Secular and tidal evolution of circumbinary systems


We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18


  1. 1.

  2. 2.

  3. 3.

    In order to reproduce the results in Naoz et al. (2011) we cannot take into account the flattening of the star (Eq. 23). The evolution is also highly chaotic, so a slightly change in the reference angles lead to different final mutual inclination.


  1. Alexander, M.E.: The weak friction approximation and tidal evolution in close binary systems. Astrophys. Space Sci. 23, 459–510 (1973). doi:10.1007/BF00645172

    ADS  Article  Google Scholar 

  2. Beaugé, C., Nesvorný, D.: Multiple-planet scattering and the origin of hot Jupiters. Astrophys. J. 751, 119 (2012). doi:10.1088/0004-637X/751/2/119. arXiv:1110.4392

  3. Bosanac, N., Howell, K.C., Fischbach, E.: Stability of orbits near large mass ratio binary systems. Celest. Mech. Dyn. Astron. 122, 27–52 (2015). doi:10.1007/s10569-015-9607-6

    ADS  MathSciNet  Article  Google Scholar 

  4. Boué, G., Fabrycky, D.C.: Compact planetary systems perturbed by an inclined companion. II. Stellar spin-orbit evolution. Astrophys. J. 789, 111 (2014). doi:10.1088/0004-637X/789/2/111. arXiv:1405.7636

  5. Boué, G., Laskar, J.: Precession of a planet with a satellite. Icarus 185, 312–330 (2006). doi:10.1016/j.icarus.2006.07.019

  6. Boué, G., Laskar, J.: Spin axis evolution of two interacting bodies. Icarus 201, 750–767 (2009). doi:10.1016/j.icarus.2009.02.001

  7. Brozović, M., Showalter, M.R., Jacobson, R.A., Buie, M.W.: The orbits and masses of satellites of Pluto. Icarus 246, 317–329 (2015). doi:10.1016/j.icarus.2014.03.015

  8. Colombo, G.: Cassini’s second and third laws. Astron. J. 71, 891–896 (1966)

    ADS  Article  Google Scholar 

  9. Correia, A.C.M.: Secular evolution of a satellite by tidal effect: application to Triton. Astrophys. J. 704, L1–L4 (2009). doi:10.1088/0004-637X/704/1/L1. arXiv:0909.4210

    ADS  Article  Google Scholar 

  10. Correia, A.C.M.: Stellar and planetary Cassini states. Astron. Astrophys. 582, A69 (2015). doi:10.1051/0004-6361/201525939

    ADS  Article  Google Scholar 

  11. Correia, A.C.M.: Cassini states for black hole binaries. Mon. Not. R. Astron. Soc 457, L49–L53 (2016). doi:10.1093/mnrasl/slv198. arXiv:1511.01890

    ADS  Article  Google Scholar 

  12. Correia, A.C.M., Laskar, J.: Tidal evolution of exoplanets. In: Exoplanets, University of Arizona Press, Tucson, pp. 534–575 (2010)

  13. 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

  14. Correia, A.C.M., Boué, G., Laskar, J.: Pumping the eccentricity of exoplanets by tidal effect. Astrophys. J. 744, L23 (2012). doi:10.1088/2041-8205/744/2/L23. arXiv:1111.5486

  15. Correia, A.C.M., Boué, G., Laskar, J., Morais, M.H.M.: Tidal damping of the mutual inclination in hierarchical systems. Astron. Astrophys. 553, A39 (2013). doi:10.1051/0004-6361/201220482. arXiv:1303.0864

  16. 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

  17. Correia, A.C.M., Leleu, A., Rambaux, N., Robutel, P.: Spin-orbit coupling and chaotic rotation for circumbinary bodies. Application to the small satellites of the Pluto–Charon system. Astron. Astrophys. 580, L14 (2015). doi:10.1051/0004-6361/201526800. arXiv:1506.06733

