A&A 528, A53 (2011) Astronomy DOI: 10.1051/0004-6361/201015867 & c ESO 2011 Astrophysics

Transit timing variations in eccentric hierarchical triple exoplanetary systems I. Perturbations on the time scale of the orbital period of the perturber

T. Borkovits1,Sz.Csizmadia2, E. Forgács-Dajka3, and T. Hegedüs1

1 Baja Astronomical Observatory of Bács-Kiskun County, 6500 Baja, Szegedi út, Kt. 766, Hungary e-mail: [borko;hege]@electra.bajaobs.hu 2 Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Planetenforschung, Rutherfordstr. 2, 12489 Berlin, Germany e-mail: [email protected] 3 Eötvös University, Department of Astronomy, 1518 Budapest, Pf. 32, Hungary e-mail: [email protected] Received 5 October 2010 / Accepted 16 December 2010 ABSTRACT

Aims. We study the long-term time scale (i.e. period comparable to the orbital period of the outer perturber object) transit timing variations (TTVs) in transiting exoplanetary systems that contain another more distant (a2 a1) planetary or stellar companion. Methods. We give an analytical form of the O−C diagram (which describes such TTVs) in a trigonometric series, which is valid for arbitrary mutual inclinations, up to the sixth order in the inner eccentricity. Results. We show that the dependence of the O−C on the orbital and physical parameters can be separated into three parts. Two of these are independent of the real physical parameters (i.e. masses, separations, periods) of a concrete system and only depend on dimensionless orbital elements, so can be analysed in general. We find that, for a specific transiting system, where eccentricity (e1) and the observable argument of periastron (ω1) are known, say, from spectroscopy, the main characteristics of that TTV, which is caused by a possible third-body can be mapped simply. Moreover, as the physical attributes of a given system only occur as scaling parameters, the real amplitude of the O−C can also be estimated for a given system, simply as a function of the m3/P2 ratio. We analyse the above-mentioned dimensionless amplitudes for different arbitrary initial parameters, as well as for two particular systems, CoRoT-9b and HD 80606b. We find in general that, while the shape of the O−C strongly varies with the angular orbital elements, the net amplitude (departing from some specific configurations) depends only weakly on these elements, but strongly on the eccentricities. As an application, we illustrate how the formulae work for the weakly eccentric CoRoT-9b and the highly eccentric HD 80606b. We also consider the question of detection, as well as the correct identification of such perturbations. Finally, we illustrate the operation and effectiveness of Kozai cycles with tidal friction (KCTF) in the case of HD 80606b. Key words. methods: analytical – planets and satellites: individual: CoRoT-9b – planets and satellites: individual: HD 80606b – methods: numerical – planetary systems – binaries: close

1. Introduction which has been the main tool for period studies by observers of variable stars (not only for eclipsing binaries) for more than The rapidly increasing number of exoplanetary systems, as well a century. Consequently, the effect of the various types of pe- as the lengthening time interval of the observations naturally riod variations (both real and apparent) for the O−Cdiagram leads to the search for perturbations in the motion of the known have already been widely studied in the past one hundred years. planets, and this can provide the possibility of detecting addi- Some of these are less relevant in the case of transiting exo- tional planetary (or stellar) components in a given system, can planets, but others are important. For example, the two clas- produce further information about the oblateness of the host star sical cases are the simple geometrical light-time effect (LITE) (or the planet), or might even refer to evolutionary effects. (of another, more distant companion), and the apsidal motion The detection and the interpretation of such perturbations effect (AME) (of both the stellar oblateness in eccentric bina- in the orbital revolution of the exoplanets usually depends on ries and the relativistic effect). LITE has been widely used for such methods and theoretical formulae, which are well known identifying other stellar companions of many variable stars (not and have been applied for a long time in the field of the close only eclipsing binaries) from the pioneering papers of Chandler eclipsing and spectroscopic binaries. We mainly refer to the (1892), Hertzsprung (1922), Woltjer (1922), and Irwin (1952)to methods developed in connection with the observed period vari- a copious number of recent applications. Owing to its small am- ations in eclipsing binaries. These period variations manifest plitude, however, it is less significant in the case of a planetary- themselves in non-linear variations of the difference between mass wide component. Nevertheless in recent years, there have the observed and the predicted time of mid-eclipses (either tran- been some efforts to discover exoplanets in this manner (see e.g. sits or occultations). In the nomenclature of exoplanet stud- Silvotti et al. 2007, V391 Pegasi; Deeg et al. 2008, CM Draconis; ies, this phenomenon is called transit timing variation (TTV). Qian et al. 2009, QS Virginis; Lee et al. 2009, HW Virginis). Plotting the observed minus the calculated mid-transit times The importance of AME in close exoplanet systems has been with respect to the cycle numbers, we get the O−Cdiagram, investigated in several papers; see e.g. Miralda-Escudé (2002), Article published by EDP Sciences A53, page 1 of 25 A&A 528, A53 (2011)

Heyl & Gladman (2007), and Jordán & Bakos (2008,andfur- Kozai cycle, some additional, tidally forced circularization may ther references thereins). This effect has been studied since the then form close systems in their recent configurations. pioneering works of Cowling (1938), and Sterne (1939), and be- Nevertheless, independently of the question, as to which sides its evident importance in the checking of the general rel- mechanism(s) is (are) the really effective ones, up to now we ativity theory, it plays also an important role in studies of the have not been able to study these mechanisms in operation, so inner mass distributions of stars (via their quadrupol moment); only the end-results have been observed. This is mainly the con- i.e. it provides observational verification of stellar models. sequence of selection effects. In order to study these phenomena Besides their similarities, there are also several differences when they are effective, one should observe the variation in the between the period variations of close binary and multiple stars, orbital elements of such extrasolar planets, as well as binary sys- and of planetary systems. For example, most of the known stellar tems that are in the period range of months to years. The easisest multiple systems form hierarchic subsystems (see e.g. Tokovinin way to carry out such observations is to monitor the TTVs of 1997). This is mainly the consequence of the dynamical stabil- these systems. However, there are only a few known transiting ity of these configurations, or put another way, the dynamical extrasolar planets in this period regime. Furthermore, although instability of the non-hierarchical stellar multiple systems. In binary stars are also known with such separations, they are not contrast, a multiple planetary system may form a stable (or at appropriate subjects for this investigation because of their non- least long-time quasi-stable) non-hierarchical configuration, as eclipsing nature. we can see in our solar system, and furthermore, the same result The continuous long-term monitoring of several hundred has been shown by numeric integrations for several known exo- stars with the CoRoT and Kepler satellites, as well as the planet systems (see e.g. Sándor et al. 2007). This means we can long-term systematic terrestrial surveys, provide an excellent op- expect several different configurations, the examination of which portunity to discover transiting exoplanets (or as by-products: has, until now, not been considered in the field of the period vari- eclipsing binaries) with the period of months. Then continuous, ations of multiple stellar systems. Examples of these phenomena long-term transit monitoring of these systems (combining the are the perturbations of both an inner planet and a companion on data with spectroscopy) may allow dynamical evolutionary ef- a resonant orbit (Agol et al. 2005) or, as a special case of the fects (i.e. orbital shrinking) to already be traced within a few latter, the possibility of Trojan exoplanets (Schwarz et al. 2009). decades. Furthermore, the larger the characteristic size of a mul- Recently, the detectability of exomoons has also been studied tiple planetary, stellar (or mixed) system the greater the ampli- (Simon et al. 2007; Kipping 2009a,b). tude of even the shorter period perturbations in the TTVs, as Furthermore, thanks to the enhanced activities related to ex- shown in detail in the discussion of Borkovits et al. (2003). trasolar planetary searches, which led to missions like CoRoT In the past few years, several papers have been pub- and Kepler that produce long-term, extraordinarily accurate data, lished on TTVs, both from theoretical aspects (e.g. Agol et al. we can expect in the near future other such observations to 2005; Holman & Murray 2005; Nesvorný & Beaugé 2010; make it possible to detect and study more phenomena never ob- Holman 2010; Cabrera 2010; Fabrycky 2010, see further ref- served earlier. For example, it is well known that neither the erences therein) and from large numbers of papers on obser- close binary stars nor the hot-Jupiter-type exoplanets could have vational aspects for individual transiting exoplanetary systems. been formed in their present positions. Different orbital shrink- Nevertheless, most (but not all) of the theoretical papers above ing mechanisms are described in the literature. Instead of listing mainly concentrate on the detectibility of additional companions them, we refer to the short summaries by Tokovinin et al. (2006), (especially super-Earths) from the TTVs. Tokovinin (2008), and Fabrycky & Tremaine (2007). Here we In this paper we consider this question in greater detail. We note only that one of the most preferred theories for the for- calculate the analytical form of the long-period1 (i.e. with a pe- mation of close binary stars, which also might have produced riod close to the orbital period of the ternary component P2), at least a portion of hot Jupiters, as well, is the combination time-scale perturbations of the O−C diagram for hierarchical of the Kozai cycles with tidal friction (KCTF). The Kozai res- (i.e. P2 P1) triple systems. (As we mainly concentrated on onance (recently and frequently referred to as Kozai cycle[s]) transiting systems with a period of weeks to months, we omitted was first described by Kozai (1962) when investigating secu- the possible tidal forces. However, our formulae can be practi- lar perturbations of asteroids. The first (theoretical) investiga- cally applied even for the closest exoplanetary systems, because tion of this phenomenon with respect to multiple stellar systems the tidal perturbations usually become effective on a notably can be found in the studies by Harrington (1968, 1969), Mazeh & Saham (1979), and Söderhjelm (1982). A higher, third-order 1 In this paper we follow the original classification of Brown (1936), theory of Kozai cycles was given by Ford et al. (2000), while which divided the periodic perturbations occurring in hierarchical sys- the first application of KCTF to explain the present configura- tems into the following three groups: tion of a close, hierarchical triple system (the emblematic Algol, – short-period perturbations. The typical period is equal to the or- itself) was presented by Kiseleva et al. (1998). According to bital period P1 of the close pair, while the amplitude has the order 2 this theory, the close binaries (as well as hot-Jupiter systems) (P1/P2) ; should have originally formed as significantly wider primordial – long-period perturbations. This group has a typical period of P2, binaries having a distant, inclined third companion. Due to the and magnitude of the order (P1/P2); 2 third object’s induced Kozai oscillation, the inner eccentricity – apse-node terms. In this group the typical period is about P2/P1, becomes cyclically so large, that around the periastron passages, and the order of the amplitude reaches unity. the two stars (or the host star and its planet) approach each other ff so closely that tidal friction may be effective, which shrinks the This classification di ers from what is used in the classical planetary perturbation theories. There, the first two groups are termed together orbit remarkably, during one or more Kozai cycles. Owing to the ff as “short period” perturbations, while the “apse-node terms” are called smaller separation, the tidal forces remain e ective on a larger “long period ones”. Nevertheless, in the hierarchical scenario, the first and larger portion of the whole revolution, and finally, they will two groups differ from each other both in period and amplitude; con- switch off the Kozai cycles, producing a highly eccentric, moder- sequently, we feel this nomenclature is more appropriate in the present ately inclined, small-separation intermediate orbit. After the last situation.

