Eclipse Timing Variations to Detect Possible Trojan Planets in Binary

Home , Double Star (satellite), HD 162020, HD 202206

arXiv:1608.00784v1 [astro-ph.EP] 2 Aug 2016 2 1 ⋆ cetd18 eebr1.Rcie 98Dcme 4 nori in 14; December 1988 Received 15. December 1988 Accepted .Schwarz R. planet Trojan systems possible binary detect in to variations timing Eclipse o.Nt .Ato.Soc. Astron. R. Not. Mon. es %frqarpl ytm,1 o systems of sys- for found triple 1% 8% systems, was stars, quadrupole binary population for 33% 4% field stars, tems, single A distance-limited for targets. 54% a about dwarf 4847 with solar-type of (2014) Tokovinin of sample by Statistics studied (2001)). were al. et Mayor stars, 0aei utpesa ytm.Tedt falplanets all J. of by maintained data Exoplanet-catalogue The and the Schneider systems. systems in multiple-star collected binary-star are in in exo- are them 1900 are exoplanets among 20 about systems, 100 know planetary exoplane- than 1200 we with more than moment compared more so- in the unique planets our At be of systems. to early architecture tary seems the the of system statistics in that lar the showed discovered Today observations (1992). was Frail the planet & Wolszczan solar by 1990s extra first The INTRODUCTION 1 a efudsprtl ntebnr aaou fexoplan- of catalogue binary the in ets separately found be can 19) ao,Ge,&Hrkp (2001); or Hartkopf double & Gies, be Mason, to (1996), confirmed ( systems are multiple percent 70 approximately c nttt o srnm,Uiest fVen,A18 Vien A-1180 Vienna, of University Astronomy, for Institute http://www.univie.ac.at/adg/schwarz/multiple.html http://exoplanet.eu E-mail:[email protected] 2 02RAS 2002 anandb .Shaz o erySnlk stars, Sun-like nearby For Schwarz. R. by maintained 1 hra h iayadmlil-trsystems multiple-star and binary the whereas ; ⋆ , ∼ .Bz´,B ukadR Zechner R. and Funk Bazsó, B. A. ´ 5 o - tr,Vrcurne al. et Verschueren stars, O-B for 75% 000 1– , ?? ytehl ftelbainamplitude libration poss the the of secondar of eccentricity inves the help the are changed of the ETVs we masses By the addition different In and planet. for stability Trojan the The binary maps using by model. of stability/ETV simulations dynamical dynamics numerical computing as did the We problem motion. investigated future tadpole body we and in planet Cheops), Kepler base a and and with ground Tess CoRoT with missions Plato, variations former such like (like of telescopes detection the space (E variations of timing probability eclipse de of the to help favourable the circumstances by the binary-star-systems study close to devoted is paper This ABSTRACT sas:iais general (stars):binaries: star – binary candidates. words: in possible Key planets of Trojan list Earth-like a amp many prepared detect that we to conclude application, can enough We large orbits. tadpole are in moving are objects 20)Pitd2 etme 08(NL (MN 2018 September 24 Printed (2002) ∼ 7 o G-M for 67% eeta ehnc ehd:nmrcl–(tr) lntr sy planetary (stars): – numerical methods: – mechanics celestial > N 4. a ¨reshnsrse1,Austria Türkenschanzstrasse 17, na, s ia om18 coe 11 October 1988 form ginal xs tbeobt eodGshasvle( value there Gascheaus that show beyond could orbits (2010) stable Sicardy and exist (2009) al. et Erdi on ra- mass rare. necessary t relatively the possibility are because the tios stars, double limits in region Trojans stability Howeverfind problem. show extended three-body also the restricted even could spatial (2013) the in al. Bazsóthat et Whereas, case. planar (1 cwr,Sui vrk(09 ol hwwt e real few a with o show study could dynamical (2009) the Dvorak Süli, Nevertheless & Schwarz, stars. like sun stricted qiiru points equilibrium star binary a in orbits planetary system: of types three distinguish slnal tbe(here stable linearly is yGsha 14)adRuh(85 htol o mass for only 0 that interval shown (1875) the been Routh in re- has and ratios circular it (1843) Gascheau the (CR3BP) by For problem interesting. three-body stricted particularly are systems ´ ii -ye lntmyobtcoet n ftetwo the of one to close orbit may binary; entire planet stars; the a two orbits T-Type: the planet of (iii) the one where orbits P-Type, planet (ii) the where S-Type, (i) − µ otosrain fpaesi iaisaefocused are binaries in planets of observations Most codn otewr fRb vrk(98 n can one (1988) Dvorak & Rabl of work the to According rmtednmclpito iwtebnr star binary the view of point dynamical the From p ≈ 23 / 0 27) σ . sashv iia ass n r re- are and masses) similar have (stars 5 ecudso hte rntalstable all not or whether show could we / A 2 T E ≈ tl l v2.2) file style X L 4 0 and . 35.Hwvr omrsuisof studies former However, 0385). 6 µ µ L = 5 6 m Toa planets). (Trojan µ 2 crit / ( m etToa lnt in planets Trojan tect 1 h irtoa motion librational the V) odetermine To TVs). + beToa planet. Trojan ible m eecpsand telescopes d iue fETVs of litudes ytm.A an As systems. tradthe and star y 2 pc missions space ,and ), trsystems star µ crit iae by tigated ulthree- full ,i the in ), µ crit stems s = o f , 2 R. Schwarz, A.´ Bazsó, B. Funk and R. Zechner

