Translated Titles

Gravitational Torque by an Oblate Saturn: Case Study of Enceladus’ Physical Libration





Key Words

重力力矩 ; 物理天平動 ; 扁圓土星 ; 土衛二 ; 內部結構 ; Gravitational Torque ; Physical Libration ; Oblate Saturn ; Enceladus ; Interior Structure



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Chinese Abstract


English Abstract

The gravitational torque of Saturn exerted on its moons plays an important role in their rotational dynamics, one of which, known as the physical libration in longitude, has been accurately measured for Enceladus (Thomas et al., 2016). This observation has been used to constrain the interior structure of Enceladus, which is believed to consist of three layers: rock core, water ocean, and ice shell. However, the formulation of the gravitational torque excludes the non-spherical shape of Saturn itself, even though Saturn is one of the most oblate planets in the Solar System and its distance to Enceladus is comparable to its mean radius. In this study, we derive the general torque equation using multipole expansion to include the effect due to the arbitrary shape of Saturn. For an oblate Saturn, the conventional torque is multiplied by a correction factor, which depends on Saturn’s oblateness and its distance to Enceladus. Given the physical parameters, the correction to the torque is about a few thousandth and the resultant correction to the estimate of Enceladus’ shell thickness is less than one percent.

Topic Category 基礎與應用科學 > 物理
理學院 > 物理學研究所
  1. References
  2. Anderson, J. D., & Schubert, G. (2007). Saturn's gravitational field, internal rotation, and interior structure. Science, 317(5843), 1384-1387.
  3. Arfken, G., Weber, H., & Harris, F. E. (2012). Mathematical Methods for Physicists. Academic Press.
  4. Baland, R. M., & Van Hoolst, T. (2010). Librations of the Galilean satellites: The influence of global internal liquid layers. Icarus, 209(2), 651-664.
  5. Baland, R. M., Yseboodt, M., & Van Hoolst, T. (2016). The obliquity of Enceladus. Icarus, 268, 12-31.
  6. Borderies, N. (1978). Mutual gravitational potential of N solid bodies. Celestial Mechanics, 18(3), 295-307.
  7. Boué, G. (2017). The two rigid body interaction using angular momentum theory formulae. Celestial Mechanics and Dynamical Astronomy, 128(2-3), 261-273.
  8. Bullen, K. E. (1975). The Earth’s density. UK: Chapman and Hall.
  9. Chao, B. F. (2017a). Dynamics of axial torsional libration under the mantle‐inner core gravitational interaction. Journal of Geophysical Research: Solid Earth, 122(1), 560-571.
  10. Chao, B. F. (2017b). Dynamics of the inner core wobble under mantle‐inner core gravitational interactions. Journal of Geophysical Research: Solid Earth, 122(9), 7437-7448.
  11. Chao, B. F., & Gross, R. S. (1987). Changes in the Earth's rotation and low‐degree gravitational field induced by earthquakes. Geophysical Journal of the Royal Astronomical Society, 91(3), 569-596.
  12. Choblet, G., Tobie, G., Sotin, C., Běhounková, M., Čadek, O., Postberg, F., & Souček, O. (2017). Powering prolonged hydrothermal activity inside Enceladus. Nature Astronomy, 1(12), 841.
  13. Comstock, R. L., & Bills, B. G. (2003). A solar system survey of forced librations in longitude. Journal of Geophysical Research: Planets, 108(E9), 5100.
  14. Cook, A. H. (1980). Interior of the Planets. New York, NY: Cambridge University Press.
  15. Dahlen, F., & Tromp, J. (1998). Theoretical Global Seismology. Princeton University Press.
  16. de Pater, I., & Lissauer, J. J. (2015). Planetary sciences. Cambridge University Press.
  17. Dumberry, M. (2008). Gravitational torque on the inner core and decadal polar motion. Geophysical Journal International, 172(3), 903-920.
  18. Efroimsky, M., & Williams, J. G. (2009). Tidal torques: a critical review of some techniques. Celestial Mechanics and Dynamical Astronomy, 104(3), 257-289.
  19. Frouard, J., & Efroimsky, M. (2017). Tides in a body librating about a spin–orbit resonance: generalisation of the Darwin–Kaula theory. Celestial Mechanics and Dynamical Astronomy, 129(1-2), 177-214.
  20. Goldreich, P. M., & Mitchell, J. L. (2010). Elastic ice shells of synchronous moons: Implications for cracks on Europa and non-synchronous rotation of Titan. Icarus, 209(2), 631-638.
  21. Griffiths, D. J. (2005). Introduction to Quantum Mechanics. Pearson Prentice Hall.
  22. Henrard, J. (2005a). Additions to the theory of the rotation of Europa. Celestial Mechanics and Dynamical Astronomy, 93(1-4), 101-112.
  23. Henrard, J. (2005b). The rotation of Io. Icarus, 178(1), 144-153.
  24. Hou, X., Scheeres, D. J., & Xin, X. (2017). Mutual potential between two rigid bodies with arbitrary shapes and mass distributions. Celestial Mechanics and Dynamical Astronomy, 127(3), 369-395.
  25. Hsu, H. W., Postberg, F., Sekine, Y., Shibuya, T., Kempf, S., Horányi, M., et al. (2015). Ongoing hydrothermal activities within Enceladus. Nature, 519(7542), 207.
