Astron. Nachr. / AN 328, No.8, 785–788 (2007) / DOI 10.1002/asna.200710789

Stability of fictitious in extrasolar systems

R. Schwarz2,⋆, R. Dvorak1, A.´ S¨uli2, and B. Erdi´ 2

1 Institute for Astronomy, University of Vienna, T¨urkenschanzstr. 17, A-1180 Wien, Austria 2 Lor´and E¨otv¨os University, Department of Astronomy, P´azm´any P´eter s´et´any 1/A, Budapest H-1117, Hungary

Received 2007 May 24, accepted 2007 May 29 Published online 2007 Sep 18

Key words celestial mechanics, stellar dynamics – planetary systems Our work deals with the dynamical possibility that in extrasolar planetary systems a terrestrial may have stable in a 1:1 mean motion resonance with a Jovian like planet. We studied the motion of fictitious Trojans around the Lagrangian points L4/L5 and checked the stability and/or chaoticity of their motion with the aid of the Lyapunov Indicators and the maximum eccentricity. The computations were carried out using the dynamical model of the elliptic restricted three-body problem that consists of a central , a gas giant moving in the habitable zone, and a massless . We found 3 new systems where the gas giant lies in the habitable zone, namely HD 99109, HD 101930, and HD 33564. Additionally we investigated all known extrasolar planetary systems where the giant planet lies partly or fully in the habitable zone. The results show that the orbits around the Lagrangian points L4/L5 of all investigated systems are stable for long times (107 revolutions).

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1 Introduction also depends on astrophysical properties (e.g. Lammer et al. 2003). Since Lagrange (1772) found the equilateral solutions of The HZ is defined as a region around a star, where the the restricted three-body problem, for a long time it was radiation for a terrestrial like planet is such that the planet believed that the solutions are only of theoretical interest. can maintain liquid on the surface and a stable atmo- But a century later, in 1906 February Max Wolf (1863– sphere. When we study the dynamical stability of extrasolar 1932) discovered the first Trojan asteroid and named it planets, we can classify 4 possible types of terrestrial plan- Achilles after one of the mythical heroes of Homer’s Il- ets in single planetary systems: iad. Today, we can speak of a Trojan cloud, because – Type 1: a gas giant (GG) is very close to the central star there are more than 2100 known objects around the equi- thus a terrestrial planet (TP) could exist with a stable L L librium points 4 and 5, in 1:1 mean motion reso- in the HZ. nance with . Since 1990 we know about Martian – Type 2: when a GG moves far away from the central Trojans, and since 2002 we also know about Trojans star (like Jupiter), then stable TPs can exist (moving in of Neptune (see also the Minor Planet Center http://cfa- the HZ) inside the orbit of the giant planet. www.harvard.edu/iau/lists/Trojans.html). Trojans play also – Type 3: when a GG itself moves in the HZ, a terrestrial- an important role in the discussions of formation theories of like satellite (like e.g. Europa in the system of Jupiter) our and other planetary systems. The first ex- could have a stable orbit (see also Domingos, Winter & trasolar planet was found in the early 1990’s by Wolszczan Yokoyama 2006). & Frail (l992) very close to a pulsar star, but the first ex- – Type 4: when a GG itself moves in the HZ, a Trojan-like trasolar planet around a star was discovered TP may move in a stable orbit around the Lagrangian by Mayor & Queloz (1995). Since that time we had obser- equilibrium points L4 or L5. vational evidence of more than 230 planets in more than 190 extrasolar planetary systems (EPS’s), but only for plan- In this article we deal with the Trojan type configuration ets from 7.5 M⊕ ( d, Rivera et al. 2005) up to (Type 4), which we know from the Trojans clouds close to several Jupiter-. 1 M⊕ is equivalent to 1 -. the equilateral equilibrium points of Jupiter. In Table 1 we When we search for life in EPSs we have to include climate show a list of possible EPSs, where the GG has an orbit with models (see Kasting, Whitmire & Reynolds 1993 and von low eccentricity (e< 0.25, with the exception of HD33564) Bloh et al. 2003) which estimate the width of the habitable and moves inside the HZ. The configurations Type 1 and zone (HZ). We emphasize that the habitability of a planet 2 were subject for a lot of investigations (e.g. Erdi´ & P´al 2003; P´al & S´andor 2003; Dvorak et al. 2003a, 2003b; Pilat- ⋆ Corresponding author: [email protected] Lohinger et al. 2006; Menou & Tabachnik 2003). There are a few dynamical studies about fictitious Trojan planets in

c 2007 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim 786 R. Schwarz et al.: Stability of fictitious Trojan planets in extrasolar systems

