Global Dynamics and Stability Limits for Planetary Systems Around HD 12661, HD 38529, HD 37124 and HD 160691

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Global dynamics and stability limits for planetary systems around HD 12661, HD 38529, HD 37124 and HD 160691.

Ludmila Kiseleva-Eggleton1 Department of Mathematics and Computer Science, School of Science, St. Mary’s College of California, Moraga, CA 94575, USA

Eric Bois & Nicolas Rambaux Observatoire de Bordeaux, 2 rue d’Observatoire, B.P.89, F-33270, Floirac, France and

Rudolf Dvorak Institute of Astronomy, University of Vienna, T¨urkenschanzstrasse 17, A-1180 Vienna, Austria

ABSTRACT In order to distinguish between regular and chaotic planetary orbits we apply a new technique called MEGNO in a wide vicinity of orbital parameters determined using standard two-body Keplerian fits for HD 12661, HD 38529, HD 37124, HD 160691 planetary systems. We show that the currently announced orbital parameters place these systems in very different situations from the point of view of dynamical stability. While HD 38529 and HD 37124 are located within large stability zones in the phase space around their determined orbits, the preliminary orbits in HD 160691 are highly unstable and there are no stability regions in the close vicinity of these orbits. The orbital parameters of the HD 12661 planets are located in a border region between quasi- periodic and chaotic dynamical regimes, so that the combination of its nominal parameters results in a chaotic orbit, while some minor changes within the margin of error of just one parameter at a time may produce regular configurations.

Subject headings: celestial mechanics, stellar dynamics - planetary systems - stars:individual (HD 12661, HD 38529, HD 37124, HD 160691)

1. Introduction (see http://www.obspm.fr/planets). Four planetary systems - HD 12661, HD 38529, HD 37124 The recent explosion in the number of newly and HD 160691 - have recently been elevated to detected extrasolar planets has brought the to- the status of multiple, following the discovery of tal number of such planets to about one hun- a second planetary companion in each of them. dred; it appears that the exact number changes In most of these systems the strong dynamical in- too rapidly to be quoted. Some of the planets form teraction between planets makes planetary orbital multiple planetary systems around their parental parameters (see Table 1), found using standard stars. At the moment the total number of sys- two-body Keplerian ﬁts, unreliable (Laughlin & tems with two or more planetary-mass compan- Chambers 2001, 2002). There is also a great uncer- ions around main sequence stars has reached 11 tainty in the determination of planetary masses. All those leave us a substantial available param- 1Also a participating guest at IGPP, LLNL, L-413, 7000 East Ave, Livermore, CA 94550, USA eter space to be explored in order to exclude the

