Gravitational potential energy and fundamental parameters of the terrestrial and giant planets
Received: August 13, 2021
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University

The basic goal of this study (as the first step) is to collect the appropriate set of the fundamental astronomic-geodetics parameters for their further use to obtain the components of the density distributions for the terrestrial and outer planets of the Solar system (in the time interval of more than 10 years). The initial data were adopted from several steps of the general way of the exploration of the Solar system by iterations through different spacecraft. The mechanical and geometrical parameters of the planets allow finding the solution of the inverse gravitational problem (as the second stage) in the case of the continued Gaussian density distribution for the Moon, terrestrial planets (Mercury, Venus, Earth, Mars) and outer planets (Jupiter, Saturn, Uranus, Neptune). This law of Gaussian density distribution or normal density was chosen as a partial solution of the Adams-Williamson equation and the best approximation of the piecewise radial profile of the Earth, including the PREM model based on independent seismic velocities. Such conclusion already obtained for the Earth’s was used as hypothetic in view of the approximation problem for other planets of the Solar system where we believing to get the density from the inverse gravitational problem in the case of the Gaussian density distribution for other planets because seismic information, in that case, is almost absent. Therefore, if we can find a stable solution for the inverse gravitational problem and corresponding continue Gaussian density distribution approximated with good quality of planet’s density distribution we come in this way to a stable determination of the gravitational potential energy of the terrestrial and giant planets. Moreover to the planet’s normal low, the gravitational potential energy, Dirichlet’s integral, and other planets’ parameters were derived. It should be noted that this study is considered time-independent to avoid possible time changes in the gravitational fields of the planets.

1. Bullard, E.C. (1954) The interior of the Earth. In: The Earth as a Planet (G.P. Kuiper, ed). Univ. of Chicago Press, 57-137.
2. Bullen, K. E. (1975) The Earth's Density. Chapman and Hall, London.
3. Cottereau, L., & Souchay, J. (2009). Rotation of rigid Venus: a complete precession-nutation model. Astronomy & Astrophysics, 507(3), 1635-1648.
4. Darwin, G. H. (1884). On the figure of equilibrium of a planet of heterogeneous density. Proceeding of the Royal Society, 36(228-231), 158-166
5. Dziewonski, A. M., & Anderson, D. L. (1981). Preliminary reference Earth model. Physics of the earth and planetary interiors, 25(4), 297-356.
6. Encyclopedia of the Solar System (2015) Editted by Tilman Spohn, Doris Breuer, Torrence V. Johnson, 3rd edition, ELSELVER
7. Gauss, C. F. (1840, 1867). Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs-und Abstossungs-Kräfte. In Werke (pp. 195-242). Springer, Berlin, Heidelberg.
8. Helled, R., Anderson, J. D., Schubert, G., & Stevenson, D. J. (2011, October). Constraining the Internal Structures of Jupiter and Saturn from Moments of Inertia Measurements: Implications for the Juno and Solstice Missions.Paper presented at the "European Planetary Science Congress", Abstracts, Vol. 6, EPSC-DPS2011-52-2, EPSC-DPS Joint Meeting.,
9. Hikida, H., & Wieczorek, M. A. (2007). Crustal thickness of the Moon: New constraints from gravity inversions using polyhedral shape models. Icarus, 192(1), 150-166.
10. Konopliv, A. S., Yoder, C. F., Standish, E. M., Yuan, D. N., & Sjogren, W. L. (2006). A global solution for the Mars static and seasonal gravity, Mars orientation, Phobos and Deimos masses, and Mars ephemeris. Icarus, 182(1), 23-50.
11. Konopliv, A. S., Park, R. S., Yuan, D. N., Asmar, S. W., Watkins, M. M., Williams, J. G., …& Zuber, M. T. (2014). High‐resolution lunar gravity fields from the GRAIL primary and extended missions. Geophysical Research Letters, 41(5), 1452-1458.
12. Konopliv, A. S., Park, R. S., & Folkner, W. M. (2016). An improved JPL Mars gravity field and orientation from Mars orbiter and lander tracking data. Icarus, 274, 253-260.
