Dynamic Elastic Tides

Abstract

This is an exploration of dynamic tides on elastic bodies. The body is thought of as a dynamical system described by its modes of oscillation. The dynamics of these modes are governed by differential equations that depend on the rheology. The modes are damped by dissipation. Tidal friction occurs as exterior bodies excite the modes and the modes act back on the tide raising body. The whole process is governed by a closed set of differential equations. Standard results from tidal theory are recovered in a two-timescale approximation to the solution of these differential equations.

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Fig. 1

Notes

  1. 1.

    We use the two argument arctangent with the property \(\psi = \mathrm{atan}(\sin (\psi ), \cos (\psi ))\), rather than the one argument arctangent with the property \(\psi = \mathrm{atan}(\tan \psi )\), in order to retain quadrant information.

References

  1. Chandler, S.C.: On the variation of latitude. Astron. J. 11, 59–61 (1891)

    ADS  Article  Google Scholar 

  2. Darwin, G.: Scientific Papers, vol. 2. Cambridge University Press, Cambridge (1908)

    Google Scholar 

  3. Dobrovolskis, A.: Spin states and climates of eccentric exoplanets. Icarus 192, 1–23 (2007)

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  5. Goldreich, P.: On the eccentricity of satellite orbits in the solar system. MNRAS 126, 257–268 (1963)

    ADS  Article  MATH  Google Scholar 

  6. Goldreich, P.: History of the lunar orbit. Rev. Geophys. Space Phys. 4, 411–439 (1966)

    ADS  Article  Google Scholar 

  7. Goldreich, P., Peale, S.J.: Spin-orbit coupling in the solar system. Astron. J. 71, 425–438 (1966)

    ADS  Article  Google Scholar 

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

    ADS  MATH  Google Scholar 

  9. Jefferys, H.: The Earth, 6th edn. Cambridge University Press, Cambridge (1976)

    Google Scholar 

  10. Kaula, W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 2, 661–685 (1966)

    ADS  Article  Google Scholar 

  11. Lamb, H.: On the vibrations of an elastic sphere. Proc. Lond. Math Soc. 13, 189–212 (1882)

    MathSciNet  MATH  Google Scholar 

  12. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn, pp. 257–259. Dover, New York, NY (1944)

    Google Scholar 

  13. Mignard, F.: The lunar orbit revisited, III. Moon Planets 24, 189–207 (1981)

    ADS  Article  MATH  Google Scholar 

  14. Newcomb, S.: On the dynamics of the earth’s rotation, with respect to the periodic variations of latitude. MNRAS 52, 336–341 (1882)

    ADS  Article  MATH  Google Scholar 

  15. Peale, S.J.: Rotation of solid bodies in the solar system. Rev. Geophys. Space Phys. 11, 767–793 (1973)

    ADS  Article  Google Scholar 

  16. Peale, S.J., Cassen, P.: Contribution of tidal dissipation to lunar thermal history. Icarus 36, 245–269 (1978)

    ADS  Article  Google Scholar 

  17. Peale, S.J., Cassen, P., Reynolds, R.: Melting of Io by tidal dissipation. Science 203, 892–894 (1979)

    ADS  Article  Google Scholar 

  18. Peale, S.J.: Resonances. In: Burns, J.A., Matthews, M.S. (eds.) Satellites, pp. 159–223. University of Arizona Press, Tucson (1986)

    Google Scholar 

  19. Poincaré, H.: Sur une forme novelle des équations de la mécanique. Comptes rendus de l’Académie des Sciences 132, 369–371 (1901)

    MATH  Google Scholar 

  20. Stacey, F.D.: Physics of the Earth, 3rd edn. Brookfield Press, Brisbane (1992)

    Google Scholar 

  21. Williams, J.G.: Lunar core and mantle. What does LLR see? Proceedings of the 16th International Workshop on Laser Ranging, pp. 101–120 (2008)

  22. Wisdom, J.: Tidal heating at arbitrary eccentricity and obliquity. Icarus 193, 637–640 (2008)

    ADS  Article  Google Scholar 

  23. Zahn, J.-P.: The dynamical tide in close binaries. Astron. Astrophys. 41, 329–344 (1975)

    ADS  Google Scholar 

Download references

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Correspondence to Jack Wisdom.

Appendix: Surface and solid spherical harmonics

Appendix: Surface and solid spherical harmonics

We will use the terms “solid spherical harmonic” and “surface spherical harmonic.” The normalized surface spherical harmonics are

$$\begin{aligned} C_l^m(\theta , \phi )= & {} P_l^m(\cos \theta ) \cos (m \phi ) \end{aligned}$$
(288)
$$\begin{aligned} S_l^m(\theta , \phi )= & {} P_l^m(\cos \theta ) \sin (m \phi ) , \end{aligned}$$
(289)

where the normalized associated Legendre polynomials satisfy

$$\begin{aligned} P_l^m(x) = N_l^m P_{lm}(x) , \end{aligned}$$
(290)

with

$$\begin{aligned} N_l^m = \left[ (2 - \delta _{m,0}) (2 l + 1) \frac{(l-m)!}{(l+m)!} \right] ^{1/2} , \end{aligned}$$
(291)

and

$$\begin{aligned} P_{lm}(x) = \frac{1}{2^l l!} (1 - x^2)^{m/2} \frac{d^{l+m}}{dx^{l+m}} \left[ (x^2 - 1)^l \right] . \end{aligned}$$
(292)

