Skip to main content
SpringerLink
Log in

Planetary Tides: Theories

  • Chapter
  • First Online:
Satellite Dynamics and Space Missions

Part of the book series: Springer INdAM Series ((SINDAMS,volume 34))

Abstract

Synthetic presentation of planetary tide theories in the simple case of a homogeneous primary rotating around an axis orthogonal to the orbital plane of the companion. The considered theories are founded on the dynamical equilibrium figure of the tidally deformed body, assumed as an ellipsoid whose rotation is delayed with respect to the motion of the companion. The orbital and rotational evolutions of the system are derived using standard physical laws. The main theory considered is the creep tide theory, a first-principles hydrodynamical theory where the dynamical tide is assimilated to a low-Reynolds-number flow and determined using a Newtonian creep law. The Darwin theories are also considered and are formally derived from the creep tide theory. The various rheologies used in Darwin theories are discussed, with emphasis on the CTL (constant time lag) and CPL (constant phase lag) theories. One introductory session is devoted to the main classical results on the hydrostatic figures of equilibrium of the celestial bodies (static tide).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For the exact definition of the words elastic and anelastic, see the online supplement to [14]. One must keep in mind, however, that the involved restoring forces are gravitational, not elastic.

  2. 2.

    The minus sign in the first term means that we are adopting the exact Physics convention: force is equal to minus the gradient of the potential.

  3. 3.

    Auxiliary first-order relations:

    $$\displaystyle \begin{aligned} \begin{array}{rcl} a&\displaystyle =&\displaystyle R_e(1+\epsilon_\rho/2) \\ b&\displaystyle =&\displaystyle R_e(1-\epsilon_\rho/2) \\ c&\displaystyle =&\displaystyle R_e(1-\epsilon_z). \end{array} \end{aligned} $$
  4. 4.

    The Cayley functions introduced here correspond to the degree 3 in a∕r—since 𝜖 ρ ∝ (a∕r)3. These functions are equivalent to the Hansen coefficients preferred by other authors and the equivalence is given by \(E^{(n)}_{q,p}=X^{-n,q}_{2-p}\) (see [8]).

  5. 5.

    We have, however, to keep in mind that these “years” are very short. In type 3 tides, “diurnal” is slower than “annual”.

  6. 6.

    Remember that in the considered case, the equations of the relative motion are \(M \ddot {\mathbf {r}} \hspace {1mm} = \hspace {1mm} (1+\frac {M}{m})\mathbf {F}\).

  7. 7.

    This does not mean that a negative mass is being assigned to void spaces; it means just that forces included in the calculation of the equilibrium figure need to be subtracted when the masses creating them are no longer there.

  8. 8.

    We avoided to overcharge this text with asterisks. We will only use them in Sect. 8 to indicate the mean anomaly and the argument of the pericenter of Diana. If the other Lagrange variational equations are used, other elements of Diana also need to be indicated. In that case, it is convenient to use the asterisk for all orbital parameters of Diana, from the beginning, and drop the asterisks only after all derivatives of the disturbing potential are calculated. Note added in proof: The presentation of the creep tide theory becomes much simpler when the new Folonier equations are used. See [28].

  9. 9.

    Remind that we are only considering the orthogonal case in which the rotation axis of the primary is perpendicular to the orbital plane.

  10. 10.

    The reader may pay attention to the opposite signs appearing in the definitions of \({\mathcal {C}}_k\) and \({\mathcal {C}}^{\prime \prime }_k\), which is often a source of mistakes in the transformation of the equations.

  11. 11.

    The results of Sect. 4 are consistent with those obtained with the creep tide theory (or with Darwin’s theory) when all lags are made equal to zero. Indeed, the expressions for \(\dot {a}\), \(\dot {e}\), \(\dot {W}\) of Sects. 8 and 9 are trigonometric series in the arguments \(\sin {}(j\ell +\sigma _k)\) and \(\sin {}(j\ell +\sigma ^{\prime \prime }_k)\), which average to zero when the lags vanish. The vanishing of the torque when the σ k vanish is less obvious. However, the auxiliary expansions given in the Online Supplement to [20], allow one to see that \( \sum _{k\in \mathbb {Z}} E_{2,k} \sin \big (2v-(2-k)\ell \big )=0\), and so that M 2 = 0 when the lags vanish.