  18. Couetdic, J., Laskar, J., Correia, A.C.M., Mayor, M., Udry, S.: Dynamical stability analysis of the HD 202206 system and constraints to the planetary orbits. Astron. Astrophys. 519, A10 (2010). doi:10.1051/0004-6361/200913635. arXiv:0911.1963

    ADS  Article  Google Scholar 

  19. Doolin, S., Blundell, K.M.: The dynamics and stability of circumbinary orbits. Mon. Not. R. Astron. Soc. 418, 2656–2668 (2011). doi:10.1111/j.1365-2966.2011.19657.x. arXiv:1108.4144

    ADS  Article  Google Scholar 

  20. Doyle, L.R., Carter, J.A., Fabrycky, D.C., Slawson, R.W., Howell, S.B., Winn, J.N., et al.: Kepler-16: a transiting circumbinary planet. Science 333, 1602 (2011). doi:10.1126/science.1210923. arXiv:1109.3432

    ADS  Article  Google Scholar 

  21. Efroimsky, M., Williams, J.G.: Tidal torques: a critical review of some techniques. Celest. Mech. Dyn. Astron. 104, 257–289 (2009). doi:10.1007/s10569-009-9204-7. arXiv:0803.3299

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. Eggleton, P.P., Kiseleva-Eggleton, L.: Orbital evolution in binary and triple stars, with an application to SS Lacertae. Astrophys. J. 562, 1012–1030 (2001). doi:10.1086/323843. arXiv:astro-ph/0104126

    ADS  Article  Google Scholar 

  23. Farago, F., Laskar, J.: High-inclination orbits in the secular quadrupolar three-body problem. Mon. Not. R. Astron. Soc. 401, 1189–1198 (2010). doi:10.1111/j.1365-2966.2009.15711.x. arXiv:0909.2287

    ADS  Article  Google Scholar 

  24. 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 

  25. Ford, E.B., Joshi, K.J., Rasio, F.A., Zbarsky, B.: Theoretical implications of the PSR B1620–26 triple system and its planet. Astrophys. J. 528, 336–350 (2000a). doi:10.1086/308167. arXiv:astro-ph/9905347

  26. Ford, E.B., Kozinsky, B., Rasio, F.A.: Secular evolution of hierarchical triple star systems. Astrophys. J. 535, 385–401 (2000b). doi:10.1086/308815

    ADS  Article  Google Scholar 

  27. Goldreich, P.: History of the lunar orbit. Rev. Geophys. Space Phys. 4, 411–439 (1966). doi:10.1029/RG004i004p00411

    ADS  Article  Google Scholar 

  28. Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1950)

    Google Scholar 

  29. Harrington, R.S.: Dynamical evolution of triple stars. Astron. J. 73, 190–194 (1968). doi:10.1086/110614

    ADS  Article  Google Scholar 

  30. Hut, P.: Stability of tidal equilibrium. Astron. Astrophys. 92, 167–170 (1980)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Kaula, W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 2, 661–685 (1964)

    ADS  Article  Google Scholar 

  32. Kennedy, G.M., Wyatt, M.C., Sibthorpe, B., Duchêne, G., Kalas, P., Matthews, B.C., et al.: 99 Herculis: host to a circumbinary polar-ring debris disc. Mon. Not. R. Astron. Soc. 421, 2264–2276 (2012). doi:10.1111/j.1365-2966.2012.20448.x. arXiv:1201.1911

  33. Kidder, L.E.: Coalescing binary systems of compact objects to (post)\(^{5/2}\)-newtonian order. V. spin effects. Phys. Rev. D 52, 821–847 (1995). doi:10.1103/PhysRevD.52.821. arXiv:gr-qc/9506022

    ADS  Article  Google Scholar 

  34. Kostov, V.B., Orosz, J.A., Welsh, W.F., Doyle, L.R., Fabrycky, D.C., Haghighipour, N., et al.: KOI-2939b: the largest and longest-period Kepler transiting circumbinary planet. arXiv:1512.00189 (2015)

