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The Dynamics of the HD 12661 Extrasolar Planetary System

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The Dynamics of the HD 12661 Extrasolar Planetary System The Dynamics of the HD 12661 Extrasolar Planetary System Adri¶anRodr¶³guezand Tabar¶eGallardo Departamento de Astronom¶³a,Facultad de Ciencias, Igu¶a4225, 11400 Montevideo, Uruguay [email protected], [email protected] ABSTRACT The main goal of this work is to analyze the possible dynamical mechanisms that dominate the motion of the HD 12661 extrasolar planetary system. By an analytical approach using the expansion of the disturbing function given by El- lis & Murray (2000) we solve the equation of motion working in a Hamiltonian formulation with the corresponding canonical variables and by means of appro- priate canonical transformations. Comparing this results with a direct numerical integration we can conclude that the system is dominated by a pure secular evolu- tion very well reproduced with a disturbing function including at least sixth order terms in the eccentricities. Because of the uncertainties in the orbital elements of the planets, we also contemplate the occurrence of mean-motion resonances in the system and analyze possible contribution from these resonant terms to the total motion. Subject headings: canonical variables, celestial mechanics, extrasolar planets, secular evolution. 1. Introduction There are some characteristics of the extrasolar planetary systems that make them dif- ferent from the solar system, like the presence of massive planets close to the central stars (i.e, small values of semimajor axis) and with large values in the orbital eccentricities. Large eccentricities and small semimajor axis are the result of early migration processes where the planets were driven by tidal interaction with the disc where they were formed (Papaloizou 2003). Also, preliminary orbital determinations suggest the existence of resonant systems like Gliese 876 and HD 82943 with two planets locked in a 2:1 mean-motion resonance, 55 Cnc (3:1), 47 UMa (7:3)(Beaug¶e& Michtchenko 2003, Hadjidemetriou 2002). These charac- teristics make classical perturbation theories originally developed for the solar system, like { 2 { the Lagrange-Laplace secular theory, very limited to deal with the dynamics of extrasolar planets. The system HD 12661 that we study here is composed by two massive planets around a central star with a relatively low semimajor axis ratio ® ' 0:29 (see Table 1). Some attempts have been done to explain the dynamical behavior of this system. In most of them, stabil- ity analysis are considered estimating the dynamical limits on the orbital parameters that provide quasi-periodic motions of the system (Go¶zdziewski2003). The di±cult to get the desired accuracy in the orbital ¯ts in most of the extrasolar planets produces very important di®erences between similar works. This is also the case for HD 12661 where there are di®er- ent results published according to the orbital elements taken. The system was supposed very a®ected by the 2:11 or 1:6 mean-motion resonance according to data published by Fischer et al. (2003) or new ¯ts in the same year respectively (Go¶zdziewskiand Maciejewski 2003, Ji et al. 2003). There are few analytical treatments applied to this particular case, unless the study by Lee & Peale (2003) (see also Nagasawa et al. 2003 for a low order treatment in a cosmogonic framework) where they give an approach by means of the so called octopole secular theory. In that formulation using Jacobi coordinates the authors use a Hamiltonian function including up to third order terms in ® and some fourth order terms in eccentricities. They ¯nd a qualitatively good description for the system but severe discrepancies between the frequencies predicted by the theory and the ones computed from the numerical integra- tion of the full equation of motion. In this paper we present a semi-analytical theory that explains the dynamical evolution of the system. The present paper is organized as follow: In section 2 we present the actual data of the system, perform a numerical integration using the orbital ¯ts by Debra Fischer (2004, private communication) and discuss the possible e®ects of the main mean motion resonances near the system. The canonical variables necessary to work in an analytical framework are presented in section 3. In section 4 we compute the value of the disturbing function to several orders in eccentricities through the Ellis & Murray (2000) expansion and investigate the contributions of possible mean-motion resonances. In section 5 we present and resolve our secular model and compare with numerical integrations. Discussion and ¯nals remarks are devoted to section 6. { 3 { 2. The HD 12661 Extrasolar System