References

  1. Alexander, M.E.: The weak friction approximation and tidal evolution in close binary systems. Astrophys. Space Sci. 23, 459–510 (1973)

    Article  Google Scholar 

  2. Barnes, R.: Tidal locking of habitable exoplanets. Celest. Mech. Dyn. Astron. 129(4), 509–536 (2017)

    Article  MathSciNet  Google Scholar 

  3. Beutler, G.: Methods of Celestial Mechanics. Springer, Berlin (2005)

    Book  Google Scholar 

  4. Brouwer, D., Clemence, M.: Methods of Celestial Mechanics. Academic Press, New York (1961)

    MATH  Google Scholar 

  5. Carone, L.: Tidal interactions of short-period extrasolar transit planets with their host stars: constraining the elusive stellar tidal dissipation factor. Dissertation, Universität zu Köln (2012)

    Google Scholar 

  6. Cayley, A.: Tables of developments of functions in the theory of elliptic motion. Mem. R. Astron. Soc. 29, 191–306 (1861)

    Google Scholar 

  7. Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven (1969). Chap. VIII

    MATH  Google Scholar 

  8. Correia, A.C.M., Boué, G., Laskar, J., Rodríguez, A.: Deformation and tidal evolution of close-in planets and satellites using a Maxwell viscoelastic rheology. Astron. Astrophys. 571, A50 (2014)

    Article  Google Scholar 

  9. Darwin, G.H.: On the influence of geological changes on the Earth’s axis of rotation. Philos. Trans. 167, 271–312 (1877)

    Article  Google Scholar 

  10. Darwin, G.H.: On the bodily tides of viscous and semi-elastic spheroids and on the ocean tides upon a yielding nucleus. Philos. Trans. 170, 1–35 (1879). Repr. Scientific Papers Vol. II, Cambridge, 1908

    Google Scholar 

  11. Darwin, G.H.: On the secular change in the elements of the orbit of a satellite revolving about a tidally distorted planet. Philos. Trans. 171, 713–891 (1880). Repr. Scientific Papers Vol. II, Cambridge, 1908

    Google Scholar 

  12. Dobbs-Dixon, I., Lin, D.N.C., Mardling, R.A.: Spin-orbit evolution of short-period Planets. Astrophys. J. 610, 464–476 (2004)

    Article  Google Scholar 

  13. Efroimsky, M.: Tidal dissipation compared to seismic dissipation: in small bodies, Earths, and super-Earths. Astrophys. J. 746, 150 (2012)

    Article  Google Scholar 

  14. Efroimsky, M.: Bodily tides near spin-orbit resonances. Celest. Mech. Dyn. Astron. 112, 283–330 (2012)

    Article  MathSciNet  Google Scholar 

  15. Efroimsky, M., Lainey, V.: Physics of bodily tides in terrestrial planets and the appropriate scales of dynamical evolution. J. Geophys. Res. 112, E12003 (2007)

    Article  Google Scholar 

  16. Efroimsky, M., Makarov, V.V.: Tidal dissipation in a homogeneous spherical body. I. Methods. Astrophys. J. 795, 6 (2014)

    Article  Google Scholar 

  17. Eggleton, P.P., Kiseleva, L.G., Hut, P.: The equilibrium tide model for tidal friction. Astrophys. J. 499, 853–870 (1998)

    Article  Google Scholar 

  18. Ferraz-Mello, S.: Earth tides in MacDonald’s model (2013). arXiv: 1301.5617 astro-ph.EP

    Google Scholar 

  19. Ferraz-Mello, S.: Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach. Celest. Mech. Dyn. Astron. 116, 109–140 (2013). arXiv: 1204.3957

    Article  MathSciNet  Google Scholar 

  20. Ferraz-Mello, S.: Tidal synchronization of close-in satellites and exoplanets: II. Spin dynamics and extension to Mercury and exoplantes host stars. Celest. Mech. Dyn. Astr. 122, 359–389 (2015). Errata: Celest. Mech. Dyn. Astr. 130, 78, 20–21 (2018). arXiv: 1505.05384

    Article  MathSciNet  Google Scholar 

  21. Ferraz-Mello, S.: On large and small tidal lags. The virtual identity of two rheophysical theories. Astron. Astrophys. 579, A97 (2015). arXiv.org/abs/1504.04609