EPSs like e.g., Nauenberg (2002), S´andor & Erdi´ (2003), Efthymiopoulos & S´andor (2005), Erdi´ & S´andor (2005), Schwarz et al. (2005), Schwarz (2005), and Schwarz et al. (2007a), all of them are aiming at to find the extension of the stable regions around the equilateral equilibrium points L4 and/or L5. One can imagine a possible formation of a planet in the 1:1 mean motion resonance as a result of an interaction with the protoplanetary disc (see Laughlin & Chambers 2002; Beaug´e2007), or as a by-product of planet formation and evolution (see Thommes 2005; Cresswell & Nelson 2006). Recently, Erdi´ et al. (2007) used the method of the relative Lyapunov indicators (RLI) for compiling a catalogue showing the stable regions, where they also used massive Trojans for their computations. In our study, we did not take into account the mass of the TP, because it is quite small compared to a Jupiter-like planet, and recent investi- gations have shown that the influence of the stability region is negligibly small (see Schwarz et al. 2005a; Dvorak et al. 2004; Erdi & S´andor 2005).

2 Dynamical models and numerical setup

We know about 230 extrasolar planets in 190 planetary sys- tems, among them 15 binary systems and 20 multiplane- tary systems (see the catalogue maintained by Jean Schnei- der1). Only 12 single-star systems have a giant planet partly or fully in the HZ (see Table 1), where the eccentricity is < 0.25 . A small eccentricity is very important for the dy- namical stability and also for the habitability. As dynamical model, we used the elliptic restricted three-body problem (ER3BP), due to the reasons given above, that there is al- most no influence of the mass of the TP on the extension of the stability zone. Therefore, we did not use the available N-body integrators, but wrote our own code on the basis of Fig. 1 Stability map of the system HD 99109 around the equi- librium point for 104 revolutions (top) and for 106 revolutions a Bulirsch-Stoer integrator, where we do not have to inte- L4 (bottom). On the x-axis we plotted the initial semi-major axis and grate the orbits of the primaries. For the analysis of the sta- on the y-axis the initial synodic longitude of the Trojan planet de- bility we used the method of the maximum eccentricity and noted by λ. The LCI is given in logarithmic scale. The dark re- the Lyapunov characteristic indicators (LCI). The LCI is the gions indicate stable orbits, initial points in the light regions result finite time approximation of the maximal Lyapunov Expo- in chaotic motion. nent (LCE), which can be found in Froeschl´eet al. (1984) and Lohinger et al. (1993). For the maximum eccentricity jan planet. The grid for the stability maps in the semi- emax we checked the largest eccentricity of the TP during its major axis was from 0.9 to 1.1 AU – with a fine stepsize motion. For larger eccentricities it becomes more probable of δa =0.0025 AU – and for the synodic longitude from 0◦ that the orbits of the test particles are more chaotic (having to 180◦ (stepsize δλ =1◦). close encounters or even collisions). The integration time was up to 106 revolutions, and the two primaries were always started from the periastron. 3 Investigations and results Our calculations show that the systems HD 99109 and HD 101930 can have stable Trojan planets for long times, but The extensive numerical computations in the ER3BP were in the case of HD 33564 all orbits are chaotic. These three accomplished only in the vicinity of L4 (L5 is exactly sym- systems were not yet investigated with respect to terres- metric in the ER3BP). We varied the initial semi-major trial Trojan planets. Additionally we checked the stability in axis and the synodic longitude2 (λ) of the massless Tro- case of HD 33564 for other initial eccentricities (e = 0.34, e = 0.24 and e = 0.14) 3 and we found that stable Trojan 1 http://exoplanets.eu planets could exist only for e =0.14. 2 The synodic longitude of the test planet is the difference between the mean orbital longitudes of the Trojan planet and the GG. 3 e=0.34 is the original value published by the observers

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Table 1 List of all EPSs with one giant planet in the HZ. The list is sorted by increasing semi-major axis of the GG, the 3 new systems are written in boldface.

mass mass a e HZ Partly Name Spec. [M⊙] [Mjup] [AU] [AU] inHZ [%] HD 101930 K1V 0.74 0.30 0.30 0.11 0.30–0.64 53 HD 93083 K3V 0.70 0.37 0.48 0.14 0.28–0.60 100 HD 134987 G5V 1.05 1.58 0.78 0.24 0.75–1.40 58 HD 17051 G0V 1.03 1.94 0.91 0.24 0.70–1.30 100 HD 28185 G5 0.99 5.70 1.03 0.07 0.70–1.30 100 HD 33564 F6V 1.25 9.1 1.1 0.34 0.99–2.12 35 HD 99109 K0 0.93 0.50 1.11 0.09 0.65–1.25 100 HD 27442 K2IVa 1.20 1.28 1.18 0.07 0.93–1.80 100 HD 188015 G5IV 1.08 1.26 1.19 0.15 0.70–1.60 100 HD 114783 K0 0.92 0.99 1.20 0.10 0.65–1.25 50 HD 20367 G0 1.05 1.07 1.25 0.23 0.75–1.40 76 HD 23079 (F8)/G0V 1.10 2.61 1.65 0.10 0.85–1.60 35

Table 2 Stable regions of EPSs for two different integration- times of 104 and 107 revolutions, sorted by the initial eccentrici- e_max ties. The columns are: the name of the EPS, the initial eccentricity, 4 7 0.54 the stable region for 10 revolutions, the stable region for 10 rev- 0.52 104 107 0.5 olutions and the difference of the stable regions for and 0.48 revolutions. 0.46 0.44 4 7 0.42 Initial 10 revs. 10 revs. 0.4 0.38 System e Stable area Stable area Diff. [%] 0.96 0.97 HD 28185 0.07 0.0644 0.0587 8.8 0.98 HD 27442 0.07 0.5031 0.3967 21.1 0.99 1 HD 99109 0.09 0.3712 0.3254 12.3 semi-major axis [AU] 1.01 80 HD 23079 0.10 0.2622 0.2099 19.9 1.02 70 60 HD 114783 0.10 0.3842 0.2566 33.2 1.03 50 1.04 HD 101930 0.11 0.1621 0.0636 60.7 40 synodic longitude [deg] HD 93083 0.14 0.1053 0.0781 25.8 HD 188015 0.15 0.2023 0.1148 43.2 Fig. 2 Cuts of the system HD 20367 around the equilibrium HD 20367 0.23 0.0833 0.0576 30.9 7 point L4 for 10 revolutions. Here we determined the stability with HD 134987 0.24 0.0299 0.0246 17.9 emax; high eccentricities (> 0.5) indicate chaotic motion. It turned HD 17051 0.24 0.0321 0.0321 0.0 out that in continuing the computation every orbit with e > 0.5 HD 33564 0.34 0.0000 0.0000 0.0 suffered from close encounters to the large planet sooner or later. mean value 22.3

In a next step we compared the results for the system HD 99109 for 104 and 106 revolutions of the primaries to direction of the semi-major axis and in the direction of the see how large is the difference in the respective size of the synodic longitude. The stability was determined with the stability region. help of emax and the LCI. A stability region (the dark egg- The results are shown in Fig. 1 upper and lower graph. shaped form in Fig. 1) can be approximated by an ellipse to We can see that the stable region is very large and has an determine their size of the stable region. Therefore, the two elongated shape. Fig. 1 (upper graph) shows a ring like cuts were taken as the main axes (semi-major and minor structure (an elongated white strip on the edge of the stable axis) of this ellipse and were used to compute the stable zone) which disappears after 106 revolutions in Fig. 1 (up- area. per graph). The difference between upper and lower graph Table 2 shows the size of the stability regions for 104 in Fig. 1 is visible but not very large, because the core of the and 107 revolutions and their difference in percent. The sys- stable region survives. With that conclusion we started new tems HD 27442, HD 99109, HD 23079, and HD 114783 7 computations up to 10 revolutions. The results are shown have the largest stable region (Table 2). We can conclude in Table 2 and for the system HD 20367 in Fig. 2. that the mean value of the difference is smaller than 23% To measure the size of the stable region we cut it exactly (the maximum value is about 60%) and we can see two through the Lagrangian point L4 in two directions: in the maxima at the initial eccentricities 0.11 and 0.15. It is also

