Secular orbital dynamics of the innermost exoplanet of the \(\upsilon\)-Andromedæ system. (English) Zbl 1527.70013
Celest. Mech. Dyn. Astron. 135, No. 3, Paper No. 28, 41 p. (2023); correction ibid. 135, No. 5, Paper No. 49, 3 p. (2023).
The \(v\)-Andromedæ system was the first ever to be discovered among the ones that host at least two exoplanets.
This article deals with the motion of \(v\)-And \({\mathbf b}\), which is the innermost planet among those discovered in the extrasolar system around the star \(v\)-Andromedæ A. The main aim is to extend the study of the stability to the orbital dynamics of \(v\)-And \({\mathbf b}\), still adopting a hierarchical approach.
The authors introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. Also, they preassign the orbits of the Super-Jupiter exoplanets \(v\)-And \({\mathbf c}\) and \(v\)-And \({\mathbf d}\) in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the well-known technique of the (so-called) frequency analysis and are injected in the equations describing the orbital dynamics of \(v\)-And \({\mathbf b}\) under the gravitational effects exerted by those two external exoplanets (that are expected to be major ones in such an extrasolar system).
In the first part of the paper, the frequency analysis is used to reconstruct the Fourier decompositions of the secular motions of the outer exoplanets \(v\)-And \({\mathbf c}\) and \(v\)-And \({\mathbf d}\). Then, in Section 3 and 4, the secular quasi-periodic restricted Hamiltonian model (with \(2 + 3/2\) degrees of freedom) is introduced and validated through the comparison with several numerical integrations of the complete four-body problem, hosting planets \({\mathbf b}\), \({\mathbf c}\), \({\mathbf d}\) of the \(v\)-Andromedæ system. The double normalization procedure allowing to perform a sort of averaging which further simplifies the model is described with a rather general approach. In Section 5, this normal form procedure is applied to the quasi-periodic restricted Hamiltonian, in such a way to derive an integrable model with 2 degrees of freedom describing the secular orbital dynamics of \(v\)-And \({\mathbf b}\). Such a simplified model is used to study \(v\)-And \({\mathbf b}\) stability domain in the parameters space of the initial values of the inclination and the longitude of node. All this computational procedure is repeated in the last section, starting from a version of the secular quasi-periodic restricted Hamiltonian model which includes also relativistic corrections; this allows to appreciate the effects on the orbital dynamics due to general relativity.
This article deals with the motion of \(v\)-And \({\mathbf b}\), which is the innermost planet among those discovered in the extrasolar system around the star \(v\)-Andromedæ A. The main aim is to extend the study of the stability to the orbital dynamics of \(v\)-And \({\mathbf b}\), still adopting a hierarchical approach.
The authors introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. Also, they preassign the orbits of the Super-Jupiter exoplanets \(v\)-And \({\mathbf c}\) and \(v\)-And \({\mathbf d}\) in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the well-known technique of the (so-called) frequency analysis and are injected in the equations describing the orbital dynamics of \(v\)-And \({\mathbf b}\) under the gravitational effects exerted by those two external exoplanets (that are expected to be major ones in such an extrasolar system).
In the first part of the paper, the frequency analysis is used to reconstruct the Fourier decompositions of the secular motions of the outer exoplanets \(v\)-And \({\mathbf c}\) and \(v\)-And \({\mathbf d}\). Then, in Section 3 and 4, the secular quasi-periodic restricted Hamiltonian model (with \(2 + 3/2\) degrees of freedom) is introduced and validated through the comparison with several numerical integrations of the complete four-body problem, hosting planets \({\mathbf b}\), \({\mathbf c}\), \({\mathbf d}\) of the \(v\)-Andromedæ system. The double normalization procedure allowing to perform a sort of averaging which further simplifies the model is described with a rather general approach. In Section 5, this normal form procedure is applied to the quasi-periodic restricted Hamiltonian, in such a way to derive an integrable model with 2 degrees of freedom describing the secular orbital dynamics of \(v\)-And \({\mathbf b}\). Such a simplified model is used to study \(v\)-And \({\mathbf b}\) stability domain in the parameters space of the initial values of the inclination and the longitude of node. All this computational procedure is repeated in the last section, starting from a version of the secular quasi-periodic restricted Hamiltonian model which includes also relativistic corrections; this allows to appreciate the effects on the orbital dynamics due to general relativity.
MSC:
| 70F15 | Celestial mechanics |
| 70F10 | \(n\)-body problems |
| 70H09 | Perturbation theories for problems in Hamiltonian and Lagrangian mechanics |
Keywords:
quasi-periodic restricted Hamiltonian; normal form; Hamiltonian perturbation theory; n-body planetary problemReferences:
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