The TrES-2 system harbors one planet which was discovered with the transit
technique. In this work we investigate the dynamical behavior of possible
additional, lower-mass planets. We identify the regions where such planets can
move on stable orbits and show how they depend on the initial eccentricity and
inclination. We find, that there are stable regions inside and outside the
orbit of TrES-2b where additional, smaller planets can move. We also show that
those planets can have a large orbital inclination which makes a detection with
the transit technique very difficult.Comment: accepted for publication in AN 330 (2009

Dvorak

Ford

Froeschle

Hanslmeier

Lichtenegger

Lohinger

Miller-Ricci

Nagy

O'Donovan

Raetz

Süli

Winn

English

arXiv

The TrES-2 system harbors one planet which was discovered with the transit technique. In this work we investigate the dynamical behavior of possible additional, lower-mass planets. We identify the regions where such planets can move on stable orbits and show how they depend on the initial eccentricity and inclination. We find, that there are stable regions inside and outside the orbit of TrES-2b where additional, smaller planets can move. We also show that those planets can have a large orbital inclination which makes a detection with the transit technique very difficult. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA. Weinhei

Freistetter, F

Suli, A

Funk, B

ELTE Digital Institutional Repository (EDIT)

arXiv:0905.1806v1  [astro-ph.EP]  12 May 2009Astron. Nachr. / AN999, No. 88, 789 – 793 (2006) /DOI please set DOI!Dynamics of the TrES-2 systemF. Freistetter1,⋆, Á. Süli2, andB. Funk31 Astrophysikalisches Institut und Universitätssternwarte Jena, Schillergäßchen 2-3, D-07745 Jena, Germany2 Department of Astronomy Eötvös Loránd University Pázmny Péter sétány 1/A 1117 Budapest, Hungary3 Institut für Astronomie, Universität Wien, Türkenschanzstrasse 17, A-1180 Wien, AustriaReceived 2008, accepted 2008Published online laterKey words TrES-2 – extrasolar planets – dynamics – transitThe TrES-2 system harbors one planet which was discovered with the transit technique. In this work we investigate thedynamical behavior of possible additional, lower-mass planets. We identify the regions where such planets can move onstable orbits and show how they depend on the initial eccentri ity and inclination. We find, that there are stable regionsinside and outside the orbit of TrES-2b where additional, smaller planets can move. We also show that those planets canhave a large orbital inclination which makes a detection with the transit technique very difficult.c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionThe planet TrES-2b was discovered in 2006 (O’Donovan et al.2006). It orbits a G0V star with approximately the samemass and radius as the Sun. The planet has a mass of 1.198MJUP and a radius of 1.22 RJUP. Its orbit is circular andhas a semi-major axis of 0.0367 AU (see table 1 for all pa-rameters and the corresponding error bars.)TrES-2b was discovered using the transit method whereone tries to observe the decrease in luminosity of a starcaused by a transiting planet. This method is very promis-ing and current and future space-mission (e.g.CoRoT andKepler) will use this method to largely increase the num-ber of known transiting planets. Although it is possible todetect quite small planets via the transit technique the dis-covery of earth-like planets is still very difficult. The premisesare better in systems were a transiting planet is already dis-covered. By investigating the variations in the transit timeof a planet it is possible to identify also very small plan-ets (Ford & Gaudi 2006; Ford & Holman 2007).It is thus desirable to have a detailed knowledge about the⋆ Corresponding author: e-mail: florian@astro.uni-jena.deTable 1 Basic data of the TrES-2 systemfrom O’Donovan et al. (2006); Winn et al. (2008)TrES-2bMass 1.198 ± 0.053MJUPSemimajor axis 0.0367 ± 0.0012 AUOrbital period 2.470621 ± 0.000017 daysEccentricity 0Radius 1.22± 0.045RJUPTime of transit 2453957.63479 ± 0.00038Inclination 83.9± 0.22dynamical properties of the systems with known transitingplanets. In this work, we investigate the gravitational influ-ence of TrES-2b on a potential second planet in the system.Depending on the orbital elements of TrES-2b we can iden-tify the zones of stability, where it could be possible to de-tect an additional planet.Our work is motivated by the very fact that for the TrES-2planet, transit time variations have been reported by O’Donova et al.