    Google Scholar 

  22. Ferraz-Mello, S., Beaugé, C., Michtchenko, T.A.: Evolution of migrating planet pairs in resonance. Celest. Mech. Dyn. Astron. 87, 99–112 (2003)

    Article  MathSciNet  Google Scholar 

  23. Ferraz-Mello, S., Rodríguez, A., Hussmann, H.: Tidal friction in close-in satellites and exoplanets. The Darwin theory re-visited. Celest. Mech. Dyn. Astron. 101, 171-201 (2008). Errata: Celest. Mech. Dyn. Astron. 104, 319–320 (2009). arXiv: 0712.1156

    Article  MathSciNet  Google Scholar 

  24. Ferraz-Mello, S., Grotta-Ragazzo, C., Ruiz, L.S.: Dissipative Forces on Celestial Mechanics, Chap. 3. Soc. Bras. Matem., Rio de Janeiro (2015)

    Google Scholar 

  25. Ferraz-Mello, S., Folonier, H., Tadeu dos Santos, M., Csizmadia, Sz., do Nascimento, J.D., Pätzold, M.: Interplay of tidal evolution and stellar wind braking in the rotation of stars hosting massive close-in planets. Astrophys. J. 807, 78 (2015). arXiv: 1503.04369

    Article  Google Scholar 

  26. Folonier, H.A.: Tide on differentiated planetary satellites. Application to Titan. Dr.Thesis, IAG/Univ. São Paulo (2016)

    Google Scholar 

  27. Folonier, H.A., Ferraz-Mello, S.: Tidal synchronization of an anelastic multi-layered satellite. Titan’s synchronous rotation. Celest. Mech. Dyn. Astron. 129, 359–396 (2017). arXiv: 1706.08603

    Google Scholar 

  28. Folonier, H.A., Ferraz-Mello, S., Andrade-Ines, E.: Tidal synchronization of close-in satellites and exoplanets: III. Tidal dissipation revisited and application to Enceladus. Celest. Mech. Dyn. Astron. 130, 78 (2018). arXiv: 1707.09229v2

    Google Scholar 

  29. Folonier, H., Ferraz-Mello, S., Kholshevnikov, K.V.: The flattenings of the layers of rotating planets and satellites deformed by a tidal potential. Celest. Mech. Dyn. Astron. 122, 183–198 (2015, online supplement). arXiv: 1503.08051

    Article  MathSciNet  Google Scholar 

  30. Goldreich, P.: On the eccentricity of satellite orbits in the Solar System. Mon. Not. R. Astron. Soc 126, 257–268 (1963)

    Article  Google Scholar 

  31. Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics. Kluwer, Dordrecht (1973)

    MATH  Google Scholar 

  32. Hut, P.: Tidal evolution in close binary systems. Astron. Astrophys. 99, 126–140 (1981)

    MATH  Google Scholar 

  33. Jeffreys, H.: The effect of tidal friction on eccentricity and inclination. Mon. Not. R. Astron. Soc. 122, 339–343 (1961)

    Article  MathSciNet  Google Scholar 

  34. Kaula, W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 3, 661–685 (1964)

    Article  Google Scholar 

  35. Laskar, J.: Large scale chaos and marginal stability in the Solar System. Celest. Mech. Dyn. Astron. 64, 115–162 (1996)

    Article  MathSciNet  Google Scholar 

  36. MacDonald, G.F.: Tidal friction. Rev. Geophys. 2, 467–541 (1964)

    Article  Google Scholar 

  37. Makarov, V.V., Efroimsky, M.: Tidal dissipation in a homogeneous spherical body. II. Three examples: Mercury, Io, and Kepler-10 b. Astrophys. J. 795, 7 (2014)

    Article  Google Scholar 

  38. Mardling, R.A., Lin, D.N.C.: On the survival of short-period terrestrial planets. Astrophys. J. 614, 955–959 (2004)

    Article  Google Scholar 

  39. Mardling, R.: Long-term tidal evolution of short-period planets with companions. Mon. Not. R. Astron. Soc. 382, 1768–1790 (2007)

