Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology


In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism taken from quantum theory. We also provide the vectorial expression of the secular tidal torque expanded in Fourier series. Applying this model to close-in exoplanets, we observe that if the relaxation time is longer than the revolution period, the phase space of the system is characterized by the presence of several spin-orbit resonances, even in the circular case. As the system evolves, the planet spin can visit different spin-orbit configurations. The obliquity is decreasing along most of these resonances, but we observe a case where the planet tilt is instead growing. These conclusions derived from the secular torque are successfully tested with numerical integrations of the instantaneous equations of motion on HD 80606 b. Our formalism is also well adapted to close-in super-Earths in multiplanet systems which are known to have non-zero mutual inclinations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Note that if the material composing the extended body was governed by the Newtonian creep rheology or by the Kelvin-Voigt one, \(\underline{k}_l(\nu )\) would have the same analytical expression but with \(\tau _e = 0\).

  2. 2.

    Our notation is very similar to that of Correia et al. (2003) and Cunha et al. (2015) but, in these papers, \(b^\mathrm {g}(\nu )\) is defined as the opposite of the imaginary part of the Love number \(\underline{k}_2(\nu )\). Thus, \(b^\mathrm {g}(\nu )\) is related to our \(b_2(\nu )\) through the relation \(b^\mathrm {g}(\nu ) = -b_2(\nu )\).


  1. Auclair-Desrotour, P., Laskar, J., Mathis, S.: Atmospheric tides in Earth-like planets. Astron. Astrophys. (2016)

  2. Correia, A.C.M., Laskar, J.: The four final rotation states of Venus. Nature 411, 767–770 (2001). doi:10.1038/35081000

    ADS  Article  Google Scholar 

  3. Correia, A.C.M., Laskar, J., de Surgy, O.N.: Long-term evolution of the spin of Venus. I. Theory. Icarus 163, 1–23 (2003). doi:10.1016/S0019-1035(03)00042-3

    ADS  Article  Google Scholar 

  4. Correia, A.C.M., Laskar, J., Farago, F., Boué, G.: Tidal evolution of hierarchical and inclined systems. Celest. Mech. Dyn. Astron. 111, 105–130 (2011). doi:10.1007/s10569-011-9368-9.arXiv:1107.0736

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. 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). doi:10.1051/0004-6361/201424211.arXiv:1411.1860

    ADS  Article  Google Scholar 

  6. Cunha, D., Correia, A.C.M., Laskar, J.: Spin evolution of Earth-sized exoplanets, including atmospheric tides and core-mantle friction. Int. J. Astrobiol. 14, 233–254 (2015). doi:10.1017/S1473550414000226.arXiv:1406.4544

    Article  Google Scholar 

  7. Darwin, G.H.: On the secular changes in the elements of the orbit of a satellite revolving about a tidally distorted planet. Philos. Trans. R. Soc. Lond. 171, 713–891 (1880)

    Article  MATH  Google Scholar 

  8. Efroimsky, M.: Bodily tides near spin-orbit resonances. Celest. Mech. Dyn. Astron. 112, 283–330 (2012a). doi:10.1007/s10569-011-9397-4.arXiv:1105.6086

    ADS  MathSciNet  Article  Google Scholar 

  9. Efroimsky, M.: Tidal dissipation compared to seismic dissipation. In small bodies, Earths, and super-Earths. Astrophys. J. 746, 150 (2012b). doi:10.1088/0004-637X/746/2/150.arXiv:1105.3936

    ADS  Article  Google Scholar 

  10. Efroimsky, M., Lainey, V.: Physics of bodily tides in terrestrial planets and the appropriate scales of dynamical evolution. J. Geophys. Res. (Planets) 112(E11), E12003 (2007). doi:10.1029/2007JE002908.arXiv:0709.1995

    ADS  Article  Google Scholar 

  11. Efroimsky, M., Makarov, V.V.: Tidal friction and tidal lagging. Applicability limitations of a popular formula for the tidal torque. Astrophys. J. 764, 26 (2013). doi:10.1088/0004-637X/764/1/26.arXiv:1209.1615