for planets in the horseshoe regime by using semi-analytical models. This paper is divided into two parts: the ﬁrst part in- vestigates the stability of Trojan planets in binary systems, whereas the second part is devoted to the possible detection of Trojan planets by the help of eclipse timing variations (ETV).

2 NUMERICAL SETUP 2.1 Models We studied the planar full three-body problem (3BP) with Figure 1. The Histogram present the mass ratio µbin of all binary numerical integrators. In this problem three finite bodies, star systems with exoplanets, taken from binary catalogue of ex- the primaries (m1 = primary star, m2 = secondary star) oplanets (http://www.univie.ac.at/adg/schwarz/multiple.html). revolve about their common center of mass, starting on circular orbits (e2 = 0), and a third body mTro = T rojan moves under their gravitational influence in the same orbit as m2 close to the equilibrium point L4 60 degree ahead of binary systems that the T-Type configuration may be not the secondary star. only of theoretical interest. Therefore we show a distribution We have regarded all the celestial bodies involved as (Fig. 1) of the mass ratio of all detected exoplanets in bi- point masses and integrated the dimensionless (the semi- naries. The most common mass ratios are 0.25 and 0.5. The major axis set to one) equations of motion for an inte- frequency of the interesting mass ratios (µcrit) for the Trojan 4 6 gration time up to Tc = 10 and 10 periods of the sec- configuration is around 3 percent. When we compare Fig. 1 7 ondary star for the stability maps and Tc = 10 periods with the statistics of the nearby Sun-like stars the value is for special cases shown in chapter 3. For our simulations we larger, it is around 10 percent (Duquennoy & Mayor 1991). used the Lie-method with an automatic step-size control to However, if one considers also systems with a brown dwarf solve the equations of motion (Hanslmeier & Dvorak 1984; as companion the number of possible candidates would be Lichtenegger 1984; Eggl & Dvorak 2010). larger (as shown in our candidate list Tab. 3), due to their substellar masses. Our work was motivated by the huge number of Tro- 2.2 Initial conditions jans which were found in the solar system around different planets (Earth, Mars, Jupiter, Uranus and Nep- The computations were accomplished only in the vicinity tune). The possibility of Trojan planets in exoplane- of L4 (L5 is symmetric in the 3BP). For the calculations of tary systems was discussed in several dynamical investi- the stability maps we varied the eccentricities of the Trojans gations like e.g. Nauenberg (2002), Laughlin & Chambers (eTro=Trojan planet) from eTro = 0.01 up to eTro = 0.5. In ´ addition, we varied the mass of the secondary object from (2002), Erdi & Sándor (2005), Dvorak et al. (2004), and −3 Schwarz et al. (2007a). Theoretical studies predict that m2 = 10 Msun (1Msun corresponds to 1 Solar mass) until Trojans are a by-product of planet formation and evolu- the end of the stability border with a map size of 235 x 40 tion, see the hydrodynamic simulations of a protoplane- values (m2 x eTro). The lower limit of m2 presents planets of tary disk, like in the work of Chiang & Lithwick (2005) and about 2 Jupiter masses as shown in our stability investiga- Beaugéet al. (2007). Whereas Pierens & Raymond (2014) tions. In case of the ETV’s we started our investigations at studied the orbital evolution of co-orbital planets embed- masses equal to substellar objects (approximately 13 Jupiter ded in a protoplanetary disc. The investigation on the pos- masses). The stability border changes because we also calcu- sibility of a migration-induced resonance locking in sys- lated the stability maps for different masses of the host star tems containing three planets, namely an Earth-like planet, m1=1, 2 and 3 Msun. For m1 = 1 Msun the stability border a super-Earth and a gas giant with one Jupiter mass lies at m2 = 0.045 Msun, for m1 = 2 Msun at m2 = 0.087 by Podlewska-Gaca & Szuszkiewicz (2014) found that the Msun and for m1 = 3 Msun at m2 = 0.130 Msun. super-Earth planet captures the Earth-like planet into co- To get a good estimation about occurring perturba- orbital motion temporarily. In case of binary systems the tions on the secondary star (to measure ETVs) we used Trojan planets with different sizes: Mars, Earth, Neptune formation or capture scenario for possible Trojan planets −3 are not yet investigated. and Jupiter, which corresponds to 10 Msun. Ford & Gaudi (2006) and Ford & Holman (2007) exam- ined the sensitivity of transit timing variations (TTVs) for 2.3 Methods a possible detection of Trojan companions. They demon- strated that this method offers the potential to detect For the analysis of the orbit we used the method of the max- terrestrial-mass Trojans using existing ground-based obser- imum eccentricity emax. In former studies we found a good vatories. Janson (2013) reported a systematic search for ex- agreement with chaos indicators (as shown in Bazsóet al. trasolar Trojan companions to 2244 known Kepler Objects, 2013) like the Lyapunov characteristic indicator (LCI), like but no Trojan candidates were found. In a theoretical work in Schwarz et al. (2007b). The emax method uses as an in- of Vokrouhlický& Nesvorný(2014), they studied the TTVs dication of stability a straightforward check based on the