  35. Kozai, Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591–598 (1962). doi:10.1086/108790

    ADS  MathSciNet  Article  Google Scholar 

  36. Lainey, V., Arlot, J.E., Karatekin, Ö., van Hoolst, T.: Strong tidal dissipation in Io and Jupiter from astrometric observations. Nature 459, 957–959 (2009). doi:10.1038/nature08108

  37. Lainey, V., Karatekin, Ö., Desmars, J., Charnoz, S., Arlot, J.E., Emelyanov, N., et al.: Strong tidal dissipation in Saturn and constraints on Enceladus’ thermal state from astrometry. Astrophys. J. 752, 14 (2012). doi:10.1088/0004-637X/752/1/14. arXiv:1204.0895

  38. Lambeck, K.: Geophysical geodesy: the slow deformations of the earth Lambeck. Clarendon Press and Oxford University Press, Oxford [England] and New York (1988)

  39. Laskar, J.: On the spacing of planetary systems. Phys. Rev. Lett. 84, 3240–3243 (2000)

    ADS  Article  Google Scholar 

  40. Laskar, J., Boué, G.: Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations. Astron. Astrophys. 522, A60 (2010). doi:10.1051/0004-6361/201014496. arXiv:1008.2947

  41. Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A.C.M., Levrard, B.: A long-term numerical solution for the insolation quantities of the Earth. Astron. Astrophys. 428, 261–285 (2004). doi:10.1051/0004-6361:20041335

    ADS  Article  Google Scholar 

  42. Lee, M.H., Peale, S.J.: Secular evolution of hierarchical planetary systems. Astrophys. J. 592, 1201–1216 (2003). doi:10.1086/375857. arXiv:astro-ph/0304454

    ADS  Article  Google Scholar 

  43. Lidov, M.L.: The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Plan. Space Sci. 9, 719–759 (1962). doi:10.1016/0032-0633(62)90129-0

    ADS  Article  Google Scholar 

  44. Lidov, M.L., Ziglin, S.L.: Non-restricted double-averaged three body problem in Hill’s case. Celest. Mech. 13, 471–489 (1976). doi:10.1007/BF01229100

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

    ADS  Article  Google Scholar 

  46. Makarov, V.V.: Equilibrium rotation of semiliquid exoplanets and satellites. arXiv:1507.07383 (2015)

  47. Marchal, C.: The Three-Body Problem. Elsevier, Amsterdam (1990)

    Google Scholar 

  48. Martin, D.V., Triaud, A.H.M.J.: Planets transiting non-eclipsing binaries. Astron. Astrophys. 570, A91 (2014). doi:10.1051/0004-6361/201323112. arXiv:1404.5360

    ADS  Article  Google Scholar 

  49. Migaszewski, C.: The generalized non-conservative model of a 1-planet system revisited. Celest. Mech. Dyn. Astron. 113, 169–203 (2012). doi:10.1007/s10569-012-9413-3. arXiv:1203.2358

    ADS  MathSciNet  Article  MATH  Google Scholar 

  50. Migaszewski, C., Goździewski, K.: The non-resonant, relativistic dynamics of circumbinary planets. Mon. Not. R. Astron. Soc. 411, 565–583 (2011). doi:10.1111/j.1365-2966.2010.17702.x. arXiv:1006.5961

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

    ADS  Article  MATH  Google Scholar 

  52. Naoz, S., Farr, W.M., Lithwick, Y., Rasio, F.A., Teyssandier, J.: Hot Jupiters from secular planet–planet interactions. Nature 473, 187–189 (2011). doi:10.1038/nature10076. arXiv:1011.2501

  53. Orosz, J.A., Welsh, W.F., Carter, J.A., Brugamyer, E., Buchhave, L.A., Cochran, W.D., et al.: The Neptune-sized circumbinary planet Kepler-38b. Astrophys. J. 758, 87 (2012a). doi:10.1088/0004-637X/758/2/87. arXiv:1208.3712