1 initial conditions which lead to dynamically un- Solar System dynamics (Tsiganis et al. 2001) stable configurations. and dynamics of extrasolar planetary systems A classical method that allows one to dis- (Pilat-Lohinger & Dvorak 2002). This integra- tinguish between regular and chaotic dynamical tion method is based on recurrence formulae and states is the method of Lyapunov Characteristic uses an automatic step-size control. The four Numbers (LCN). Let us note that chaotic in the systems under study were integrated over evolu- Poincaré sense means that the dynamical behav- tionary times of up to 107 yrs. Our study revealed ior is not quasi-periodic, and does not necessarily many interesting and rather unexpected dynami- mean that the system will desintegrate during cal features in all of these systems. It is remarkable any limited period of time. The estimation of that being discovered almost simultaneously they LCN usually requires computations over long evo- represent a real ‘dynamical zoo’ which we describe lutionary time, sometimes much longer than the below. lifetime of the system studied. 2.1. HD 12661 In our previous papers (see for example, Go´zdziewski et al. 2001a, 2001b) we showed that a In HD 12661 the two planets are rather close to new method developed by Cincotta & Simó (2000) each other. In fact, for this system the ratio of the and called MEGNO (the acronym of Mean Expo- periastron distance of the outer orbit to the apoas- nential Growth of Nearby Orbits), can be success- tron distance of the inner orbit matches exactly fully applied to the studies of dynamical stability the corresponding critical value for three-body dy- of extrasolar planetary systems. This method is namical stability derived by Eggleton & Kiseleva based on the same ideas as LCN but is converg- (1995) 2 for less extreme mass ratios (triple stars). ing about 100 times faster, and more sensitive. This places HD 12661 in a very interesting zone For example its application to the Gliese 876 sys- of critical parameters where small changes can tem (Go´zdziewski et al. 2001b) clearly identified lead to totally different dynamical regimes. Fig.1 the exact location of the 2:1 mean motion reso- shows the changes in the semi-major axis ab of nance and its width. MEGNO helped to show the innermost planet in HD 12661 over 1 million (Go´zdziewski & Maciejewski 2001) that while the years. In the top panel, which represents the sys- orbital parameters of the HD 82943 system derived tem with nominal initial orbital parameters (Ta- from the Keplerian fit lead to an unstable self- ble 1), the semi-major axis displays dynamical be- destructing system, there are some small changes havior with several irregular and therefore unpre- in this fit which lead to stable configurations. dictable jumps from one ”mean” value of ab to In this work we apply the MEGNO technique to another. We call such an orbit chaotic. This phe- all four new planetary systems. We are especially nomenon of unpredictibilty is well known in dy- interested in a comparative study of the global dynamical systems since Poincaré and it is a charac- namics of these systems, as they are likely to rep- teristic of chaos in such systems. The lower panel resent the different types of dynamical behavior. shows totally different behavior of ab for the same Taking into account the rather preliminary nature planetary system with only a very small difference of all orbital fits, we hope that this paper will pro- in initial value of ac: ac=2.63 AU instead of vide a useful guide for available stable orbital pa- 2.61 AU. This ∆ac =0.02 AU is well within the rameters (such as a, e and ω) for more sophis- error of the ac determinaton (see, for example, ticated self-consistent fits (Laughlin & Chambers Fischer et al. 2001). The planetary orbits in this 2001, 2002; Marcy et al. 2002). ‘new’ HD 12661 system are quasi-periodic with no sign of unpredictable variations over time, and the 2. Results. amplitude of fluctuations for all its orbital parameters are much smaller than for the nominal HD All our results obtained with MEGNO were 12661. These results are in a very good agreement confirmed by direct integration of the equa- with MEGNO tests: for the system with ac=2.63 tions of motion using the Lie-series method (e.g. AU the MEGNO indicator (see Go´zdziewski Hanslmeier & Dvorak, 1984) which has been al- ready used for many numerical simulations of 2Note that in their Eq2 the second sign should be ”-”

2 Table 1 Orbital parameters of new planetary systems.

a a a a a a Name Mp sin i(MJ ) M ( M ) a(AU) P (days) e ω(deg) ∗ HD 12661b 2.47 1.07 0.85 263.6 0.347 291.73 HD 12661c 1.66 1.07 2.61 1407.0 0.220 148.27 HD 37124b 0.86 0.91 0.54 155 0.10 97.0 HD 37124c 1.01 0.91 2.95 1942 0.40 265.0 HD 38529b 0.67 1.39 0.12 14.3 0.312 89.173 HD 38529c 11.3 1.39 3.51 2189.5 0.340 9.424 HD 160691b 1.7(1.6?b) 1.08 1.5(1.4?b) 638(603?b) 0.31(0.37?b) 316.0 HD 160691c 1.0? 1.08 2.3? 1300? 0.8? 99?

aData from the Extrasolar Planets Encyclopedia website: http://www.obspm.fr/planets bFrom the sum of two Keplerian ﬁts (see Jones et al. 2002)