c 2002 RAS, MNRAS 000, 1–?? On the detection of Trojan planets in binaries 3

maximum value of the Trojans eccentricity reached during Table 1. Number of stable orbits of the stability maps for Tc = the total integration time. If the Trojan’s orbit becomes 106 periods and for diﬀerent masses of the primary body (central parabolic (emax > 1) the system is considered to be un- star) and third body (Trojan planet) presented in the second −5 stable. The emax is deﬁned as follows: column starting with Earth mass (mTro = 0.30049 · 10 Msun), −4 Neptune mass (mTro = 0.51684 · 10 Msun), and Jupiter mass max −3 e = max(e). (1) (mTro = 0.9547 · 10 Msun). t6Tc

To analyse the orbital behaviour of the third body m3 mass of mass of number of percent of (e.g. tadpole, horseshoe, or satellite), we used the amplitude the primary the Trojan stable orbits stable orbits of libration, as done in the work of Freistetter (2006) and [Msun] body Schwarz, Funk, & Bazsó(2013). 1 Earth 734 21.839 The classification of the orbit was done by checking the 1 Neptune 729 21.690 libration amplitude σ which is defined as the difference be- 1 Jupiter 639 19.012 tween the mean longitude of the Trojan and the secondary star (λTro − λ2). λTro, λ2 are given by λTro = ̟ + M, 2 Earth 1523 22.947 λ2 = ̟2 + M2 were ̟, ̟2 are the longitudes of ascending 2 Neptune 1517 22.857 2 Jupiter 1404 21.154 node of the Trojan body and of the m2 and MTro, M2 are the mean anomaly of the third body respectively of the sec- 3 Earth 2316 23.363 ondary star. As for the emax we determined the maximum 3 Neptune 2305 23.252 libration amplitude σmax for the total integration time: 3 Jupiter 2168 21.870