A53, page 2 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. longer time scale.) This work is a continuation and extension meanings. Furthermore, in Eq. (3) we write the true anomaly as of the previous paper of Borkovits et al. (2003). In that paper we v = u − ω.ToevaluateEq.(3) we first have to express the pertur- formulated the long-period perturbations of an (arbitrarily ec- bations in the orbital elements with respect to u. Assuming that centric and inclined) distant companion to the O−Cdiagramfor the orbital elements (except of u) are constant, the first term of a circular inner orbit. (Our formula is more general than that of the right-hand side (r.h.s.) yields the following closed solution Agol et al. 2005. For the coplanar case the two results become ⎡ ⎛ ⎞ ⎤ identical.) Now we extend the results to the case of an eccen- P ⎢ ⎜ 1 − e ∓ cos ω ⎟ e cos ω ⎥ P = ⎣⎢2arctan⎝⎜ ⎠⎟± (1 − e2)1/2 ⎦⎥ , tric inner binary (formed either a host star with its planet or two I,II 2π 1 + e 1 ∓ sin ω 1 ∓ e sin ω stars). As shown, our formulae have a satisfactory accuracy even (4) for high eccentricity, such as e1 = 0.9. In the period regime of a few months, the tidal forces are ineffective, so we expect eccen- for the two types of minima, respectively. (Here P denotes the tric orbits. This is especially valid in the case of the predecessor anomalistic or Keplerian period which is considered to be con- systems of hot Jupiters, in which case the formation theory men- stant.) Instead of the exact forms above, widely used is the ex- tioned above predicts very high eccentricities. pansion (as in this paper), which, up to the fifth order in e,is In the next section we give a very brief summary of our calculations. (A somewhat detailed description can be found in P 1 3 1 P = P E + ∓ π ± 2e cos ω + e2 + e4 sin 2ω Borkovits et al. 2003, 2007; nevertheless for self-consistency of I,II s 2π 2 4 8 the present paper we provide a brief overview.) In Sect. 3 we 1 1 5 3 discuss our results, while in Sect. 4 we illustrate the results with ∓ e3 + e5 cos 3ω − e4 sin 4ω ± e5 cos 5ω , both analytical and numerical calculations on two individual sys- 3 8 32 40 tems, CoRoT-9b and HD 80606b. Finally in Sect. 5 we draw (5) conclusions from our results, and, furthermore, we compare our where P is the sidereal (or eclipsing) period of, for example, method and results with those of Nesvorný & Morbidelli (2008), s the first cycle, and E the cycle-number. In the present case the and Nesvorný (2009). orbital elements cease to remain constant. Nevertheless, as can be seen from Eqs. (8)–(14), their variation on the P timescaleis 2. Analytical investigations 2 related to P1/P2  1, which allows linearization of the problem; 2.1. General considerations and equations of the problem i.e., in such a case, Eq. (5) is formally valid in the same form, but e, ω,andPs are no longer constant. Then a further integra- As is well known, at the moment of the mid-transit (which for tion of Eq. (5) with respect to v gives the analytical form of the an eccentric orbit usually does not coincide with half the whole 2 perturbed O−ContheP2 time scale. To this, as a next step, we transit event) have to calculate the long-period and apse-node perturbations π in u. Some of these arise simply from the similar perturbations u ≈± + 2kπ, (1) 2 of the orbital elements (or directly of the e cos ω, e sin ω func- tions), while others (we refer to them as direct perturbations in where u is the true longitude measured from the intersection of u) come from the variations in the mean motion (for more de- the orbital plane and the plane of sky, and k is an integer. (Since, tails see Borkovits et al. 2007)2. (In other words this means that traditionally the positive z-axis is directed away from the ob- in such a case P will no longer be constant.) server, the primary transit occurs at u = −π/2.) An exact equal- At this point, to avoid any confusion, we emphasize that dur- ity is valid only if the binary has a circular orbit, or if the orbit ing our calculations we use different sets of the angular orbital is seen edge-on exactly. The correct inclination dependence of elements. As we are interested in a phenomenon that primarily the occurrence of the mid-eclipses can be found in Gimènez & depends on the relative positions of the orbiting celestial bod- Garcia-Pelayo (1983). Nevertheless, the observable inclination ies with respect to the observer, the angular elements (i.e. u, ω, i, of a potentially month-long period transiting extrasolar planet Ω ◦ ) should be expressed in the “observational” frame of reference should be close to i = 90 , if the latter condition is to be sat- with the plane of the sky as the fundamental plane, and u,aswell isfied. Due to its key role in the transits we use u as our inde- as ω, is measured from the intersection of the binary’s orbital pendent time-like variable, instead of the usual variables. It is plane with that plane, while Ω is measured along the plane of known from the textbooks of celestial mechanics that the sky from an arbitrary origin. On the other hand, the physical c variations in the motion of the bodies depend on their positions u˙ = 1 − Ω˙ cos i, ρ2 relative to each other, and it is consequently more beneficial (and 1 convenient) to express the equations of perturbations of the or- 1/2 −3/2 2 −3/2 2 = μ a (1 − e ) (1 + e cos v) − Ω˙ cos i;(2)bital elements in a different frame of reference (we refer to it as the dynamical frame), determined only by the relative positions consequently, the moment of the Nth primary minimum (transit) of the bodies. The fundamental plane of this frame of reference after an epoch t0 can be calculated as is the invariable plane of the triple system, i.e. the plane per- t 2Nπ−π/2 pendicular to the net angular momentum vector of the complete N a3/2 (1 − e2)3/2 du dt = , triple system. In this frame of reference, the longitude of the 1/2 2 2 − μ [1 + e cos(u − ω)] ρ1 t0 π/2 1 − Ω˙ cos i ascending node (h) gives the arc between the sky and the corre- ⎛ c1 ⎞ 2 sponding intersection of the two orbital planes measured along a3/2 (1 − e2)3/2 ⎜ ρ ⎟ ≈ ⎝⎜ + 1 Ω˙ i⎠⎟ u. the invariable plane, while the true longitude (w) and the argu- 1/2 2 1 cos d (3) μ [1 + e cos(u − ω)] c1 ment of periastron (g) are measured from that ascending node, along the respected orbital plane. To avoid a further confusion, In the equations above, c1 denotes the specific angular momen- tum of the inner binary, ρ1 is the radius vector of the planet with 2 The minus signs on the r.h.s. of Eqs. (10) and (11) of Borkovits et al. respect to its host star, and the orbital elements have their usual (2007) should be replaced by plus.

A53, page 3 of 25 A&A 528, A53 (2011)

is of the order P1/P2, so can be neglected. For the long-period perturbations of the orbital elements of the close orbit we get da 1 = 0, (8) dv2 de1 = − 2 1/2 + AL(1 e1) e1(1 e2 cos v2) dv2 2 × 1 − I sin 2g1

1 2 − (1 + I) sin(2v2 + 2g2 − 2g1) 2 1 + (1 − I)2 sin(2v + 2g + 2g ) , (9) 2 2 2 1 dg1 = − 2 1/2 + AL(1 e1) (1 e2 cos v2) dv2 3 1 × I2 − 5 3 Fig. 1. The spatial configuration of the system. 2 3 + 1 − I cos 2g1 + cos(2v2 + 2g2) the relative inclination of the orbits to the invariable plane is de- 5 ff noted by j. The meaning and the relation between the di erent 1 2 + (1 + I) cos(2v2 + 2g2 − 2g1) elements can be seen in Fig. 1, and are also listed in Appendix A. 2 1 2 2.2. Long-period perturbations + (1 − I) cos(2v2 + 2g2 + 2g1) 2 From now on we refer to the orbital elements of the inner binary dh1 (i.e. the pair formed by either a host star and the inner planet, − cos j1, (10) dv2 or two stellar mass objects) by subscript 1, while those of the dh sin i wider binary (i.e. the orbit of the third component around the 1 = −A (1 − e2)−1/2 m (1 + e cos v ) ff L 1 2 2 centre of mass of the inner system) by subscript 2. The di er- dv2 sin j1 ential perturbation equations of the orbital elements are listed in 2 3 3 × 1 + e2 I Borkovits et al. (2007) . In order to get the long-period terms of 5 2 1 the perturbation equations, the usual method involves averaging ≈ −e2I cos 2g the equations for the short-period ( P1) variables, which is usu- 1 1 ally the mean anomaly (l1) or the true anomaly (v1) of the inner 2 3 − 1 + e2 I cos(2v + 2g ) binary, but in our special case it is the true longitude u1.This 5 2 1 2 2 means that we get the variation in the orbital elements averaged over an eclipsing period. Furthermore, in the case of the aver- +1 2 + + − e1(1 I)cos(2v2 2g2 2g1) aged equations we change the independent variable from u1 to 2 (the averaged) v2, by the use of 1 2 − e (1 − I)cos(2v2 + 2g2 + 2g1) , ⎛ ⎞ 2 1 du ρ2 c ⎜ ρ2 ⎟ 1 ≈ 2 1 ⎜ − 1 Ω˙ ⎟ d j1 − ⎝1 1 cos i1⎠ , (6) = A − e2 1/2 i + e v dv c ρ2 c L(1 1) sin m(1 2 cos 2) 2 2 1 1 dv2 2 3 from which, after averaging, we get × 1 + e2 sin(2v + 2g ) 5 2 1 2 2 1/2 2 + 2 du μ ρ dΩ e1I sin 2g1 1 ≈ 1 2 − 1 cos i 3/2 1 1 dv2 a 2 dv2 − 2 + + − 1 μ2a2(1 − e ) e1(1 I)sin(2v2 2g2 2g1) 2 2 − 2 3/2 P2 (1 e ) dΩ1 1 ≈ 2 − i . + 2 − + + 2 cos 1 (7) e1(1 I)sin(2v2 2g2 2g1) , (11) P1 (1 + e2 cos v2) dv2 2 dω1 = − 2 1/2 + In the following we omit the overlining. Furthermore, as one AL(1 e1) (1 e2 cos v2) can see from Eq. (13) below, the second term on the r.h.s. of (7) dv2 3 1 × I2 − 3 In that paper, from practical considerations, we did not restrict our- 5 3 selves to the most usual representation of the perturbation equations, i.e. the Lagrange equations with the perturbing function; rather, we used the 2 3 + 1 − I cos 2g1 + cos(2v2 + 2g2) somewhat more general form that expresses the perturbations with the 5 three orthogonal components of the perturbing force. Nevertheless, as 1 2 long as only conservative, three-body terms are considered, the two rep- + (1 + I) cos(2v2 + 2g2 − 2g1) resentations are perfectly equivalent. (In the denominator of Eqs. (14) 2 Ω and (15) for e and ω, a closing bracket is evidently missing, and further- +1 − 2 + + − d 1 Ω e (1 I) cos(2v2 2g2 2g1) cos i1, (12) more, the equation for should be divided by for the correct result.) 2 dv2 A53, page 4 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. Ω d 1 = − 2 −1/2 sin im + = + 31 2 + 23 4 AL(1 e1) (1 e2 cos v2) f2(e) 1 e e . (18) dv2 sin i1 51 48 2 3 × 1 + e2 I cos(ω − g ) Furthermore, im denotes the mutual inclination of the two orbital 5 2 1 1 1 planes, while 1 3 2 + 1 + e (1 − I)cos(2v + 2g − ω + g ) I = cos im, (19) 5 2 1 2 2 1 1 and m123 stands for the total mass of the system. Formal integra- −1 + 3 2 + + + − 1 e1 (1 I)cos(2v2 2g2 ω1 g1) tion of the first five of the equations above (i.e. those that refer 5 2 to the orbital elements in the dynamical system, Eqs. (8)−(11)) − 2 + e1I cos(ω1 g1) reproduce the results of Söderhjelm (1982). 1 Strictly speaking, some of the equations above are only valid + e2(1 + I)cos(2v + 2g − ω − g ) 2 1 2 2 1 1 when both orbits are non-circular, and the orbital planes are in- clined to each other and/or to the plane of the sky. For exam- 1 2 − e (1 − I)cos(2v + 2g + ω + g ) , (13) ple, if the outer orbit is circular, neither v2 nor g2 has any mean- 2 1 2 2 1 1 ing. Nevertheless, their sum i.e. v2 + g2 is meaningful, as before. di1 = − 2 −1/2 + Similarly, although the derivative of g1 hasnomeaninginthe AL(1 e1) sin im(1 e2 cos v2) case of a circular inner orbit, the derivatives of e cos g , e sin g dv2 1 1 1 1 2 3 (or e1 cos ω1, e1 sin ω1), i.e. the so-called Lagrangian elements × − 1 + e2 I sin(ω − g ) can be calculated correctly. (Instead of the “pure” derivatives of 5 2 1 1 1 e1 and g1 (or ω1) these occur directly in the O−C.) Similar re- 1 3 defenitions can be done in the case of coplanarity. For practical + 1 + e2 (1 − I)sin(2v + 2g − ω + g ) 5 2 1 2 2 1 1 reasons and for the sake of clarity, we therefore retain the orig- inal formulations, even in such cases, when it is formally not +1 + 3 2 + + + − valid. 1 e1 (1 I)sin(2v2 2g2 ω1 g1) 5 2 As one can see, there are some terms on the r.h.s. of these 2 +e I sin(ω1 + g1) equations that do not depend on v2. Primarily, these terms give the so-called apse-node time-scale contribution to the variation −1 2 + + − − e1(1 I)sin(2v2 2g2 ω1 g1) in the orbital elements and the TTVs. Nevertheless, to get a cor- 2 rect result for the long-term behaviour of the orbital elements, 1 2 these terms must nevertheless be retained in our long-term for- + e (1 − I)sin(2v2 + 2g2 + ω1 + g1) . (14) 2 1 mulae. These terms were calculated for tidally distorted triplets in Borkovits et al. (2007). Such formulae (after omitting the tidal terms) are also valid for the present case in the low mutual Finally, the direct term is 2 3 inclination (i.e. approx. I > 5 ) domain. (Formulae valid for arbitrary mutual inclinations will be presented in a subsequent dλ1 = − 2 1/2 + paper.) AL(1 e1) (1 e2 cos v2) dv2 Carrying out the integrations, all the orbital elements on the 4 1 r.h.s. of these equations with the exception of v2 are considered × I2 − f (e ) 5 3 1 1 as constants. This can be justified for two reasons. First, as one can see, the amplitudes of the long-period perturbative terms 51 2 2 + 1 − I e f (e )cos2g (AL) remain small for P2 P1 (which is especially valid for 20 1 2 1 1 the case where the host star is orbited by two planets, when 4 m  m 4 + 1 − I2 f (e )cos(2v + 2g ) 3 123) , and second, although the amplitude of the apse- 5 1 1 2 2 node perturbative terms can reach unity, the period is usually so long (∼P2/P , see e.g. Brown 1936) that its contribution can be +51 + 2 2 + − 2 1 (1 I) e1 f2(e1)cos(2v2 2g2 2g1) 5 40 safely ignored during one revolution of the outer object .The 51 + (1 − I)2e2 f (e )cos(2v + 2g + 2g ) , (15) 40 1 2 1 2 2 1 where 4 This assumption is analogous to that of the classical low-order pertur- bation theory where the squares of the orbital element changes are ne- 2 15 m P glected, as they should be proportional to (mperturber/mstar) , but with the = 3 1 − 2 −3/2 difference that here the small parameter is the ratio of the semi-major AL (1 e2) , (16) 8 m123 P2 axes a1/a2 instead of the mass ratio. Consequently, this assumption re- mains valid even if the perturber is a sufficiently distant stellar-mass object. and 5 This is not necessarily true in the presence of other perturbative ef- fects. Nevertheless, in such cases one can assume that if there is a phys- ical effect producing apse-node time scale perturbations having a period 25 2 15 4 95 6 f1(e) = 1 + e + e + e , (17) comparable to the period of the long-period perturbations arising from 8 8 64 some other sources, then the amplitudes of these long-period perturba- tions are usually so small with respect to the amplitudes of the apse- node perturbations that their effect can be neglected.

