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Sensitivity of the Euler–Poinsot Tensor Values to the Choice of the Body Surface Triangulation Mesh

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Abstract

The inertial characteristics of celestial bodies can be calculated using their triangle partitions based on photometric observations. Such partitions can be refined along with the accumulation of necessary information. In this regard, the question arises to what extent the approximations of the inertial characteristics of celestial bodies, in particular, the approximations of the components of the Euler–Poinsot tensor of different orders, are susceptible to the choice of such partitions. Such components enter into the expansion of the gravitational potential in harmonic polynomials. In this paper, for some small celestial bodies, a comparison of such coefficients is carried out as coarse partitions are replaced with finer ones.

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Notes

  1. †December 9, 2019.

REFERENCES

  1. P. Appell, Leçons sur l’attraction et la fonction potentielle: professées à la Sorbonne en 1890–1891 (G. Carrie, Paris, 1892).

    MATH  Google Scholar 

  2. H. Poincaré, Théorie du potentiel newtonien: Leçons professées à la Sorbonne pendant le premier semestre 1894–1895 (Gauthier-Villars, Paris, 1899).

  3. G. N. Duboshin, Theory of Gravitation (Fizmatlit, Moscow, 1961) [in Russian].

    MATH  Google Scholar 

  4. L. N. Sretenskii, Theory of Newtonian Potentials (Gostekhizdat, Moscow, 1946) [in Russian].

    Google Scholar 

  5. V. V. Beletskii, Motion of an Artificial Satellite about Its Center of Mass (Nauka, Moscow, 1965; Israel Program for Scientific Translations, 1966).

  6. G. N. Duboshin, Celestial Mechanics: Basic Problems and Methods (Fizmatlit, Moscow, 1968; Defense Tech. Inf. Center, Fort Belvoir, 1969).

  7. V. A. Antonov, E. I. Timoshkova, and K. V. Kholshevnikov, Introduction to the Theory of Newtonian Potentials (Fizmatlit, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  8. G. N. Doubochine, “Sur le développement de la fonction des forces dans le problème de deux corps finis,” Celestial Mech. 14, 239–281 (1976).

    Article  MathSciNet  Google Scholar 

  9. A. R. Dobrovolskis, “Inertia of any polyhedron,” Icarus 124 (2), 698–704 (1996).

    Article  Google Scholar 

  10. B. Mirtich, “Fast and accurate computation of polyhedral mass properties,” J. Graphics Tools 1 (2), 31–50 (1996).

    Article  Google Scholar 

  11. F. A. Sludskii, Master’s Dissertation in Astronomy (Universitetskaya (Katkov K), Moscow, 1863).

  12. R. A. Werner, “The gravitational potential of a homogeneous polyhedron or don’t cut corners,” Celestial Mech. Dyn. Astron. 59 (3), 253–278 (1994).

    Article  Google Scholar 

  13. R. A. Werner and D. J. Scheeres, “Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia,” Celestial Mech. Dyn. Astron. 65 (3), 313–344 (1996).

    MATH  Google Scholar 

  14. A. A. Burov and V. I. Nikonov, “Computation of attraction potential of asteroid (433) Eros with an accuracy up to the terms of the fourth order,” Dokl. Phys. 65, 164–168 (2020).

    Article  Google Scholar 

  15. V. I. Nikonov, Gravitational Fields of Small Celestial Bodies (Belyi Veter, Moscow, 2020) [in Russian].

    Google Scholar 

  16. A. A. Burov and V. I. Nikonov, “Inertial characteristics of higher orders and dynamics in a proximity of a small celestial body,” Russ. J. Nonlinear Dyn. 16 (2), 259–273 (2020).

    MathSciNet  MATH  Google Scholar 

  17. R. A. Werner, “Spherical harmonic coefficients for the potential of a constant-density polyhedron,” Comput. Geosci. 23 (10), 1071–1077 (1997).

    Article  Google Scholar 

  18. D. Liao-McPherson, W. D. Dunham, and I. Kolmanovsky, “Model predictive control strategies for constrained soft landing on an asteroid,” AIAA/AAS Astrodynamics Specialist Conference, September 13–16, 2016, Long Beach, California (2016).

  19. P. C. Thomas, J. Joseph, B. Carcich, et al., “Eros: Shape, topography, and slope processes,” Icarus 155 (1), 18–37 (2002).

    Article  Google Scholar 

  20. M. T. Zuber, D. E. Smith, A. F. Cheng, et al., “The shape of 433 Eros from the NEAR-Shoemaker laser rangefinder,” Science 289, 2097–2100 (2000).

    Article  Google Scholar 

  21. J. K. Miller, A. S. Konopliv, P. G. Antreasian, et al., “Determination of shape, gravity, and rotational state of asteroid 433 Eros,” Icarus 155 (1), 3–17 (2002).

    Article  Google Scholar 

  22. T. L. Farnham, “Shape model of asteroid 21 Lutetia, RO-A-OSINAC/OSIWAC-5-LUTETIA-SHAPE-V1.0” (NASA Planetary Data System, 2013).

    Google Scholar 

  23. C. Capanna, L. Jorda, P. Gutierrez, and S. Hviid, “MSPCD SHAP2 Cartesian plate model for comet 67P/C-G 6K PLATES, RO-C-MULTI-5-67P-SHAPE-V1.0:CG_MSPCD_SHAP2_006K_CART” (NASA Planetary Data System and ESA Planetary Science Archive, 2015).

  24. C. Capanna, L. Jorda, P. Gutierrez, and S. Hviid, “MSPCD SHAP2 Cartesian plate model for comet 67P/C-G 12K PLATES, RO-C-MULTI-5-67P-SHAPE-V1.0:CG_MSPCD_SHAP2_012K_CART” (NASA Planetary Data System and ESA Planetary Science Archive, 2015).

  25. C. Capanna, L. Jorda, P. Gutierrez, and S. Hviid, “MSPCD SHAP2 Cartesian plate model for comet 67P/C-G 24K PLATES, RO-C-MULTI-5-67P-SHAPE-V1.0:CG_MSPCD_SHAP2_024K_CART” (NASA Planetary Data System and ESA Planetary Science Archive, 2015).

  26. C. Capanna, L. Jorda, P. Gutierrez, and S. Hviid, “MSPCD SHAP2 Cartesian plate model for comet 67P/C-G 48K PLATES, RO-C-MULTI-5-67P-SHAPE-V1.0:CG_MSPCD_SHAP2_048K_CART” (NASA Planetary Data System and ESA Planetary Science Archive, 2015).

  27. C. Capanna, L. Jorda, P. Gutierrez, and S. Hviid, “MSPCD SHAP2 Cartesian plate model for comet 67P/C-G 98K PLATES, RO-C-MULTI-5-67P-SHAPE-V1.0:CG_MSPCD_SHAP2_098K_CART” (NASA Planetary Data System and ESA Planetary Science Archive, 2015).

  28. X. Wang, Y. Jiang, and Sh. Gong, “Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies,” Astrophys. Space Sci. 353 (1), 105–121 (2014).

    Article  Google Scholar 

  29. Y. Jiang, H. Baoyin, and H. Li, “Collision and annihilation of relative equilibrium points around asteroids with a changing parameter,” Mon. Not. R. Astron. Soc. 452 (4), 3924–3931 (2015).

    Article  Google Scholar 

  30. Y. Jiang and H. Baoyin, “Annihilation of relative equilibria in the gravitational field of irregular-shaped minor celestial bodies,” Planet. Space Sci. 161, 107–136 (2018).

    Article  Google Scholar 

  31. S. Aljbaae, T. G. G. Chanut, V. Carruba, et al., “The dynamical environment of asteroid 21 Lutetia according to different internal models,” Mon. Not. R. Astron. Soc. 464 (3), 3552–3560 (2017).

    Article  Google Scholar 

  32. R. A. Werner, “The solid angle hidden in polyhedron gravitation formulations,” J. Geodesy 91, 307–328 (2017).

    Article  Google Scholar 

  33. A. P. Markeev, Libration Points in Celestial Mechanics and Astrodynamics (Fizmatlit, Moscow, 1978) [in Russian].

    Google Scholar 

  34. V. K. Abalakin, “On the stability of libration points of a rotating gravitating ellipsoid,” Byull. Inst. Teor. Astron. 6 (8), 543–549 (1957).

    MathSciNet  Google Scholar 

  35. Yu. V. Batrakov, “Periodic motion of a particle in the gravitational field of a rotating triaxial ellipsoid,” Byull. Inst. Teor. Astron. 6, 524–542, (1957).

    MathSciNet  Google Scholar 

  36. S. G. Zhuravlev, “Instability of libration points of a rotating gravitating ellipsoid,” in Collected Research Papers of Postgraduate Students (Univ. Druzhby Narodov, Moscow, 1968), No. 1, pp. 169–183 [in Russian].

  37. S. G. Zhuravlev, “Stability of the libration points of a rotating triaxial ellipsoid,” Celestial Mech. 6, 255–267 (1972).

    Article  Google Scholar 

  38. S. G. Zhuravlev, “About the stability of the libration points of a rotating triaxial ellipsoid in a degenerate case,” Celestial Mech. 8 (1), 75–84 (1973).

    Article  Google Scholar 

  39. S. G. Zhuravlev, “Stability of the libration points of a rotating triaxial ellipsoid in the three-dimensional case,” Astron. Zh. 51 (16), 1330–1334 (1974).

    MATH  Google Scholar 

  40. I. I. Kosenko, “On libration points near a gravitating and rotating triaxial ellipsoid,” J. Appl. Math. Mech. 45 (1), 18–23 (1981).

    Article  MathSciNet  Google Scholar 

  41. I. I. Kosenko, “Libration points in the problem of a triaxial gravitating ellipsoid: Geometry of the stability domain,” Kosm. Issled. 19 (2), 200–209 (1981).

    Google Scholar 

  42. I. I. Kosenko, “Nonlinear analysis of the stability of the libration points of a triaxial ellipsoid,” J. Appl. Math. Mech. 49 (1), 17–24 (1985).

    Article  MathSciNet  Google Scholar 

  43. I. I. Kosenko, “On a power series expansion of the gravitational potential of an inhomogeneous ellipsoid,” J. Appl. Math. Mech. 50 (2), 142–146 (1986).

    Article  MathSciNet  Google Scholar 

  44. I. I. Kosenko, “On the stability of points of libration of an inhomogeneous triaxial ellipsoid,” J. Appl. Math. Mech. 51 (1), 1–5 (1987).

    Article  MathSciNet  Google Scholar 

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Funding

The research by V.I. Nikonov was supported by a Russian Federation Presidential grant (no. MK-1712.2019.1) for the state support of scientific research of young Russian scientists: candidates and doctors of sciences, and was supported in part by the Russian Foundation for Basic Research (project no. 18-01-00335).

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Authors and Affiliations

  1. Dorodnitsyn Computing Center, Russian Academy of Sciences, 119991, Moscow, Russia

    A. A. Burov & V. I. Nikonov

  2. National Research University Higher Scholl of Economics, 101000, Moscow, Russia

    A. A. Burov & V. I. Nikonov

Authors
  1. A. A. Burov
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  2. V. I. Nikonov
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Correspondence to A. A. Burov or V. I. Nikonov.

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Translated by E. Chernokozhin

APPENDIX: TABLES WITH CALCULATION RESULTS

APPENDIX: TABLES WITH CALCULATION RESULTS

Table 1.   Asteroid (433) Eros: vertices \({v}\), faces  f, volume \(V\) (in km3), and the principal central moments of inertia (in km2), related to the volume
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Table 2.   Asteroid (21) Lutetia: vertices \({v}\), faces  f, volume \(V\) (in km3), and the principal central moments of inertia (in km2), related to the volume
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Table 3.   Comet (67P) Churyumov–Gerasimenko: vertices \({v}\), faces  f, volume \(V\) (in km3), and the principal central moments of inertia (in km2), related to the volume
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Table 4.   Asteroid (433) Eros: components of tensors \(J_{3}^{'}\) (in km3) and \(J_{4}^{'}\) (in km4)
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Table 5.   Asteroid (433) Eros: components of tensors \(J_{3}^{'}\) (in km3) and \(J_{4}^{'}\) (in km4) (continuation)
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Table 6. Asteroid (21) Lutetia: components of tensors \(J_{3}^{'}\) (in km3) and \(J_{4}^{'}\) (in km4)
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Table 7.   Asteroid (21) Lutetia: components of tensors \(J_{3}^{'}\) (in km3) and \(J_{4}^{'}\) (in km4) (continuation)
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Table 8.   Comet (67P) Churyumov–Gerasimenko: components of tensors \(J_{3}^{'}\) (in km3) and \(J_{4}^{'}\) (in km4)
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Table 9.   Comet (67P) Churyumov–Gerasimenko: components of tensors \(J_{3}^{'}\) (in km3) and \(J_{4}^{'}\) (in km4) (continuation)
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Table 10.   Asteroid (21) Lutetia: comparison of the coordinates of libration points calculated within Model 5 using different approximations of the potential (in km)
Full size table

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Burov, A.A., Nikonov, V.I. Sensitivity of the Euler–Poinsot Tensor Values to the Choice of the Body Surface Triangulation Mesh. Comput. Math. and Math. Phys. 60, 1708–1720 (2020). https://doi.org/10.1134/S0965542520100061

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