et al. 2001a,b) converges to the characteristic for triple stars near the stability limit. The HD 12661 regular systems value 2, while for the nominal pa- system probably represents a real case of a plane- rameters of HD 12661 6, which indicates tary system with chaotic orbits, whose inevitable the presence of chaos in this≈ system. Let us note existence was recently claimed by Murray & Hol- we do not claim that such a system will desinte- man (2001). grate after any particular period of time. We just point out that its dynamical state over a long time 2.2. HD 37124 and HD 38529 is unpredictable. The orbits in both these systems are solidly sta- Fig.2 shows the stability maps for initial ab, ac ble and quasi-peiodic with no chaotic features for and ωc of the HD 12661 system in a rather large a wide range of orbital parameters including high vicinity of the nominal values of these orbital pa- relative inclinations between the two planetary or- rameters . One can see that the parameter maps bits. Let us note that all results presented in this themselves look rather chaotic, as they display paper were obtained for almost coplanar systems unsystematically located zones of initial orbital unless it is indicated overwise. When applied to parameters leading to regular orbits ( 2, ≈ the planetary orbits of the HD 38529 system with marked with the ﬁlled squares), leading to chaotic diﬀerent variations of its orbital parameters, the orbits (marked with open triangles when is MEGNO indicator almost always converges between 2 and 5, and by for larger values of to 2, with a few exceptions for very large eccentric- ) in this phase space.× This particular struc- ities ec of the outer planet (see Fig. 3c). Direct N- ture of the phase space is typical for regions close body intergations of the equations of motion per- to the border between stable and unstable mo- formed for selected samples also showed a quasi- tion (e.g. Dvorak et al. 1989). In such regions periodic behaviour for all orbital parameters con- minor changes in each of the parameters may dra- sidered. The same is true for HD 37124: this sys- maticaly transfer the system from a regular into tem become unstable when ec approached 0.7 (see a chaotic dynamical state. This behavior is not Fig. 3b), but such values of ec are well outside the observed in the motion of the planets in our So- vicinity of nominal orbital parameters. However, lar system (e.g. Murray & Holman 2001), but it the orbital parameters of planets in both systems, is a well-known phenomenon for the dynamics of especially the planetary eccentricities (usually an asteroids (Wisdom 1983). Kiseleva et al. (1994a, eccentricity is the orbital parameter most sensi- 1994b) found a similar phenomenon for simulated

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0.846 0 200 400 600 800 1000 Fig. 2.— Quasi-periodic (ﬁlled squares) and chaotic orbital parameters for the HD 12661 plan- Fig. 1.— Variations of semi-major axis ab of the etary system. Each panel shows MEGNO results innermost planet in HD 12661 over one million (see text for details) when two parameters were years: in the top panel the semi-major axis ac of varied while others where kept constant and equal the outer planet is 2.61 AU and in the lower panel to the nominal parameters of HD 12661 (Table ac = 2.63 AU. All other parameters are the same 1). The nominal parameters are indicated by an in both panels. arrow. Cases that produce systems with strong dynamical chaos are marked with the symbol. Some minor deviations from the nominal× values produce regular systems.

4 tive to external perturbations), are not unaffected a few stability zones in a parameter space which by the presence of the other planetary compan- includes together with the parameters listed above ion. For example, the eccentricity of the planet also parameters not determined from observations HD 37124b changes periodically between 0.1 and such as the relative inclination ir between the two 0.43. Even in the HD 38529 system where the ratio planetary orbits, Ω and L. Most of these stable of the two planetary periods is over 153 and there- configurations are associated with 2:1 mean mo- fore the planets are not expected to have much tion resonance. We are going to discuss in detail influence on each other’s dynamics, the eccentric- the possible combinations of parameters resulting ity of the very close innermost planet (Pb =14.3 in stable orbits in HD 160991 in our forthcoming days) fluctuates within the interval [0.292, 0.368] paper (Bois et al., in preparation). for a coplanar model with nominal parameters. min The massive (Mc =11.3 M ) outer planet in 3. Comparative stability analysis and con- this system is also subject to small eccentricity clusions flutuations between 0.332 and 0.345. Fig. 3 presents a comparative visualisation of It can be interesting to consider a case where our MEGNO stability analysis for all four new the two planetary planes in HD 38529 have high planetary systems. For this visual presentation we relative inclination. MEGNO tests showed that choose to consider orbital stability as a function of the orbits in this systems remain stable and quasi- both orbital eccentricities simply because the val- periodic for all relative inclinations, including or- ues of e and e can only be changed between 0 and thogonal orbits with i =90o. Taking into account b c r 1, and therefore do not require any scaling (as for the sub-stellar minimal mass of planet C, com- example would semi-major axes) for different sys- pletely independent formation scenarios for com- tems. All other orbital parameters are the nominal panions B and C are possible. Therefore a highly ones from Table 1. It is easy to see the differ- non-coplanar configuration would not be unlikely ences in dynamical status of the four systems and for this system. The innermost planet B is close to identify the ranges of eccentricities which allow enough to the star for tidal friction to become im- stable planetary orbits in each of them. We also portant when the ‘Kozai effect’ (Kozai 1962) in- produced simular maps for other pairs of orbital duces large fluctuations on the orbital eccentricity parameters. Such ranges of stable parameters can (e =1 for i =90o). This may significantly max r be very useful for the improvement of preliminary affect the dynamics of this system (see, for exam- orbits in the new planetary systems discussed in ple, Kiseleva et al. 1998; Eggleton et al. 1998; this paper. We plan to present detailed dynam- Eggleton & Kiseleva-Eggleton 2001a, 2001b; Wu ical analyses of each system taking into account & Murray, in preparation). We are going to inves- angular orbital parameters not constrained by ob- tigate these effects in a future paper. servational data (ir, L, Ω), as well as sin i and the 2.3. HD 160991 resulting different planetary masses in our future papers. Our simulations showed that for HD 160691 both sets of initial orbital parameters taken from The authors thank the John Templeton Foun- Table 1 produce highly unstable self-destructing dation for its support of this project. RD did systems. However, the orbital parameters of both this work in the framework of the Austrian FWF planetary orbits are rather speculative at the mo- project P14375-TPH. LK-E and RD thank the ment: even the existence of the second planetary Bordeaux Observatory for its hospitality. companion is not yet fully confirmed (Jones et al. 2002). Therefore we looked for stable configura- REFERENCES tions among wide ranges of orbital parameters. We found no combinations of two semi-major axes []Cincotta, P. M. & Simó, C., 2000, A&ASS, 147, (or orbital periods) without simultaneous changes 20 in other orbital parameters such as e , e , ω ,and b c b []Eggleton, P. & Kiseleva, L., 1995, ApJ, 455, 640 ωc given in Table 1 which would produce stable orbits. But using MEGNO we were able to identify

