Astrophys. J., 699, 23-30 (2009/July-1)
A quasi-stationary solution to Gliese 436b's eccentricity.
BATYGIN K., LAUGHLIN G., MESCHIARI S., RIVERA E., VOGT S. and BUTLER P.
Abstract (from CDS):
We investigate the possibility that the large orbital eccentricity of the transiting Neptune-mass planet Gliese 436b (Gl 436b) is maintained in the face of tidal dissipation by a second planet in the system. We find that the currently observed configuration can be understood if Gl 436b and a putative companion have evolved to a quasi-stationary fixed point in which the planets' orbital apses are co-linear and in which secular variations in the orbital eccentricities of the two planets have been almost entirely damped out. In our picture, the two planets are currently experiencing a long period of gradual orbital circularization. Specifically, if Gl 436b has a tidal Q ∼ 300,000, similar to both the Jovian Q and to the upper limit for the Neptunian Q, then this circularization timescale can be of order τ ∼ 8 Gyr given the presence of a favorably situated perturber. We adopt an octopole-order secular theory based on a Legendre expansion in the semimajor axis ratio a1/a2 to delineate well-defined regions of (Pc, Mc, ec) space that can be occupied by a perturbing companion. This description includes the leading-order effects of general relativity, and retains accuracy for perturbing companion planets that have high eccentricity. We incorporate the evolutionary effect of tidal dissipation into our secular model of the system, and solve the resulting initial value problems for a large sample of the allowed configurations. We find a locus of apsidally aligned configurations that are (1) consistent with the currently published radial velocity data, (2) consistent with the current lack of observed transit timing variations (TTVs), (3) subject to rough constraint on dynamical stability, and which (4) have damping timescales consistent with the current multi-Gyr age of the star. We then polish the stationary configurations derived from secular theory with full numerical integrations, and compute the TTVs and radial velocity half-amplitudes induced by the resulting configurations. We present our results in the form of candidate companion planets to Gl 436b. For these candidates, radial velocity half-amplitudes, Kc, are of order 3 m/s, and the maximum amplitude of orbit-to-orbit TTVs are of order Δt = 1 s to Δt = 5 s. For the particular example case of a perturber with orbital period, Pc= 40 d, mass, Mc= 8.5 M⊕, and eccentricity, ec= 0.58, we confirm our semianalytic calculations with a full numerical three-body integration of the orbital decay that includes tidal damping and spin evolution. Additionally, we discuss the possibility of many-perturber stationary configurations, utilizing modified Laplace-Lagrange secular theory. We then perform a proof-of-concept tidally dissipated numerical integration with three planets, which shows the system approaching a triply circular state.
methods: analytical - methods: numerical - planetary systems
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