    ADS  Article  Google Scholar 

  54. Orosz, J.A., Welsh, W.F., Carter, J.A., Fabrycky, D.C., Cochran, W.D., Endl, M., et al.: Kepler-47: a transiting circumbinary multiplanet system. Science 337, 1511 (2012a). doi:10.1126/science.1228380. arXiv:1208.5489

    ADS  Article  Google Scholar 

  55. Penev, K., Jackson, B., Spada, F., Thom, N.: Constraining tidal dissipation in stars from the destruction rates of exoplanets. Astrophys. J. 751, 96 (2012). doi:10.1088/0004-637X/751/2/96. arXiv:1205.1803

    ADS  Article  Google Scholar 

  56. Plavchan, P., Güth, T., Laohakunakorn, N., Parks, J.R.: The identification of 93 day periodic photometric variability for YSO YLW 16A. Astron. Astrophys. 554, A110 (2013). doi:10.1051/0004-6361/201220747. arXiv:1304.2398

  57. Poincaré, H.: Leçons de Mécanique Céleste, Tome I. Gauthier-Villars, Paris (1905)

  58. Singer, S.F.: The origin of the moon and geophysical consequences. Geophys. J. R. Astron. Soc. 15, 205–226 (1968)

    Article  Google Scholar 

  59. Skumanich, A.: Time scales for CA II emission decay, rotational braking, and lithium depletion. Astrophys. J. 171, 565 (1972). doi:10.1086/151310

    ADS  Article  Google Scholar 

  60. Smart, W.M.: Celestial Mechanics. Longmans, Green, London, New York (1953)

    Google Scholar 

  61. Touma, J.R., Tremaine, S., Kazandjian, M.V.: Gauss’s method for secular dynamics, softened. Mon. Not. R. Astron. Soc. 394, 1085–1108 (2009). doi:10.1111/j.1365-2966.2009.14409.x. arXiv:0811.2812

    ADS  Article  Google Scholar 

  62. Verrier, P.E., Evans, N.W.: High-inclination planets and asteroids in multistellar systems. Mon. Not. R. Astron. Soc. 394, 1721–1726 (2009). doi:10.1111/j.1365-2966.2009.14446.x. arXiv:0812.4528

    ADS  Article  Google Scholar 

  63. Ward, W.R.: Tidal friction and generalized Cassini’s laws in the solar system. Astron. J. 80, 64–70 (1975)

    ADS  Article  Google Scholar 

  64. Ward, W.R., Hamilton, D.P.: Tilting Saturn I. Analytic model. Astron. J. 128, 2501–2509 (2004). doi:10.1086/424533

    ADS  Article  Google Scholar 

  65. Welsh, W.F., Orosz, J.A., Carter, J.A., Fabrycky, D.C., Ford, E.B., Lissauer, J.J., et al.: Transiting circumbinary planets Kepler-34 b and Kepler-35 b. Nature 481, 475–479 (2012). doi:10.1038/nature10768. arXiv:1204.3955

    ADS  Article  Google Scholar 

  66. Winn, J.N., Albrecht, S., Johnson, J.A., Torres, G., Cochran, W.D., Marcy, G.W., et al.: Spin-orbit alignment for the circumbinary planet host Kepler-16 A. Astrophys. J. 741, L1 (2011). doi:10.1088/2041-8205/741/1/L1. arXiv:1109.3198

  67. Yoder, C.F.: Astrometric and geodetic properties of Earth and the Solar System. Global Earth Physics: A Handbook of Physical Constants, pp. 1–31. American Geophysical Union, Washington D.C. (1995)

Download references


We acknowledge support from PNP-CNRS, and from CIDMA strategic Project UID/MAT/04106/2013.

Author information



Corresponding author

Correspondence to Alexandre C. M. Correia.