www.an-journal.org c 2007 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim 788 R. Schwarz et al.: Stability of fictitious Trojan planets in extrasolar systems visible that for higher eccentricities the size of the stabil- in the framework of a Schr¨odinger grant of the FWF (J2619-N16). ity region differs less for the two timescales, but we have to Thanks also go to the ’Wissenschaftlich-technische Zusammenar- check this in future with more examples. Finally, we want beit Osterreich-Ungarn’¨ project A12-2004: Dynamics of extraso- to remark that the size of the stable region depends also on lar planetary Systems. the mass ratio of the primaries and on the influence of sec- ondary resonances (see in the case of our Solar system see References Robutel, Gabern & Jorba 2005). Beaug´e, C., S´andor, Zs., Erdi,´ B., S¨uli, A.:´ 2007, A&A 463, 359 von Bloh, W., Cuntz, M., Franck, S., Bounama, C.: 2003, AsBio 4 Conclusions 3, 681 Cresswell, P., Nelson, R.P.: 2006, A&A 450, 833 In our work we investigated fictitious Trojan planets in 12 Domingos, R.C., Winter, O.C., Yokoyama, T.: 2006, MNRAS 373, single planetary extrasolar systems, by using the ER3BP as 1227 dynamical model. All these systems have one GG moving Dvorak, R., Pilat-Lohinger, E., Schwarz, R., Freistetter, F.: 2004, in the HZ, with low eccentricities ( e < 0.25). We found A&A 426, 37 Dvorak, R., Pilat-Lohinger, E., Funk, B., Freistetter, F.: 2003a, three new systems; namely HD 99109, HD 101930, and HD A&A 398, L1 33564, where we checked also the size of the stability re- Dvorak, R., Pilat-Lohinger, E., Funk, B., Freistetter, F.: 2003b, gion. Table 2 shows that there is no stable region around L4 A&A 410, L13 in the system HD 33564 due to the high mass-ratio and the Efthymiopoulos, C., S´andor, Zs.: 2005, MNRAS 364, 253 eccentricity of the published orbital elements. But we found Erdi,´ B., P´al, A.: 2003, in: F. Freistetter, R. Dvorak, B. Erdi´ (eds.), that HD 33564 is stable for an initial eccentricity of 0.14. Celestial Mechanics, p. 3 ´ HD 101930 has a small stable region (see Table 2), while Erdi, B., S´andor, Zs.: 2005, CeMDA 92, 113 ´ ´ HD 99109 has a very large stable region with an elliptic Erdi, B., Fr¨ohlich, G., Nagy, I., S´andor, Zs.: 2007, in: A. S¨uli, F. Freistetter, A. P´al (eds.), Celestial Mechanics, p. 85 shape (see Fig. 1 upper and lower graph). Froeschl´e, C.: 1984, CeMDA 34, 95 By using the cuts, we estimated the stable region for all Jones, B.W., Sleep, P.N.: 2002, A&A 393, 1015 systems. The results reveal that only one system HD 33564 Kasting, J.F., Whitmire, D.P., Reynolds, R.T.: 1993, Icar 101, 108 is not stable, whereas the systems HD 27442, HD 99109, Lammer, H., Kulikov, Yu., Penz, T., Leitner, M., Biernat, H.K., and HD 114783 have the largest stable region. Another in- Erkaev, N.V.: 2005, in: R. Dvorak, S. Ferraz-Mello (eds.), A teresting result is that when we look at the difference (of the Comparison of the Dynamical Evolution of Planetary Systems largeness of the stable region) for the different integration 92, p. 273 Laughlin, G., Chambers, J.E.: 2002, AJ 124, 592 times (104 and 107 revolutions), we found two maxima of Lohinger, E., Froeschl´e, Ch., Dvorak, R.: 1993, CMDA 56, 315 the difference at the initial eccentricities of 0.11 and 0.15 Mayor, M., Queloz, D.: 1995, Nature 378, 355 (we have to check this result in future investigations). We Menou, K., Tabachnik, S.: 2003, AJ 583, 473 can additionally conclude that this difference has a mean Nauenberg, M.: 2002, AJ 124, 2332 value of about 23 percent (maximum value at 60 %). This P´al, A., S´andor, Zs.: 2003, in: F. Freistetter, R. Dvorak, B. Erdi´ means that even for an integration time 1000 times longer, (eds.), Trojans and Related Topics, p. 25 less than a quarter of the stable orbits of the short-term inte- Pilat-Lohinger, E., Suli, A., Freistetter, F., Dvorak, R., Schwarz, gration survived, these results were confirmed by Schwarz R., Funk, B.: 2006, European Planetary Science Congress, p. 717 et al. (2007b). Although we determine the stable area by Rivera, E.J., Lissauer, J.J., Butler, R.P., et al.: 2005, ApJ 634, 625 using the results of integrations for 107 revolutions, an inte- 4 Robutel, P., Gabern, F., Jorba, A.: 2005, CeMDA 92, 53 gration time of 0.1% (10 revolutions) gives already a good S´andor, Zs., Erdi,´ B.: 2003, CeMDA 86, 301 first approximation. When we think of the errors of the de- Schwarz, R., Pilat-Lohinger, E., Dvorak, R., Erdi,´ B., S´andor, Zs.: termination of the orbital elements from the observations, 2005, AsBio 5, 579 a mean error of 23% for this estimate is satisfying. Much Schwarz, R.: 2005, PHD thesis, University of Vienna, online work have to be done to check the accuracy of the deter- database: http://media.obvsg.at/dissdb ´ ´ mined stable areas by using the cuts. Schwarz, R., Dvorak, R., Pilat Lohinger, E., S¨uli, A., Erdi, B.: 2007a, A&A 462, 1165 Acknowledgements. For the realization of this study we thank the Schwarz, R., Dvorak, R., S¨uli, A.,´ Erdi,´ B.: 2007b, A&A, in prepa- Austrian Science Foundation (=FWF, project P16024-N05) and ration the ISSI Institute in Bern, which supported us in the framework of Thommes, E.W.: 2005, ApJ 626, 1033 the ISSI team “Evolution of habitable planets”. RS did this work Wloszczan, A., Frail, D.: 1992, Nature 355, 145

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