A53, page 5 of 25 A&A 528, A53 (2011)

final result of such an analysis is an analytical form of the TTVs, and i.e. the long-period O−Cdiagram: u = ω − g ; (25) P 4 5 6 m1 1 1 O−C = 1 A (1 − e2)1/2 + e2 ∓ e sin ω P2 2π L 1 5 2 1 5 1 1 i.e., um1 is the angular distance of the intersection of the two or- 1 1 × I2 − M + 1 − I2 S(2v + 2g ) bits from the plane of the sky, or, in other words, the longitude 3 2 2 2 of the (dynamical) ascending node of the inner orbit along the orbital plane, measured from the sky. Here we give the result up + 51 2 ∓ − e1 cos 2g1 2e1 sin(ω1 2g1) to the first order in the inner eccentricity, while a more extended 20 result up to the sixth order in the inner eccentricity can be found 2 1 2 in Appendix B, where we give also the perturbation equations × 1 − I M + 1 + I S(2v2 + 2g2) m m 2 directly for the e1 cos nω1, e1 sin nω1 expressions. The upper signs refer to the exoplanetary transits (primary minima), and 51 2 − e sin 2g1 ∓ 2e1 cos(ω1 − 2g1) IC(2v2 + 2g2) the lower signs to the secondary occultations (secondary min- 20 1 5 2 ima). Furthermore, we assumed the formally second-order 2 e1, 2 51 2 and e1 terms to be first order, as their values exceed e1 for + cot i1 sin im − (1 ∓ 2e1 sin ω1) cos um1I 20 5 medium eccentricities. We also included the pure geometrical 1 light-time contribution, shown in the last row. Here c denotes × M− S(2v + 2g ) 2 2 2 the speed of light. The minus sign arises because this term re- flects the motion of the inner pair around the common centre of 1 + (1 ∓ 2e sin ω )sinu C(2v + 2g ) mass, whose true longitude differs by π from the one of the third 5 1 1 m1 2 2 component. In Eq. (21) the mean anomaly of the outer body (l2) 1 − e2 sin(v + ω ) appears because of (the constant part of) the difference between − m3 a2 sin i2 2 2 2 the anomalistic P and sideral (eclipsing, or transiting) Ps period m123 c 1 + e2 cos v2 + O 2 included into the first term on the r.h.s. of Eq. (5). (e1), (20) where 3. Discussion of the results M = 1 + e cos v dv 2 2 2 As one can easily see, the result for a circular inner orbit (i.e. = = v2 − l2 + e2 sin v2 (21) e1 0) is identical with the formula (46) of Borkovits et al. 3 1 (2003). Consequently, the discussion in Sect. 4 of that paper is = 3e sin v − e2 sin 2v + e3 sin 3v + O(e4) 2 2 4 2 2 3 2 2 2 also valid. Nevertheless, as one can also see in Figs. 2, 3,asig- 3 9 53 nificant inner eccentricity produces notably higher amplitudes = 3e 1 − e2 sin l + e2 sin 2l + e3 sin 3l + O(e4), 2 8 2 2 4 2 2 24 2 2 2 and is consequently easier to detect. Furthermore, some other (22) attributes of the TTVs also change drastically. For example, in contrast to the previously studied coplanar (I = ±1), circular in- = and ner orbit (e1 0) case (Borkovits et al. 2003; Agol et al. 2005), the dynamical term does not disappear even if the outer orbit is 1 circular (e = 0), as long as the inner orbit is eccentric. Another S(2v + x) = sin(2v + x) + e sin(v + x) + e sin(3v + x) 2 2 2 2 2 3 2 2 important feature for an eccentric inner orbit is that the ampli- − 13 tude, the phase, and the shape of the O C variations cease to = 1 − 4e2 sin(2l + x) + e − 1 − e2 sin(l + x) depend simply on the physical (i.e. relative) positions of the ce- 2 2 2 8 2 2 lestial bodies, but also on the orientation of the orbit with respect 7 207 1 17 to the observer. These last elements (i.e. e sin ω , e cos ω ,and + 1− e2 sin(3l +x) +e2 − sin x+ sin(4l +x) 1 1 1 1 3 56 2 2 2 4 4 2 their combinations) can also be determined from (a) radial ve- locity measurements, (b) the shape of the transit light curves, 1 169 + 3 − + + + O 4 c ff e2 sin(l2 x) sin(5l2 x) (e2), (23) and ( ) the time delay between the two di erent eclipsing events. 24 24 (Naturally, the last one requires the detection of the secondary 1 occultations or minima.) While the relative, i.e., physical angular C(2v2 + x) = cos(2v2 + x) + e2 cos(v2 + x) + e2 cos(3v2 + x) 3 parameters (i.e. periastron distances from the intersection of the two orbits, g , g , and mutual inclination i ) cannot be aquired = 1 − 4e2 cos(2l + x) 1 2 m 2 2 from other, generally used methods (e.g. light-curve, or simple 13 radial velocity curve analysis), these observational geometrical + e − 1 − e2 cos(l + x) 2 8 2 2 parameters could be determined from other sources of informa- tion, and then can be simply built into such a fitting algorithm, 7 207 + 1 − e2 cos(3l + x) which was described in Borkovits et al. (2003), or could be in- 3 56 2 2 cluded in procedures like those presented by Pál (2010). 1 17 In the following we omit terms multiplied by cot i , because + e2 − cos x + cos(4l + x) 1 2 4 4 2 for the considered relatively longer period transiting systems i ≈ 90◦ holds. In Borkovits et al. (2003) we only considered 1 169 1 + 3 − − + + + O 4 triple stellar systems, where the three masses were usually nearly e2 cos(l2 x) cos(5l2 x) (e2), (24) 24 24 equal, and the inner period was a few days. In such cases, the

A53, page 6 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I.

8.0 20.0 ω =0 i =0 AM1 ω =0 i =0 g =0 A1 (e2=0.3) 1 m A 1 m 2 A (e =0.3) 7.0 S1 2 2 AM6 A3 (e2=0.3) AS6 A1 (e2=0.7) 6.0 15.0 A2 (e2=0.7) A3 (e2=0.7) 5.0

4.0 10.0

3.0 Fourier coefficients

2.0 5.0

1.0

0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Inner eccentricity (e1) Inner eccentricity (e1)

2.5 6.0 ω A ω A (e =0.3) 1=90 im=0 M1 1=90 im=0 g2=0 1 2 AS1 A2 (e2=0.3) A A (e =0.3) M6 5.0 3 2 2.0 AS6 A1 (e2=0.7) A2 (e2=0.7) A3 (e2=0.7) 4.0 1.5 3.0 1.0

Fourier coefficients 2.0

0.5 1.0

0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Inner eccentricity (e1) Inner eccentricity (e1)

3.5 ω A ω A (e =0.3) 1=121 g1=90 im=39.23 M1 1=121 g1=90 im=39 g2=0 1 2 AS1 10.0 A2 (e2=0.3) 3.0 AM6 A3 (e2=0.3) AS6 A1 (e2=0.7) A (e =0.7) 8.0 2 2 2.5 A3 (e2=0.7)

2.0 6.0

1.5

Fourier coefficients 4.0 1.0

2.0 0.5

0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Inner eccentricity (e1) Inner eccentricity (e1)

4.0 6.0 ω A ω A (e =0.3) 1=0 im=90 M1 1=0 g1=45 im=90 g2=0 1 2 AS1 A2 (e2=0.3) 3.5 A A (e =0.3) M6 5.0 3 2 AS6 A1 (e2=0.7) 3.0 A2 (e2=0.7) A3 (e2=0.7) 4.0 2.5

2.0 3.0

1.5 Fourier coefficients 2.0 1.0 1.0 0.5

0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Inner eccentricity (e1) Inner eccentricity (e1)

Fig. 2. Left panels: the dependence on the inner eccentricity (e1) of the amplitudes AM, AS of the functions M, S describing the long-period dynamical part of O−C for specific values of other orbital elements. To compare the M and S terms, the former should be multiplied by e2.The thin lines (indexed by “1”) refer to the first-order approximation, and the thick ones (index “6”) to the sixth one. Right panels: the corresponding A1,2,3 amplitudes of the trigonometric representation (Eq. (32)) of the O−Cfortwodifferent outer eccentricities (e2 = 0.3ande2 = 0.7) (g2 was set to 0◦).

A53, page 7 of 25 A&A 528, A53 (2011)

e =0 e =0 g1=7 im=0 g2=90 1 g1=7 im=90 g2=90 1 0.6 e1=0.11 0.6 e1=0.11 e1=0.5 e1=0.5 e1=0.9 e1=0.9 0.4 0.4

0.2 0.2

O-C in days 0.0 O-C in days 0.0

-0.2 -0.2

-0.4 -0.4

0 20 40 60 80 100 0 20 40 60 80 100 Years Years

e =0 e =0 0.6 g1=7 im=30 g2=45 1 0.6 g1=337 im=60 g2=45 1 e1=0.11 e1=0.11 e1=0.5 e1=0.5 0.4 e1=0.9 0.4 e1=0.9

0.2 0.2

0.0 0.0 O-C in days O-C in days -0.2 -0.2

-0.4 -0.4

-0.6 -0.6

-0.8 -0.8 0 20 40 60 80 100 0 20 40 60 80 100 Years Years 0.6 0.6 e =0 e =0 g1=307 im=30 g2=45 1 g1=307 im=90 g2=45 1 e1=0.11 e1=0.11 e1=0.5 e1=0.5 0.4 e1=0.9 0.4 e1=0.9

0.2 0.2

0.0 0.0 O-C in days O-C in days

-0.2 -0.2

-0.4 -0.4

0 20 40 60 80 100 0 20 40 60 80 100 Years Years 0.8 0.8 e =0 e =0 g1=277 im=0 g2=45 1 g1=277 im=90 g2=45 1 e1=0.11 e1=0.11 e =0.5 e =0.5 0.6 1 0.6 1 e1=0.9 e1=0.9

0.4 0.4

0.2 0.2 O-C in days O-C in days

0.0 0.0

-0.2 -0.2

-0.4 -0.4 0 20 40 60 80 100 0 20 40 60 80 100 Years Years

Fig. 3. Transit timing variatons caused by a hypothetical P2 = 10 000 day-period m3 = 1 M mass, moderately eccentric e2 = 0.3 third companion for CoRoT-9b, at different initial orbital elements for four different initial inner eccentricities (e1 = 0, 0.11, 0.5, 0.9). The various initial elements ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ of each panel are as follows. Panel a): g1 = 7 , g2 = 90 , im = 0 ; b): the same, but for im = 90 ; c): g1 = 7 , g2 = 45 , im = 30 ; d): g1 = 337 , ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ g2 = 45 , im = 60 ; e): g1 = 307 , g2 = 45 , im = 30 ; f): same as previous, but for im = 90 ; g): g1 = 277 , g2 = 45 , im = 0 ; h):sameas ◦ previous, but for im = 90 . (For better comparison the curves are corrected for the different average transit periods, and zero point shifts.)

A53, page 8 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. light-time term dominates. The amplitude of the light-time effect where (up to third order in e2) is simply −3/2 = − 2 2 + 2 + A1 1 e2 e2 AS AM 2ASAM cos φ, (33)