Appendix 1: Oblate spheroid potential

The gravitational potential of an oblate body of mass \(m_i\) symmetric about its rotation axis \(\hat{\mathbf {s}}\) is given by (e.g. Goldstein 1950):

$$\begin{aligned} V_{i} (\mathbf {r}) = - G \frac{m_i}{r} \left[ 1 - J_{2,i} \left( \frac{R_i}{r}\right) ^2 P_2 (\hat{\mathbf {r}} \cdot \hat{\mathbf {s}}_i) \right] , \end{aligned}$$

where we neglected terms in \((R_i/r)^3\). The gravity field coefficient \(J_{2,i}\) is obtained from the principal moments of inertia \(A_i= B_i\) and \(C_i\) as \(J_{2,i} = (C_i-A_i) / m_iR_i^2\). When the asymmetry in the body mass distribution results only from its rotation, \(J_{2,i}\) is given by expression (2). The main term in the above expression is responsible for the orbital motion (Eq. 4), while the contribution in \(J_{2,i}\) is responsible for a perturbation of this motion, since \( J_{2,i} (R_i/r)^2 \ll 1 \). Thus, retaining only the terms in \(J_{2,i}\), the resulting perturbing potential energy of a system composed of three oblate bodies is given by:

$$\begin{aligned} U_R = U_{R,0} + U_{R,1} + U_{R,2}, \end{aligned}$$

where we have for the planet

$$\begin{aligned} U_{R,2} = \sum _{i= 0,1} m_iV_{2} (\mathbf {r}_{2i}) = \sum _{i= 0,1} G \frac{m_im_2}{r_{2i}} J_{2,2} \left( \frac{R_2}{r_{2i}}\right) ^2 P_2 (\hat{\mathbf {r}}_{2i} \cdot \hat{\mathbf {s}}_2), \end{aligned}$$

and for each star (\(i= 0, 1\))

$$\begin{aligned} U_{R,i} = G \frac{m_0m_1}{r_1} J_{2,i} \left( \frac{R_i}{r_1}\right) ^2 P_2 (\hat{\mathbf {r}}_1\cdot \hat{\mathbf {s}}_i) + G \frac{m_im_2}{r_{2i}} J_{2,i} \left( \frac{R_i}{r_{2i}}\right) ^2 P_2 (\hat{\mathbf {r}}_{2i} \cdot \hat{\mathbf {s}}_i). \end{aligned}$$

We also have (Fig. 1)

$$\begin{aligned} \mathbf {r}_{2i} = \mathbf {r}_2+ \delta _i\mathbf {r}_1, \end{aligned}$$

where \( \delta _0= m_1/ m_{01}\) and \( \delta _1= - m_0/ m_{01}\). Since we assume that \(r_1\ll r_2\), we can write

$$\begin{aligned} \frac{P_2 (\hat{\mathbf {r}}_{2i} \cdot \hat{\mathbf {s}}_j)}{r_{2i}^3}\approx & {} \frac{P_2 (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {s}}_j)}{r_2^3} + \frac{3 \delta _i}{2 r_2^3} \frac{r_1}{r_2} \left[ \hat{\mathbf {r}}_1\cdot \hat{\mathbf {r}}_2- 5 (\hat{\mathbf {r}}_1\cdot \hat{\mathbf {r}}_2) (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {s}}_j)^2\right. \nonumber \\&\left. + 2 (\hat{\mathbf {r}}_1\cdot \mathbf {s}_j) (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {s}}_j) \right] , \end{aligned}$$

where we neglected terms in \((r_1/ r_2)^2\), that is, we neglect terms in \(J_{2,i} (R_i/r_i)^2 (r_1/ r_2)^2\) in the potential energy. Replacing in expressions (96) and (95) we get for the planet

$$\begin{aligned} U_{R,2} = G \frac{m_2m_{01}}{r_2} J_{2,2} \left( \frac{R_2}{r_2} \right) ^2 P_2 (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {s}}_2), \end{aligned}$$

since \(m_0\delta _0+ m_1\delta _1= 0\), and for each star (\(i= 0,1\); \(j=1-i\))

