Abstract
We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincaré canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the Keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the mean anomalies on the secular dynamics. Finally, we set up a simple criterion that is useful to discriminate between three different categories of planetary systems: (i) secular systems (HD 11964, HD 74156, HD 134987, HD 163607, HD 12661 and HD 147018); (ii) systems near a mean-motion resonance (HD 11506, HD 177830, HD 9446, HD 169830 and \(\upsilon \) Andromedae); (iii) systems really close to or in a mean-motion resonance (HD 108874, HD 128311 and HD 183263).
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Notes
Let us note that the Jacobi variables are less suitable for our purpose, as they require a Taylor expansion in the planetary masses.
We recall that, as shown by Poisson, the semi-major axes are constant up to the second order in the masses. Here we expand around their initial values, but we could also have taken their average values over a long-term numerical integration (see, e.g., Sansottera et al. 2013).
For sake of completeness, we check that computing the “averaged” initial conditions using the generating functions \(\chi _1\) and \(\chi _2\), as in the approximation at order two in the masses, does not influence neither qualitatively nor quantitatively the results.
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Acknowledgments
The work of A.-S. L. is supported by an FNRS Postdoctoral Research Fellowship. The work of M. S. is supported by an FSR Incoming Post-doctoral Fellowship of the Académie universitaire Louvain, co-funded by the Marie Curie Actions of the European Commission.
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Appendix: Secular Hamiltonian for the \(\upsilon \) Andromedae up to order 6
Appendix: Secular Hamiltonian for the \(\upsilon \) Andromedae up to order 6
We report here the expansion of the secular Hamiltonian \(\mathcal{H }^{(\mathrm{sec})}\) (Eq. (10)) of the \(\upsilon \) Andromedae extrasolar system up to degree \(6\) in \(({\varvec{\xi }},{\varvec{\eta }})\,\). In particular, we show both the approximations at order one and two in the masses to highlight the differences. A detailed description of the \(\upsilon \) Andromedae system is given in Sect. 5. As this system is near the 5:1 mean-motion resonance, the main difference between the two secular approximations affects terms that are at least of order 6 in the canonical secular variables.
\(\xi _1\) | \(\xi _2\) | \(\eta _1\) | \(\eta _2\) | First order | Second order |
|---|---|---|---|---|---|
0 | 0 | 0 | 0 | \(-3.8449638957147059 \times 10^{+0}\) | \(-3.8490132363346130 \times 10^{+0}\) |
2 | 0 | 0 | 0 | \(-4.7203675679835364 \times 10^{-4}\) | \(-4.7442843563932181 \times 10^{-4}\) |
1 | 1 | 0 | 0 | \( 1.9765062410537654 \times 10^{-4}\) | \( 1.9478085580405423 \times 10^{-4}\) |
0 | 2 | 0 | 0 | \(-1.2594397524843563 \times 10^{-4}\) | \(-1.2389253809188814 \times 10^{-4}\) |
0 | 0 | 2 | 0 | \(-4.7203675679835364 \times 10^{-4}\) | \(-4.7442843563932181 \times 10^{-4}\) |
0 | 0 | 1 | 1 | \( 1.9765062410537654 \times 10^{-4}\) | \( 1.9478085580405423 \times 10^{-4}\) |
0 | 0 | 0 | 2 | \(-1.2594397524843563 \times 10^{-4}\) | \(-1.2389253809188814 \times 10^{-4}\) |