m3 a2 sin i2 1/2 −3/2 1 1 A = 1 − e2 cos2 ω = − 2 2 + 4 2 + 2 LITE 2 2 A2 1 e2 AS e AM e ASAM cos φ, (34) m123 c 16 2 2 2 1/3 m3 Gm123 sin i2 2/3 1/2 −3/2 = − 2 2 2 1 2 1 4 2 2 2 P2 1 e2 cos ω2 A = − e e A + e A + e ASAM m 4π2 c 3 1 2 2 S 2 M 2 cos φ, (35) 123 3 9 3 m 1/2 ≈ 1.1 × 10−4 3 sin i P2/3 1 − e2 cos2 ω , (26) sin φ 2/3 2 2 2 2 φ = arctan , (36) 1 AM m123 cos φ + AS sin φ φ = arctan , (37) 2 AM where masses should be expressed in solar units, and P2 in days. cos φ + 1 e2 4 2 AS − The amplitude of the dynamically forced O C and, conse- sin φ quently, the detectability limit of such perturbations depends on φ3 = arctan , (38) cos φ + 1 e2 AM almost all the dynamical, as well as geometrical, variables. We 9 2 AS can therefore give only some limits on the detectability limit. and Nevertheless, for an easier, and somewhat general study, we sep- arate the physical and geometrical variables from each other and, = 2 + 2 furthermore, also separate the elements of the inner orbit from AS α β , (39) those of the outer perturber (with the exception of the mutual in- AM = 3γ, (40) clination). To do this, we introduce the following quantities, all of which depend on eccentricity (e1), the two types of periastron and, furthermore, arguments (g , ω ), and mutual inclination (i ), via its cosine: 1 1 m ∗ = − 2 3/2 AL AL 1 e2 . (41) 2 5 3 For a circular outer orbit (e = 0) α = + e2 (1 + cos 2g ) ∓ e sin ω + sin(ω − 2g ) 2 5 4 1 1 1 5 1 1 1 A1 = A3 = 0, (42) 2 2 5 2 −I + e (1 − cos 2g ) A2 = AS. (43) 5 4 1 1 As one can see, both sets of amplitudes, i.e. AM,S and A1,2,3 ∓ 3 − − − 2 1/2 e1 sin ω1 sin(ω1 2g1) 1 e1 , (27) are independent of the real physical parameters of the current 5 exoplanetary system. Furthermore, the dependence on the ele- 1/2 5 ments of the outer orbit (i.e. e , g ) appears only in the A β = − 1 − e2 2I e2 sin 2g ∓ e cos(ω − 2g ) , (28) 2 2 1,2,3 1 4 1 1 1 1 1 Fourier amplitudes. The masses (or more accurately mass ra- tios) and the periods and period ratios (and indirectly the physi- 4 5 1 1 γ = − + e2 cos 2g − ∓ 2e sin(ω − 2g ) − sin ω cal size of the system) only occur as a scaling parameter. In this 15 2 1 1 3 1 1 1 5 1 way, the following general statements are valid for all hierarchi- 2 4 5 2 cal triple systems (as long as the initial model assumptions are +I + e (1 − cos 2g1) − 5 2 1 valid). To get the real, i.e. physical values for the O C ampli- tudes in a given system, one must multiply the general system- ∓ 3 − − − 2 1/2 independent, dimensionless amplitudes by the system specific 2e1 sin ω1 sin(ω1 2g1) 1 e1 . (29) 5 P2 number 15 1 m3 . 16π P2 m1+m2+m3 Considering first the (nearly) coplanar case, i.e. when I ≈ Then ±1, then the (half-)amplitude of the two sinusoidals becomes ≈ 8 − 2 1/2 + 25 2 ∓ 3 P e2AM0 1 e1 1 e1 e1 sin ω1 e2, (44) − ≈ 1 M + 2 + 2S + 5 8 2 O Cdyn AL γ α β (2v2 φ) , (30) 2π 2 1/2 5 25 AS ≈ 2 1 − e e 1 ∓ e sin ω + e2. (45) 0 1 1 2 1 1 16 1 where This illustrates the above-mentioned fundamental differences to the previously-studied coplanar, e1 = 0 case (Borkovits et al. β 2003; Agol et al. 2005), as according to our new result for ec- φ = 2g + arctan (31) 2 α centric inner orbits, the dynamical term does not disappear even if the outer orbit is circular (e2 = 0). Furthermore, the ampli- tudes strongly depend on the orientation of the orbital axis with or, in trigonometric form, respect to the observer. When the apsidal line coincides (more ◦ or less) with the line of sight (i.e. ω1 = ±90 ), there can be very significant differences both in the shape and amplitude of P ∗ − O−C ≈ 1 A A sin(nv + φ ), (32) the primary (transit) and secondary (occultation) O Ccurves. dyn π L n 2 n 2 n However, when the apsidal line lies nearly in the sky, then A53, page 9 of 25 A&A 528, A53 (2011) these differences disappear. Another interesting feature of the for the cases shown in Fig. 2. Moreover, the numerically gener- ◦ ω1 = ±90 configuration is that for eclipse events that occur ated sample O−CcurvesofFig.3 also suggest this conclusion. around apastron, there is a full square under the square-root sign Nevertheless, there are several exceptions. There are particular in the S-term, with the root of e1 = 0.8, which means that this configurations where one of the two amplitudes, or even both of term would disappear in this situation. Nevertheless, for such them (and consequently, the corresponding terms), vanish. Such high eccentricity, the first order approximation is far from being a situation is discussed in Sect. 4.1. We return to this question valid, as is illustrated in Fig. 2. in detail in all of Sect. 4, where the dependence of the ampli- For highly inclined (I ≈ 0) orbits the (half-)amplitudes are tudes on other parameters will also be studied for the systems of as follows: CoRoT-9b and HD 80606b. Returning now to the LITE amplitude and comparing it with ≈ 4 − 2 1/2 − − 25 2 − e2AM90 1 e1 1 e1 (1 3cos2g1) the dynamical case, we see that an increment of P2 (keeping 5 8 5/3 P1 constant) results in an increment of ALITE/AP2−dyn by P . 3 2 ± e sin ω − 5sin(ω − 2g ) e , (46) Consequently, for more distant systems, the pure geometrical ef- 2 1 1 1 1 2 fect tends to exceed the dynamical one, as is the case in all but one (λ Tau, see e.g. Söderhjelm 1975) of the known hierarchi- 2 2 1/2 25 2 AS ≈ 1 − e 1 + e (1 + cos 2g ) cal eclipsing triple stellar systems. Nevertheless, as we illustrate 90 5 1 8 1 1 in this paper (see Fig. 5), we have a good chance of finding the 3 5 opposite situation in some of the recently discovered transiting ∓ e sin ω + sin(ω − 2g ) , (47) 2 1 1 3 1 1 exoplanetary systems. We can make the following crude esti- mation. Consider a system with a solar-like host star, and two S and the phase of the -term is simply approximately Jupiter-mass companions) (m123 ≈ m1 = 1 M , −3 −3 m = m = 10 M ), choosing A = 10 day for the case = 2 3 φ 2g2. (48) of a certain detection, then the LITE term for the most ideal 6 case gives P2 ≥ 10 d ≈ 2700 y. Alternatively, after setting In this case another parameter, namely, the periastron distance −2 −4 m3 = 10 M and allowing A = 10 for detection limit, the of the inner planet from the intersection of the two orbital planes 3 result is P2 ≥ 10 d ≈ 2.7 y. Similarly, for two Jupiter-mass (g ) also plays an important role. − 1 planet, in the coplanar case, for small e ,theA = 10 3 day limit Finally we also mention a very specific case, the maxi- 1 gives the mum eccentricity phase of the Kozai mechanism driven e-cycles. During this phase, cos 2g1 takes one definite value, namely − = − 3 2 3/2 2 cos 2g1 1. Furthermore, the mutual inclination of the two P2 ≤ 2 × 10 m3e2 1 − e P (51) orbits here reaches its minimum. The actual value depends on 2 1 both the maximum mutual inclination im or, more strictly, on condition for the detectability of a third companion by its long- j1 and the minimal inner eccentricity e1. Nevertheless, in the case of an initially almost circular inner orbit, the minimum mu- term dynamical perturbations. For the perpendicular case the tual inclination is almost independent of its maximum value and same limit is ≈ ≈ ◦ ≈ ≈ takes j1( im) 39.23 (or its retrograde counterpart, j1( im) ◦ 2 = −3/2 140.77), i.e. I 3/5. For this scenario ≤ 1 × 3 − 2 + 2 2 P2 10 m3 1 e2 1 e2 P1. (52) 2 3 ≈ 16 − 2 1/2 − 25 2 ± 9 e2AMKozai,90 1 e1 1 e1 e1 sin ω1 e2, (49) 25 16 4  We note that for m3 m1 the above equations are linear for m3, = − 2 1/2 4 − 5 2 ± 34 so it is very easy to give the limiting period P2 in the function αKozai,90 1 e1 e1 e1 sin ω1 , 25 3 25 of m3. Nevertheless, we emphasize again that there are so many √ ff 1/2 2 terms with di erent periodicities and phases that these equations β = ∓ 1 − e2 15e cos ω . (50) give only a very crude first estimation. For P = 1, 10, 100 days, Kozai,90 1 5 1 1 1 (at zero outer eccentricity) gives P2 ≤ 0.5, 50, 5000 days, re- In order to get a better overview of the parameter dependence of spectively, the formulae above, we investigate the AM,S,aswellastheA1,2,3, These results refer to the total amplitude of the O−Ccurve, amplitudes graphically. Due to the complex dependence of these i.e. the variation in the transit times during a complete revolution amplitudes on many parameters, it is difficult to give general of the distant companion, which can take as long as several years statements. Therefore we investigate only the dependence of the or decades. Naturally, the perturbations in the transit times could amplitudes on the inner eccentricity, while the effects of other be observed within a much shorter period, from the variation in parameters are considered in Sect. 4 for specific systems, where the interval between consecutive minima. This estimation can some of the parameters can be fixed. be calculated, e.g., from Eq. (30) or even directly from Eq. (15). Figure 2 shows the inner eccentricity (e1) dependence of the According to the meaning of the O−C curve, the transiting or amplitudes for some specific values of the other parameters. In eclipsing period (P) between two consecutive (let us assume the general, one sees that the e1 increase with amplitude, but this nth and (n + 1)th) minima is growth usually remains within one order of magnitude. Although the shapes of the individual O−C curves may differ significantly, Pn = tn+1 − tn = (O−C) (tn+1) − (O−C) (tn) the net amplitudes vary over a narrow range. The graphs also P 1/2 suggest another, perhaps suprising, fact that the amplitude of ≈ P + 1 A 1 − e2 Δv (1 + e cos v ) − s L 1 2n 2 2n O CP2dyn only weakly depends on the mutual inclination (im). 2π This is especially valid for medium inner eccentricities, since 2 2 × γ + 2 α + β cos(2v2n + φ) , (53) in this case all AM,S amplitudes have similar values, at least

A53, page 10 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. where up to third order in e2 Table 1. The initial parameters of the transiting planetary subsytems. 1 Δv ≈ Δl 1 + 2e2 + 2e cos v + e2 cos 2v + 3e3 cos 3v System m m P e ω i 2 2 2 2 2 2 2 2 2 2 1 2 1 1 1 1 d ◦ ◦ CoRoT − 9b 0.99 M 0.0008 M 95.2738 0.11 217 89.99 d ◦ ◦ P HD 80606b 0.97 M 0.0038 M 111.4357 0.93 121 89.32 ≈ 2π (1 + δv2) P2 P1 Notes. The parameters are taken from Deeg et al. (2010)andPont et al. ≈ 2π (1 + δv2); (54) (2009). P2 i.e., we approximated the variation of the mean anomaly of the (Deeg et al. 2010). Consequently, tidal forces (including the pos- outer companion (l2) by the constant value of sible rotational oblateness) can safely be neglected in this sys- tem. Furthermore, thanks to the relatively large absolute separa- P1 Δl2 ≈ 2π · (55) tion of the planet from its star, we can expect a large amplitude P 2 signal from the perturbations of a hypothetical, distant (but not (Here, and in the following text we neglect the pure geometri- too distant) additional companion. To illustrate this possibility, cal LITE contribution.) Then the variation in the length of the and, furthermore, to check our formulae, we both calculated and consecutive transiting periods becomes plotted the amplitudes with the measured parameters of this spe- / cific system, and carried out short-term numeric integrations for 2 1 2 15 m P4 1 − e comparison. The physical parameters and some of the orbital el- ΔP ≈− π 3 1 1 4 m 3 3/2 ements of CoRoT-9b were taken from Deeg et al. (2010). These 123 P2 1 − e2 ◦ 2 data are listed in Table 1. (We added 180 to the ω1 published in that paper, as the spectroscopic ω1 refers to the orbit of the host × + 9 2 − 2 − 9 3 γ 3e2 1 e2 sin v2 6e2 sin 2v2 e2 sin 3v2 star around the common centre of mass of the star-planet double 2 2 ff ◦ system, so it di ers by 180 from the observational argument 2 2 2 of periastron of the relative orbit of the transiting planet around +2 α + β 2 1 + 8e sin(2v2 + φ) 2 its host star.) In the case of the numerical integrations, the inner 7 9 planets were started from periastron and the outer one from its + e 1 + e2 sin(v + φ) 2 2 2 2 2 apastron. Fixing the observationally aquired data, AM,S only depend 13 9 2 + e2 1 + e sin(3v2 + φ) on two parameters, namely g1 and im. In the left panels of Fig. 4, 2 2 2 = ◦ = ◦ we plotted the im versus AM,S graphs for g1 69 and g1 158 . 1 19 We find that the amplitudes reach their extrema around these pe- + e3 sin(−v + φ) + e3 sin(5v + φ) . (56) 4 2 4 2 2 riastron arguments; i.e. for other g1 values, results occur within the areas limited by these lines6. In the middle and right panels, Comparing (the amplitude of) this result with Eqs. (1, 2) of the effect of the two other free parameters, i.e. outer eccentricity Holman & Murray (2005), one can see that the power of the (e ), and dynamical (relative) argument of periastron (g )were ff 2 2 period ratio di ers. (Furthermore, in the original paper π stands also considered. The middle panels show A1,2 for e2 = 0.3, the ◦ at the same place, i.e. in the numerator as above but later, in right panels for e2 = 0.7, for both g2 = 0 (upper panels), and ◦ Holman 2010, this was declared as an error, and it was put into g = 90 (bottom panels). (AS, plotted in the left panel is identi- ff 2 the denominator.) There is a principal di erence in the back- cal to A2 for e2 = 0, in which case A1,3 = 0.) ground. Holman & Murray (2005) state that they “estimate the These figures clearly show again that the amplitudes, variation in transit intervals between successive transits”, but and consequently, the actual full amplitude of the dynamical what they really calculate is the departure of a transiting period O−C curve remains within one order of magnitude over a wide (i.e. the interval between two successive transits) from a con- range of orbital parameters. This once more verifies the very stant mean value. This quantity was estimated in our Eq. (53). crude estimations given by Eqs. (51), (52). As a consequence, for In other words, the Lagrangian perturbation equations gives the a given system we can very easily estimate the expected ampli- instantaneous period of the perturbed system, so its integration tude of the O−C variations induced by an additional companion. gives the time left between two consecutive minima. The first For a planet-mass third body, the derivative of the perturbation equation gives the instantaneous period variation, and so the variation in the length of two con- P2 ≈ 1 m3 1 secutive transiting period can be deduced by integrating the lat- Adyn (57) 2π P2 mhost−star ter. As a consequence, the best possibility for detecting the TTV, hence for the presence of some perturber, occurs when the ab- gives a likely estimation at least in magnitude. For example, for solute value of the second derivative of the O−Cdiagram,or CoRoT-9b, practically Eq. (56) (the period variation during a revolution) is P2 maximum. We illustrate this statement in the next section for 1 1 2 −1 A = ≈ 1459d M specific systems. 2π mhost−star ≈ 2 −1 1.39d MJ ; (58) 4. Case studies i.e., a Jupiter-mass additional planet could produce 0d.001 half- 4.1. CoRoT-9b amplitude already from the distance of the Mars. (Of course, if

CoRoT-9b is a transiting giant planet that revolves around 6 Strictly speaking this argument is true only if negative values are also its host star approximately at the distance of Mercury allowed for the coefficients.