    ADS  Article  Google Scholar 

  12. Fabrycky, D.C., Lissauer, J.J., Ragozzine, D., Rowe, J.F., Steffen, J.H., Agol, E., et al.: Architecture of Kepler’s multi-transiting systems. II. New investigations with twice as many candidates. Astrophys. J. 790, 146 (2014). doi:10.1088/0004-637X/790/2/146.arXiv:1202.6328

    ADS  Article  Google Scholar 

  13. Ferraz-Mello, S.: Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach. Celest. Mech. Dyn. Astron. 116, 109–140 (2013). doi:10.1007/s10569-013-9482-y.arXiv:1204.3957

    ADS  MathSciNet  Article  Google Scholar 

  14. Ferraz-Mello, S.: The small and large lags of the elastic and anelastic tides. The virtual identity of two rheophysical theories. Astron. Astrophys. 579, A97 (2015). doi:10.1051/0004-6361/201525900.arXiv:1504.04609

    ADS  Article  Google Scholar 

  15. Figueira, P., Marmier, M., Boué, G., Lovis, C., Santos, N.C., Montalto, M., et al.: Comparing HARPS and Kepler surveys. The alignment of multiple-planet systems. Astron. Astrophys. 541, A139 (2012). doi:10.1051/0004-6361/201219017.arXiv:1202.2801

    ADS  Article  Google Scholar 

  16. Frouard, J., Quillen, A.C., Efroimsky, M., Giannella, D.: Numerical simulation of tidal evolution of a viscoelastic body modelled with a mass-spring network. Mon. Not. R. Astron. Soc. 458, 2890–2901 (2016). doi:10.1093/mnras/stw491.arXiv:1601.08222

    ADS  Article  Google Scholar 

  17. Gimbutas, Z., Greengard, L.: A fast and stable method for rotating spherical harmonic expansions. J. Comput. Phys. 228, 5621–5627 (2009)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. Henning, W.G., O’Connell, R.J., Sasselov, D.D.: Tidally heated terrestrial exoplanets: viscoelastic response models. Astrophys. J. 707, 1000–1015 (2009). doi:10.1088/0004-637X/707/2/1000.arXiv:0912.1907

    ADS  Article  Google Scholar 

  19. Jeffreys, H.: The Earth. Its Origin, History and Physical Constitution. Cambridge University Press, Cambridge (1976)

    Google Scholar 

  20. Kaula, W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. Space Phys. 2, 661–685 (1964). doi:10.1029/RG002i004p00661

    ADS  Article  Google Scholar 

  21. Lambeck, K.: Geophysical Geodesy—The Slow Deformations of the Earth. Oxford University Press, Oxford (1988)

    Google Scholar 

  22. Laskar, J., Boué, G., Correia, A.C.M.: Tidal dissipation in multi-planet systems and constraints on orbit fitting. Astron. Astrophys. 538, A105 (2012). doi:10.1051/0004-6361/201116643. arXiv:1110.4565

    ADS  Article  MATH  Google Scholar 

  23. MacDonald, G.J.F.: Tidal friction. Rev. Geophys. Space Phys. 2, 467–541 (1964). doi:10.1029/RG002i003p00467

    ADS  Article  Google Scholar 

  24. Makarov, V.V., Efroimsky, M.: No pseudosynchronous rotation for terrestrial planets and Moons. Astrophys. J. 764, 27 (2013). doi:10.1088/0004-637X/764/1/27. arXiv:1209.1616

    ADS  Article  Google Scholar 

  25. Mignard, F.: Multiple expansion of the tidal potential. Celest. Mech. 18, 287–294 (1978). doi:10.1007/BF01230169

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. Mignard, F.: The evolution of the lunar orbit revisited. I. Moon Planets 20, 301–315 (1979). doi:10.1007/BF00907581