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eb (HD 38529) eb (HD 160691)

Fig. 3.— Stability maps in the eb ec parameter space for all four new planetary systems. The symbols are the same as in Fig. 2 (ﬁlled squares− indicate stable regular orbits).

6 []Eggleton, P. P., Kiseleva, L. G. & Hut, P., 1998, []Wisdom, J.: 1983, Icarus, 56,51 ApJ, 499, 853 []Eggleton, P. P. & Kiseleva-Eggleton, L., 2001a, ApJ, 562, 1012 []Go´zdziewski, K., Bois, E., Maciejewski, A. J. & Kiseleva-Eggleton, L., 2001a, A&A, 378, 569 []Go´zdziewski, K., Bois, E., Maciejewski, A. J., 2001b, MNRAS, 332, 839 []Go´zdziewski, K. & Maciejewski, A. J., 2001, ApJ, 563, L81 []Fischer, D. A. et al., 2001,ApJ, 551, 1107 []Hanslmeier & A., Dvorak, R.: 1984, A&A, 132, 203 []Jones, H. R. A. et al., 2002, MNRAS, submitted (astro-ph/0206216) []Kiseleva, L. G., Eggleton, P. P. & Anosova, J. P., 1994a, MNRAS, 264, 161 []Kiseleva, L. G., Eggleton, P. P. & Orlov, V. V., 1994b, MNRAS, 270, 936 []Kiseleva, L. G., Eggleton, P. P. & Mikkola, S., 1998, MNRAS, 300, 292 []Kiseleva-Eggleton, L. & Bois, E.,2001, ApJ, 553, L73 []Kiseleva-Eggleton, L. & Eggleton, P. P., 2001b, APS Conf. Ser., 229, 91 []Kozai, Y. 1962, AJ, 67, 591 []Laughlin, G. & Chambers, J. E., 2001, ApJ, 551, L109 []Laughlin, G. & Chambers, J. E., 2002, AJ, 124, 592 []Marcy, G. W. et al., 2002, ApJ, submitted (astro- ph/0207294)

[]Murray, N. & Holman, M.: 2001, Nature, 410, 773 []Pilat-Lohinger & E., Dvorak R.: 2002, Ce- lest.Mech., 82, 143 []Tsiganis, K., Dvorak, R. & Pilat-Lohinger, E., 2000, A&A, 354, 1091 AAS LATEX macros v5.0.