  26. Hubbard, W. B. (1984). Planetary Interiors. New York, NY: Van Nostrand Reinhold Company Inc.
  27. Iess, L., Folkner, W. M., Durante, D., Parisi, M., Kaspi, Y., Galanti, E., et al. (2018). Measurement of Jupiter’s asymmetric gravity field. Nature, 555(7695), 220.
  28. Iess, L., Stevenson, D. J., Parisi, M., Hemingway, D., Jacobson, R. A., Lunine, J. I., et al. (2014). The gravity field and interior structure of Enceladus. Science, 344(6179), 78-80.
  29. Jackson, J. D. (1975). Classical Electrodynamics. New York, NY: John Wiley & Sons.
  30. Jara-Orué, H. M., & Vermeersen, B. L. (2014). The forced libration of Europa’s deformable shell and its dependence on interior parameters. Icarus, 229, 31-44.
  31. Kaula, W. M. (2000). Theory of Satellite Geodesy. Mineola, NY: Dover Publications.
  32. Lemasquerier, D., Grannan, A. M., Vidal, J., Cébron, D., Favier, B., Le Bars, M., & Aurnou, J. M. (2017). Libration‐driven flows in ellipsoidal shells. Journal of Geophysical Research: Planets, 122(9), 1926-1950.
  33. Liu, H. S., & Chao, B. F. (1991). The Earth's equatorial principal axes and moments of inertia. Geophysical Journal International, 106(3), 699-702.
  34. Maciejewski, A. J. (1995). Reduction, relative equilibria and potential in the two rigid bodies problem. Celestial Mechanics and Dynamical Astronomy, 63(1), 1-28.
  35. McKinnon, W. B. (2015). Effect of Enceladus's rapid synchronous spin on interpretation of Cassini gravity. Geophysical Research Letters, 42(7), 2137-2143.
  36. Moritz, H. (1990). The Figure of the Earth. Wichmann.
  37. Murray, C. D., & Dermott, S. F. (1999). Solar System Dynamics. New York, NY: Cambridge University Press.
  38. Nayar, K. G., Sharqawy, M. H., & Banchik, L. D. (2016). Thermophysical properties of seawater: a review and new correlations that include pressure dependence. Desalination, 390, 1-24.
  39. Nimmo, F., Bills, B. G., & Thomas, P. C. (2011). Geophysical implications of the long‐wavelength topography of the Saturnian satellites. Journal of Geophysical Research: Planets, 116(E11001).
  40. Porco, C. C., Helfenstein, P., Thomas, P. C., Ingersoll, A. P., Wisdom, J., West, R., et al. (2006). Cassini observes the active south pole of Enceladus. Science, 311(5766), 1393-1401.
  41. Rappaport, N., Bertotti, B., Giampieri, G., & Anderson, J. D. (1997). Doppler measurements of the quadrupole moments of Titan. Icarus, 126(2), 313-323.
  42. Roncoli, R. B. (2005). Lunar constants and models document. JPL D-32296.
  43. Rotenberg, M., Bivins, R., Metropolis, N., & Wooten Jr., J. K. (1959). The 3-j and 6-j Symbols. The MIT Press.
  44. Schubert, G., Anderson, J. D., Travis, B. J., & Palguta, J. (2007). Enceladus: Present internal structure and differentiation by early and long-term radiogenic heating. Icarus, 188(2), 345-355.
  45. Stacey, F. D., & Davis, P. M. (2008). Physics of the Earth. New York, NY: Cambridge University Press.
  46. Tajeddine, R., Rambaux, N., Lainey, V., Charnoz, S., Richard, A., Rivoldini, A., & Noyelles, B. (2014). Constraints on Mimas’ interior from Cassini ISS libration measurements. Science, 346(6207), 322-324.
  47. Thomas, P. C. (2010). Sizes, shapes, and derived properties of the saturnian satellites after the Cassini nominal mission. Icarus, 208(1), 395-401.
  48. Thomas, P. C., Tajeddine, R., Tiscareno, M. S., Burns, J. A., Joseph, J., Loredo, T. J.,et al. (2016). Enceladus’s measured physical libration requires a global subsurface ocean. Icarus, 264, 37-47.
  49. Thornton, S. T., & Marion, J. B. (2008) Classical Dynamics of Particles and Systems. Cengage Learning.
  50. Tough, R. J., & Stone, A. J. (1977). Properties of the regular and irregular solid harmonics. Journal of Physics A: Mathematical and General, 10(8), 1261.
  51. Van Hoolst, T., & Dehant, V. (2002). Influence of triaxiality and second-order terms in flattenings on the rotation of terrestrial planets: I. Formalism and rotational normal modes. Physics of the Earth and Planetary Interiors, 134(1-2), 17-33.
  52. Van Hoolst, T., Baland, R. M., & Trinh, A. (2013). On the librations and tides of large icy satellites. Icarus, 226(1), 299-315.
  53. Van Hoolst, T., Baland, R. M., & Trinh, A. (2016). The diurnal libration and interior structure of Enceladus. Icarus, 277, 311-318.
  54. Van Hoolst, T., Rambaux, N., Karatekin, Ö., & Baland, R. M. (2009). The effect of gravitational and pressure torques on Titan's length-of-day variations. Icarus, 200(1), 256-264.
  55. Van Hoolst, T., Rambaux, N., Karatekin, Ö., Dehant, V., & Rivoldini, A. (2008). The librations, shape, and icy shell of Europa. Icarus, 195(1), 386-399.
  56. Xu, S., Crossley, D., & Szeto, A. M. K. (2000). Variations in length of day and inner core differential rotation from gravitational coupling. Physics of the Earth and Planetary Interiors, 117(1-4), 95-110.