A53, page 11 of 25 A&A 528, A53 (2011)

2.0 1.0 4.5 A (g =69) A (g =69) ω A (g =69) e =0.11 ω =217 M 1 e =0.11 ω =217 e =0.3 g =0 1 1 e1=0.11 1=217 e2=0.7 g2=0 1 1 1 1 AS (g1=69) 1 1 2 2 A2 (g1=69) 4.0 A2 (g1=69) AM (g1=158) A1 (g1=158) A1 (g1=158) A (g =158) A (g =158) A (g =158) S 1 0.8 2 1 3.5 2 1 1.5 3.0 0.6 2.5 1.0 2.0 0.4

Fourier coefficients Fourier coefficients 1.5 0.5 0.2 1.0 0.5

0.0 0.0 0.0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

Mutual inclination (im) Mutual inclination (im) Mutual inclination (im)

2.0 1.0 A (g =69) ω A (g =69) ω A (g =69) e =0.11 ω =217 M 1 e1=0.11 1=217 e2=0.3 g2=90 1 1 4.5 e1=0.11 1=217 e2=0.7 g2=90 1 1 1 1 AS (g1=69) A2 (g1=69) A2 (g1=69) A (g =158) A (g =158) A (g =158) M 1 1 1 4.0 1 1 AS (g1=158) 0.8 A2 (g1=158) A2 (g1=158) 1.5 3.5

0.6 3.0 1.0 2.5 0.4 2.0 Fourier coefficients Fourier coefficients 1.5 0.5 0.2 1.0 0.5

0.0 0.0 0.0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

Mutual inclination (im) Mutual inclination (im) Mutual inclination (im)

◦ ◦ Fig. 4. Left panels: the dependence on the mutual inclination (im) of the amplitudes AM, AS for CoRoT-9b, for g1 = 69 and g1 = 158 . For other g1 values the corresponding curves run within the area limited by these lines. (The two left panels are identical.) Middle and right panels:the corresponding A1,2 amplitudes of the trigonometric representation (Eq. (32)) of the O−Cfortwodifferent outer eccentricities e2 = 0.3(middle) ◦ ◦ and e2 = 0.7(right); and outer dynamical (relative) argument of pericentrum arguments g2 = 0 (up)andg2 = 90 (down). For the sake of clarity, we did not plot the small A3 coefficients. (For e2 = 0.0theAS amplitudes of the two identical left panels are equal to A2, while A1 = A3 = 0.) e1 and ω1 are known, a more precise estimation can be provided of this paper, we suppose that in these situations the discrep- easily using the formulae of the present paper.) ancies come from the higher order perturbative terms. As was Nevertheless, while the net amplitudes usually vary in a nar- shown by Söderhjelm (1982)andFord et al. (2000, among oth- ers), the higher order contributions are the most significant for row range, the dominances of the two main terms (with period ◦ 1 e1 ∼ 0, im = (0, 1) × 180 . Although these authors only con- P2 and 2 P2) can alternate and, although it is not shown, their rel- ative phase can also vary, so the shape of the actual curves may sidered the secular, or apse-node term perturbations, the same show great diversity, as is illustrated in Figs. 3, 5.Furthermore, might be the case for the long-period ones. This might ex- for specific values of the parameters, one or other of the am- plain the better correspondence in the perpendicular configura- plitudes might disappear. At CoRoT-9b particularly interesting tion than in the coplanar one. We suppose there is some similar ◦ ◦ = ◦ is the g1 = 69 im ∼ 45 configuration since in this case both reason in the im 46 case. As in this situation the first order AM and AS disappear very close to each other. This means that contributions almost disappear, whereas the small higher order the dynamical O−C almost disappears for specific e2, g2 val- terms can also be more significant. Furthermore, numeric inte- ues. This possibility warns us that from the absence of TTV in a grations in this case show a disappearence of the identity be- + ◦ given system one cannot automatically exclude the presence of tween the g1, g1 180 initial conditions, which also suggests an additional planet (which should be observed according to its a significance of the higher order terms, because in these terms parameters). we can expect the appearance of trigonometric functions with g1 This can be seen clearly in Fig. 5, where we plotted the and 3g1 in their arguments. corresponding O−C curves for coplanar, perpendicular, and the As expected, the highly eccentric-distant-companion sce- ◦ ◦ above-mentioned interesting im = 40 and 46 configurations. In nario produces the largest amplitude TTV, at least when a com- this figure we plotted O−C curves obtained from both numeri- plete revolution is considered. Nevertheless, on a shorter time cal integrations of the three-body motions and analytical calcu- scale, the length of the observing window necessary for the de- lations with our sixth-order formula. (According to Fig. 2,for tection depends highly on the phase of the curve. To illustrate the small inner eccentricity of CoRoT-9b, e1 = 0.11, the first- this, in Fig. 6 we plotted the first, and second 8 years of the three order approximation would have given practically the same re- primary transit O−C curves for both the coplanar, and the per- sults.) Comparing the analytical and the numerical curves, the pendicular cases, shown in the first and last rows of Fig. 5.The best similarity can be seen in the perpendicular case (last row). transiting periods for each curve were calculated in the usual In the coplanar case some minor discrepancies can be observed observational manner; i.e., the time interval between the first both in the shape and amplitude, while the discrepancies are ex- (some) transits were used. In the e2 = 0.7 cases, according to ◦ pressed more in the particular im = 46 case, where for high Fig. 6, it would be unlikely to detect the perturbations within outer eccentricity (right panel in the third row) our solution fails. the first eight years, despite that they show the largest (total) (Nevertheless, the total amplitude of the analytical curve is sim- amplitude. The same conclusion can be drawn for the coplanar ilar to the numerical one in this case, too.) Although a thorough e2 = 0.3 case. The most certain detection would be possible analysis of the sources of the discrepancies is beyond the scope in the two smallest amplitude, circular e2 = 0 configurations.

A53, page 12 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I.

0.0015 0.0025 e =0.11 ω =217 g =69 i =0 e =0 g =90 num. e =0.11 ω =217 g =69 i =0 e =0.3 g =90 num. 0.0080 e =0.11 ω =217 g =69 i =0 e =0.7 g =90 num. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. LITE-0.0006d 0.0020 LITE-0.0015d LITE-0.0065d 0.0060 0.0010 0.0015 0.0040 0.0010 0.0005 0.0020 0.0005 0.0000

O-C in days O-C in days 0.0000 O-C in days 0.0000 -0.0020 -0.0005 -0.0040 -0.0005 -0.0010 -0.0015 -0.0060

-0.0010 -0.0020 -0.0080 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years 0.0015 0.0025 0.0080 e =0.11 ω =217 g =69 i =40 e =0 g =90 num. e =0.11 ω =217 g =69 i =40 e =0.3 g =90 num. e =0.11 ω =217 g =69 i =40 e =0.7 g =90 num. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. LITE-0.0006d 0.0020 LITE-0.0015d 0.0060 LITE-0.0065d

0.0010 0.0015 0.0040 0.0010 0.0020 0.0005 0.0005 0.0000

O-C in days O-C in days 0.0000 O-C in days 0.0000 -0.0020 -0.0005 -0.0040 -0.0005 -0.0010 -0.0015 -0.0060

-0.0010 -0.0020 -0.0080 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years 0.0015 0.0025 0.0080 e =0.11 ω =217 g =69 i =46 e =0 g =90 num. e =0.11 ω =217 g =69 i =46 e =0.3 g =90 num. e =0.11 ω =217 g =69 i =46 e =0.7 g =90 num. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. LITE-0.0006d 0.0020 LITE-0.0015d 0.0060 LITE-0.0065d

0.0010 0.0015 0.0040 0.0010 0.0020 0.0005 0.0005 0.0000

O-C in days O-C in days 0.0000 O-C in days 0.0000 -0.0020 -0.0005 -0.0040 -0.0005 -0.0010 -0.0015 -0.0060

-0.0010 -0.0020 -0.0080 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years 0.0015 0.0025 0.0080 e =0.11 ω =217 g =69 i =90 e =0 g =90 num. e =0.11 ω =217 g =69 i =90 e =0.3 g =90 num. e =0.11 ω =217 g =69 i =90 e =0.7 g =90 num. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. 1 1 1 m 2 2 anal. LITE-0.0007d 0.0020 LITE-0.0015d 0.0060 LITE-0.0065d

0.0010 0.0015 0.0040 0.0010 0.0020 0.0005 0.0005 0.0000

O-C in days O-C in days 0.0000 O-C in days 0.0000 -0.0020 -0.0005 -0.0040 -0.0005 -0.0010 -0.0015 -0.0060

-0.0010 -0.0020 -0.0080 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years

Fig. 5. Sample of transit timing variations caused by a hypothetical P2 = 10 000 day-period m3 = 0.005 M (≈5 MJ) mass third companion for ◦ ◦ ◦ ◦ CoRoT-9b at four different initial mutual inclinations (im = 0 ,40,46,90, from up to down), and three different initial outer eccentricities ◦ ◦ (e2 = 0, 0.3, 0.7, from left to right). The dynamical (relative) arguments of periastrons are set to g1 = 69 , g2 = 90 . The curves show the sum of the geometrical LITE, and the dynamical terms obtained both from numeric integrations, and analytic calculations up to sixth order in e1.The pure LITE contributions are also plotted separately. Note that the vertical scale of the individual columns (i.e. different e2-s) are different.

Nevertheless, if the observations start at those phases plotted in stress that, although we plotted the O−C curves with continuous the right panels (8−16 years), the pictures completely differ. In lines, in reality they would contain only 3−4 points during the this latter interval, the circular cases produce the smallest cur- phase of the seemingly abrupt jump, which is a further compli- vature O−C-s, and the discrepancy from the linear trend reaches cating factor with regards to “certain” detection. 0d.001 days (which can be considered as a limit for certain de- In the sample runs above, a moderately hierarchic scenario tection) occurring towards the end of the interval. On the other a2 ≈ 22a1 was studied. To get some picture about the lower limit hand, in the case of the highly eccentric configurations, the “mo- of the validity of our low-order, hierarchical approximation, we ment of truth” comes after some years. Nevertheless, we have to carried out other integrations for less hierarchic configurations.

A53, page 13 of 25 A&A 528, A53 (2011)

0.0006 0.016 e =0.11 ω =217 g =69 i =0 g =90 e2=0.7 e =0.11 ω =217 g =69 i =0 g =90 e2=0.7 1 1 1 m 2 e =0.3 1 1 1 m 2 e =0.3 2 0.014 2 0.0004 e2=0.0 e2=0.0

0.0002 0.012

0.0000 0.010

0.008 -0.0002

O-C in days O-C in days 0.006 -0.0004 0.004 -0.0006 0.002 -0.0008 0.000 -0.0010 0 1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 16 Years Years 0.0030 ω e =0.7 ω e =0.7 e1=0.11 1=217 g1=69 im=90 g2=90 2 e1=0.11 1=217 g1=69 im=90 g2=90 2 e2=0.3 0.002 e2=0.3 0.0025 e2=0.0 e2=0.0 0.000 0.0020

-0.002 0.0015

-0.004 O-C in days 0.0010 O-C in days

0.0005 -0.006

0.0000 -0.008

-0.0005 -0.010 0 1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 16 Years Years Fig. 6. The first and the second 8 years of O−C-s plotted in the first and the last rows of Fig. 5. The periods of the individual curves were set equal to the respective initial transiting periods.

In Fig. 7 we show the results of some of these runs, which were last row. In this case the numerical integration shows very high carried out with the same initial conditions as used in the middle amplitude apse-node scale variations that do not occur in the an- panel of the first and last rows (coplanar and perpendicular cases, alytical curve. To better compare the analytical and numerical respectively) of Fig. 5,butforP2 = 1000 days (i.e. a2 ≈ 4.8a1), long-term variations in this case, we removed the apse-node ef- P2 = 2000 days (a2 ≈ 7.6a1), and P2 ≈ 952.7days(a2 ≈ 4.6a1), fect from the numerical curve by using a quadratic term; i.e., the in which last case the two planets orbit in 1:10 mean motion (blue) O−C curve was calculated in the form of resonance. In the left panels of Fig. 7 we plot 20-year-long inter- − = + + 2 vals, while in right ones show century-long time scales. As one O C c0 c1E c2E , (59) can see, on this time scale, apse-node effects already reach or ex- where E is the cycle number. As one can see, this quadratic ceed the magnitude of the long period ones. This naturally arises (blue) curve shows similar agreement with the analytical curve, from the fact that the typical time scales of these high-amplitude which was found in the similar a2/a1 ratio non-resonant case. 2 perturbations are proportional to P2/P1; i.e., in the present cases Consequently, we can state that our long term formulae are ca- these are ∼100× faster than in the previously investigated case. pable of producing the same accuracy even around mean-motion For the sake of a better comparison, the analytical curves in the resonances. right panels were calculated by including these apse-node terms, Nevertheless, from these few arbitrary trial runs we cannot although these terms will only be presented in a forthcoming pa- make general statements about the limits of our approximations. per. Turning back to the 20-year-long integrations, one can see A detailed discussion of this point is postponed to a forthcoming that the limit of the validity of the present approximation in- paper, where we will include the apse-node time scale terms. deed strongly depends on the mutual inclination. While for the ◦ While CoRoT-9b served as an illustration for the TTVs im = 90 configurations the long period analytical results are in in the low inner eccentricity case, our next sample exoplanet remarkably good agreement with the numerical curves even for HD 80606b represents the extremely eccentric case. a2 < 5a1 (left panels of third and forth rows), for the coplanar ◦ (im = 0 ) case our approximation is clearly insufficient for such 4.2. HD 80606b low a2/a1 ratios, and even for the doubled outer period case (i.e. ≈ a2 8a1) the amplitude of the analytical curve is highly under- The high-mass gas giant exoplanet HD 80606b features an al- estimated. most 4-month long period and an extremely eccentric orbit We also investigated the case of the 1:10 mean-motion res- around its solar-type host star. It was discovered spectroscopi- onance. Our results for the perpendicular case are plotted in the cally by Naef et al. (2001). Recently both secondary occultation

A53, page 14 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I.

ω num. ω num. 0.04 e1=0.11 1=217 g1=69 im=0 e2=0.3 g2=90 e1=0.11 1=217 g1=69 im=0 e2=0.3 g2=90 d anal. 0.04 d anal. P2=1000 P2=1000 0.03

0.02 0.02

0.01 0.00

O-C in days 0.00 O-C in days

-0.01 -0.02

-0.02 -0.04 -0.03

0 5 10 15 20 0 20 40 60 80 100 Years Years 0.015 0.015 ω ω e1=0.11 1=217 g1=69 im=0 e2=0.3 g2=90 num. e1=0.11 1=217 g1=69 im=0 e2=0.3 g2=90 num. anal. anal. d d P2=2000 P2=2000 0.010 0.010

0.005 0.005 O-C in days O-C in days 0.000 0.000

-0.005 -0.005

-0.010 -0.010 0 5 10 15 20 0 20 40 60 80 100 Years Years 0.020 0.04 ω ω e1=0.11 1=217 g1=69 im=90 e2=0.3 g2=90 num. e1=0.11 1=217 g1=69 im=90 e2=0.3 g2=90 num. d anal. d anal. 0.015 P2=1000 0.03 P2=1000

0.010 0.02

0.005 0.01

O-C in days 0.000 O-C in days 0.00

-0.005 -0.01

-0.010 -0.02

-0.015 -0.03 0 5 10 15 20 0 20 40 60 80 100 Years Years 0.03 1.50 ω ω e1=0.11 1=217 g1=69 im=90 e2=0.3 g2=90 num. e1=0.11 1=217 g1=69 im=90 e2=0.3 g2=90 num. anal. anal. 0.02 d d P2=952.7 num. (quadratic) P2=952.7 1.00 0.01

0.00 0.50

-0.01 O-C in days O-C in days 0.00 -0.02

-0.03 -0.50 -0.04

-0.05 -1.00 0 5 10 15 20 0 20 40 60 80 100 Years Years Fig. 7. Checking the validity of hierarchical approximation for closer systems. In the first two rows the initial conditions were set to be the same as at the uppermost middle panel of Fig. 5, with the exception of P2 = 1000 (first row) and 2000 days (second row), i.e. a2/a1 ≈ 4.8and≈7.6, respectively, while the third and last rows have initial conditions similar to the bottom middle panel of Fig. 5, with the exception of P2 = 1000 (third row) and 952.7days(last row). This latter illustrates the case of a 1:10 mean-motion resonance. The left panels represent a 20-year-long time scale, while the right ones show the TTV behaviour during a century. In the left panel of the last row the blue (quadratic) curve shows the O−C curve calculated by including a quadratic term. See text for details.