σmax = max(σ). (2) t6Tc In the case of a large libration amplitude the Trojan may become a horseshoe orbit or may be ejected out of the Trojan region after a close approach with the secondary ◦ body. The value of σmax should be smaller than 180 for which is divided into 3 different graphs, showing different tadpole orbits. masses of the primary star (m1). Whereas, the upper graph As indicated by the results of the stability analysis, an depicts the stability map for m1 = 1Msun, the middle one exoplanet in a T-type orbit may be stable in the vicinity depicts m1 = 2Msun and the lower one m1 = 3Msun. We can of a binary star system. The gravitational perturbation of conclude that the number of stable orbits do not differ much, this planet may affect the motions of the two stars and cause as summarized in Tab. 1, where we also show the studies for their orbits to deviate from Keplerian. In an eclipsing binary, different masses of the Trojan planet (from Earth mass up these deviations result in variations in the time and duration to Jupiter mass). In principle the number of stable orbits do of the eclipse. not change very much for the different masses of the Tro- We determined the ETVs by calculating the perturbed jan. Nevertheless, the shape of the stable region (gaps) are case, where the planet induced constant rate of apsidal pre- changing when we change the mass of the secondary star. cession is removed by a linear fit. We also took into account The gaps are mainly caused by secondary resonances which the long-term effects caused by the binaries motion around were investigated in the work of Schwarz et al. (2012) for the system center of mass. large mass ratios and in Schwarz, Funk, & Bazsó(2013) for low mass ratios. In addition, one can see that the Trojan body can have larger initial eccentricity for larger mass of the primary star m1, this is especially well visible for small 3 STABILITY masses of m2. The maximum stable eccentricity eini = 0.245 First of all we investigated the stability of the Lagrangian is valid for m2 = 0.01 (which corresponds to a binary-like point L4 in the planar three-body problem to determine the star system with a brown dwarf) and for different masses stability limits and to investigate whether the orbits are still of central star (m1) MTro = 1 MJup. µcrit = 0.04060 for in Trojan motion or not, by the help of σmax. Our studies 1 MSun, µcrit = 0.08185 for 2 MSun and µcrit = 0.12365 showed that all stable orbits stay in tadpole motion. In addi- for 3 MSun. Also the limit of the stable region increases tion we used different masses of the primary and secondary with a more massive central star: for m1 = 1Msun the limit star, which might be useful for future observations of differ- amounts m2 = 0.041, for m1 = 2Msun the limit amounts ent stars. We computed the stability maps by changing the m2 = 0.082 and for m1 = Msun we have the largest value mass of the secondary and the eccentricity of the Trojan. m2 = 0.12. The stability limit changes because of the differ- An example is shown in Fig. 2, for the 3 different masses of ent mass ratio µ = m2/(m1 +m2) as we know from the circu- the primary star (1, 2 and 3 Msun). lar restricted three-body problem. In addition, we made long As one can see the stability maps do not start at zero (x term integrations, represented in Fig. 3 by the help of cuts and y axes in Fig. 2). That means the initial mass of the sec- of the stability maps shown in Fig. 2 (cuts for eTro = 0.035). ondary body is 0.001 Msun ∼ MJup and eTro = 0.005. How- The cuts were made for the different masses of the planets, as ever, we focus on the stability region from m2 > 10MJup, shown in the chapter of the initial conditions. We presented because lower masses (planet-like masses) of the secondary only the case of MTro = 1MNeptune, because the other cases are not interesting for binary (like) star-systems and initially are similar. The goal of these cuts was to see whether the circular orbits (eTro = 0) do not produce detectable ETV stability-island for high mass ratios is still stable for a long signals. An example of the stability maps is given in Fig. 2, time, which is the case.

c 2002 RAS, MNRAS 000, 1–?? 4 R. Schwarz, A.´ Bazs´o, B. Funk and R. Zechner

MTro = Neptune

1.0 m1 = 1 MSun 4 M1 = 1 MSun, Trojan = Jupiter 10 yrs 107 yrs 0.4 1.0 0.5 max. ecc. 0.9 0.0 0.8 0 0.01 0.02 0.03 0.04 0.3 0.7 1.0 m1 = 2 MSun 0.6 104 yrs 7 0.5 10 yrs 0.2 0.5 max. ecc. 0.4 0.0 0.3 0 0.02 0.04 0.06 0.08 initial eccentricity 0.1 0.2 maximum eccentricity 1.0 m1 = 3 MSun 0.1 104 yrs 107 yrs 0.005 0.0 0.5 max. ecc. 0.001 0.01 0.02 0.03 0.04 0.0 mass secondary [M ] 0 0.03 0.06 0.09 0.12 Sun mass secondary [MSun]

