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A Dynamical Analysis of the HD 37124 Planetary System

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A Dynamical Analysis of the HD 37124 Planetary System A&A 398, 315–325 (2003) Astronomy DOI: 10.1051/0004-6361:20021602 & c ESO 2003 Astrophysics A dynamical analysis of the HD 37124 planetary system K. Go´zdziewski Torun ´ Centre for Astronomy, N. Copernicus University, Gagarin Str. 11, 87-100 Toru´n, Poland Received 29 July 2002 / Accepted 7 October 2002 Abstract. In this paper we study the stability of the HD 37124 planetary system. Using initial conditions found by Butler et al. (2002) we estimate the dynamical bounds on orbital parameters that provide stable (quasi-periodic) motions of the system. The stability analysis has been performed with the help of the MEGNO technique. This method, invented by Cincotta & Sim´o (2000), makes it possible to distinguish efficiently between chaotic and regular dynamics of a dynamical system. The MEGNO analysis helps us to confirm the result of Butler et al. (2002) who found that the critical factor for system stability is the eccentricity of the outer planet. This parameter is not well constrained by the current set of observations. For coplanar configurations, the limiting value of the eccentricity, providing quasi-periodic motion of the system, is roughly equal to 0.55. This value is typically slightly smaller when the system inclination increases (and companions’ masses grow) but for i 25◦ it can be as low as 0.25. The system is located in a wide stable zone in the plane of eccentricities. The dynamics are insensitive to the initial phases of the planets. If the eccentricity of the outer planet is close to the current fit value 0.4, then the system is regular over wide ranges of the relative inclination of the planets. We analyze whether a telluric planet can survive orbitally in the habitable zone of the system. This zone is covered by the investigated region of semi-major axes between 0.6 AU and 2.4 AU. The dynamics in this zone strongly depend on the eccentricity of the outer companion. For moderately low values of this parameter, ec 0.1–0.2, the habitable zone is a dynamical analogue of the Asteroid belt in the Solar system, and the habitable zone of the 47 UMa system. Moreover, in this instance, relatively wide stable areas exist. For increasing eccentricity of the outer planet the stable zones shrink rapidly, and if the parameter is larger than 0.4 they practically disappear. We describe an alternative integration of the MEGNO indicator in the planetary N-body problem, based on the symplectic leapfrog scheme and the tangent map by Mikkola & Innanen (1999). Key words. celestial mechanics, stellar dynamics – methods: numerical, N-body simulations – planetary systems – stars: individual: HD 37124 1. Introduction 4.8 m s−1. In this work we conduct a preliminary study of the orbital stability of the HD 37124 system using the 2-Keplerian The search for extrasolar planetary systems conducted in the fit published in (Butler et al. 2002). The data are reproduced in past years by the California & Carnegie Planet Search Team Table 1. We consider them as formal osculating elements at the has recently revealed new multi-planetary systems (Butler et al. epoch of the periastron passage of the outer planet. Because 2002). Our work is devoted to the system orbiting the star the orbital fit is uncertain, we try to recover some qualitative HD 37124. The variability of its radial velocity was detected dynamical properties of the HD 37124 planetary system, and at the beginning of the survey initiated by Mayor and Queloz to set dynamical constraints on undetermined or badly fixed in 1994 (Udry 2002). A first planet in 155 day orbit was found orbital parameters. in the frame of the Lick survey (Vogt et al. 2000). The second planet in 2000 day orbit has been reported by Butler et al. The recent discoveries of the second companion of (2002). The discovery of the second planet has been also an- HD 37124, the system of 47 UMa (Fischer et al. 2002b), 55 Cnc nounced by the Geneva Planet Search Team (Udry 2002). (Marcy et al. 2002), HD 12661, HD 38529 (Fischer et al. The observational time span of HD 37124 is short with 2002a), and (possibly) HD 160169 (Jones et al. 2002), revealed respect to the orbital period of the outer companion. During exosystems that can be considered more or less analogues of the years 1996–2002 only 30 observations have been collected. the Solar system. In these systems, except 47 UMa, which is The orbital fit to the radial velocity curve of HD 37124 is not by far the closest dynamical relative of the Solar system, the well constrained with these data (Butler et al. 2002). A dou- inner planets’ semi-major axes are much smaller than 