13. Konopliv, A. S. (2016). Private communication.
14. Lemoine, F. G., Goossens, S., Sabaka, T. J., Nicholas, J. B., Mazarico, E., Rowlands, D. D., ... & Zuber, M. T. (2014). GRGM900C: A degree 900 lunar gravity model from GRAIL primary and extended mission data. Geophysical research letters, 41(10), 3382-3389.
15. Marchenko, A. N. (2000). Earth's radial density profiles based on Gauss' and Roche's distributions. Bolletino di Geodesia e Scienze Affini, 59(3), 201-220.
16. Marchenko, A. N. (2009). The Earth's global density distribution and gravitational potential energy. In Observing our Changing Earth (pp. 483-491). Springer, Berlin, Heidelberg.
17. Marchenko, A. N., & Zayats, A. S. (2011). Estimation of the gravitational potential energy of the earth based on different density models. Studia Geophysica et Geodaetica, 55(1), 35-54.
18. Margot, J. L., Peale, S. J., Solomon, S. C., Hauck, S. A., Ghigo, F. D., Jurgens, R. F., ... & Campbell, D. B. (2012). Mercury's moment of inertia from spin and gravity data. Journal of Geophysical Research: Planets, 117(E12).
19. Mescheryakov, G. A. (1977). On the unique solution of the inverse problem of the potential theory. Reports of the Ukrainian Academy of Sciences. Kiev, Series A, No. 6, 492-495 (in Ukrainian)
20. Mocquet, A., Rosenblatt, P., Dehant, V., & Verhoeven, O. (2011). The deep interior of Venus, Mars, and the Earth: A brief review and the need for planetary surface-based measurements. Planetary and Space Science, 59(10), 1048-1061.
21. Moritz, H. (1990). The figure of the Earth: theoretical geodesy and the Earth's interior. Karlsruhe: Wichmann.
22. Neumann, G. A., Zuber, M. T., Wieczorek, M. A., Head, J. W., Baker, D. M., Solomon, S. C., ... & Kiefer, W. S. (2015). Lunar impact basins revealed by Gravity Recovery and Interior Laboratory measurements. Science advances, 1(9), e1500852.
23. Rubincam, D. P. (1979). Gravitational potential energy of the Earth: A spherical harmonic approach. Journal of Geophysical Research: Solid Earth, 84(B11), 6219-6225.
24. Rappaport, N. J., Konopliv, A. S., Kucinskas, A. B., & Ford, P. G. (1999). An improved 360 degree and order model of Venus topography. Icarus, 139(1), 19-31.
25. Rivoldini, A., Van Hoolst, T., & Verhoeven, O. (2009). The interior structure of Mercury and its core sulfur content. Icarus, 201(1), 12-30.
26. Seidelmann, P. K., Abalakin, V. K., Bursa, M., Davies, M. E., De Bergh, C., Lieske, J. H., Oberst, J., Simon, J. L., Standish, E. M., Stooke, P., Thomas, P. C. (2002). Report of the IAU/IAG working group on cartographic coordinates and rotational elements of the planets and satellites: 2000. Celest. Mech. Dyn. Astron. 82(1), 83-110.
27. Thomson, W., & Tait, P. G. (1883). Treatise on Natural Philosophy. Vol. 2, Cambridge University Press
28. Tikhonov, A. N. & Arsenin, V. Y. (1974). Methods of the solution of ill-posed problems, Nauka, Moscow, 1974 (in Russian)
29. Wermer, J. (1981). Potential theory, Springer, Berlin Heidelberg New York, 1981
30. Wieczorek, M. A., Neumann, G. A., Nimmo, F., Kiefer, W. S., Taylor, G. J., Melosh, H. J., ... & Zuber, M. T. (2013). The crust of the Moon as seen by GRAIL. Science, 339(6120), 671-675. DOI: 10.1126/science.1231530
31. Yoder, C. F. (1995). Venus' free obliquity. Icarus, 117(2), 250-286.
32. Zuber, M. T., Solomon, S. C., Phillips, R. J., Smith, D. E., Tyler, G. L., Aharonson, O., ... & Zhong, S. (2000). Internal structure and early thermal evolution of Mars from Mars Global Surveyor topography and gravity. Science, 287(5459), 1788-1793.