Figure 3. Cuts of the stability maps for a mass of the Trojan

M1 = 2 MSun, Trojan = Jupiter body of MTro = 1MNeptune for low eccentricity (eTro = 0.05 and 4 7 0.4 1.0 an integration time Tc = 10 compared with Tc = 10 periods. 0.9 0.8 0.3 0.7 0.6 Holman & Murray (2005) and Agol & Steffen (2007). The 0.2 0.5 feasibility of the detection of extrasolar planets by the 0.4 partial occultation on eclipsing binaries was investigated 0.3 initial eccentricity 0.1 by Schneider & Chevreton (1990). Therefore, we set our 0.2 maximum eccentricity goal to show which planet sizes for the T-Type configu- 0.1 0.005 0.0 ration are detectable in the ETV signal of the secondary star with current observational equipment. In order to ap- 0.001 0.02 0.04 0.06 0.08 mass secondary [M ] proximate the detectability of possible extrasolar planets Sun in T-type motion by means of ETVs we used the work of Sybilsky, Konacki & Kozlowski (2010). In principle they determined the sensitivity of the eclipse timing technique M1 = 3 MSun, Trojan = Jupiter 0.4 1.0 to circumbinary planets (exoplanets in P-Type motion) for 0.9 space-based photometric observations. They showed that 0.8 the typical photometric error (detectable timing amplitude 0.3 0.7 dT) for CoRoT is about dT = 4 sec for a brightness (L) 0.6 of 12 [mag] and dT = 16 sec for L=15.5 [mag]. Kepler 0.2 0.5 has a dT = 0.5 sec for L=9 [mag] and a dT = 4 sec for 0.4 0.3 L=14.5 [mag]. Future space missions will support the effort initial eccentricity 0.1 0.2 maximum eccentricity to detect new planets and smaller planets like for exam- 0.1 ple: PLATO (Planetary Transits and Oscillations of stars) 0.005 0.0 will monitor relatively nearby stars to hunt for Sun-Earth 0.001 0.03 0.06 0.09 0.12 analogue systems (Rauer et al. 2014). TESS (Transiting Ex-

mass secondary [MSun] oplanet Survey Satellite) space mission is dedicated to detect nearby Earth or super-Earth-size planets on close-in ◦ Figure 2. Stability map for L4 in the planar (i = 0 ) three-body orbits around the brightest M dwarfs (Ricker et al. 2014). problem for a Trojan planet with 1 MJup, for 1 Msun (upper CHEOPS (Characterising ExOPlanets Satellite) will exam- graph) 2 Msun (middle graph) and 3 Msun (lower graph). L4 is ine transiting exoplanets of known bright and nearby host stable in the light, unstable in the dark region for the emax and stars (Broeg et al. 2013). For our investigations we will use the LCI. The grey scale presents the values of emax, where the as detection criterion the photometric precision of CoRoT dark regions represent small values emax (stable orbits) and light dTcrit = 16sec. regions depicts large values of e (unstable orbits). max In our studies we investigated the amplitude of the ETV signals, therefore we varied the mass of the secondary body; starting with a binary like system m2 = 13MJup (13 Jupiter 4 ECLIPSE TIMING VARIATIONS masses corresponds to the lowest mass of a brown dwarf). When we are looking for extrasolar planets in T-Type con- In addition, we varied the eccentricity of the Trojan planet figuration we are interested in the ETV signal of the sec- whereas the secondary body moves in a circular orbit. We ondary star caused by an additional planet. This method used the stability analysis for the study of the ETV signals, is particularly important in the case when the planet’s or- which is presented in Fig. 2 for different masses of the Tro- bit may not be in the line of sight. However, such planets jan body MJup, for 1 Msun (upper graph) 2 Msun (middle cause perturbations in the orbit of the transiting star, lead- graph) and 3 Msun (lower graph). Whereas, m1 ∼ 1Msun ing to detectable ETVs. These were investigated for TTVs represent Sun-like stars and m1 > 1 Msun depicts brighter in several articles like e.g. Miralda-Escudé& Adams (2005); stars. But we did not take into account less massive stars

c 2002 RAS, MNRAS 000, 1–?? On the detection of Trojan planets in binaries 5

Table 2. Detectable timing amplitude dT for different masses M1 = 1 MSun, Trojan = Earth 0.4 of the primary star (first column) and different type of the ETV amplitude dT [sec] Trojan bodies are given in the second column starting with 500 −6 200 Mars mass (mTro = 0.032271 · 10 Msun) then Earth mass 100 −5 0.3 50 (mTro = 0.30049 · 10 Msun), Neptune mass (mTro = 0.51684 · emax < 0.4 −4 −3 10 Msun), and Jupiter mass (mTro = 0.9547 · 10 Msun). The third column represents the minimum dT and the last column 0.2 depicts the percentage of all possible detectable signals (dTcrit) within the stable region, presented as bold line in Fig. 4 and Fig. 5.