1 AU, ble Keplerian solution found by these authors allows for the and the outer planet in an orbit (3,5) AU plays the dynamical eccentricity of the outer planet to vary from 0.3 to 0.8, giv- role of Jupiter. A common feature of these systems is that they ing residuals consistent with the estimated velocity uncertainty do not possess giant planets at distances 1 AU, which roughly coincide with habitable zones of Sun-like stars (Franck et al. e-mail: [email protected] 2000). This leads to a natural question: can an Earth-like planet Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021602 316 K. Go´zdziewski: Stability of the HD 37124 planetary system form and survive in an orbit confined to such zone? A positive long-term integrations. Instead, we explore the space of initial answer to this question is important for selecting targets of ter- conditions in order to find regions in the phase space, which are restrial planets’ search missions. In this context, the system of filled up with quasi-periodic, stable evolutions. This allows us 47 UMa has been studied by many authors (Jones et al. 2001; to specify whether chaotic motions are correlated with changes Laughlin et al. 2002; Th´ebault et al. 2002; Noble et al. 2002; of orbital elements that appear in a relatively short period of 4 Go´zdziewski 2002). A common conclusion of these works is MEGNO integrations, of the order of NP = 10 orbital periods that stable orbits are possible in the habitable zone of 47 UMa, of the outer companion, Pc. Such short time-scale, TM = NPPc, but the creation of terrestrial planets in the dynamical environ- is typically required to derive reliable estimates of MEGNO, ment created by the two planets is rather unlikely. It does not and guarantees computational efficiency of the tool. The anal- seem to be a rule in the case of new planetary systems. Butler ysis of the short-term orbital dynamics is also very helpfull for et al. (2002) announced the results of their investigations of interpreting the MEGNO results. 55 Cnc. They found a wide and stable habitable zone in this Most of the MEGNO integrations in this paper are driven system. In the present paper we search for a possibility of sta- by a Bulirsh-Stoer integrator. We used the ODEX code (Hairer ble orbits in the habitable zone of HD 37124. & Wanner 1995). The relative and absolute accuracies of the The orbital parameters space of this exosystem is ex- integrator have been set to 10−14 and 5 × 10−16, respectively. plored with the help of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) (Cincotta & Sim´o 2000). This 2. MEGNO signature of the initial condition technique seems to be well designed for studying planetary dy- namics in the framework of the N-body problem. The basic First we examined the stability of the 2-Keplerian fit, found idea of our approach relies on a rapid determination whether by Butler et al. (2002). The results of this test are illustrated a given initial condition leads to a quasi-periodic or irregular in Figs. 1a,c, where a number of MEGNO graphs, derived for (chaotic) motion of the planetary system. We applied it already, randomly selected initial variations1, are plotted as functions of studying the global dynamics of υ Andr (Go´zdziewski et al. time. We observe that for the integration time span TM,the 2001), HD 82943 (Go´zdziewski & Maciejewski 2001), GJ 876 character of MEGNO convergence is not always quite clear. (Go´zdziewski et al. 2002), and, recently, to the 47 UMa sys- In particular, this concerns MEGNO variations shown in two tem (Go´zdziewski 2002). These papers contain an explanation plots, labeled by (1) and (2), in Fig. 1c. of our motivations and technical aspects of the calculations. Integrations continued over 10TM do not give clear answers Summarizing the idea of the calculations briefly, MEGNO Y for all choices of the initial variations, either. Actually, the two is calculated during numerical integration of the equations of Y-graphs, labeled with (1) and (2) in Fig. 1c, seem to be motion and the respective variational equations. As was shown slowly divergent. We have observed this kind of MEGNO be- by Cincotta & Sim´o (2000), if Y(t) converges to 2, and af- havior in many runs. Note, that oscillations of Y are corre- ter a chosen integration time tmax we have |Y(tmax) − 2| <, lated with variations of the eccentricities (compare Figs. 1a,c (where is a small quantity, of the order of 10−2–10−3), then and Fig. 2). In all examined runs the energy integral grows lin- − the tested initial condition leads to a quasi-periodic, stable so- early with time over a few orders of magnitude (from 10 13 − lution of the system; otherwise it corresponds to chaotic behav- to 10 9) in the time span of integrations.
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