A53, page 15 of 25 A&A 528, A53 (2011)

(with the Spitzer space telescope, Laughlin et al. 2009), and pri- moreover, mary transit (Moutou et al. 2009; Fossey et al. 2009) have been m m detected. A thorough analysis of the collected data around the T = 6 2 k(1)R5 + 1 k(2)R5 , February 2009 primary transit led to the conclusion that there is m 2 1 m 2 2 1 ⎡ 2 ⎤ a significant spin-orbit misalignment in the system, i.e. the or- m ⎢ ρ 2 R ⎥ = 2 5 ⎢ (1) + (2) 1 1 ⎥ bital plane of HD 80606b fails to coincide with the equatorial 6 R1 ⎣k2 k2 ⎦ , (67) plane of its host star (Pont et al. 2009). These facts suggest that m1 ρ2 R2 this planet might be seen at an instant close to the maximum ec- (1) 5 2 (2) 5 2 k2 R1ωz k2 R2ωz centricity phase of a Kozai cycle induced by a distant, inclined R = 1 + 2 third companion (cf. Wu & Murray 2003; Fabrycky & Tremaine Gm1 Gm2 2007). Although HD 80606 itself forms a binary with HD 80607, 3k(1) 3k(2) = 2 v2 + 2 v2 . (68) due to the large separation, one may expect an additional, not so e1 e2 distant companion for an effective Kozai mechanism. The very 4πGρ1 4πGρ2 distant companion HD 80607 may nevertheless play an indirect (1) In these equations m1, R1, ρ , k , ωz ,andve refer to the mass, role in the initial triggering of the Kozai mechanism, by the effect 1 2 1 1 radius, average density, first apsidal motion constant, uni-axial described in Takeda et al. (2009). The most important parame- rotational angular velocity, and equatorial rotational velocity of ters of the system are summarized in Table 1. the host star, respectively, while subscript 2 denotes the same Due to its very high eccentricity, the periastron distance of quantities for the inner planet. Also, the rotational term is only = − ≈ this planet is q1 a1(1 e1) 0.03 AU, at which distance the valid for non-aligned rotation, and consequently only provides a tidal forces and especially tidal dissipation should be effective. crude estimation for HD 80606b. Nevertheless, for the present (As a comparison we note that this separation corresponds to the purpose it seems satisfactory. First consider the third-body term. semi-major axis of an approx. 2 day period orbit.) Furthermore, Substituting cos 2g = −1, I2 = 3/5, we can also expect a significant relativistic contribution to the apsidal motion. Although such circumstances do not invalidate 2 − − 3π m3 P1 2 3/2 2 1/2 ff g˙ rd = 1 − e 1 − e the following conclusions (as the time scales of these e ects are 3 2 m P 2 1 significantly longer than the ones investigated), we nevertheless ⎡ 123 2 ⎤ ⎢ ⎥ provide a quantitive estimation of their effects. Furthermore, the ⎢ L 1 − e2 ⎥ × ⎢ 2 ± 3 + 2 1 1 ⎥ effect of dissipation will also be considered numerically at the ⎢3e1 1 4e1 ⎥ , (69) ⎣ 5 L 1 − e2 ⎦ end of this section. 2 2 The apsidal advance speed, averaged for one orbital revolu- from which the last term is usually omittable in hierarchical sys- tion, can be written in the following form: tems, since C1  C2, which is more expressively true for high e1. For example, in the present situation

= + − g˙1 A B cos 2g1, (60) C1 ≈ − 2 1/2 0.06 1 e2 , (70) C2 where the non-zero contributions of the third-body, tidal, and so for HD 80606b we find that relativistic terms are as follows: − ◦ 2 3/2 −1 Δg3rd ≈ 0.43 × 1 − e century . (71) 2 − = P1 − 2 1/2 2 − 1 − 2 + 2 + 3 2 C1 (This result is in excellent correspondence with Fig. 11.) A3rd AL 1 e1 I 1 e1 1 e1 I , P2 5 5 2 C2 There are several uncertainties in the calculation of the tidal k (61) contribution. While the 2 constant is relatively well-known for − ordinary stars, it is very ambigous for exoplanets. Furthermore, P1 2 1/2 2 2 2 C1 B rd = AL 1 − e 1 − e − I − e I , (62) we know nothing about the rotational velocity of HD 80606b. 3 P 1 1 1 C 2 2 For the numeric integrations, according to the tables of Claret 3 2 1 4 5T 1 + e + e R & Gimènez (1992) we thus set k(1) = 0.02 for the host star, A = 2 1 8 1 + , 2 tidal 5 5 2 (63) (2) 2 2 2 = 2a1 1 − e a 1 − e and assume k2 0.2 for its planet, which is the same order 1 1 1 of magnitude as for Jupiter and for WASP-12b (Campo et al. Gm −1 = 1 2011). The stellar rotation is set to Vrot = 1.8kms (i.e. Arel 6π , (64) −1 c2a (1 − e2) P = 27d.5; ω  = 0.228 day )(Fischer & Valenti 2005), while 1 1 rot z1 for the planet we take (arbitrarily) a one-day rotation period, i.e. −1 ω  = 6.283 day . With these values, the classical tidal contri- z2 where bution to the apsidal motion becomes ◦ −1 m m Δgtidal ≈ 0.007 century , (72) C = 1 2 Gm a 1 − e2 , 1 m 12 1 1 12 which can be neglected. 2 = L1 1 − e , (65) Finally, the relativistic contribution is estimated to be 1 m m Δ ≈ ◦ −1 = 12 3 − 2 grel 0.06 century , (73) C2 Gm123a2 1 e2 , m123 i.e. it is smaller by one magnitude than the third-body term, so = − 2 L2 1 e2, (66) does also not play any important role. A53, page 16 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I.

12.0 12.0 ω A1 ω A1 e1=0.93 1=121 g=90 im=39.23 g2=0 e1=0.93 1=301 g=90 im=39.23 g2=0 A2 A2 A A 10.0 3 10.0 3

8.0 8.0

6.0 6.0

Fourier coefficients 4.0 Fourier coefficients 4.0

2.0 2.0

0.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Outer eccentricity (e2) Outer eccentricity (e2)

Fig. 8. The A1,2,3 amplitudes for HD 80606b for primary transits (left), and secondary occultations (right), supposing that the planet is seen at the instant of the maximum phase of a Kozai-cycle.

Returning to the P2 time scale variations of the TTVs, we body case, while the curvature of the highly eccentric curve is carried out our calculations and integration runs with the sup- so small that it needs almost seven years to exceed the 0d.001 position that a second, similar mass giant planet is the source difference, which could promise certain detection. (Of course, of this comet-like orbit, which is seen at the instant of the max- this is a non-realistic ideal case, where all the transits are mea- imum eccentricity phase of the Kozai-cycle. Consequently, we sured without any observational error.) Around periastron (right ◦ ◦ set g1 = 90 ,andim = 39.23. With these values from the sixth- panel), the situation is completely different, similar to the case order formula, we get AM = −0.38, AS = 1.69 for primary of CoRoT-9b. transits, and AM = −1.06, AS = 1.72 for secondary occulta- − We also carried out numerical integrations to investigate the tions. Consequently, for primary transits, the O C is evidently possible orbital evolution of HD 80606b in the presence of such dominated by the S-term. (The negative AM indicates a simple ◦ a perturber. We have shown that both relativistic and tidal ef- 180 phase shift.) In Fig. 8 the A1,2,3 amplitudes are plotted as fects can be omitted in the present configuration of the sys- a function of the outer eccentricity (e2). Due to the more than S tem. However, this assumption is only correct in those cases four and a half-times larger primary transit amplitude, the g2 where tidally forced dissipation is not considered. Consequently, dependence of the amplitudes here are weak, and therefore we = ◦ we have integrated the dynamical evolution of the system both show only A1,2,3-s for g2 0 .InFig.9 we present both the ff − with tidal e ects (both dissipative and conservative tidal terms) numerically generated short-term O C curves and the analyti- and without them (i.e. in the frame of pure three, point-mass cally calculated cases up to the sixth order in eccentricity (see gravitational interactions). The equations of the motion (includ- Appendix B)forthreedifferent eccentricities (e2 = 0, 0.3, 0.7) − ing tidal and dissipative terms and stellar rotation) are given of the outer perturber’s orbit. We plotted the O C-s for both pri- in Borkovits et al. (2004), where the description of our in- mary transits and secondary occultations. Also shown are the tegrator can also be found. In the dissipative case, our dissi- corresponding primary minus secondary curves. pation constant was selected in such a way that it produced d −7 As one can see, the sixth-order formulae gave satisfactory Δt1 ≈ 2.16 × 10 tidal lag-time for the host-star, and Δt1 ≈ d −4 7 results even for such high eccentricities, although the analytical 5.21×10 for the planet, which are equivalent to Q1 ≈ 4.1×10 , 4 amplitudes are somewhat overestimated. Nevertheless, a more Q2 ≈ 1.7 × 10 dissipation parameters, respectively. Figure 11 detailed analysis shows that the accuracy of our formulae for shows the variation in most of the orbital elements (both dy- such high inner eccentricities strongly depends on the other or- namical and observational) for 100 and 1 million years in the bital parameters. This is illustrated even in the present situation, dynamically most-excited e2 = 0.7 case. Thin lines represent where, for the secondary-occultation curves, (which correspond the dissipative case, while thick curves show the point-mass re- ◦ to ω1 = 301 ), the discrepancies are clearly larger. From this sult. The century-long left panel demonstrates that, apart from point of view, the e2 = 0 case (first row) is even more interest- the shrinking semi-major axis, there is no detectable variation in ing, because in this situation, where the only one non-vanishing the orbital elements during such a short time interval. And, of amplitude is A2 = AS, the similar AS values would lead to al- course, if this is true for the extremum of the Kozai cycle, it is most identical O−C curves (compare the corresponding dashed more expressly valid for other situations, as this phase produces analytical curves). However, according to the numerical results the fastest orbital element variations. Furthermore, on such a (see the solid curves), this, in fact, is not the case. short time scale, the orbital variations in a point-mass or non- Considering the case of fastest possible detection of the am- Keplerian, tidal framework are indistinguishable. (Again, we do plitudes, in Fig. 10 we also plotted the first and second 8 years of not take the semi-major axis into account.) This provides further − verification of the effects discussed previously, namely that we the three primary transit O C curves shown in Fig. 9.Thetran- ff siting periods for each curves were calculated in the usual ob- have neglected the tidal and relativistic e ects completely and servational manner, i.e. with the time interval between the first considered all the orbital elements as constant. (some) transits. According to Fig. 10, in the first eight years, Now, we consider orbital shrinking due to dissipation. As i.e. after the apastron of the outer body, the fastest detection one can see, in the present situation the decrease in a1 dur- −4 would be possible in the smallest (total) amplitude circular third ing the first 100 years is Δa1 ≈ 6 × 10 R . Converting this

A53, page 17 of 25 A&A 528, A53 (2011)

0.004 0.006 ω num. ω num. ω num. e1=0.93 1=121 g1=90 e2=0 e1=0.93 1=121 g1=90 e2=0 e1=0.93 1=121 g1=90 e2=0 anal. 49.646 anal. anal. Primary transit LITE - 0.006d Secondary occultation Primary - secondary minima 0.002 0.004 49.644 0.000 0.002 49.642 -0.002 0.000 O-C in days O-C in days O-C in days 49.640 -0.004 -0.002 49.638 -0.006 -0.004 49.636 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years 0.006 0.008 e =0.93 ω =121 g =90 e =0.3 num. e =0.93 ω =121 g =90 e =0.3 num. e =0.93 ω =121 g =90 e =0.3 num. 1 1 1 2 49.646 1 1 1 2 1 1 1 2 Primary transit anal. Secondary occultation anal. Primary - secondary minima anal. 0.004 LITE - 0.007d 0.006 49.644 0.002 0.004 49.642 0.000 0.002 49.640

O-C in days -0.002 O-C in days O-C in days 0.000 49.638 -0.004 -0.002 49.636 -0.006 -0.004 49.634 -0.008 -0.006 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years 49.670 e =0.93 ω =121 g =90 e =0.7 num. e =0.93 ω =121 g =90 e =0.3 num. e =0.93 ω =121 g =90 e =0.7 num. 0.030 1 1 1 2 anal. 1 1 1 2 anal. 0.030 1 1 1 2 anal. Primary transit LITE - 0.02d Secondary occultation Primary - secondary minima 49.660 0.020 0.020 49.650 0.010 0.010 49.640

O-C in days 0.000 O-C in days O-C in days 0.000 49.630 -0.010 -0.010 49.620 -0.020 -0.020 49.610 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Years Years Years

Fig. 9. Transit timing variatons caused by a hypothetical P2 = 10 000 day-period m3 = 0.005 M (≈5 MJ) mass third companion for HD 80606b at ◦ the maximum eccentricity phase of the induced Kozai-cycles. Top: e2 = 0; middle: e2 = 0.3; bottom: e2 = 0.7(g2 = 0 in all cases); left panels: primary transits; middle panels: secondary occultations; right panels: primary minus secondary difference. See text for further details. (For better comparison the curves are corrected for the different average transit periods, and zero point shifts.)