    ADS  Article  MATH  Google Scholar 

  27. Ogilvie, G.I., Lin, D.N.C.: Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477–509 (2004). doi:10.1086/421454.arXiv:astro-ph/0310218

    ADS  Article  Google Scholar 

  28. Peltier, W.R.: The impulse response of a Maxwell Earth. Rev. Geophys. Space Phys. 12, 649–669 (1974). doi:10.1029/RG012i004p00649

    ADS  Article  Google Scholar 

  29. Remus, F., Mathis, S., Zahn, J.P., Lainey, V.: Anelastic tidal dissipation in multi-layer planets. Astron. Astrophys. 541, A165 (2012). doi:10.1051/0004-6361/201118595.arXiv:1204.1468

    ADS  Article  Google Scholar 

  30. Singer, S.F.: The origin of the Moon and geophysical consequences*. Geophys. J. R. Astron. Soc. 15(1–2), 205–226 (1968). doi:10.1111/j.1365-246X.1968.tb05759.x

    Google Scholar 

  31. Tremaine, S., Dong, S.: The statistics of multi-planet systems. Astronom. J. 143, 94 (2012). doi:10.1088/0004-6256/143/4/94.arXiv:1106.5403

    ADS  Article  Google Scholar 

  32. Varshalovich, D., Moskalev, A., Khersonskii, V.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)

    Google Scholar 

Download references


GB is grateful to Dan Fabrycky for the fruitful discussions which lead to this work. We acknowledge support from CIDMA strategic project UID/MAT/04106/2013.

Author information



Corresponding author

Correspondence to Gwenaël Boué.


Appendix 1: Spherical harmonic

By convention, Legendre associated polynomials are defined as

$$\begin{aligned} P_{l,m}(x) = \frac{1}{2^l l!} (1-x^2)^{m/2} \frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l, \end{aligned}$$

with the symmetry

$$\begin{aligned} P_{l,-m} (x) = (-1)^m \frac{(l-m)!}{(l+m)!}P_{l,m}(x). \end{aligned}$$

The Schmidt semi-normalized spherical harmonics are defined as

$$\begin{aligned} Y_{l,m}(\theta ,\phi ) = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_{l,m}(\cos \theta ) \mathrm {e}^{\mathrm {i}m\phi } \end{aligned}$$

with the symmetry

$$\begin{aligned} Y_{l,-m}(\theta ,\phi ) = (-1)^m \bar{Y}_{l,m}(\theta ,\phi ) . \end{aligned}$$

Using the complex Cartesian coordinate system as defined in Varshalovich et al. (1988), for any unit vector \(\hat{x}\), we have

$$\begin{aligned}&\displaystyle Y_{0,0}(\hat{\mathbf{x}}) = 1 , \end{aligned}$$
$$\begin{aligned}&\displaystyle Y_{1,0}(\hat{\mathbf{x}}) = \hat{x}_0 , \end{aligned}$$
$$\begin{aligned}&\displaystyle Y_{1,1}(\hat{\mathbf{x}}) = \hat{x}_+ , \end{aligned}$$
$$\begin{aligned}&\displaystyle l\, Y_{l,0}(\hat{\mathbf{x}}) = (2l-1)\hat{x}_0 Y_{l-1,0}(\hat{\mathbf{x}}) - (l-1)Y_{l-2,0}(\hat{\mathbf{x}}), \end{aligned}$$
$$\begin{aligned}&\displaystyle \sqrt{l+m} Y_{l,m}(\hat{\mathbf{x}}) = \sqrt{l-m}\, \hat{x}_0 Y_{l-1,m}(\hat{\mathbf{x}}) + \sqrt{2(l+m-1)} \hat{x}_+ Y_{l-1,m-1}. \end{aligned}$$