4 | 0 | 0 | 0 | \( 1.4338305925091211 \times 10^{-4}\) | \( 1.4383176583648995 \times 10^{-4}\) |
3 | 1 | 0 | 0 | \( 4.3147125112054390 \times 10^{-4}\) | \( 4.5045949999300181 \times 10^{-4}\) |
2 | 2 | 0 | 0 | \(-7.4883810227863515 \times 10^{-4}\) | \(-7.5868060326294868 \times 10^{-4}\) |
2 | 0 | 2 | 0 | \( 2.8676611850182422 \times 10^{-4}\) | \( 2.8766295710810365 \times 10^{-4}\) |
2 | 0 | 1 | 1 | \( 4.3147125112054390 \times 10^{-4}\) | \( 4.5045950531661890 \times 10^{-4}\) |
2 | 0 | 0 | 2 | \(-5.1509576520639426 \times 10^{-4}\) | \(-5.2857952689768168 \times 10^{-4}\) |
1 | 3 | 0 | 0 | \( 3.3341302753346505 \times 10^{-4}\) | \( 3.2386648540035952 \times 10^{-4}\) |
1 | 1 | 2 | 0 | \( 4.3147125112054390 \times 10^{-4}\) | \( 4.5045950531661890 \times 10^{-4}\) |
1 | 1 | 1 | 1 | \(-4.6748467414448156 \times 10^{-4}\) | \(-4.6020211054162414 \times 10^{-4}\) |
1 | 1 | 0 | 2 | \( 3.3341302753346505 \times 10^{-4}\) | \( 3.2386626581402522 \times 10^{-4}\) |
0 | 4 | 0 | 0 | \(-9.4514514989701095 \times 10^{-5}\) | \(-9.0913589478614943 \times 10^{-5}\) |
0 | 2 | 2 | 0 | \(-5.1509576520639426 \times 10^{-4}\) | \(-5.2857952689768124 \times 10^{-4}\) |
0 | 2 | 1 | 1 | \( 3.3341302753346505 \times 10^{-4}\) | \( 3.2386626581402554 \times 10^{-4}\) |
0 | 2 | 0 | 2 | \(-1.8902902997940219 \times 10^{-4}\) | \(-1.8182716424233877 \times 10^{-4}\) |
0 | 0 | 4 | 0 | \( 1.4338305925091211 \times 10^{-4}\) | \( 1.4383176583649006 \times 10^{-4}\) |
0 | 0 | 3 | 1 | \( 4.3147125112054390 \times 10^{-4}\) | \( 4.5045949999300165 \times 10^{-4}\) |
0 | 0 | 2 | 2 | \(-7.4883810227863515 \times 10^{-4}\) | \(-7.5868060326294889 \times 10^{-4}\) |
\(\xi _1\) | \(\xi _2\) | \(\eta _1\) | \(\eta _2\) | First order | Second order |
|---|---|---|---|---|---|
0 | 0 | 1 | 3 | \( 3.3341302753346505 \times 10^{-4}\) | \( 3.2386648540035947 \times 10^{-4}\) |
0 | 0 | 0 | 4 | \(-9.4514514989701095 \times 10^{-5}\) | \(-9.0913589478614848 \times 10^{-5}\) |
6 | 0 | 0 | 0 | \( 8.0737006151169034 \times 10^{-5}\) | \( 1.3917499875750025 \times 10^{-4}\) |
5 | 1 | 0 | 0 | \(-1.4728781329895123 \times 10^{-4}\) | \(-5.7065127472031446 \times 10^{-4}\) |
4 | 2 | 0 | 0 | \(-3.8625662439607426 \times 10^{-4}\) | \( 8.8051997114562091 \times 10^{-4}\) |
4 | 0 | 2 | 0 | \( 2.4221101845350710 \times 10^{-4}\) | \( 4.1753226095492534 \times 10^{-4}\) |
4 | 0 | 1 | 1 | \(-1.4728781329895123 \times 10^{-4}\) | \(-5.7066077231623241 \times 10^{-4}\) |
4 | 0 | 0 | 2 | \( 4.4154338663068642 \times 10^{-5}\) | \( 3.1718226339201008 \times 10^{-4}\) |
3 | 3 | 0 | 0 | \( 1.1715817984811095 \times 10^{-3}\) | \(-9.4757874129889675 \times 10^{-4}\) |
3 | 1 | 2 | 0 | \(-2.9457562659790246 \times 10^{-4}\) | \(-1.1413685872968715 \times 10^{-3}\) |
3 | 1 | 1 | 1 | \(-8.6082192611828580 \times 10^{-4}\) | \( 1.1266453489623077 \times 10^{-3}\) |
3 | 1 | 0 | 2 | \( 9.0270698998234857 \times 10^{-4}\) | \(-7.3916625373992911 \times 10^{-4}\) |
2 | 4 | 0 | 0 | \(-1.0274689835474350 \times 10^{-3}\) | \( 8.0672276683552634 \times 10^{-4}\) |
2 | 2 | 2 | 0 | \(-3.4210228573300561 \times 10^{-4}\) | \( 1.1977229419632242 \times 10^{-3}\) |
2 | 2 | 1 | 1 | \( 1.7093314154786317 \times 10^{-3}\) | \(-1.3644052823847199 \times 10^{-3}\) |
2 | 2 | 0 | 2 | \(-1.4654674698375437 \times 10^{-3}\) | \( 1.2083393139899557 \times 10^{-3}\) |
