Secular Evolution of Hierarchical Planetary Systems
166 citations recorded by the source corpus.
We 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 ϖ1 - ϖ2 as the relevant phase-space variables (where e1,2 are the orbital eccentricities of the inner and outer orbits, respectively, and ϖ1,2 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 ϖ1 - ϖ2 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 , 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 ϖ1 - ϖ2 circulates. The HD 12661 system is the first extrasolar planetary system found to have ϖ1 - ϖ2 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 ϖ1 - ϖ2 is large. The libration of ϖ1 - ϖ2 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.
Subfields: Astronomy and Astrophysics, Instrumentation
View Paper: https://www.semanticscholar.org/paper/a98fdea2fc118ff760556d7a5d5c56583ad872fa
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Astronomy and Astrophysics
In Instrumentation
Citations in the corpus
- A ~7.5 M⊕ Planet Orbiting the Nearby Star, GJ 876 (2005, background): “As explained in Lissauer & Rivera (2001) and Lee & Peale (2003), Jacobi coordinates are the most natural system for expressing multiple-planet fits to radial velocity data.”
- TEN NEW AND UPDATED MULTIPLANET SYSTEMS AND A SURVEY OF EXOPLANETARY SYSTEMS (2008, methodology): “The radial velocity parameters were converted into initial conditions using a Jacobi coordinate system (Lee & Peale 2003).”
- Long-term tidal evolution of short-period planets with companions (2007, background): “Equations (4) to (7) are given to octopole order (corresponding to the fourth power in ap/ac); see also Lee & Peale (2003) and Ford & Rasio (2000).”
- Secular Dynamics in Hierarchical Three-Body Systems (2011, methodology): “The octupole-level Hamiltonian and equations of motion were previously derived by Harrington (1968, 1969); Sidlichovsky (1983); Marchal (1990); Krymolowski and Mazeh (1999); Ford et al. (2000a); Blaes et al. (2002) and Lee and Peale (2003).”
- SECULAR CHAOS AND THE PRODUCTION OF HOT JUPITERS (2010, methodology): “The dissipative part of the tidal effect is handled in a way that differs from the standard treatment of Lee & Peale (2003).”
- ECCENTRICITY GROWTH AND ORBIT FLIP IN NEAR-COPLANAR HIERARCHICAL THREE-BODY SYSTEMS (2013, background): “Lee & Peale (2003) considered the case when the mutual inclination is zero, and they showed that the eccentricity can oscillate due to the octupole effects.”
- Debris disc stirring by secular perturbations from giant planets (2009, background): “This eliminates variations in eccentricity (of order 4µ/ √ α) on the Keplerian timescale (Lee & Peale 2003), which exceed the forced secular eccentricity at large distances from the star.”