The last two Eqs. (32d) and (32e) allow to recursively compute all spherical harmonics of order \(m\ge 0\). Those with \(m<0\) are deduced from the symmetry relation (31). Up to the degree 3 included, we have

$$\begin{aligned} \left\{ \begin{array}{ll} Y_{2,0} &{}= \displaystyle \frac{1}{2}(3 \hat{x}_0^2 -1) \\ Y_{2,1} &{}= \displaystyle \sqrt{3} \hat{x}_0 \hat{x}_+ \\ Y_{2,2} &{}= \displaystyle \frac{\sqrt{6}}{2} \hat{x}_+^2 \end{array} \right. , \qquad \left\{ \begin{array}{ll} Y_{3,0} &{}= \displaystyle \frac{5}{2}\hat{x}_0^3 - \frac{3}{2}\hat{x}_0 \\ Y_{3,1} &{}= \displaystyle \frac{\sqrt{6}}{4} \left( 5 \hat{x}_0^2 \hat{x}_+ - \hat{x}_+\right) \\ Y_{3,2} &{}= \displaystyle \frac{\sqrt{30}}{2} \hat{x}_0 \hat{x}_+^2 \\ Y_{3,3} &{}= \displaystyle \frac{\sqrt{10}}{2} \hat{x}_+^3 \end{array} \right. . \end{aligned}$$

Appendix 2: Ladder operators

Regular solid harmonics \(x^l Y_{l,m}(\hat{\mathbf{x}})\) and irregular ones \(Y_{l,m}(\hat{\mathbf{x}})/x^{l+1}\) are eigenvectors of each component of the gradient operator \(\varvec{\nabla } = (\nabla _+, \nabla _0, \nabla _-)\) and of the angular momentum operator \(\mathbf{J} = (J_+, J_0, J_-)\). The respective eigenvalues can be found in (e.g., Varshalovich et al. 1988). We have

$$\begin{aligned} \nabla _+ \left( x^{l}Y_{l,m}(\hat{\mathbf{x}})\right)&= -\,\sqrt{\frac{(l-m-1)(l-m)}{2}} \,x^{l-1}Y_{l-1,m+1}(\hat{\mathbf{x}}) \nonumber \\ \nabla _0 \left( x^{l}Y_{l,m}(\hat{\mathbf{x}})\right)&= +\,\sqrt{(l+m)(l-m)} \,x^{l-1} Y_{l-1,m}(\hat{\mathbf{x}}) \nonumber \\ \nabla _- \left( x^{l}Y_{l,m}(\hat{\mathbf{x}})\right)&= -\,\sqrt{\frac{(l+m-1)(l+m)}{2}} \,x^{l-1}Y_{l-1,m-1}(\hat{\mathbf{x}}), \end{aligned}$$
$$\begin{aligned} \nabla _+ \left( \frac{1}{x^{l+1}}Y_{l,m}(\hat{\mathbf{x}})\right)&= -\,\sqrt{\frac{(l+m+1)(l+m+2)}{2}} \frac{1}{x^{l+2}}Y_{l+1,m+1}(\hat{\mathbf{x}}) \nonumber \\ \nabla _0 \left( \frac{1}{x^{l+1}}Y_{l,m}(\hat{\mathbf{x}})\right)&= -\,\sqrt{(l+m+1)(l-m+1)} \frac{1}{x^{l+2}} Y_{l+1,m}(\hat{\mathbf{x}}) \nonumber \\ \nabla _- \left( \frac{1}{x^{l+1}}Y_{l,m}(\hat{\mathbf{x}})\right)&= -\,\sqrt{\frac{(l-m+1)(l-m+2)}{2}} \frac{1}{x^{l+2}}Y_{l+1,m-1}(\hat{\mathbf{x}}), \end{aligned}$$


$$\begin{aligned} J_+ \bigg ( f(x) Y_{l,m}(\hat{\mathbf{x}}) \bigg )&= -\,\sqrt{\frac{l(l+1)-m(m+1)}{2}} f(x) Y_{l,m+1}(\hat{\mathbf{x}}) \nonumber \\ J_0 \bigg ( f(x) Y_{l,m}(\hat{\mathbf{x}}) \bigg )&= m f(x) Y_{l,m}(\hat{\mathbf{x}}) \nonumber \\ J_- \bigg ( f(x) Y_{l,m}(\hat{\mathbf{x}}) \bigg )&= +\,\sqrt{\frac{l(l+1)-m(m-1)}{2}} f(x) Y_{l,m-1}(\hat{\mathbf{x}}), \end{aligned}$$

where f(x) is any function of the modulus \(x=\Vert \vec {x}\Vert \).

