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Article: Secular evolution of hierarchical planetary systems

TitleSecular evolution of hierarchical planetary systems
Authors
KeywordsCelestial Mechanics
Planetary Systems
Planets And Satellites: General
Issue Date2003
PublisherInstitute of Physics Publishing Ltd. The Journal's web site is located at http://iopscience.iop.org/2041-8205
Citation
Astrophysical Journal Letters, 2003, v. 592 n. 2 I, p. 1201-1216 How to Cite?
AbstractWe investigate the dynamical evolution of coplanar, hierarchical, two-planet systems where the ratio of the orbital semimajor axes α = a1/a2 is small. Hierarchical two-planet systems are likely to be ubiquitous among extrasolar planetary systems. We show that the orbital parameters obtained from a multiple-Kepler fit to the radial velocity variations of a host star are best interpreted as Jacobi coordinates and that Jacobi coordinates should be used in any analyses of hierarchical planetary systems. An approximate theory that can be applied to coplanar, hierarchical, two-planet systems with a wide range of masses and orbital eccentricities is the octopole-level secular perturbation theory, which is based on an expansion to order α3 and orbit averaging. It reduces the coplanar problem to 1 degree of freedom, with e1 (or e2) and ω̄1 - ω̄2 as the relevant phase-space variables (where e1,2 are the orbital eccentricities of the inner and outer orbits, respectively, and ω̄1,2 are the longitudes of periapse). The octopole equations show that if the ratio of the maximum orbital angular momenta, λ = L1/L2 ≈ (m1/m2)α1/2, for given semimajor axes is approximately equal to a critical value λcrit, then libration of ω̄1 - ω̄2 about either 0° or 180° is almost certain, with possibly large amplitude variations of both eccentricities. From a study of the HD 168443 and HD 12661 systems and their variants using both the octopole theory and direct numerical orbit integrations, we establish that the octopole theory is highly accurate for systems with α≲0.1 and reasonably accurate even for systems with α as large as 1/3, provided that α is not too close to a significant mean-motion commensurability or above the stability boundary. The HD 168443 system is not in a secular resonance, and its ω̄ 1 - ω̄2 circulates. The HD 12661 system is the first extrasolar planetary system found to have ω̄1 - ω̄2 librating about 180°. The secular resonance means that the lines of apsides of the two orbits are on average antialigned, although the amplitude of libration of ω̄1 - ω̄2 is large. The libration ω̄1 - ω̄2 and the large amplitude variations of both eccentricities in the HD 12661 system are consistent with the analytic results on systems with λ ≈ λcrit. The evolution of the HD 12661 system with the best-fit orbital parameters and sin i = 1 (i is the inclination of the orbital plane from the plane of the sky) is affected by the close proximity to the 11:2 mean-motion commensurability, but small changes in the orbital period of the outer planet within the uncertainty can result in configurations that are not affected by mean-motion commensurabilities. The stability of the HD 12661 system requires sin i > 0.3.
Persistent Identifierhttp://hdl.handle.net/10722/151107
ISSN
2015 Impact Factor: 5.487
2015 SCImago Journal Rankings: 3.369
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorLee, MHen_US
dc.contributor.authorPeale, SJen_US
dc.date.accessioned2012-06-26T06:17:08Z-
dc.date.available2012-06-26T06:17:08Z-
dc.date.issued2003en_US
dc.identifier.citationAstrophysical Journal Letters, 2003, v. 592 n. 2 I, p. 1201-1216en_US
dc.identifier.issn2041-8205en_US
dc.identifier.urihttp://hdl.handle.net/10722/151107-
dc.description.abstractWe investigate the dynamical evolution of coplanar, hierarchical, two-planet systems where the ratio of the orbital semimajor axes α = a1/a2 is small. Hierarchical two-planet systems are likely to be ubiquitous among extrasolar planetary systems. We show that the orbital parameters obtained from a multiple-Kepler fit to the radial velocity variations of a host star are best interpreted as Jacobi coordinates and that Jacobi coordinates should be used in any analyses of hierarchical planetary systems. An approximate theory that can be applied to coplanar, hierarchical, two-planet systems with a wide range of masses and orbital eccentricities is the octopole-level secular perturbation theory, which is based on an expansion to order α3 and orbit averaging. It reduces the coplanar problem to 1 degree of freedom, with e1 (or e2) and ω̄1 - ω̄2 as the relevant phase-space variables (where e1,2 are the orbital eccentricities of the inner and outer orbits, respectively, and ω̄1,2 are the longitudes of periapse). The octopole equations show that if the ratio of the maximum orbital angular momenta, λ = L1/L2 ≈ (m1/m2)α1/2, for given semimajor axes is approximately equal to a critical value λcrit, then libration of ω̄1 - ω̄2 about either 0° or 180° is almost certain, with possibly large amplitude variations of both eccentricities. From a study of the HD 168443 and HD 12661 systems and their variants using both the octopole theory and direct numerical orbit integrations, we establish that the octopole theory is highly accurate for systems with α≲0.1 and reasonably accurate even for systems with α as large as 1/3, provided that α is not too close to a significant mean-motion commensurability or above the stability boundary. The HD 168443 system is not in a secular resonance, and its ω̄ 1 - ω̄2 circulates. The HD 12661 system is the first extrasolar planetary system found to have ω̄1 - ω̄2 librating about 180°. The secular resonance means that the lines of apsides of the two orbits are on average antialigned, although the amplitude of libration of ω̄1 - ω̄2 is large. The libration ω̄1 - ω̄2 and the large amplitude variations of both eccentricities in the HD 12661 system are consistent with the analytic results on systems with λ ≈ λcrit. The evolution of the HD 12661 system with the best-fit orbital parameters and sin i = 1 (i is the inclination of the orbital plane from the plane of the sky) is affected by the close proximity to the 11:2 mean-motion commensurability, but small changes in the orbital period of the outer planet within the uncertainty can result in configurations that are not affected by mean-motion commensurabilities. The stability of the HD 12661 system requires sin i > 0.3.en_US
dc.languageengen_US
dc.publisherInstitute of Physics Publishing Ltd. The Journal's web site is located at http://iopscience.iop.org/2041-8205en_US
dc.relation.ispartofAstrophysical Journal Lettersen_US
dc.subjectCelestial Mechanicsen_US
dc.subjectPlanetary Systemsen_US
dc.subjectPlanets And Satellites: Generalen_US
dc.titleSecular evolution of hierarchical planetary systemsen_US
dc.typeArticleen_US
dc.identifier.emailLee, MH:mhlee@hku.hken_US
dc.identifier.authorityLee, MH=rp00724en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1086/375857en_US
dc.identifier.scopuseid_2-s2.0-0042917705en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0042917705&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume592en_US
dc.identifier.issue2 Ien_US
dc.identifier.spage1201en_US
dc.identifier.epage1216en_US
dc.identifier.isiWOS:000184408100045-
dc.publisher.placeUnited Kingdomen_US
dc.identifier.scopusauthoridLee, MH=7409119699en_US
dc.identifier.scopusauthoridPeale, SJ=6602697819en_US

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