(2006), Rätz et al. (2008a) and Rätz et al. (2008b).2 Numerical SetupThe majority of planetary systems consist of a star and agiant planet orbiting in an eccentric orbit around it. TrES-2 is such a single-planet system and can be modeled bythe elliptic restricted three-body problem, a relatively sim-ple dynamical model, where two massive bodies, called theprimaries, move in elliptic orbits around their common cen-ter of mass, and a third body of negligible mass moves un-d r their gravitational influence (for details see Szebehely(1967)). For the computation of the stability maps we variedboth the orbital elements of the giant and the test particlesand for more than 350 thousands initial conditions the dy-namical character of the massless particle representing anearth-like planet was determined. To compute the stabilitymaps, the method of the maximum eccentricity (ME) andthe Lyapunov characteristic indicator (LCI) were used asmain tools for stability investigations of massless terrest ial-like planets:– To indicate stability, the ME method uses a straightfor-ward check based on the eccentricity. This action-likevariable shows the probability of orbital crossing andclose encounter of two bodies and therefore its valuec© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim790 F. Freistetter et al.: Dynamics of the TrES-2 systemprovides information on the stability of orbits. This sim-ple check was already used in several stability investi-gations, and was found to be a powerful indicator of thestability character of orbits (Dvorak et al. 2003; Nagy et al.2006; Süli et al. 2005).– As a complementary tool, we computed also the LCI, awell-known chaos indicator. The LCI is the finite timeapproximation of the largest Lyapunov exponent (LCE),which is described in detail in e.g. Froeschle (1984) andLohinger et al. (1993).The mass ofµ of the TrES-2b system is approximately0.0022097 solar masses which can be rounded to 0.002.The time span of the integration was105 revolutions of theprimaries. For the integration of the system we applied anefficient variable-time-step algorithm, known as Bulirsch-Stoer integration method. The most important feature of theBulirsch-Stoer algorithm for N-body simulations is that itis capable of keeping an upper bound on the local errorsintroduced due to taking finite time-steps by adaptively re-ducing the step size when interactions between the particlesincrease in strength. The parameterǫ which controls the ac-curacy of the integration was determined by preliminary testruns and was set to10−8.The giant planet’s orbital elements are: semi-major axisa1, eccentricitye1, argument of periastronω1, and meananomalyM1. The semi-major axis of the giant isa1 =0.0367 AU, and its eccentricity was 0. Since the giant re-volves in a circular orbit,ω1 is undefined and the meananomaly alone determines the position of the giant alongthe orbit.M1 was varied between0◦ and180◦ with ∆M1 =45◦.The semi-major axis of the test particles ranges from0.014 to 0.147 AU for stability maps in thea− i plane andup to 0.183 AU for stability maps in thea − e plane. Theinner border for the test particles is very close to the star.Al-though earth-mass planets could exist here, due to the largetemperatures they would certainly not be earth-like planets.The step-size∆a = 1.8·10−4 AU, ∆e = 0.02 and∆i = 1◦.The upper limit of the grid ine is 0.5 and50◦ in i.To investigate the resonances in detail we also appliedthe restricted three body problem. In this case the integra-tion of the Newtonian equations of motion was undertakenwith a different method. We used the so calledLie inte-gration (Hanslmeier & Dvorak 1984; Lichtenegger 1984)),which implements an automatic step-size and is, because ofthe recurrence of the Lie-terms, a very precise integrationmethod. Again the integration time was105 revolutions ofthe primaries. For the calculations we took into account theresonances up to the 7:4 mean motion resonance for the in-ner and the outer region. For the motion in resonances theinitial starting position is quite important, therefore wehavechosen 8 starting positions for the fictitious planets in a cer-tain resonance with the mean anomaly M ranging from0◦to 315◦ with ∆M = 45◦.3 Stability RegionsFigure 1 shows the stability plots for thea− e plane. We in-vestigated the small region between the star and the knownplanet and the region outside the orbit of TrES-2b up tosemimajor-axes of 0.18 AU. The first 5 pictures show thestability region for different initial values of the mean ano-maly of TrES-2b whereas for the last picture we took theglobal maximum eccentricity that was obtained for any valueof M1.The overall shape of the stability is of course not influencedby the mean anomaly; it looks the same in all cases. Onecan see that the border of the inner region between the starand TrES-2b is almost independent of the initial eccentric-ity of the test particle. For initial semimajor axes smallerthan 0.025 and initial eccentricities smaller than 0.25 thetest particles stay stable on moderate eccentric orbits withe < 0.25. Particles with larger initial eccentricities can stillstay on stable (eccentric) orbits further away from the planet(a < 0.02 AU).The border of the