Appendix 3: Rotation and Wigner matrices

Let a vector \(\vec {x}\) and two coordinate systems \(\mathcal{B}\) and \(\mathcal{B}'\) such that \(\mathbf{x}\) and \(\mathbf{x}'\) are the coordinates of \(\vec {x}\) in \(\mathcal{B}\) and \(\mathcal{B}'\), respectively. Let us further assume that \(\mathbf{x}\) and \(\mathbf{x}'\) are related to each other by a rotation of the form

$$\begin{aligned} \mathbf{x} = {{\mathsf {R}}}_3(\alpha ) {{\mathsf {R}}}_2(\beta ) {{\mathsf {R}}}_3(\gamma ) \mathbf{x}', \end{aligned}$$

where \({{\mathsf {R}}}_3\) and \({{\mathsf {R}}}_2\) are the matrices of rotation around the third and the second axes, respectively. Wigner D matrix \({\mathsf {D}}^l_{m,m'}(\alpha ,\beta ,\gamma )\) is defined such that (e.g., Varshalovich et al. 1988)

$$\begin{aligned} Y_{l,m}(\hat{\mathbf{x}}') = \sum _{m'=-l}^l {\mathsf {D}}^l_{m',m}(\alpha ,\beta ,\gamma ) Y_{l,m'}(\hat{\mathbf{x}}). \end{aligned}$$

Each element \({\mathsf {D}}^l_{m,m'}(\alpha ,\beta ,\gamma )\) can be written as (e.g., Varshalovich et al. 1988)

$$\begin{aligned} {\mathsf {D}}^l_{m,m'}(\alpha ,\beta ,\gamma ) = \mathrm {e}^{-\mathrm {i}m \alpha } {\mathsf {d}}^l_{m,m'}(\beta ) \mathrm {e}^{-\mathrm {i}m'\gamma }, \end{aligned}$$

where \({\mathsf {d}}^l_{m,m'}(\beta )\) is the Wigner d matrix. The inverse \({\mathsf {D}}^l_{m,m'}(-\gamma ,-\beta ,-\alpha )\) is given by the adjoint \(\bar{{\mathsf {D}}}^l_{m',m}(\alpha ,\beta ,\gamma )\) of \({\mathsf {D}}^l_{m,m'}(\alpha ,\beta ,\gamma )\):

$$\begin{aligned} {\mathsf {D}}^l_{m,m'}(-\gamma ,-\beta ,-\alpha ) = \mathrm {e}^{\mathrm {i}m'\alpha } {\mathsf {d}}^l_{m',m}(\beta )\mathrm {e}^{\mathrm {i}m\gamma }. \end{aligned}$$

The convention 3-2-3 of the rotation (Eq. 37) is such that \({\mathsf {d}}^l_{m,m'}(\beta )\) is a real function. Wigner d matrix possesses many symmetries, among which (e.g., Varshalovich et al. 1988)

$$\begin{aligned} {\mathsf {d}}^l_{m,m'}(\beta ) = (-1)^{m-m'}{\mathsf {d}}^l_{-m,-m'}(\beta ) = (-1)^{m-m'}{\mathsf {d}}^l_{m',m}(\beta ) = {\mathsf {d}}^l_{-m',-m}(\beta ). \end{aligned}$$

