar
Michel, P., Küppers, M., and Carnelli, I. The Hera mission: European component of the ESA-NASA AIDA mission to a binary asteroid. 42, B1.1–42–18 July (2018).Google Scholar
Cheng, A. F., Michel, P., Jutzi, M., Rivkin, A. S., Stickle, A., Barnouin, O., Ernst, C., Atchison, J., Pravec, P., and Richardson, D. C. Asteroid Impact & Deflection Assessment mission: Kinetic impactor. Planetary and Space Science 121, 2735 February (2016).CrossRefGoogle Scholar
Oh, D. Y., Collins, S., Drain, T., Hart, W., Imken, T., Larson, K., Marsh, D., Muthulingam, D., Snyder, J. S., Trofimov, D., Elkins-Tanton, L. T., Johnson, I., Lord, P., and Pirkl, Z. Development of the Psyche mission for NASA’s Discovery program. September (2019).Google Scholar
Werner, R. and Scheeres, D. Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celestial Mech Dyn Astr 65(3) (1997).CrossRefGoogle Scholar
Russell, R. and Wittick, P. Mascon Models for Small Body Gravity Fields. August (2017).Google Scholar
Balmino, G. Gravitational potential harmonics from the shape of an homogeneous body. Celestial Mech Dyn Astr 60(3), 331364 November (1994).CrossRefGoogle Scholar
Inc, W. R.Mathematica, Version 12.3.1.Google Scholar
Montenbruck, O., Gill, E., and Lutze, F. Satellite Orbits: Models, Methods, and Applications. Appl. Mech. Rev. 55(2), B27 (2002).CrossRefGoogle Scholar
González, D. DaniGlez/polygrav, June (2020). original-date: 2020-06-18T18:16:52Z.Google Scholar
Werner, R. A. The gravitational potential of a homogeneous polyhedron or don’t cut corners. Celestial Mech Dyn Astr 59(3), 253278 July (1994).CrossRefGoogle Scholar