Some authors considered the possibility of this system being strongly a®ected by a mean motion resonance. In particular Lee & Peale (2003) suggested that discrepancies between their octopole model and the direct numerical integration could be due to the 2:11 resonance. Di®erent orbital determinations according to di®erent data sets and methods (Lee & Peale 2003, Go¶zdziewski& Maciejewski 2003) lead to substantial di®erences in the semimajor axis of the exterior planet. On the contrary, the semimajor axis of the internal planet seems to be well determined according to the successive orbital determinations. For illustration purposes at Figure 1 we show the approximate position of the most important exterior mean motion 2=3 resonances deduced from the simple relation a = a1(p=(p + q)) together with the most probable location of the exterior planet according to Table 1. The parameter p is the degree of the resonance and is taken positive for interior resonances and negative for exterior ones. The parameter q, always positive, is the order. The width of each resonance is not indicated, we refer to Go¶zdziewski& Maciejewski (2003) for a numerical estimation of the width of the most important resonances. The exterior planet is very near the 2:13 and 3:19 resonances, but both are very high order resonances. In an expansion of the disturbing function for quasi planar orbits the most important resonant terms are of order q in the eccentricities. Then that terms are of order 11 and 16 in eccentricity respectively and for small and medium eccentricity values as in this particular system in principle we cannot expect those terms to became relevant for the dynamical evolution of the system. But, in spite of not being important for the approximate description of the system, some of those slight e®ects could be detected with numerical methods like MEGNO (Cincotta & Sim¶o2000) as was clearly shown by Go¶zdziewski& Maciejewski (2003). We will discuss more profoundly this point at section 4. We perform a numerical integration of the HD 12661 extrasolar planetary system sup- posed coplanar, with sin i = 1 and with a central star mass of 1:07M¯. The main goal is to compare the results with an analytical model to ¯nd the dynamical mechanisms that dom- inate the system. The orbital elements of HD 12661 had recently changed from their last values (Fischer et al. 2003). The astrocentric osculating orbital elements used here to make the numerical integrations are presented at Table 2 and were deduced from Table 1 assuming the radial velocity ¯t for the planet c better emulate an orbit referred at the baricenter of the star and planet b as suggested by Lissauer & Rivera (2001). The numerical integrator used was MERCURY (Chambers 1999). { 4 { Mean motion resonances in the vecinity of the exterior planet 18 17 3:20 16 3:19 15 14 3:17 13 3:16 2:15 12 11 2:13 10 probable location of the exterior planet 9 2:11 q : order of the resonance 8 7 6 1:7 5 1:6 4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 semimajor axis (AU) Fig. 1.| Approximate localization of the exterior mean motion resonances jp + qj : jpj near the exterior planet. Higher order resonances should have less dynamical signatures. It should be kept in mind that each resonance covers some range ¢a that is not indicated here. The semimajor axis of the exterior planet was taken from Table 1 which is approximately equal to its mean semimajor axis. Planet P (days) K(m/s) e !(deg) T p(MJD) m(MJup) a(AU) HD 12661b 262:37 75:02 0:34 296:6 9952:267 2:33 0:821 Error(+/-) 0.32 0.85 0:02 2:6 3:6 0:02 0:001 HD 12661c 1681:7 29:9 0:06 102:8 10314:221 1:83 2:834 Error(+/-) 14.2 0.93 0:02 77:9 641:7 0:06 0:016 Table 1: Parameters P , K, e, ! and Tp are from two-Kepler ¯ts by Debra Fischer (2004, private communication). The errors were estimated by a Monte Carlo method. Parameters { 5 { m and a were deduced following formulae 2,4,6 and 8 from Lee & Peale (2003) assuming sin i = 1 and a stellar mass of 1:07M¯. Planet a(AU) e !(deg) M(deg) HD 12661b 0:821 0:34 296:6 136:7 Error(+/-) 0.001 0.02 2:6 4:9 HD 12661c 2:855 0:066 102:1 0:7 Error(+/-) 0.016 0.02 77:9 137 Table 2: Astrocentric osculating orbital elements deduced from Table 1 for the epoch JD 2450314.22 The Figures 2 and 3 show the time evolution of the astrocentric semi-major axes and eccentricities for the two planets. We can see the periodic evolution in eccentricities show- ing an almost perfect coupling between them, i.e both elements oscillates with the same period. This is a simple consequence of the conservation of the total angular momentum of the system, which is approximately equal to the angular momentum of the planets because the angular momentum of the star with respect to the barycenter is negligible.
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