$$\begin{aligned} U_{R,i}= & {} G \frac{m_0m_1}{r_1} J_{2,i} \left( \frac{R_i}{r_1}\right) ^2 \left[ P_2 (\hat{\mathbf {r}}_1\cdot \hat{\mathbf {s}}_i) + \frac{m_2}{m_j} \left( \frac{r_1}{r_2} \right) ^3 P_2 (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {s}}_i) \right] \nonumber \\\approx & {} G \frac{m_0m_1}{r_1} J_{2,i} \left( \frac{R_i}{r_1}\right) ^2 P_2 (\hat{\mathbf {r}}_1\cdot \hat{\mathbf {s}}_i), \end{aligned}$$

since terms in \(m_2/ m_j(r_1/r_2)^3\) can also be neglected.

Appendix 2: Tidal potential

The tidal potential of a body of mass \(m_i\) when deformed by another body of mass \(m'\) at the position \(\mathbf {r}'\) is given by (e.g. Kaula 1964):

$$\begin{aligned} V_{i} (\mathbf {r}, \mathbf {r}', m') = - k_{2,i} \frac{G m' R_i^5}{r^3 r'^3} P_2 (\hat{\mathbf {r}} \cdot \hat{\mathbf {r}}'), \end{aligned}$$

where we neglected terms in \((R_i/r)^4 (R_i/r')^4\). The resulting perturbing potential energy of a system composed of three bodies is given by:

$$\begin{aligned} U_T = U_{T,0} + U_{T,1} + U_{T,2}, \end{aligned}$$

where we have for the planet

$$\begin{aligned} U_{T,2} = \sum _{i,j=0,1} m_iV_{2} (\mathbf {r}_{2i}, \mathbf {r}_{2j}', m_j) = \sum _{i,j=0,1} - k_{2,2} \frac{G m_im_jR_2^5}{r_{2i}^3 r_{2j}'^3} P_2 (\hat{\mathbf {r}}_{2i} \cdot \hat{\mathbf {r}}_{2j}'), \end{aligned}$$

and for each star (\(i= 0,1\); \(j=1-i\))

$$\begin{aligned} U_{T,i} = m_j\left[ V_{i} (\mathbf {r}_1, \mathbf {r}_1', m_j) + V_{i} (\mathbf {r}_1, \mathbf {r}_{2i}', m_2) \right] + m_2\left[ V_{i} (\mathbf {r}_{2i}, \mathbf {r}_1', m_j) + V_{i} (\mathbf {r}_{2i}, \mathbf {r}_{2i}', m_2) \right] .\nonumber \\ \end{aligned}$$

Neglecting the tidal interactions with the external body \(m_2\), i.e., neglecting terms in \(m_2/ m_j(r_1/r_2)^3\), the above potential can be simplified as

$$\begin{aligned} U_{T,i} \approx m_iV_{i} (\mathbf {r}_1, \mathbf {r}_1', m_j) = - k_{2,i} \frac{G m_j^2 R_i^5}{r_1^3 r_1'^3} P_2 \left( \hat{\mathbf {r}}_1\cdot \hat{\mathbf {r}}_1'\right) . \end{aligned}$$

Using expression (97) we can rewrite

$$\begin{aligned} \frac{P_2 (\hat{\mathbf {r}}_{2i} \cdot \hat{\mathbf {r}}_{2j}')}{r_{2i}^3 r_{2j}'^3}\approx & {} \frac{P_2 (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_2')}{r_2^3 r_2'^3} + \frac{3 \delta _i}{2 r_2^3 r_2'^3} \frac{r_1}{r_2} \left[ \hat{\mathbf {r}}_1\cdot \hat{\mathbf {r}}_2- 5 (\hat{\mathbf {r}}_1\cdot \hat{\mathbf {r}}_2) (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_2')^2 + 2 (\hat{\mathbf {r}}_1\cdot \hat{\mathbf {r}}_2')(\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_2') \right] \nonumber \\&+ \frac{3 \delta _j}{2 r_2^3 r_2'^3} \frac{r_1'}{r_2'} \left[ \hat{\mathbf {r}}_1' \cdot \hat{\mathbf {r}}_2' - 5 (\hat{\mathbf {r}}_1' \cdot \hat{\mathbf {r}}_2')(\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_2')^2 + 2 (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_1')(\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_2')\right] , \end{aligned}$$