stability region outside the orbit of TrES-2b shows a different behavior. For almost circular orbits thewhole range between 0.05 an 0.18 AU shows stable mo-tion. This region however is shrinking when the initial ec-centricity is increased. The gravitational influence of TrES-2b reaches out to 0.12 AU; from this point on the stabilityregion is independent from the initial eccentricity.The influence ofmean-motion resonances with TrES-2b is also visible in the figures. Depending on the initialvalue ofM1, one can observe additional stable regions cor-responding to the position of the resonances. They are moststable forM1 = 90◦; an initial value of 45◦ shows fewerstable resonances; for 0◦, 135◦ and 180◦ almost all reso-nances become unstable (for a detailed discussion of thestability of resonances see section 4).The orbital inclination of a planet plays an importantrole when one wants to observe a transit. We know that -as viewed from the Earth - TrES-2b passes the visible diskof the star. Normally one would expect any additional plan-ets to orbit almost in the same plane as TrES-2b. But fromour solar system we know that small objects like asteroidscan have substantial orbital inclinations and even the dwarf-planet Pluto has an orbit that is inclined about 17◦ towardsthe ecliptic. Knowledge about the possible inclinations thatadditional planets in the TrES-2 system can have is thus cru-cial for the search of transits.We thus also investigated the stability regions in thea − iplane; the results can be seen in figure 2. For different ini-tial values of the eccentricity we plotted the maximum ec-centricity in the region between 0.014 and 0.147 AU in thesemimajor axis and 0◦ and 50◦ in the inclination.Figure 2 shows clearly that the borders of the stability re-gions behave almost indepently from the initial inclination.For an initial eccentricity of 0.01 the whole region exteriorto TrES-2b between 0.05 and 0.147 AU shows ordered mo-tion on circular orbits for all initial values of the inclination(with the exception of the 1:2 and 1:3 resonance). The in-c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.orgAstron. Nachr. / AN (2006) 791Fig. 1 Stability plots in thea − e plane for different initial values ofM showing the maximum eccentricity (ME).From top left to bottom left: M=0◦, M=45◦, M=90◦, M=135◦, M=180◦. Bottom right: combination of all figures using themaximum ME over all values of M.terior region is also stable up to 0.03 AU although here theorbits become more eccentric for initial inclinations of 30◦and larger (with the exception of the 2:1 resonance) wherethe test particles can maintain almost circular orbits evenfor larger inclinations. If the initial eccentricity is increased,the maximum eccentricity increases for all initial values ofa and i. The border of the exterior stability region movesinward with increasing initiale from 0.05 AU to≈ 0.115AU for e = 0.5 (again, at some mean motion resonancesthe orbits can remain stable).Figure 3 shows, how the borders of the stability regiondepend on the initial eccentricity. The change of the in-ner border with the initial eccentricity is almost constantwhereas the outer border changes linearly (with a slope ofapproximately0.1). The difference betweeni = 0◦ andi = 40◦ is very small and only significant for small initialeccentricities.4 Stability of ResonancesResonances play an important role concerning the possi-bility of additional planets. On the one hand, a resonantconfiguration can be more unstable and thus make it pos-sible to exclude larger planets (see e.g. Miller-Ricci et al.(2008a,b)). On the other hand, resonances can stabilize themotion of planets under certain conditions and we knowabout planetary systems that are in a mean-motion reso-nance (see e.g. Ferraz-Mello et al. (2005) and references therein).We thus decided to perform a detailed study of the sta-bility inside the mean motion resonances. The stability plotswww.an-journal.org c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim792 F. Freistetter et al.: Dynamics of the TrES-2 systemFig. 2 Stability plots in thea − i plane for different initial values ofe showing the maximum eccentricity (ME). Fromtop left to bottom right: e=0.01, e=0.1, e=0.2, e=0.3, e=0.4, e=0.5.presented in section 3 give an overview on the stability prop-erties - but to decide whether a resonance is really stable ornot, one has to use special initial conditions. In this simu-lation the test particles were placed directly inside the res-onance. Since the initial mean anomaly can have an impor-tant influence on the stability of a resonance (see figure 1)we performed calculations for 8 initial values ofM of thetest particle between 0◦ and 315◦.Figure 4 shows the results of this study. We investigatedall major inner and outer resonances with the known planet.For the outer resonances (figure 4 top) we can see, that allresonances