initial eccentricity Trojan 0.1

mass of type of minimum of percent lie the primary Trojan dT [h,m,s] in the dTcrit 0 [Msun] bodies 0.01 0.02 0.03 0.04 mass secondary [MSun] 1 Mars 4.3s 40

1 Earth 24.2s 100 M1 = 2 MSun, Trojan = Earth 1 Neptune 7.2m 100 0.4 ETV amplitude dT [sec] 1 Jupiter 1.82h 100 500 200 100 2 Mars 3.5s 15 0.3 50 16 2 Earth 10.4s 89 emax < 0.4 2 Neptune 2.1m 100 2 Jupiter 0.82h 100 0.2 3 Mars 2.6s 3 3 Earth 6.0s 77

initial eccentricity Trojan 0.1 3 Neptune 1.5m 100 3 Jupiter 0.39h 100 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 mass secondary [MSun] (m1), because of the critical mass ratio µcrit, which means that the secondary star would be a planetary object instead M1 = 3 MSun, Trojan = Earth of a brown dwarf. As one can see the eccentricity of pos- 0.4 ETV amplitude dT [sec] sible Trojan planets is limited by their eccentricity, that 500 200 means Trojan planets can not have larger eccentricities than 100 0.3 50 eTro = 0.2 (for m2 > 0.01). This is also well visible in Fig. 4. 16 and Fig. 5 were we present contour plots for terrestrial-like emax < 0.4 planets. Fig. 4 shows Trojans with Earth mass and Fig. 5 0.2 shows Trojans with Mars mass. The most important con- tours are the stability limit (outer border) and the border

with dTcrit = 16s. In case of an Earth-like Trojan planet initial eccentricity Trojan 0.1 with m1 = Msun all stable orbits produce a detectable ETV signal (dTcrit > 16s). The figures 4 and 5 showed that with 0 larger mass of the central star (m1 = 2, 3MSun) the stabil- 0.01 0.03 0.05 0.07 0.09 0.11 0.13 ity limits increase, because of that Trojans can have larger mass secondary [MSun] eccentricities. However, the number of detectable ETV sig- 1 ◦ nals shrinks with larger m and amplitude dT have smaller Figure 4. ETV map for L4 in the planar (i = 0 ) three-body values too, as concluded in Tab. 2. This is mainly important problem for possible Trojan planets with one Earth mass, for for terrestrial like Trojan planets. In case of MTro is equal 1 Msun (upper graph) 2 Msun (middle graph), and 3 Msun (lower to Jupiter or Neptune mass all stable orbits produce a de- graph). The different lines presents the values of the amplitude tectable ETV signal. As represented in Tab. 2, where the of the ETV signal dT in [s]. The outer (bold) line represent the third column show the minimum amplitude of the ETV sig- stability border. nal dT and fourth column depicts the number of detectable ETV signals within the stable region. We can summarize conditions. In case of double stars it is very difficult to find that possible terrestrial-like planets are detectable in binary candidates, because the important parameters like semi- star systems by the help of eclipse timing variations with major axis a and mass ratio µ of the primary bodies are restrictions. not or only partly given. Maybe this problem will change in future, because of the data of the space mission GAIA3, which will make a survey of one billion stars with many 5 LIST OF CANDIDATES FOR BINARY binary star systems Eyer et al. (2012) and binary-like sys- (LIKE) SYSTEMS tems de Bruijne (2012). For our search we used the ’Wash- As an application we prepared a list of possible candidates, which fulfil the most important dynamical and observational 3 Global Astrometric Interferometer for Astrophysics

c 2002 RAS, MNRAS 000, 1–?? 6 R. Schwarz, A.´ Bazs´o, B. Funk and R. Zechner

M1 = 1 MSun, Trojan = Mars Table 3. List of candidates to detect possible Trojan planets 0.4 ETV amplitude dT [sec] in binary-like systems. The list is sorted according to the semi- 500 major axis (a2) of the brown dwarf. All systems were detected by 200 100 TTVs except for those marked with a star (which were detected 0.3 50 16 by radial velocity). emax < 0.4

a m1 µ 6 µ 0.2 Name mass in [AU] crit [MJup][MSun]