    Article  Google Scholar 

  40. Melchior, P.: The Tides of the Planet Earth. Pergamon Press, Oxford (1983)

    Google Scholar 

  41. Mignard, F.: The evolution of the lunar orbit revisited - I. Moon and Planets 20, 301–315 (1979)

    Article  Google Scholar 

  42. Moulton, F.R.: An Introduction to Celestial Mechanics. Macmillan, New York (1914)

    MATH  Google Scholar 

  43. Ogilvie, G.I., Lin, D.N.C.: Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477–509 (2004)

    Article  Google Scholar 

  44. Oswald, P.: Rheophysics: The Deformation and Flow of Matter. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  45. Peale, S.J.: Origin and evolution of the natural satellites. Annu. Rev. Astron. Astrophys. 37(1), 533–602 (1999)

    Article  Google Scholar 

  46. Ragazzo, C., Ruiz, L.S.: Viscoelastic tides: models for use in Celestial Mechanics. Celest. Mech. Dyn. Astron. 128, 19–59 (2017)

    Article  MathSciNet  Google Scholar 

  47. Ray, R.D., Eanes, R.J., Lemoine, F.G.: Constraints on energy dissipation in the Earth’s body tide from satellite tracking and altimetry. Geophys. J. Int. 144, 471–480 (2001)

    Article  Google Scholar 

  48. Remus, F., Mathis, S., Zahn, J.P., Lainey, V.: The surface signature of the tidal dissipation of the core in a two-layer planet. Astron. Astrophys. 573, A23 (2015)

    Article  Google Scholar 

  49. Rodríguez, A., Ferraz-Mello, S., Michtchenko, T.A., Beaugé, C., Miloni, O.: Tidal decay and orbital circularization in close-in two-planet systems. Mon. Not. R. Astron. Soc. 415, 2349–2358 (2011)

    Article  Google Scholar 

  50. Singer, S.F.: The origin of the Moon and geophysical consequences. Geophys. J.R. Astron. Soc. 15, 205–22 (1968)

    Article  Google Scholar 

  51. Sommerfeld, A.: Lectures on Theoretical Physics, vol. 2. Mechanics of Deformable Bodies. Academic Press, New York (1950)

    MATH  Google Scholar 

  52. Taylor, P.A., Margot, J.-L.: Tidal evolution of close binary asteroid systems. Celest. Mech. Dyn. Astron. 108, 315–338 (2010)

    Article  MathSciNet  Google Scholar 

  53. Tisserand, F.: Traité de Mécanique Céleste, tome II. Gauthier-Villars, Paris (1891)

    Google Scholar 

  54. Williams, J.G., Boggs, D.: Tides on the Moon: Theory and determination of dissipation. J. Geophys. Res. Planets 120, 689–724 (2015)

    Article  Google Scholar 

  55. Williams, J.G., Efroimsky, M.: Bodily tides near the 1:1 spin-orbit resonance. Correction to Goldreich’s dynamical model. Celest. Mech. Dyn. Astron. 114, 387–414 (2012)

    Article  Google Scholar 

  56. Yoder, C.F., Peale, S.J.: The tides of Io. Icarus 47, 1–35 (1981)

    Article  Google Scholar 

Download references

Acknowledgements

This text follows closely the course given at the University of São Paulo in 2017 and it is a pleasure to thank Profs. J.A.S. de Lima and T.A.Michtchenko and Drs. H.A.Folonier and E.Andrade-Ines, for their many comments. Thanks are also due to G.O.Gomes for reading the entire manuscript and for the suggestions improving the clarity. The investigation on the creep tide theory is funded by the National Research Council, CNPq, grant 302742/2015-8, by FAPESP, grants 2014/13407-4, 2016/13750-6 and 2017/10072-0 and by the INCT Inespaço procs. FAPESP 2008/57866-1 and CNPq 574004/2008-4.

Author information

Authors and Affiliations

  1. Instituto de Astronomia Geofísica e Ciências Atmosféricas, Universidade de São Paulo, São Paulo, Brazil

    Sylvio Ferraz-Mello

Authors
  1. Sylvio Ferraz-Mello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvio Ferraz-Mello .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Giulio Baù

  2. Department of Mathematics, University of Rome Tor Vergata, Roma, Italy

    Alessandra Celletti

  3. Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi, Iasi, Romania

    Cătălin Bogdan Galeș

  4. Department of Mathematics, University of Pisa, Pisa, Italy

    Giovanni Federico Gronchi

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ferraz-Mello, S. (2019). Planetary Tides: Theories. In: Baù, G., Celletti, A., Galeș, C., Gronchi, G. (eds) Satellite Dynamics and Space Missions. Springer INdAM Series, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-20633-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation