Dynamics of possible Trojan planets in binary systems

Transcription

1 Mon. Not. R. Astron. Soc. 398, (2009) doi:1111/j x Dynamics of possible Trojan planets in binary systems R. Schwarz, 1 Á. Süli 1 and R. Dvorak 2 1 Department of Astronomy, Eötvös University, H-1117 Budapest, Pázmány Péter sétány 1/A, Hungary 2 Institute for Astronomy, University of Vienna, A-1180 Vienna, Türkenschanzstrasse 17, Austria Accepted 2009 June 15. Received 2009 June 9; in original form 2009 May 25 ABSTRACT This paper is devoted to the dynamical stability of possible Trojan planets in binaries and in binary systems where one of the substellar companions is not larger than a brown dwarf. Using numerical integrations, we investigated how the size of the stable region around the Lagrangian point L 4 depends on the mass parameter and the eccentricity of the secondary star. An additional goal of this work was to create a catalogue of all possible candidates, which could be useful for future observations to detect such objects. Key words: celestial mechanics. 1 INTRODUCTION The first extrasolar planet was discovered in the early 1990s by Wolszczan & Frail (1992). Today we know about 350 extrasolar planets in more than 270 planetary systems, but only 32 are in binary systems (shown in the catalogue maintained by Jean Schneider at However, 70 per cent of all stars in the solar neighbourhood are members of binary or multiple systems: 67 per cent for GM stars (Mayor et al. 2001); 75 per cent for OB stars (Verschueren, David & Brown 1996; Mason, Gies & Hartkopf 2001), and they are very common in our Galaxy (Fabricius et al. 2002). This fact leads to the conclusion that the knowledge of the dynamics of extrasolar planets in binary systems will be very essential in the future. We distinguish three dynamical types of stable motion of extrasolar planets in binary star systems, see also Rabl & Dvorak (1988): (i) S-type: a planet orbits around one of the two stars; (ii) P-type: a planet stays in an orbit around both stars; (iii) T-type: a planet may orbit close to one of the equilibrium points L 4 and L 5 (Fig. 1). In our Solar system there exists an asteroid cloud around L 4 /L 5 in the 1:1 mean motion resonance with Jupiter. These asteroids are moving on orbits similar to that of Jupiter but they are either 60 ahead or 60 behind with respect to Jupiter. Objects found orbiting at the L 4 and L 5 points are called Trojans after the three large asteroids Agamemnon, Achilles and Hector. However, the Lagrangian points are only stable for a mass parameter μ = m 2 /(m 2 + m 1 ) < 1/25, where m 1 and m 2 are the masses of the primaries, m 2 being lower than m 1. So it seemed that binaries may not have a population of Trojan asteroids or a Trojan planet, because the masses of the stars involved are comparable. However, after a careful search in the catalogue of physical multiple stars schwarz@astro.univie.ac.at (Tokovinin 1996) in fact we found that there are a few double stars which have mass parameters below this stability limit. Out of the six binaries two candidates are well-known binary systems, their masses and orbital elements are given in Table 1. For the other four systems we only know the mass parameters. To observe Trojan planets in the future, we think that the transit method is the most effective. Therefore, the investigation of eclipsing binaries could be very promising, because the period of the binaries is very small (Schneider, private communication). There are also binary systems where the stars are far away from each other, like in HD (a 100 au). From the dynamical point of view they are also interesting, but the observation time would be far too long. There exist several dynamical investigations about Trojan planets in extrasolar planetary systems (e.g. Dvorak et al. 2004; Érdi &Sándor 2005). Theoretical studies predict that Trojans are likely a frequent by-product of planet formation and evolution. Hydrodynamic simulations of protoplanetary discs have been undertaken by Laughlin & Chambers (2002), Chiang & Lithwick (2005) and Beaugé et al. (2007). Ford & Gaudi (2006) and Ford & Holman (2007) examined the sensitivity of transit timing observations for detecting Trojan companions of extrasolar planets. They demonstrate that this method offers the potential to detect terrestrial mass Trojans using existing ground-based observatories. As mentioned before, extrasolar planets in double stars could occur more frequently as currently known. Caton, Davis & Kluttz (2000) and Davis et al. (2001) consider light curves of eclipsing binaries for possible photometric signatures of planets that may exist at the L 4 and L 5 Lagrange points of known stellar systems. They investigated 18 systems and in a few cases they found anomalies in the light curves, but Trojan planets are not confirmed yet. In this project we will study the dynamical stability of T-type planets in double stars. This paper is divided into three parts: first we discuss the stability of the Lagrangian points, the second part investigates the stability of Trojan planets in binary systems, whereas the third part is devoted to systems with substellar companions. C 2009 The Authors. Journal compilation C 2009 RAS

2 2086 R. Schwarz, A. Su li and R. Dvorak Figure 2. Stability in the Lagrangian point L4 depending on the eccentricity and the mass parameter. 53 ξ Uma is not presented, because the initial eccentricity is too high. Table 1. Orbital elements of six binary candidates, with μ < 1/25, m denotes the mass of the star expressed in solar mass units (M ). In the upper part systems with known orbital parameters, whereas in the lower part systems with unknown orbital parameters are listed. Name m (M ) μ a (au) e 51 θ Vir 51 θ Vir ξ Uma 53 ξ Uma HD SB 152 SB 667 SB T H E S TA B I L I T Y O F T H E L AG R A N G I A N POINT The first step was to analyse the stability of possible Trojan planets for different eccentricities and similar mass parameters of the formerly mentioned candidates. We started the Trojan planet exactly in the Lagrangian point L4, i.e. ω = ω1 + 60, e = e1 (shown in Fig. 1). The argument of pericentre ω is measured between the x-axis and the radius vector of the pericentre. Hereafter the orbital elements of the Trojan will be denoted without subscript, while those of the secondary with subscript 1, i.e. e1 is the notation for the binary s eccentricity. The eccentricity was varied between 0 e with a grid size of e = 0.01 and the mass parameter μ between < μ < and μ = , which corresponds to 0.5 Jupiter masses (M J ), if the star has a mass equal to the Sun. The orbits were integrated for 10 periods of the secondary. More details about the numerical setup are shown in the next section. The results for binary systems are shown in Fig. 2, where the 10-base logarithms of the Lyapunov characteristic indicator (LCI) values were plotted in the (μ e) parameter space. We can see that binary systems may lie in the stable region except 53 ξ Uma, because its eccentricity is too large. It is also visible that the stable region extends for the mass parameters μ up to C Figure 3. Stability of the Lagrangian point L4 depending on the eccentricity and the mass parameter. e = 0.04 and for 0.03 μ up to e = 0.05 (see Fig. 2, black area). This seems to be inconsistent with the theoretical work, but it should be noted that in the study of e.g. Marchal (1990) only first order stability study was performed, whereas our investigation determines a kind of effective stability obtained with long-time integrations. Our results are in good agreement with those of E rdi & Sa ndor (2005). The next step was to extend our investigation to analyse the stability of possible Trojan planets for larger eccentricities and smaller mass parameters (shown in Fig. 3), which is necessary for the systems with substellar companions. Therefore, we calculated eccentricities up to 0.7 with a grid size e = 0.01 and the mass parameter μ starting at μ = (corresponding to 0.5 M J if the central star has 1 M ) with a grid size of μ = The results are presented in Fig. 3, which includes the potential candidates given in Table 3. C 2009 RAS, MNRAS 398, The Authors. Journal compilation Figure 1. The schematic figure shows the configuration of the restricted three-body problem (RTBP) of a binary system with a possible Trojan planet in the Lagrangian point L4. a and ω are the variations with respect to the semimajor axis and the argument of pericentre.

3 Dynamics of possible Trojan planets in binary systems BINARY SYSTEMS 3.1 Numerical setup To study the structure of the phase space of the binary systems we applied the model of the planar elliptic restricted three-body problem (ER3BP). We integrated the dimensionless (the semimajor axis was set to one) equations of motion. An obvious advantage of using such equations is that the results are independent of the exact value of the semimajor axis of the secondary. Let the unit of mass be the sum of the primaries, i.e. m 1 + m 2 = 1 and let the unit of time be chosen such that k 2 (m 1 + m 2 ) = 1, where k is the gravitational constant. The orbital plane of the primaries was used as the reference plane, in which the line connecting the primaries at t = 0 defines a reference x-axis. The values of ω = 0 and the mean anomaly M = 0 places the test particle on the x-axis at a distance of a(1 e) from the barycentre, where a is the semimajor axis and e is the eccentricity of the test particle. For all investigations we used the above defined ER3BP as the dynamical model. To compute the stability maps, the method of the maximum eccentricity (ME) and the LCI were used as tools for stability investigations of the massless bodies representing possible Trojan planets. The ME method uses as an indication of stability a straightforward check based on the eccentricity. This action-like variable shows the probability of orbital crossing and close encounter of two bodies and therefore its value provides information on the stability of orbits. This simple check was already used in several stability investigations, and was found to be a powerful indicator of the stability character of orbits (Dvorak et al. 2003; Süli, Dvorak & Freistetter 2005; Nagy, Süli & Érdi 2006). In this work we define ME as follows: ME = max (e). t [0,10 6 ]T C As a complementary tool, we computed also the LCI, a wellknown chaos indicator. The LCI is the finite time approximation of the largest Lyapunov exponent, which is described in detail in e.g. Lohinger, Froeschlé & Dvorak (1993). The definition of the LCI is given by LCI(t,x 0,ξ 0 ) = 1 log ξ(t), t where x 0 is the initial condition of the orbit and ξ(t) is the solution of the first-order variational equations. The function LCI (t, x 0, ξ 0 ) measures the mean rate of divergence of the orbits. The two methods are not equivalent, however, they complement one another. For example, the ME of the Earth is small, indicating stability, although we know from numerical experiments that in fact the Earth is moving on a chaotic trajectory with a small but nonzero Lyapunov exponent. Therefore, the ME detects macroscopic instability (which may even result in an escape from the system), whereas the LCI is capable of indicating microscopic instability. For the analysis of the stability we used both, the ME and the LCI. 3.2 Initial conditions As was mentioned above, we found only two binaries (see upper part of Table 1) with mass parameter less than the critical value (μ c 0.04) and with known orbital elements. Additionally, we found four binary systems with μ<μ c but without known orbital elements, namely HD , SB 152, SB 667 and SB 831 (see lower part of Table 1). For the integration of the system we applied an efficient variable time-step algorithm, the BulirschStoer integration method. The Table 2. The size of the stable region expressed by the stable area, for the different mass parameters belonging to the binary system candidates. μ e Extension Extension Stable area in a in ω ( ) A most important feature of this algorithm for simulations is that it is capable of keeping an upper bound on the local errors introduced due to taking finite time-steps by adaptively reducing the step size when interactions between the particles increase in strength. The parameter ɛ which controls the accuracy of the integration was set to The orbits were integrated for 10 6 periods of the binary. 3.3 Results We examined the size of the stable regions around the equilateral equilibrium point L 4. Thus we varied the semimajor axis a between 0.9 and 1.1 and the argument of pericentre ω between 0 and 180 with a step-size of a = and ω = 2 (see also Fig. 1). The mean anomaly was set to zero. We were especially interested in the mass parameters μ = 0.026, 0.030, (see Table 1), which we found for the binary candidates for possible Trojan planets in these systems. For these three mass parameters we calculated the orbits for three different eccentricities e = 0, and The results correspond well to those shown in Fig. 2 and are summarized in Table 2 which shows the extension of the stable area (A). To calculate the size of the stable region, we measured the extension of the semimajor axis a and the argument of pericentre ω. One argument used in the papers by Schwarz et al. (2007a,b) was that the whole stable area resembles an ellipse and therefore we used its area as stable region (the elliptic shape is also well visible in Fig. 4). The dimensionless stable area A was calculated using the formula for its area A = aωπ. Table 2 shows the stable region for the calculated masses and eccentricities. (i) For the mass parameter μ = with the three initial eccentricities mentioned above there exists a stable region around L 4. This is evident from Fig. 4, where the extensions of the stable regions are presented. Hence it follows that systems with this mass parameter, such as the systems HD , SB 667 and SB 831 (see Table 1), do have quite good prospects for hosting T-type planets. (ii) In case of the mass parameter μ = (e.g. 51 θ Vir) we found that orbits with the initial eccentricities e = and 0.05 are not stable, whereas for e = 0 the stable region is very small as shown in Table 2. (iii) The orbits for μ = with the eccentricities e = 0and are stable. The stable regions are similar for e = 0 to those of μ = 0.030, but in case of e = it is much smaller (see Table 2). We can conclude that for the binary system 53 ξ Uma the initial eccentricity is too large for stable Trojan planets and only the system 51 θ Vir can host stable Trojan planets in circular orbits. The stable zones are also influenced by secondary resonances between the libration periods of the massless body. Érdi et al. (2007) C 2009 The Authors. Journal compilation C 2009 RAS, MNRAS 398,

4 2088 R. Schwarz, Á. Süli and R. Dvorak found that secondary resonances also act up to large mass parameters. The 6:5 (μ = ) and the 4:3 (μ = ) resonances have a significant effect on the size of the stable zone around L 4. It is evident that the system 51 θ Vir is very close to the 6:5 resonance and the systems 53 ξ Uma and SB 152 are not far from the 4:3 resonance shown in Fig. 2. Figure 4. Stable regions for possible Trojan planets with a mass parameter of μ = 0.03, for different initial eccentricities: e = 0 (upper graph), (middle graph) and 0.05 (lower graph). The stability maps were created using the value of log 10 (LCI); dark regions depict stable motion and light regions chaotic ones. Note that the Lagrangian point L 4 is located at ω = 60 and a = 1. 4 SYSTEMS WITH SUBSTELLAR COMPANIONS In this section we are concentrating on double stars where the central star hosts a substellar companion. This is a brown dwarf, which is larger than a planet but does not have sufficient mass to convert hydrogen into helium via nuclear fusion as stars do. That is why they are also called failed stars, which form in the same way as true stars. The upper mass for a brown dwarf is about 0.08 M. The lower mass limit is somewhat arbitrary as there is no obvious point of transition between a high-mass planet and a low-mass brown dwarf. Generally it is taken to be about 0.01 M. In the extrasolar catalogue we found 25 systems where the mass of the secondary is larger than 10 M J. However, there are only eight candidates which fulfil the dynamical conditions for possible Trojan planets. That is, from the analytical point of view it is necessary that the mass parameter of the central star and the secondary does not exceed Numerical investigations showed us that the initial eccentricity should not exceed (e.g. Schwarz et al. 2005, 2007a). The candidates are given in Table 3, where all systems have one substellar companion, except HD which hosts an additional extrasolar planet. 4.1 Numerical setup The basic dynamical model was again the ER3BP for most of the candidates in Table 3, except for HD for which we used the restricted four-body model 1 (see Table 3) and integrated the dimensionless equations of motion. The integration time was generally set to 10 5 periods of the secondary, which corresponds to approximately 1 Myr in most cases. For the close substellar companions XO 3 b, HD b and CoRoT-Exo3 b (see Deleuil et al. 2008) the integration time was 10 4 yr, because of a< (see Table 3). Again we started the Trojan planet with the same initial eccentricity as the substellar companion and varied the semimajor axis a and the argument of pericentre to measure the size of the stable area around the Lagrangian point L 4. For the analysis of the stability we used the ME method. The grid size was a = and ω = 2, in cases of the very close substellar companions, like XO 3 b, HD b and CoRoT-Exo3 b, we changed the grid size to a = and ω = 1. To take possible observational errors into account we varied the initial eccentricity e ini of the substellar companion in the following manner. Let e 2 denotes the observed values, which are given in Table 3. This value was changed by ±, resulting in e 1 = e ini and in e 3 = e ini +. The size of the stable region is expressed through the stable area A as explained in the previous section. 1 This means that the dynamical model consists of the three masses, namely the star + brown dwarf + gas giant of the system, and massless fictitious Trojan planets. C 2009 The Authors. Journal compilation C 2009 RAS, MNRAS 398,

5 Dynamics of possible Trojan planets in binary systems 2089 Table 3. Orbital elements of all binaries with substellar candidates, sorted by increasing mass of the secondary. Name Spectral m star m planet sin (i) μ a (au) e ω ( ) type (M ) (M J ) Del G6 III Del b NGC NGC b XO 3 F5 V XO 3 b HD K2 V 0.8 HD b HD K2 II 4.5 HD b HD G HD b HD c NGC 4349 No NGC 4349 No127 b CoRoT-Exo3 G0 V 1.37 CoRoT-Exo3 b a [AU] argument of pericenter [deg] e=0 e=0.08 e=8 Figure 5. Stable area around the Lagrangian point L 4 for the system 18 Del for three different initial eccentricities e 1,2,3 (e 2 = 0.08). 4.2 Results We decided to depict our results through Figs 58, which present the structure of the stable region for the different candidates with different initial eccentricities (e 1,2,3 ). These figures give a good overview of the systems that can host a Trojan planet. In Table 4 we show the size of the stable region for every system. (a) Systems with unstable Lagrangian points It is well visible that the candidates HD , HD and XO 3 are not in the stable region (Fig. 3, white region), because the eccentricity is too large, even for all initial eccentricities including possible observational errors. (b) Systems with stable Lagrangian points (i) CoRoT-Exo3 b has a small stable region, if the secondary is on a circular orbit e 2 (see Table 4). That could be caused by the former mentioned secondary resonance shown in the work of Érdi et al. (2007). The actual value of the system CoRoT-Exo3 b is very close to μ = , where the 2:1 resonance is acting a [AU] argument of pericenter[deg] e=1 e=1 e=1 Figure 6. Stable area around the Lagrangian point L 4 for the system NGC for three different initial eccentricities e 1,2,3 (e 2 = 1). (ii) The remaining candidates 18 Del (Fig. 5), NGC (Fig. 6), HD (Fig. 7) and NGC 4349 No127 (Fig. 8) have stable regions for all initial eccentricities (e 1,2,3 ). The system 18 Del shows the largest stable region (see Fig. 5 and Table 4) but the initial eccentricity shows large fluctuations of ME for the initial e 3.The systems NGC (Fig. 6) and NGC 4349 No127 (Fig. 8) also have quite large stable regions. (iii) Most interesting is that NGC 4349 No127 has a similar size of the stable area for e 2,3 (visible in Table 4). The results presented for HD (Fig. 7) show a smaller stable area for e 3 (Table 4) and large fluctuations of ME for all initial eccentricities. 5 CONCLUSIONS This paper is dedicated to the dynamical stability of possible Trojan planets in binary systems or a star with a substellar companion. The theory of the ER3BP shows that the equilateral equilibrium points (L 4 /L 5 ) are only stable for a mass parameter of the primary bodies μ<0.04. Therefore, it seemed at first view that binaries cannot have C 2009 The Authors. Journal compilation C 2009 RAS, MNRAS 398,

6 2090 R. Schwarz, Á. Süli and R. Dvorak a [AU] argument of pericenter [deg] e=8 e=8 e=8 Figure 7. Stable area around the Lagrangian point L 4 for the system HD for three different initial eccentricities e 1,2,3 (e 2 = 8) a [AU] argument of pericenter [deg] e=0.09 e=9 e=9 Figure 8. Stable area around the Lagrangian point L 4 for the system NGC 4349 No127 for three different initial eccentricities e 1,2,3 (e 2 = 9). Table 4. The size of the stable region expressed by the stable area, for the different systems. Name + e ini Extension Extension Stable area of a of ω ( ) (au 2 ) 18 Del e e e NGC e e e X0 3 e HD e e e NGC 4349 No127 e e e CoRoT-Exo3 e a population of Trojan asteroids or planets. Nevertheless, we found six systems in the catalogue of physical multiple stars (Tokovinin 1996) which have mass parameters below this stability limit. Our examination showed that three of them have relatively large stable regions, namely HD , SB 667 and SB 831. In these cases, the probability of a Trojan planet is quite large (shown in Table 2). In the case of the substellar companions we found 25 systems with brown dwarfs. 18 systems fulfil the stability limit of μ<0.04, but only eight candidates have promising initial conditions for stable Trojan planets (shown in Table 3). The systems 18 Del, NGC 24233, HD and NGC 4349 No127 have the best prospects of Trojan planets for all initial eccentricities (e 1,2,3 ). We compiled a catalogue of all possible candidates that, due to their stability properties, can host Trojan terrestrial planets which can serve as a guideline for future observations. Finally, we need to say that our investigation was confined to the plane case. Therefore, the possible inclinations of the Trojan orbits will be the subject of a further study of this problem. ACKNOWLEDGMENTS We thank P. Klagyivik for helpful suggestions and M. Lendl and P. Reegen for their comments to improve the presentation of the paper. RS did this work in the framework of the MOEL grant of the ÖFG (Project MOEL 309). ÁS acknowledges support from the Hungarian Scientific Research Fund, grant no. OTKA A017/09, and RD wishes to thank the FWF project P18930 for their support. REFERENCES BeaugéC.,Sándor Z., Érdi B., Süli Á., 2007, A&A, 463, 359 Caton D. B., Davis S. A., Kluttz K. A., 2000, Am. Astron. Soc. Meeting, 32, 1416 Chiang E. I., Lithwick Y., 2005, ApJ, 628, 520 Davis S. A., Caton D. B., Kluttz K. A., Wohlman K. D., Stamilio R. J., Hix K. B., 2001, Am. Astron. Soc. Meeting, 33, 1303 Deleuil M. et al., 2008, A&A, 491, 889 Dvorak R., Pilat-Lohinger E., Funk B., Freistetter F., 2003, A&A, 398, L1 Dvorak R., Pilat-Lohinger E., Schwarz R., Freistetter F., 2004, A&A, 426, L37 Érdi B., Sándor Z., 2005, Celest. Mech. Dyn. Astron., 92, 113 Érdi B., Nagy I., Sándor Z., Süli Á., Fröhlich G., 2007, MNRAS, 381, 33 Fabricius C., Hog E., Makarov V. V., Mason B. D., Wycoff G. L., Urban S. E., 2002, A&A, 384, 180 Ford E. B., Gaudi B. S., 2006, ApJ, 652, 137 Ford E. B., Holman M. J., 2007, ApJ, 664, 51 Laughlin G., Chambers J. E., 2002, AJ, 124, 592 Lohinger E., Froeschlé C., Dvorak R., 1993, Celest. Mech. Dyn. Astron., 56, 315 Marchal C., 1990, The Three-Body Problem. Elsevier, Amsterdam, p. 49 Mason B. D., Gies D. R., Hartkopf W. I., 2001, ASSL, 264, 37 Mayor M., Udry S., Halbwachs J.-L., Arenou F., 2001, in Zinnecker H., Mathieu R. D., eds, IAU Symp. 200, The Formation of Binary Stars. Astron. Soc. Pac., San Francisco, p. 45 Nagy I., Süli Á., Érdi B., 2006, MNRAS, 370, L19 Rabl G., Dvorak R., 1988, A&A, 191, 385 Schwarz R., Pilat-Lohinger E., Dvorak R., Érdi B., Sándor Z., 2005, Astrobiology, 5, 579 Schwarz R., Dvorak R., Süli Á., Érdi B., 2007a, A&A, 474, 1023 Schwarz R., Dvorak R., Süli Á., Érdi B., 2007b, Astron. Nachr., 328, 785 Süli Á., Dvorak R., Freistetter F., 2005, MNRAS, 363, 241 Tokovinin A. A., 1996, A&A, 124, 75 Verschueren W., David M., Brown A. G. A., 1996, in Milone E. F., Mermilliod J. C., eds, ASP Conf. Ser. Vol. 90, The Origins, Evolution, and Destinies of Binary Stars in Clusters. Astron. Soc. Pac., San Francisco, p. 131 Wolszczan A., Frail D., 1992, Nat, 355, 155 This paper has been typeset from a TEX/LATEX file prepared by the author. C 2009 The Authors. Journal compilation C 2009 RAS, MNRAS 398,