−3 into a period variation suggests ΔP1 ≈ 10 d. This gives a the point-mass case. This manifests itself not only as different −6 −1 P˙ 1 ≈ 3 × 10 day cycle rate for one transiting period. From periods and maximum eccentricities, but also in how the argu- the point of view of an eclipsing binary observer, this is an ment of (dynamical) periastron (g1) shows both circulation and incredibly high value. For comparison, a typical, secular period libration, alternately (cf. Ford et al. 2000). Nevertheless, the de- variation rate measured for many of the eclipsing binaries is tailed investigation of the apse-node time scale behaviour of the about a 10−9−10−11 day cycle−1. Such a high rate would produce TTV, as well as other orbital elements and observables, will be 0d.001 departure in transit time during ∼26 cycles, i.e. during ap- the subject of a subsequent paper. proximately eight years. This period variation ratio is close to the ratio produced by typical long-term perturbations within a few years. This illustrates that, if the data-length is significantly 5. Conclusions shorter than the period of the long-term periodic perturbations, We have studied the long-term P time scale TTVs in transiting then the effect has arisen from such perturbations, and other ef- 2 exoplanetary systems, which feature an additional, more distant fects of i.e. orbital shrinking can overlap each other and can as (a a ) planetary, or else stellar, companion. We gave the an- a result, be misinterpreted easily. (The question of such kinds 2 1 alytical form of the O−C diagram, which describes such TTVs. of misinterpretations or false identifications were considered in Our result is an extension of our previous work Borkovits et al. general in Sect. 2 of Borkovits et al. 2005.) (2003) for arbitrary orbital elements of both the inner transit- Finally, we consider the right panel of Fig. 11, which il- ing planet and the outer companion. We showed that the depen- lustrates the frequently mentioned Kozai-mechanism (with and dence of the O−C on the orbital and physical parameters can be without tidal friction) in operation. In the present situation, be- separated into three parts. Two of these are independent of the cause of the high outer eccentricity and along with the rela- real physical parameters (i.e. masses, separations, periods) of a tively weak hierarchicity of the system (i.e. a1/a2 ≈ 0.05), the concrete system and only depend on dimensionless orbital ele- higher order terms of the perturbation function are also signifi- ments, and so can be analysed in general. For the two other kinds cant, which results in very different consecutive cycles even in of parameters, which are amplitudes and phases of trigonometric

A53, page 18 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I.

0.006 0.035 e =0.93 ω =121 g =90 i =39.23 g =0 e2=0.7 e =0.93 ω =121 g =90 i =39.23 g =0 e2=0.7 1 1 1 m 2 e =0.3 1 1 1 m 2 e =0.3 0.005 2 0.030 2 e2=0.0 e2=0.0 0.004 0.025

0.003 0.020

0.002 0.015

O-C in days 0.001 O-C in days 0.010

0.000 0.005

-0.001 0.000

-0.002 -0.005

-0.003 -0.010 0 1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 16 Years Years Fig. 10. The first and second 8 years of the left panels of Fig. 9. The periods of the individual curves was set equal to the each initial transiting periods. functions, we separated the orbital elements of the inner from the and Nesvorný (2009) beyond the evident non-hierarchical con- outer planet. The practical importance of such a separation is that figurations is similarly (or even better) applicable in the nearly in the case of any actual transiting exoplanets, if eccentricity (e1) coplanar case, too, especially when the inner orbit is nearly cir- and the observable argument of periastron (ω1) are known, e.g., cular (a special case where the first-order hierarchical approx- from spectroscopy, then the main characteristics of any, caused imation becomes insufficient, see e.g. Ford et al. 2000), but in by a possible third body, TTVs can be mapped simply by the other cases, our approach could give a faster and simpler method, variation in two free parameters (dynamical, relative argument as long as the hierarchical assumption is satisfied. Furthermore, of periastron, g1, and mutual inclination, im),whichthencan since in the hierarchical approximation, the principal small pa- be refined by the use of the other derived parameters, including rameter in the perturbation equations is the ratio of the separa- two additional parameters for the possible third body (eccen- tions instead of the mass ratio, these formulae are valid for stellar tricity, e2, and dynamical, relative argument of periastron, g2). mass objects as well, and can also be applied to planets orbiting Moreover, as the physical attributes of a given system only oc- an S-type orbit in binary stars, or even for hierarchical triple stel- cur as scaling parameters, the real amplitude of the O−C can lar systems. also be estimated for a given system, simply as a function of the We analysed the above-mentioned dimensionless amplitudes m3/P2 ratio. for different arbitrary initial parameters, as well as for two con- At this point it would be beneficial to compare our re- crete systems CoRoT-9b and HD 80606b. We found in general sults with the conclusions of Nesvorný & Morbidelli (2008) that, while the shape of the O−C strongly varies with the an- and Nesvorný (2009). These authors have investigated the same gular orbital elements, the net amplitude (departing from some problem, i.e. the fast detection of outer perturbing planets and specific configurations) depends only weakly on these elements, the determination of their orbital and physical parameters from but strongly on the eccentricities. Nevertheless, we found some = ◦ their perturbations on the transit timing of the inner planet, by situations around im 45 for the specific case of CoRoT-9b, the help of the analytical description of the perturbed transit- where the O−C almost disappeared. ing O−C curve. For the mathematical description, they use the We used CoRoT-9b and HD 80606b for case studies. Both explicit perturbation theory of Hori (1966)andDeprit (1969), giant planets revolve on several monthly orbits. The former has based on canonical transformations and on the use of Lie-series. an almost circular orbit, while the latter has a comet-like, ex- This theory does not require the hierarchical assumption; i.e., tremely eccentric orbit. These large-period systems are ideal for the α = a1/a2 parameter, although less than unity, is not re- searching for other perturbing components, since the amplitude − 2 quired to be small. This suggests a somewhat greater generality of the O C is multiplied by P1/P2 and consequently, as the mag- of the results, so it is applicable systems similar to our solar sys- nitude of the perturbations determined by P1/P2,thesameam- tem. Nevertheless, perhaps a small disadvantage to this method plitude perturbations cause a more detectable effect, if the char- over the hierarchical approximation is that, for large mutual acteristic size of the system (i.e. P1)islarger. inclinations, the number of terms in the perturbation function We also considered the questions of both detection and the needed for a given accuracy grows very fast, while our formu- correct identification of such perturbations. We emphasize again lae have the same accuracy even for the highest mutual incli- that the O−Ccurveisaveryeffective tool for detecting any nations. There is also a similar discomfort in the case of high period variations, thanks to its cumulative nature. Nevertheless, eccentricities, in which case the general formulae of Nesvorný some care is necessary. First, it has to be kept in mind that the (2009) are more sensitive to these parameters than in the hi- detectability of a period variation depends on the curvature of erarchical case. For example, we could reproduce satisfactory the O−C curve. (If the plotted O−C is simply a straight line, accuracy even for e1 = 0.9, in which case the classical formu- with any given slope, it means that the period is constant, which lae are divergent. (Although we concentrate only on the long- is known with an error equal to that slope.) This implies that period perturbations, the same is valid for the apse-node time the interval needed to detect the period variation coming from scale variations, as will be illustrated in the next paper.) As a a periodic phenomenon depends more strongly on which phase conclusion, the method used by Nesvorný & Morbidelli (2008) is observed than on the amplitude of the total variation. (We

A53, page 19 of 25 A&A 528, A53 (2011)

96.5962 96 a1 a1 96.5959 93

96.5956 90

0.9301 1.0 e1 e1

0.9299 0.5

0.9297 0.0

91.2 360 g1 g1

90.6 180

90.0 0 360 ω 121.01 ω 1 1 120.99 180

120.97 0 360 h 37.0 h 1 1 36.3 180

35.6 0 360 Ω 45.0 Ω 1 1 44.6 180

44.2 0

36.724 j j 75 1 1 36.718 55

36.712 35

i 89.8 i 110 1 1 89.6 65

89.4 20

39.242 90 im im

39.234 65

39.226 40 0 20 40 60 80 100 0 2 4 6 8 10 time [in years] time [in 105 years]

Fig. 11. Dynamical evolution of the orbital elements of HD 80606b in the presence of a hypothetical P2 = 10 000 day-period m3 = 0.005 M (≈5 MJ) mass third companion. The initial orbital elements correspond to the last row of Fig. 9. Red curves: tidal effects and dissi- pation are considered; blue curves: three point-mass model. (Relativistic contributions are omitted.) Note, the semi-major axis (a1)isgivenin R .

illustrated this possibility for both systems sampled.) We also Acknowledgements. This research made use of NASA’s Astrophysics Data illustrated that, in the case of a very eccentric third compan- System Bibliographic Services. We thank Drs. John Lee Greenfell and Imre ion, the fastest rate period variation lasts a very short interval, Barna Bíró for the linguistic corrections. We thank for the referee and the lan- guage editor for their valuable suggestions and comments. which consists of only a few transit events. This emphasizes the importance of observing all possible transit (and, of course ,oc- cultation) events, with great accuracy. Another question is the References possible misinterpretation of the O−C diagram. For example, as Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567 was shown, in the HD 80606b system we can expect a secu- Borkovits, T., Érdi, B., Forgács-Dajka, E., & Kovács, T. 2003, A&A, 398, 1091 lar period change due to tidal dissipation. On a short time scale Borkovits, T., Forgács-Dajka, E., & Regály, Zs. 2004, A&A, 426, 951 of decades it may happen that this change is of the same order Borkovits, T., Elkhateeb, M. M., Csizmadia, Sz., et al. 2005, A&A, 441, 1087 as the one expected from periodic perturbations of a third com- Borkovits, T., Forgács-Dajka, E., & Regály, Zs. 2007, A&A, 473, 191 panion. These two types of perturbations could be separated in Brown, E. W. 1936, MNRAS, 97, 62 ff Campo, C. J., Harrington, J., Hardy, R. A., et al. 2011, ApJ, 727, 125 two di erent ways. One way is simply a question of time. In the Claret, A., & Gimènez, A. 1991, A&AS, 96, 255 case of an enough long observing window, a periodic pertuba- Cabrera, J. 2010, EAS Publ. Ser., 42, 109 tion would separate from a secular one, which would in turn pro- Chandler, S. C. 1892, AJ, 11, 113 duce a parabola-like O−C continuously. But, there is also a faster Cowling, T. G. 1938, MNRAS, 98, 734 Deeg, H. J., Ocana, B., Kozhevnikov, V. P., et al. 2008, A&A, 480, 563 possibility. In the case that not only primary transits, but also Deeg, H. J., Moutou, C., Erikson, A., et al. 2010, Nature, 464, 384 secondary occultations, are observed with similar frequency and Deprit, A. 1969, Cel. Mech., 1, 12 accuracy, then, on subtracting the two (primary and secondary) Fabrycky, D. 2010, [arXiv:1006.3834] O−C curves from each other, the secular change – i.e., the effect Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298 of dissipation – would disappear, since it is similar for primary Fischer, D. A., & Valenti, J. 2005, ApJ, 622, 1102 Ford, E. B., Kozinsky, B., & Rasio, F. A. 2000, ApJ, 535, 385 transits and secondary occultations. This also makes it desirable Fossey, S. J., Waldman, I. P., & Kipping, D. M. 2009, MNRAS, 396, L16 to collect as many occultation observations as possible too. Gimènez, A., & Garcia-Pelayo, J. M. 1983, Ap&SS, 92, 203

A53, page 20 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I.

Harrington, R. S. 1968, AJ, 73, 190 while the second comes from the relation of the inclinations to Harrington, R. S. 1969, Cel. Mech., 1, 200 the orbital angular momenta. The trivial case is Hertzsprung, E. 1922, B.A.N., 1, 87 Heyl, J. S., & Gladman, B. J. 2007, MNRAS, 377, 1511 im = j1 + j2, (A.9) Holman, M. J. 2010, EAS Pub. Ser., 42, 39 Holman, M. J., & Murray, N. W. 2005, Science, 307, 1288 while the non-trivial case comes from the fact that Hori, G. 1966, PASJ, 18, 287 CC Irwin, J. B. 1952, ApJ, 116, 211 cos j = 1 , (A.10) Jordán, A., & Bakos, G. Á. 2008 ApJ, 685, 543 1 CC1 Kiseleva, L. G., Eggleton, P. P., & Mikkola, S. 1998, MNRAS, 300, 292 | × | Kipping, D. M. 2009a, MNRAS, 392, 181 C C1 sin j1 = , (A.11) Kipping, D. M. 2009b, MNRAS, 396, 1797 CC1 Kozai, Y. 1962, AJ, 67, 591 Laughlin, G., Deming, D., Langton, J., et al. 2009, Nature, 457, 562 i.e. Lee, J. W., Kim, S.-L., Kim, C.-H., et al. 2009, AJ, 137, 3181 C1 C2 Mazeh, T., & Saham, J. 1979, A&A, 77, 145 cos j1 = + I, (A.12) Miralda-Escudé, J. 2002, ApJ, 564, 1019 C C Moutou, C., Hebrard, G., Bouchy, F., et al. 2009, A&A, 498, L5 C2 Naef, D., Latham, D. W., Mayor, M., et al. 2001, A&A, 375, L27 sin j1 = sin im, (A.13) Nesvorný, D., 2009, ApJ, 701, 1116 C Nesvorný, D., & Beaugé, A. 2010, ApJ, 709, L44 where C1,2 represents the orbital angular momentum vector of Nesvorný, D., & Morbidelli, A. 2008, ApJ, 688, 636 the two orbits, C the net orbital angular momentum vector, and Pál, A. 2010, MNRAS, 409, 975 Pont, F., Hébrard, G., Irwin, J. M., et al. 2009, A&A, 502, 695 m1m2 Qian, S.-B., Liao, W.-P., Zhu, L.-Y., et al. 2009, MNRAS, 401, L34 = − 2 C1 Gm12a1 1 e1 , (A.14) Sándor, Zs., Süli, Á., Érdi, B., Pilat-Lohinger, E., & Dvorak, R. 2007, MNRAS, m12 375, 1495 m m = 12 3 − 2 Schwarz, R., Süli, Á., Dvorak, R., & Pilat-Lohinger, E. 2009, CeMDA, 104, 69 C2 Gm123a2 1 e2 , (A.15) Simon, A., Szatmáry, K., & Szabó, Gy. M. 2007, A&A, 470, 727 m123 Silvotti, R., Schuh, S., Janulis, R., et al. 2007, Nature, 449, 189 C = C1 cos j1 + C2 cos j2. (A.16) Söderhjelm, S. 1975, A&A, 42, 229 Söderhjelm, S. 1982, A&A, 107, 54 (The rotational angular momenta were neglected.) At this point Söderhjelm, S. 1984, A&A, 141, 232 we notice that for hierarchical systems it is a very reasonable to Sterne, T. E. 1939, MNRAS, 99, 451 assume Takeda, G., Kita, R., & Rasio, F. A. 2009, IAU Symp., 253, 181 Tokovinin, A. A. 1997, A&AS, 124, 75 http://www.ctio.noao.edu/ C cos j = const., (A.17) atokovinin/stars/index.php (continuously refreshed) 1 1 Tokovinin, A. A. 2008, MNRAS, 389, 925 (since the first-order, doubly averaged Hamiltonian of such sys- Tokovinin, A. A., Thomas, S., Sterzik, M., & Udry, S. 2006, A&A, 450, 681 Woltjer, J., Jr. 1922, B.A.N., 1, 93 tems does not contain its conjugated variable, h1), which prop- Wu, Y., & Murray, N. 2003, ApJ, 589, 605 erty, together with the constancy of the semi-major axis, con- nects the eccentricity (e1) with the dynamical inclination ( j1). Further relations can be written by the use of the several identities of spherical triangles, for example Appendix A: Relation between the observable, cos im = cos i1 cos i2 + sin i1 sin i2 cos(Ω2 − Ω1), (A.18) and the dynamical orbital elements. sin im sin um2 = sin i1 sin(Ω2 − Ω1), (A.19) First we emphasize that some of the relations between the ele- sin im sin um1 = sin i2 sin(Ω2 − Ω1), (A.20) ments, given in this section, are strictly valid only for the case sin im cos um2 = − cos i1 sin i2 + sin i1 cos i2 cos(Ω2 − Ω1), shown in Fig. 1. According to the actual orientations of the or- (A.21) Ω − Ω bital planes, the spherical triangle (with sides um1, um2, 1 2) = − Ω − Ω couldbeorientedindifferent ways, and so some precise discus- sin im cos um1 cos i2 sin i1 sin i2 cos i1 cos( 2 1), (A.22) sion is necessary, which is omitted in the present appendix. and similar ones can be written for the two smaller spherical The relation between the pericentrum arguments is triangles. Finally, in hierarchical systems, usually C2 C1,so we can take the following as a first approximation: ω1 = g1 + um1, (A.1) ◦ j1 = im, (A.23) ω2 = g2 + um2 + 180 . (A.2) j2 = 0, (A.24) Consequently, the true longitudes measured from the sky (u)and i0 = i2, (A.25) from the intersection of the two orbital planes (w)are h1 = um2. (A.26)