where we neglected terms in \((r_1/ r_2)^2\), that is, we neglect terms in \((R_2/r_2)^6 (r_1/ r_2)^2\) in the potential energy. Replacing in expression (103) we get for the planet

$$\begin{aligned} U_{T,2} = - k_{2,2} \frac{G m_{01}^2 R_2^5}{r_2^3 r_2'^3} P_2 (\hat{\mathbf {r}}_2\cdot \hat{\mathbf {r}}_2'), \end{aligned}$$

since \(m_0^2 \delta _0+ m_0m_1(\delta _0+ \delta _1) + m_1^2 \delta _1= 0\).

Appendix 3: Averaged quantities

For completeness, we gather here the average formulae that are used in the computation of secular equations. Let \(F(\mathbf {r},\dot{\mathbf {r}})\) be a function of a position vector \(\mathbf {r}\) and velocity \(\dot{\mathbf {r}}\), its averaged expression over the mean anomaly (M) is given by

$$\begin{aligned} \left\langle {F}\right\rangle _{M} = \frac{1}{2\pi }\int _0^{2\pi } F(\mathbf {r}, \dot{\mathbf {r}})\, \mathrm{d}M\ . \end{aligned}$$

Depending on the case, this integral is computing using the eccentric anomaly (E), or the true anomaly (v) as an intermediate variable. The basic formulae are

$$\begin{aligned} \mathrm{d}M= & {} \frac{r}{a}\mathrm{d}E = \frac{r^2}{a^2\sqrt{1-e^2}}\mathrm{d}v\ ,\nonumber \\ \mathbf {r}= & {} a(\cos E-e)\, \hat{\mathbf {e}} + a\sqrt{1-e^2}(\sin E)\, \hat{\mathbf {k}} \times \hat{\mathbf {e}}\ ,\nonumber \\ \mathbf {r}= & {} r\cos v\, \hat{\mathbf {e}} + r\sin v\, \hat{\mathbf {k}} \times \hat{\mathbf {e}}\ ,\nonumber \\ \dot{\mathbf {r}}= & {} \frac{na}{\sqrt{1-e^2}}\, \hat{\mathbf {k}} \times (\hat{\mathbf {r}} + \mathbf {e})\ ,\nonumber \\ r= & {} a (1-e\cos E) = \frac{a(1-e^2)}{1+e\cos v}\ , \end{aligned}$$

where \(\hat{\mathbf {k}}\) is the unit vector of the orbital angular momentum, and \(\mathbf {e}\) the Laplace–Runge–Lenz vector (Eq. 6). We have then

$$\begin{aligned} \left\langle {\frac{1}{r^3}}\right\rangle = \frac{1}{a^3(1-e^2)^{3/2}}\ , \quad \mathrm{and } \quad \left\langle {\frac{\mathbf {r} {}{\mathbf {r}}^t}{r^5}}\right\rangle = \frac{1}{2a^3(1-e^2)^{3/2}} \left( 1-\hat{\mathbf {k}} {}{\hat{\mathbf {k}}}^t \right) , \end{aligned}$$

where \({}{\mathbf {u}}^t\) denotes the transpose of any vector \(\mathbf {u}\). This leads to

$$\begin{aligned} \left\langle {\frac{1}{r^3}P_2(\hat{\mathbf {r}} \cdot \hat{\mathbf {u}})}\right\rangle = -\frac{1}{2a^3(1-e^2)^{3/2}} P_2(\hat{\mathbf {k}} \cdot \hat{\mathbf {u}})\ , \end{aligned}$$