2 | 0 | 4 | 0 | \( 2.4221101845350710 \times 10^{-4}\) | \( 4.1753226095492669 \times 10^{-4}\) |
2 | 0 | 3 | 1 | \(-2.9457562659790246 \times 10^{-4}\) | \(-1.1413685872968726 \times 10^{-3}\) |
2 | 0 | 2 | 2 | \(-3.4210228573300561 \times 10^{-4}\) | \( 1.1977229419632145 \times 10^{-3}\) |
2 | 0 | 1 | 3 | \( 9.0270698998234857 \times 10^{-4}\) | \(-7.3916701878536670 \times 10^{-4}\) |
2 | 0 | 0 | 4 | \(-4.3799848629010854 \times 10^{-4}\) | \( 4.0161436876523369 \times 10^{-4}\) |
1 | 5 | 0 | 0 | \( 3.8723701961918303 \times 10^{-4}\) | \(-2.8749294965146893 \times 10^{-4}\) |
1 | 3 | 2 | 0 | \( 9.0270698998234857 \times 10^{-4}\) | \(-7.3916701878535239 \times 10^{-4}\) |
1 | 3 | 1 | 1 | \(-1.1789409945146532 \times 10^{-3}\) | \( 8.1021818828667783 \times 10^{-4}\) |
1 | 3 | 0 | 2 | \( 7.7447403923836607 \times 10^{-4}\) | \(-5.7498741718758071 \times 10^{-4}\) |
1 | 1 | 4 | 0 | \(-1.4728781329895123 \times 10^{-4}\) | \(-5.7066077231623220 \times 10^{-4}\) |
1 | 1 | 3 | 1 | \(-8.6082192611828580 \times 10^{-4}\) | \( 1.1266453489623106 \times 10^{-3}\) |
1 | 1 | 2 | 2 | \( 1.7093314154786317 \times 10^{-3}\) | \(-1.3644052823847082 \times 10^{-3}\) |
1 | 1 | 1 | 3 | \(-1.1789409945146532 \times 10^{-3}\) | \( 8.1021818828667317 \times 10^{-4}\) |
1 | 1 | 0 | 4 | \( 3.8723701961918303 \times 10^{-4}\) | \(-2.8749285273268994 \times 10^{-4}\) |
0 | 6 | 0 | 0 | \(-6.9390702058934342 \times 10^{-5}\) | \( 6.9301447295937329 \times 10^{-5}\) |
0 | 4 | 2 | 0 | \(-4.3799848629010854 \times 10^{-4}\) | \( 4.0161436876524556 \times 10^{-4}\) |
0 | 4 | 1 | 1 | \( 3.8723701961918303 \times 10^{-4}\) | \(-2.8749285273267357 \times 10^{-4}\) |
0 | 4 | 0 | 2 | \(-2.0817210617680304 \times 10^{-4}\) | \( 2.0789708715975659 \times 10^{-4}\) |
0 | 2 | 4 | 0 | \( 4.4154338663068642 \times 10^{-5}\) | \( 3.1718226339199647 \times 10^{-4}\) |
0 | 2 | 3 | 1 | \( 9.0270698998234857 \times 10^{-4}\) | \(-7.3916625373990645 \times 10^{-4}\) |
0 | 2 | 2 | 2 | \(-1.4654674698375437 \times 10^{-3}\) | \( 1.2083393139899626 \times 10^{-3}\) |
0 | 2 | 1 | 3 | \( 7.7447403923836607 \times 10^{-4}\) | \(-5.7498741718756510 \times 10^{-4}\) |
0 | 2 | 0 | 4 | \(-2.0817210617680304 \times 10^{-4}\) | \( 2.0789708715975702 \times 10^{-4}\) |
0 | 0 | 6 | 0 | \( 8.0737006151169034 \times 10^{-5}\) | \( 1.3917499875750112 \times 10^{-4}\) |
\(\xi _1\) | \(\xi _2\) | \(\eta _1\) | \(\eta _2\) | First order | Second order |
|---|---|---|---|---|---|
0 | 0 | 5 | 1 | \(-1.4728781329895123 \times 10^{-4}\) | \(-5.7065127472031587 \times 10^{-4}\) |
0 | 0 | 4 | 2 | \(-3.8625662439607426 \times 10^{-4}\) | \( 8.8051997114559945 \times 10^{-4}\) |
0 | 0 | 3 | 3 | \( 1.1715817984811095 \times 10^{-3}\) | \(-9.4757874129888428 \times 10^{-4}\) |
0 | 0 | 2 | 4 | \(-1.0274689835474350 \times 10^{-3}\) | \( 8.0672276683552298 \times 10^{-4}\) |
0 | 0 | 1 | 5 | \( 3.8723701961918303 \times 10^{-4}\) | \(-2.8749294965147088 \times 10^{-4}\) |
0 | 0 | 0 | 6 | \(-6.9390702058934342 \times 10^{-5}\) | \( 6.9301447295938372 \times 10^{-5}\) |
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Libert, AS., Sansottera, M. On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems. Celest Mech Dyn Astr 117, 149–168 (2013). https://doi.org/10.1007/s10569-013-9501-z
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DOI: https://doi.org/10.1007/s10569-013-9501-z