between 0.107 AU (1:5) and 0.053 AU (4:7) arestable for all values ofM . From 0.044 AU (3:4) to 0.041(5:6) the resonances are unstable for all values ofM . Inbetween lie the 3:5 and 2:3 resonance where the stabilitydepends on the mean anomaly.The inner resonances (figure 4 bottom) show a more orderedbehavior: from 0.011 AU (6:1) to 0.026 AU (5:3) all reso-nances are stable; the resonances close to the planet, from0.028 AU (3:2) to 0.032 AU (5:4) are unstable (with the ex-ception of the 3:2 resonance forM = 0◦).5 ConclusionsWe investigated the dynamical stability of the TrES-2 plane-tary system in order to identify regions where an additionalsmaller planet could exist. Our results can be summarizedas follows:– There is broad range of possible orbits for additionalplanets on almost circular orbits. Such a planet couldalso exist inside the orbit of TrES-2b. Only the regionc© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.orgAstron. Nachr. / AN (2006) 793 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4 stability border (AU)initial eccentricityinner border, i=0 deginner border, i=40 degouter border, i=40 degouter border, i=0 degFig. 3 Borders of the stability region and their dependence on the initial eccentricity. The borders are shown for i=0◦ andi=40◦.between approximately0.03 and0.05 AU is excludeddue to the gravitational perturbations of TrES-2b.– Additional planets can also exist on moderately eccen-tric orbits. Here, the excluded region increases with in-creasing initial eccentricity. Planets with eccentric orbits(e > 0.4) can only exist at a distance from the star largerthan 0.095 AU.– Concerning the stability, a possible inclination of the or-bit of an additional planets plays a less important rolethan its eccentricity. Planets on almost circular orbitsand a semimajor axis larger than 0.05 AU can have anorbital inclination up to 50◦ without becoming dynam-ically unstable; planets inside the orbit of TrES-2b canbe stable up to 35◦.– Many of the major mean-motion resonances in the sys-tem are stable. Only close to the existing planet, theycan destabilize the motion of a planet. It is thus possiblefor planets to exist in a resonant configuration.According to our results, the possibility to detect an ad-ditional planet in the TrES-2 system are good - at least froma dynamical point of view. There exists a wide range of re-gions where such a planet could exist on a stable orbit. Ourresults also showed that additional planets can have a sub-stantial orbital inclination. If this is the case, a search withthe transit technique would fail, but a detection by the radialvelocity technique would be possible. In this case precisemeasurements of the transit time of TrES-2b could be usedto help identify such a planet.Future works will be dedicated to determine the limits of themasses the additional planets can have. It will then be pos-sible to make much better predictions for the observabilityof possible transits.ReferencesDvorak, R., Pilat-Lohinger, E., Funk, B., & Freistetter, F.2003, A&A, 398, L1Ferraz-Mello, S., Michtchenko, T. A., Beaugé, C., & Calle-gari, Jr., N. 2005, in LNP Vol. 683: Chaos and Stabilityin Planetary Systems, 219–+Ford, E. B. & Gaudi, B. S. 2006, ApJ, 652, L137Ford, E. B. & Holman, M. J. 2007, ApJ, 664, L51Froeschle, C. 1984, Celestial Mechanics, 34, 95Hanslmeier, A. & Dvorak, R. 1984, A&A, 132, 203Lichtenegger, H. 1984, Celestial Mechanics, 34, 357Lohinger, E., Froeschle, C., & Dvorak, R. 1993, CelestialMechanics and Dynamical Astronomy, 56, 315Miller-Ricci, E., Rowe, J. F., Sasselov, D., et al. 2008a, ApJ,682, 586Miller-Ricci, E., Rowe, J. F., Sasselov, D., et al. 2008b, ApJ,682, 593Nagy, I., Süli,Á., & Érdi, B. 2006, MNRAS, 370, L19O’Donovan, F. T., Charbonneau, D., Mandushev, G., et al.2006, ApJ, 651, L61Rätz, S., Mugrauer, M., & Schmidt, T. 2008a, in Proceed-ings of the International Astronomical Union, IAU Sym-posium, Vol. 253Rätz, S., Mugrauer, M., & Schmidt, T. 2008b, Astronomis-che Nachrichten, submittedSüli, Á., Dvorak, R., & Freistetter, F. 2005, MNRAS, 363,241Szebehely, V. 1967, Theory of orbits. The restricted prob-lem of three bodies (New York: Academic Press)Winn, J. N., Asher Johnson, J., Narita, N., et al. 2008, ArXive-prints, 0804.2259www.an-journal.org c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim794 F. Freistetter et al.: Dynamics of the TrES-2 systemFig. 4 Stability of Resonances. The figure shows the stability of the resonances for different initial values ofM . Theupper figure shows the exterior resonances with TrES-2b; thelow r figure the interior resonances. White points indicatestable motion; black points show unstable motion.c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org

Dynamics of the TrES-2 system

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Dynamics of the TrES-2 system

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