A few of the \(P_{lm}\) are

$$\begin{aligned} P_{20}(\cos \theta )= & {} \frac{3}{2} (\cos \theta )^2 - \frac{1}{2} \end{aligned}$$
(293)
$$\begin{aligned} P_{21}(\cos \theta )= & {} 3 \sin \theta \cos \theta \end{aligned}$$
(294)
$$\begin{aligned} P_{22}(\cos \theta )= & {} 3 (\sin \theta )^2 . \end{aligned}$$
(295)

For convenience we introduce

$$\begin{aligned} X_l^m(\theta , \phi )= & {} C_l^m(\theta , \phi ) \quad m \ge 0 \end{aligned}$$
(296)
$$\begin{aligned}= & {} S_l^{-m}(\theta , \phi ) \quad m < 0 \end{aligned}$$
(297)

The \(X_l^m\) (\(C_l^m\) and \(S_l^m\)) are orthonormal in that

$$\begin{aligned} \delta _{l l'} \delta _{mm'} = \frac{1}{4\pi } \int _0^\pi \int _0^{2\pi } X_l^m(\theta , \phi ) X_{l'}^{m'}(\theta , \phi ) \sin \theta d\theta d\phi \end{aligned}$$
(298)

The solid spherical harmonics are

$$\begin{aligned} \tilde{X}_l^m\left( \frac{x}{R}, \frac{y}{R}, \frac{z}{R}\right) = \left( \frac{r}{R} \right) ^l X_l^m(\theta , \phi ) , \end{aligned}$$
(299)

where \(r^2 = x^2 + y^2 + z^2\), \(\cos \theta = z/r\), \(\phi = \mathrm{atan}(y, x)\). The solid harmonics satisfy the orthogonality relations

$$\begin{aligned}&\frac{\delta _{ll'} \delta _{mm'}}{2l+3} \end{aligned}$$
(300)
$$\begin{aligned}&\quad = \frac{1}{4\pi R^3} \int _0^a \int _0^\pi \int _0^{2\pi } \tilde{X}_l^m\left( \frac{x}{R}, \frac{y}{R}, \frac{z}{R}\right) \tilde{X}_{l'}^{m'}\left( \frac{x}{R}, \frac{y}{R}, \frac{z}{R}\right) r^2 \sin \theta dr d\theta d\phi \quad \quad \quad \end{aligned}$$
(301)
$$\begin{aligned}&\quad = \frac{1}{4\pi a^3} \int _V \tilde{X}_l^m \tilde{X}_{l'}^{m'} dV , \end{aligned}$$
(302)

where the V is the sphere of radius R, and the last line introduces an abbreviated notation. Note that whenever we write a \(\tilde{X}_l^m\) without arguments, they are assumed to be (x / Ry / Rz / R).

It is convenient to introduce

$$\begin{aligned} \bar{X}_l^m = \frac{1}{\sqrt{2l+1}} \tilde{X}_l^m . \end{aligned}$$
(303)

The first few \(\bar{X}_l^m\) are

$$\begin{aligned} \bar{X}_0^0(x, y, z)= & {} 1 \end{aligned}$$
(304)
$$\begin{aligned} \bar{X}_1^0(x, y, z)= & {} z \end{aligned}$$
(305)
$$\begin{aligned} \bar{X}_1^1(x, y, z)= & {} x \end{aligned}$$
(306)
$$\begin{aligned} \bar{X}_1^{-1}(x, y, z)= & {} y \end{aligned}$$
(307)
$$\begin{aligned} \bar{X}_2^0(x, y, z)= & {} (2 z^2 - x^2 - y^2)/2 \end{aligned}$$
(308)
$$\begin{aligned} \bar{X}_2^1(x, y, z)= & {} \sqrt{3} x z \end{aligned}$$
(309)
$$\begin{aligned} \bar{X}_2^2(x, y, z)= & {} \sqrt{3} (x^2 - y^2)/2 \end{aligned}$$
(310)
$$\begin{aligned} \bar{X}_2^{-1}(x, y, z)= & {} \sqrt{3} y z \end{aligned}$$
(311)
$$\begin{aligned} \bar{X}_2^{-2}(x, y, z)= & {} \sqrt{3} x y . \end{aligned}$$
(312)

Let \(\tilde{X}_l^m\) be a solid spherical harmonic. It is a homogeneous function of degree l in the coordinates (x / Ry / Rz / R). Euler’s theorem tells us that

$$\begin{aligned} \left( \mathbf {x} \cdot \nabla \right) \tilde{X}_l^m (x/R, y/R, z/R) = l \tilde{X}_l^m (x/R, y/R, z/R). \end{aligned}$$
(313)

A little calculation shows that

$$\begin{aligned} \nabla ^2 ( \mathbf {x} \tilde{X}_l^m(x/R, y/R, z/R) ) = 2 \nabla \tilde{X}_l^m (x/R, y/R, z/R), \end{aligned}$$
(314)

and

$$\begin{aligned} \nabla ^2 ( r^k \tilde{X}_l^m (x/R, y/R, z/R)) = k ( k + 2l + 1) r^{k-2} \tilde{X}_l^m (x/R, y/R, z/R) . \end{aligned}$$
(315)

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Wisdom, J., Meyer, J. Dynamic Elastic Tides. Celest Mech Dyn Astr 126, 1–30 (2016). https://doi.org/10.1007/s10569-016-9682-3

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Keywords

  • Tides
  • Tidal evolution
  • Tidal friction
  • Lags
  • Orbital evolution
  • Wobble damping
  • Dynamical astronomy