Wigner d matrix can be constructed recursively using the hereinabove symmetries, the following initialization (e.g., Varshalovich et al. 1988)

$$\begin{aligned} {\mathsf {d}}^0_{0,0}(\beta )&= 1\,,\ {\mathsf {d}}^1_{0,0}(\beta ) = \cos \beta \,,\ {\mathsf {d}}^1_{1,-1}(\beta ) = \frac{1-\cos \beta }{2} \,,\ {\mathsf {d}}^1_{1,0}(\beta ) = -\frac{\sin \beta }{\sqrt{2}} \,,\ {\mathsf {d}}^1_{1,1}(\beta )\nonumber \\&= \frac{1+\cos \beta }{2} \end{aligned}$$

and the recurrence relation (Gimbutas and Greengard 2009)

$$\begin{aligned} \begin{aligned} {\mathsf {d}}^l_{m,m'}(\beta ) =&+\sqrt{\frac{(l+m')(l+m'-1)}{(l+m)(l+m-1)}} {\mathsf {d}}^1_{1,1}(\beta ) {\mathsf {d}}^{l-1}_{m-1,m'-1}(\beta ) \\&- \sqrt{\frac{(l+m')(l-m' )}{(l+m)(l+m-1)}} \sin (\beta ) {\mathsf {d}}^{l-1}_{m-1,m'}(\beta ) \\&+ \sqrt{\frac{(l-m')(l-m'-1)}{(l+m)(l+m-1)}} {\mathsf {d}}^1_{1,-1}(\beta ) {\mathsf {d}}^{l-1}_{m-1,m'+1}(\beta ) \end{aligned} \end{aligned}$$

which also implies

$$\begin{aligned} {\mathsf {d}}^l_{l,l}(\beta ) = {\mathsf {d}}^1_{1,1}(\beta ) {\mathsf {d}}^{l-1}_{l-1,l-1}(\beta ) \qquad \text {and} \qquad {\mathsf {d}}^l_{l,-l}(\beta ) = {\mathsf {d}}^1_{1,-1}(\beta ) {\mathsf {d}}^{l-1}_{l-1,1-l}(\beta ). \end{aligned}$$

The algorithm is the following:


For completeness, we also provide the explicit terms at order \(l=2\) in Table 2.

Appendix 4: Time derivatives

Let a function \(f(\vec {x}, t)\) developed in spherical harmonics as

$$\begin{aligned} f(\vec {x}, t) = \sum _{l,m} \bar{z}_{l,m}(t) Y_{l,m}(\hat{\mathbf{x}}) \end{aligned}$$
Table 2 Explicit Wigner d matrix \({\mathsf {d}}^2_{m,m'}(\beta )\)

in the inertial frame \(\mathcal{F}_0\), and as

$$\begin{aligned} f(\vec {x}, t) = \sum _{l,m} \bar{Z}_{l,m}(t) Y_{l,m}(\hat{\mathbf{X}}) \end{aligned}$$

in the body frame \(\mathcal{F}_p\). For any constant vector \(\vec {x}\) in \(\mathcal{F}_p\), we have

$$\begin{aligned} \dot{\mathbf{x}} = {\varvec{\omega }} \times \mathbf{x} \qquad \mathrm {and} \qquad \dot{\mathbf{X}} = \mathbf{0} , \end{aligned}$$

with respect to the frame \(\mathcal{F}_p\). By consequence, in \(\mathcal{F}_p\), on the one hand,

$$\begin{aligned} \dot{f}(\vec {x}, t) = \sum _{l,m} \left( \dot{\bar{z}}_{l,m}(t) Y_{l,m}(\hat{\mathbf{x}}) + \bar{z}_{l,m}(t) \dot{\mathbf{x}}\cdot \varvec{\nabla } Y_{l,m}(\hat{\mathbf{x}}) \right) , \end{aligned}$$

and on the other hand,

$$\begin{aligned} \dot{f}(\vec {x}, t) = \sum _{l,m} \dot{\bar{Z}}_{l,m}(t) Y_{l,m}(\hat{\mathbf{X}}). \end{aligned}$$

But given that the time derivative of \(\mathbf{x}\) is \(\dot{\mathbf{x}} = \varvec{\omega }\times \mathbf{x}\), we get

$$\begin{aligned} \dot{\mathbf{x}}\cdot \varvec{\nabla } = ({\varvec{\omega }} \times \mathbf{x}) \cdot \varvec{\nabla } = \mathrm {i}({\varvec{\omega }} \cdot \mathbf{J}) \end{aligned}$$

where \(\mathbf{J} = -\mathrm {i}\mathbf{x} \times \varvec{\nabla }\) is the angular momentum operator and where, by construction of the scalar product (Varshalovich et al. 1988),

$$\begin{aligned} {\varvec{\omega }} \cdot \mathbf{J} = - \omega _{+} J_{-} + \omega _0 J_0 - \omega _{-} J_{+}. \end{aligned}$$

We then define a matrix \({\mathsf {J}}({\varvec{\omega }})\) of size \((2l+1)\times (2l+1)\) such that

$$\begin{aligned} ({\varvec{\omega }} \cdot \mathbf{J}) Y_{l,m}(\hat{\mathbf{x}}) = \sum _{m'=-l}^l [{\mathsf {J}}({\varvec{\omega }})]^l_{m',m} Y_{l,m'}(\hat{\mathbf{x}}), \end{aligned}$$

where all non-zero coefficients are

$$\begin{aligned} \begin{array}{lll} \left[ {\mathsf {J}}({\varvec{\omega }})\right] ^l_{m-1,m} &{}=&{} - \sqrt{\frac{l(l+1)-m(m-1)}{2}} \omega _+, \\ \left[ {\mathsf {J}}({\varvec{\omega }})\right] ^l_{m,m} &{}=&{} m\, \omega _0 , \\ \left[ {\mathsf {J}}({\varvec{\omega }})\right] ^l_{m+1,m} &{}=&{} + \sqrt{\frac{l(l+1)-m(m+1)}{2}} \omega _-. \end{array} \end{aligned}$$

Combining Eqs. (10), (4547), and (49), we obtain

$$\begin{aligned} \sum _{m'} {\mathsf {D}}^l_{m,m'}\dot{\bar{Z}}_{l,m'} = \dot{\bar{z}}_{l,m} + \mathrm {i}\sum _{m'}\,[{\mathsf {J}}({\varvec{\omega }})]^l_{m,m'} {\bar{z}}_{l,m'} . \end{aligned}$$

Appendix 5: Fourier transform

Let two functions \(f(\vec {x}, t)\) and \(g(\vec {x}, t)\) expanded in spherical harmonics as \(f = \sum _l f_l\) and \(g=\sum _l g_l\) with

$$\begin{aligned} f_l(\vec {x}, t) = \sum _{m=-l}^l \bar{Z}_{l,m}(t) Y_{l,m}(\hat{\mathbf{X}}) \qquad \text {and} \qquad g_l(\vec {x}, t) = \sum _{m=-l}^l \bar{Z}'_{l,m}(t) Y_{l,m}(\hat{\mathbf{X}}) \end{aligned}$$

in the frame \(\mathcal{F}_p\) and

$$\begin{aligned} f_l(\vec {x}, t) = \sum _{m=-l}^l \bar{z}_{l,m}(t) Y_{l,m}(\hat{\mathbf{x}}) \qquad \text {and} \qquad g_l(\vec {x}, t) = \sum _{m=-l}^l \bar{z}'_{l,m}(t) Y_{l,m}(\hat{\mathbf{x}}) \end{aligned}$$

in \(\mathcal{F}_0\). Let \(\alpha \), \(\beta \), and \(\gamma =\omega t\) be the three angles such that

$$\begin{aligned} \mathbf{x} = {\mathsf {R}}_3(\alpha ) {\mathsf {R}}_2(\beta ) {\mathsf {R}}_3(\gamma ) \mathbf{X}. \end{aligned}$$