for any unit vector \(\hat{\mathbf {u}}\). In the same way,

$$\begin{aligned} \left\langle {r^2}\right\rangle = a^2\left( 1+\frac{3}{2}e^2\right) \ , \quad \mathrm{and } \quad \left\langle {\mathbf {r} {}{\mathbf {r}}^t}\right\rangle = a^2\frac{1-e^2}{2} \left( 1-\hat{\mathbf {k}} {}{\hat{\mathbf {k}}}^t\right) + \frac{5}{2} a^2 \mathbf {e} {}{\mathbf {e}}^t\ , \end{aligned}$$


$$\begin{aligned} \left\langle {r^2 P_2(\hat{\mathbf {r}} \cdot \hat{\mathbf {u}})}\right\rangle = -\frac{a^2}{2}\Big ((1-e^2)P_2(\hat{\mathbf {k}} \cdot \hat{\mathbf {u}}) - 5 e^2 P_2(\hat{\mathbf {e}} \cdot \hat{\mathbf {u}})\Big )\ . \end{aligned}$$

The other useful formulae are

$$\begin{aligned} \left\langle {\frac{1}{r^6}}\right\rangle= & {} \frac{1}{a^6} f_1(e)\ , \end{aligned}$$
$$\begin{aligned} \left\langle {\frac{1}{r^8}}\right\rangle= & {} \frac{1}{a^8\sqrt{1-e^2}} f_2(e)\ , \end{aligned}$$
$$\begin{aligned} \left\langle {\frac{\mathbf {r} {}{\mathbf {r}}^t}{r^8}}\right\rangle= & {} \frac{\sqrt{1-e^2}}{2a^6} f_4(e) \left( 1-\hat{\mathbf {k}} {}{\hat{\mathbf {k}}}^t\right) +\frac{6+e^2}{4a^6(1-e^2)^{9/2}} \mathbf {e} {}{\mathbf {e}}^t\ , \end{aligned}$$
$$\begin{aligned} \left\langle {\frac{\mathbf {r}}{r^8}}\right\rangle= & {} \frac{5}{2}\frac{1}{a^7\sqrt{1-e^2}}\, f_4(e) \mathbf {e}\ , \end{aligned}$$
$$\begin{aligned} \left\langle {\frac{\mathbf {r}}{r^{10}}}\right\rangle= & {} \frac{7}{2}\frac{1}{a^9(1-e^2)}\, f_5(e) \mathbf {e}\ , \end{aligned}$$
$$\begin{aligned} \left\langle {\frac{(\mathbf {r} \cdot \dot{\mathbf {r}}) \mathbf {r}}{r^{10}}}\right\rangle= & {} \frac{n}{2a^7 \sqrt{1-e^2}} f_5(e)\, \hat{\mathbf {k}} \times \mathbf {e}\ , \end{aligned}$$

where the \(f_i(e)\) functions are given by expressions (46)–(50).

Finally, for the average over the argument of the pericenter (\(\omega \)), we can proceed in an identical manner:

$$\begin{aligned} \left\langle {\mathbf {e} {}{\mathbf {e}}^t}\right\rangle _\omega = \frac{1}{2 \pi } \int _0^{2 \pi } \mathbf {e} {}{\mathbf {e}}^t \, d \omega = \frac{e^2}{2} \left( 1 - \mathbf {k} {}{\mathbf {k}}^t \right) , \end{aligned}$$

which gives

$$\begin{aligned} \left\langle { \left( \mathbf {e} \cdot \hat{\mathbf {u}} \right) \mathbf {e}}\right\rangle _\omega = \frac{e^2}{2} \Big ( \hat{\mathbf {u}} - ( \mathbf {k} \cdot \hat{\mathbf {u}} ) \mathbf {k} \Big ). \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Correia, A.C.M., Boué, G. & Laskar, J. Secular and tidal evolution of circumbinary systems. Celest Mech Dyn Astr 126, 189–225 (2016).

Download citation


  • Extended body
  • Dissipative forces
  • Planetary systems
  • Rotation