WASP-18 b 10.43 0.020 1.24 0.00796

initial eccentricity Trojan 0.1 KELT-1 b 27.38 0.025 1.335 0.01919 XO-3 b 11.79 0.045 1.41 0.00791

0 CoRoT-27 b 10.39 0.048 1.05 0.00935 0.01 0.02 0.03 0.04 CoRoT-3 b 21.77 0.057 1.41 0.01452 mass secondary [MSun] HD 162020 b* 14.4 0.074 0.75 0.01799 Kepler-39 b 18 0.155 1.1 0.01537

M1 = 2 MSun, Trojan = Mars Kepler-27 c 13.8 0.191 0.65 0.01985 0.4 ETV amplitude dT [sec] HD 114762 b* 10.98 0.353 0.84 0.01232 500 HD 202206 b* 17.4 0.830 1.13 0.01448 200 100 0.3 50 16 emax < 0.4 masses larger then 10 Jupiter masses. We classiﬁed them as binary like systems, because these objects are brown dwarfs. 0.2 When we look at mass ratios which fulﬁl the criterion that µ 6 µcrit, the list of candidates decreases to 68 candidates. Concerning to the condition that the distance to the central

initial eccentricity Trojan 0.1 star should be not too far (that means a 6 1 AU because of long observation time) we found only 10 candidates. The 0 data of all planets are taken from the Exoplanet-catalogue 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 mass secondary [MSun] maintained by J. Schneider.

M1 = 3 MSun, Trojan = Mars 0.4 ETV amplitude dT [sec] 6 CONCLUSIONS 200 100 50 We have carried out a parameter study for a variety of bi- 0.3 16 4 naries with diﬀerent mass-ratios of the primary bodies and e < 0.4 max diﬀerent orbital elements of the Trojan planet. The stabil-

0.2 ity analysis has shown that there exist a stable region for planets – gas giants as well as terrestrial like planets – in T-type motion for low eccentricities eTro 6 0.25. Our re-

initial eccentricity Trojan 0.1 sults have shown that binary-like systems (the secondary body is a brown dwarf) and binary systems with low-mass

0 stars are good candidates to discover new planets by eclipse 0.01 0.03 0.05 0.07 0.09 0.11 0.13 timing of the secondary star, agreeing with the work of mass secondary [MSun] Schneider & Doyle (1995). At this point we have to concede that from an observational point of view M-stars are better ◦ Figure 5. ETV map for L4 in the planar (i = 0 ) three-body candidates than brown dwarfs. Nevertheless, ETV signals of problem for possible Trojan planets with one Mars mass, for brown dwarfs would be detectable too. We found that for all 1 Msun (upper graph) 2 Msun (middle graph), and 3 Msun (lower stable initial conditions an Earth-like Trojan planet would graph). The diﬀerent lines presents the values of the amplitude produce detectable ETV signals if the primary star has 1 of the ETV signal dT in [s]. The outer (bold) line represents the Msun. This is also valid for Jupiter and Neptune type Tro- stability border. jan planets, but also for not Sun-like stars (m1 = 2Msun and m1 = 3Msun). Unexpectedly, we found that even Tro- jan planets with Mars mass would be detectable, but under ington Visual Double Star Catalogue’ (Mason et al. 2001). favourable circumstances. As an application we prepared a We found 24 systems with a mass ratio smaller than the crit- list of possible candidates for the detection possible Trojan ical one (µ 6 0.04). But we found only one system where the planets of binary and binary-like systems. In case of bina- semi-major axis from the secondary star a is given, namely ries we found 24 candidates, but for only a single system Antares (α Sco) with a = 2.9 AU (Worley & Heintz 1983) (Antares, α Sco) the semi-major axis a is constrained. In and µ ∼ 0.016. Finally, we would like to mention that the addition, we made a list of candidates for binary-like stars distance of the secondary star should not be larger than which is given in Tab. 3 where we found 10 candidates. 1 AU, because of the long observation time span. We can conclude that possible terrestrial-like planets At the moment we know about 1900 exoplanets in more are detectable in binary star systems by the help of eclipse than 1200 planetary systems, among them 93 planets have timing variations with restrictions. However, future space

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