We have then

$$\begin{aligned} z_{l,m}(t) = \sum _{m'=-l}^l \bar{{\mathsf {D}}}^l_{m,m'}(t) Z_{l,m'}(t) \qquad \text {and} \qquad z'_{l,m}(t) = \sum _{m'=-l}^l \bar{{\mathsf {D}}}^l_{m,m'}(t) Z'_{l,m'}(t) \end{aligned}$$


$$\begin{aligned} {\mathsf {D}}^l_{m,m'} (t) = {\mathsf {D}}^l_{m,m'}(0) \mathrm {e}^{-\mathrm {i}m' \omega t}. \end{aligned}$$

Let us further assume that the two functions are related to each other in \(\mathcal{F}_p\) by

$$\begin{aligned} f_l(\vec {x}, t) = h_l(t) * g_l(\vec {x}, t) \qquad \text {for all } l , \end{aligned}$$

where \(h_l(t) \in \mathbb {R}\) is a real distribution. The symbol \(*\) denotes the convolution product. As the convolution is done with respect to time, the orthogonality of the spherical harmonics implies that for all l and m,

$$\begin{aligned} Z_{l,m}(t) = h_l(t) * Z'_{l,m}(t). \end{aligned}$$

Combining Eqs. (52) and (53), we get

$$\begin{aligned} z_{l,m}(t)&= \sum _{m'=-l}^l \sum _{m''=-l}^l \int _{-\infty }^ \infty \bar{{\mathsf {D}}}^l_{m,m'}(t) h_l(t-t') {{\mathsf {D}}}^l_{m'',m'}(t') z'_{l,m''}(t')\,dt' \nonumber \\&= \sum _{m''=-l}^l {\mathsf {h}}^l_{m,m''}(t) * z'_{l,m''}(t), \end{aligned}$$


$$\begin{aligned} {\mathsf {h}}^l_{m,m''}(t) = \sum _{m'=-l}^l \bar{{\mathsf {D}}}^l_{m,m'}(0) h_l(t) \mathrm {e}^{\mathrm {i}m' \omega t} {\mathsf {D}}^l_{m'',m'}(0). \end{aligned}$$

In particular, if the rotation axis \(\vec {\omega }\) is aligned with the third axis of \(\mathcal{F}_0\) and \(\mathcal{F}_p\), i.e., if \(\alpha =\beta =0\), \({\mathsf {h}}^l_{m,m''}(t)\) is diagonal and we obtain

$$\begin{aligned} z_{l,m}(t) = \left( h_l(t)\mathrm {e}^{\mathrm {i}m \omega t}\right) * z'_{l,m}(t) \qquad \text {if } \quad \alpha =\beta =0. \end{aligned}$$

Taking the Fourier transform of Eqs. (54) and (55), we get

$$\begin{aligned} \underline{z}_{l,m}(\nu )= & {} \sum _{m''=-l}^l \underline{{\mathsf {h}}}^l_{m,m''}(\nu ) \underline{z}'_{l,m''}(\nu ) \quad \text {} \\ \end{aligned}$$


$$\begin{aligned} \underline{{\mathsf {h}}}^l_{m,m''}(\nu )= & {} \sum _{m'=-l}^l \bar{{\mathsf {D}}}^l_{m,m'}(0) \underline{h}_l(\nu -m'\omega ) {\mathsf {D}}^l_{m'',m'}(0), \end{aligned}$$

on the one hand, and

$$\begin{aligned} \underline{z}_{l,m}(\nu ) = \underline{h}_l(\nu - m\omega ) \underline{z}'_{l,m}(\nu ) \qquad \text {if}\quad \alpha =\beta =0, \end{aligned}$$

on the other.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Boué, G., Correia, A.C.M. & Laskar, J. Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology . Celest Mech Dyn Astr 126, 31–60 (2016).

Download citation


  • Restricted problems
  • Extended body
  • Dissipative forces
  • Planetary systems
  • Rotation
  • HD 80606 b