u1 = v1 + ω1, (A.3) In Fig. 1, for practical reasons, we chose the intersection of u = v + ω , (A.4) the invariable plane and the sky for the arbitrary starting point 2 2 2 Ω Ω w = v + g , (A.5) of the observable nodes ( 1, 2). Traditionally, in astrometry, 1 1 1 these quantities are measured from north to east on the sky; = − u1 um1, (A.6) nevertheless, such a practical choice is valid since the Ω-s only ◦ w2 = v2 + g2 + 180 , (A.7) appear in the equations in the form of their differences, and derivatives. (Nodes cannot be determined from photometry and = u2 − um2. (A.8) spectroscopy, as both the light curves and the radial velocity There are two relations between the mutual inclination (im)and curves are invariant for any orientation of the orbital planes pro- the two dynamical inclinations ( j1,2). One of these is trivial, jected onto the sky.)

A53, page 21 of 25 A&A 528, A53 (2011)

Appendix B: The derivatives used for calculating the long-period dynamical O–C up to sixth order in e1 The derivatives of the indirect terms are as follows: de cos ω 3 1 1 1 = A (1 + e cos v ) (1 − e2)1/2 − e sin ω I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 1 3 2 2 2 2 2 −e1 sin(ω1 − 2g1) 1 − I + 1 + I cos(2v2 + 2g2) − − + } e1 cos(ω1 2g1)2 I sin(2v2 2g2) sin i cot i 2 3 + m 1 + e2 u − e2 g + u e ω I − v + g 1/2 1 1 cos m1 1 cos(2 1 m1) 1 sin 1 1 cos(2 2 2 2) 1 − e2 5 2 1 2 3 + 1 + e2 sin u + e2 sin(2g + u ) e sin ω sin(2v + 2g ) , (B.1) 5 2 1 m1 1 1 m1 1 1 2 2

2 de f3(e1)sin2ω1 6 1 1 = A (1 + e cos v ) (1 − e2)1/2 e2 f cos 2ω I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 3 1 3 2 2 2 1 + 2e2 f cos(2ω − 2g ) − e4 f cos(2ω + 2g ) 1 − I2 + 1 + I2 cos(2v + 2g ) 1 4 1 1 6 1 5 1 1 2 2 1 − 2e2 f sin(2ω − 2g ) + e4 f sin(2ω + 2g ) 2I sin(2v + 2g ) 1 4 1 1 6 1 5 1 1 2 2 sin i cot i 4 3 + m 1 f − + e2 u + e2 g + u e2 ω I − v + g 1/2 3 1 1 cos m1 2 1 cos(2 1 m1) 1 cos 2 1 1 cos(2 2 2 2) 1 − e2 5 2 1 4 3 − 1 + e2 sin u + 2e2 sin(2g + u ) e2 cos 2ω sin(2v + 2g ) , (B.2) 5 2 1 m1 1 1 m1 1 1 2 2

3 de f6(e1)cos3ω1 9 1 1 = A (1 + e cos v ) (1 − e2)1/2 − e3 f sin 3ω I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 6 1 3 2 2 2 3 − 3e3 f sin(3ω − 2g ) − e5 sin(3ω + 2g ) 1 − I2 + 1 + I2 cos(2v + 2g ) 1 7 1 1 8 1 1 1 2 2 3 − 3e3 f cos(3ω − 2g ) + e5 cos(3ω + 2g ) 2I sin(2v + 2g ) 1 7 1 1 8 1 1 1 2 2 sin i cot i 6 3 + m 1 f 1 + e2 cos u − 3e2 cos(2g + u ) e3 sin 3ω I 1 − cos(2v + 2g ) 1/2 6 1 m1 1 1 m1 1 1 2 2 1 − e2 5 2 1 6 3 + 1 + e2 sin u + 3e2 sin(2g + u ) e3 sin 3ω sin(2v + 2g ) , (B.3) 5 2 1 m1 1 1 m1 1 1 2 2

4 de f8(e1)sin4ω1 12 1 1 = A (1 + e cos v ) (1 − e2)1/2 e4 f cos 4ω I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 8 1 3 2 2 2 3 + 4e4 f cos(4ω − 2g ) − e6 cos(4ω + 2g ) 1 − I2 + 1 + I2 cos(2v + 2g ) 1 5 1 1 5 1 1 1 2 2 3 − 4e4 f sin(4ω − 2g ) + e6 sin(4ω + 2g ) 2I sin(2v + 2g ) 1 5 1 1 5 1 1 1 2 2 sin i cot i 8 3 + m 1 f − + e2 u + e2 g + u e4 ω I − v + g 1/2 8 1 1 cos m1 4 1 cos(2 1 m1) 1 cos 4 1 1 cos(2 2 2 2) 1 − e2 5 2 1 8 3 − 1 + e2 sin u + 4e2 sin(2g + u ) e4 cos 4ω sin(2v + 2g ) , (B.4) 5 2 1 m1 1 1 m1 1 1 2 2 A53, page 22 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. 5 de cos 5ω1 1 1 = A (1 + e cos v ) (1 − e2)1/2 −3e5 sin 5ω I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 1 1 3 2 2 2 −5e5 sin(5ω − 2g ) 1 − I2 + 1 + I2 cos(2v + 2g ) 1 1 1 2 2 −5e5 cos(5ω − 2g )2I sin(2v + 2g ) 1 1 1 2 2 sin i cot i 3 + m 1 + e2 u − e2 g + u e5 ω I − v + g 1/2 2 1 1 cos m1 5 1 cos(2 1 m1) 1 sin 5 1 1 cos(2 2 2 2) 1 − e2 2 1 3 + 2 1 + e2 sin u + 5e2 sin(2g + u ) e5 sin 5ω sin(2v + 2g ) , (B.5) 2 1 m1 1 1 m1 1 1 2 2 6 de sin 6ω1 18 1 1 = A (1 + e cos v ) (1 − e2)1/2 e6 cos 6ω I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 1 3 2 2 2 +6e6 cos(6ω − 2g ) 1 − I2 + 1 + I2 cos(2v + 2g ) 1 1 1 2 2 −6e6 sin(6ω − 2g )2I sin(2v + 2g ) 1 1 1 2 2 sin i cot i 12 3 + m 1 f − 1 + e2 cos u + 6e2 cos(2g + u ) e4 cos 6ω I 1 − cos(2v + 2g ) 1/2 8 1 m1 1 1 m1 1 1 2 2 1 − e2 5 2 1 12 3 − 1 + e2 sin u + 6e2 sin(2g + u ) e6 cos 6ω sin(2v + 2g ) . (B.6) 5 2 1 m1 1 1 m1 1 1 2 2

3/2 m The direct terms are coming from (a e cos(nv)andΩ˙ cos i1) are as follows: de cos v 2 1 1 1 = A (1 + e cos v ) (1 − e2)1/2 − 1 + e2 I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 2 1 3 2 2 2 dir −e2 cos 2g 1 − I2 + 1 + I2 cos(2v + 2g ) 1 1 2 2 − 2 + + e1 sin 2g12I sin(2v2 2g2) ..., (B.7) ⎛ ⎞ ⎜ 2 ⎟ ⎜de f3(e1)cos2v⎟ 3 1 1 1 ⎝⎜ 1 ⎠⎟ = A (1 + e cos v ) (1 − e2)1/2 e2 1 + e2 + e4 I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 4 1 18 1 3 2 2 2 dir 1 53 3 − e2 1 − e2 − e4 cos 2g 1 − I2 + 1 + I2 cos(2v + 2g ) 5 1 12 1 2 1 1 2 2 1 53 3 − e2 1 − e2 − e4 sin 2g 2I sin(2v + 2g ) + ..., (B.8) 5 1 12 1 2 1 1 2 2 ⎛ ⎞ ⎜ 3 ⎟ ⎜de f6(e1)cos3v⎟ 3 1 1 ⎝⎜ 1 ⎠⎟ = A (1 + e cos v ) (1 − e2)1/2 − e4 1 + e2 I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 5 1 2 1 3 2 2 2 dir 1 49 + e2 1 + e2 − e4 cos 2g 1 − I2 + 1 + I2 cos(2v + 2g ) 5 1 1 16 1 1 2 2 1 49 + e2 1 + e2 − e4 sin 2g 2I sin(2v + 2g ) + ..., (B.9) 5 1 1 16 1 1 2 2 ⎛ ⎞ ⎜ 4 ⎟ ⎜de f8(e1)cos4v⎟ 1 1 ⎝⎜ 1 ⎠⎟ = A (1 + e cos v ) (1 − e2)1/2 e6 I2 − + 1 − I2 cos(2v + 2g ) dv L 2 2 1 2 1 3 2 2 2 dir 2 3 − e4 1 + e2 cos 2g 1 − I2 + 1 + I2 cos(2v + 2g ) 5 1 4 1 1 2 2 2 3 − e4 1 + e2 sin 2g 2I sin(2v + 2g ) + ..., (B.10) 5 1 4 1 1 2 2 ⎛ ⎞ 5 ⎜de cos 5v⎟ 1 ⎝⎜ 1 ⎠⎟ = A (1 + e cos v ) (1 − e2)1/2 e6 cos 2g 1 − I2 + 1 + I2 cos(2v + 2g ) dv L 2 2 1 2 1 1 2 2 2 dir 1 + e6 sin 2g 2I sin(2v + 2g ) + ..., (B.11) 2 1 1 2 2 A53, page 23 of 25 A&A 528, A53 (2011)

⎛ ⎞ 6 ⎜de cos 6v⎟ ⎝⎜ 1 ⎠⎟ = 0, (B.12) dv2 dir where +... refer to those terms which come from the normal force component, and will be cancelled by equal but opposite in sign direct Ω˙ -terms. Furthermore, from ⎛ ⎞ ⎜ 3/2 ⎟ ⎜ 3/2 1 − e2 ⎟ ⎜ − da 1 ⎟ P1 6 9 7 1 ⎜μ 1/2 ⎟ = A (1 + e cos v ) (1 − e2)1/2 e2 1 + e2 + e4 I2 − + 1 − I2 cos(2v + 2g ) ⎝⎜ 2 ⎠⎟ L 2 2 1 1 1 1 2 2 dv2 (1 + e1 cos v) 2π 5 16 16 3 dir 21 9 57 + e2 1 + e2 + e4 cos 2g 1 − I2 + 1 + I2 cos(2v + 2g ) 20 1 14 1 112 1 1 2 2 21 9 57 + e2 1 + e2 + e4 sin 2g 2I sin(2v + 2g ) , (B.13) 20 1 14 1 112 1 1 2 2 and from Ω˙ -term: ⎛ ⎞ ⎜ 3/2 ⎟ ⎜ Ω 1 − e2 ⎟ ⎜ d 1 ⎟ = + sin im cot i1 −2 + 3 2 + 2 + − + ⎜ cos i1 ⎟ AL (1 e2 cos v2) 1 e cos um1 e cos(um1 2g1) I 1 cos(2v2 2g2) ⎝ v + 2 ⎠ − 2 1/2 1 1 d 2 (1 e1 cos v) (1 e1) 5 2 dir 2 3 − 1 + e2 sin u + e2 sin(u + 2g ) sin(2v + 2g ) − ... (B.14) 5 2 1 m1 1 m1 1 2 2

By the use of equations above, and integrating for v2, finally we get:

P 1/2 4 6 1 1 (O − C) = 1 A 1 − e2 f (e ) + K (e ,ω ) I2 − M + 1 − I2 S(2v + 2g ) v2 2π L 1 5 1 1 5 1 1 1 3 2 2 2 51 1 1 + e2 f (e )cos2g + 2K (e ,ω ,g ) + e2K (e ,ω ,g ) 1 − I2 M + 1 + I2 S(2v + 2g ) 20 1 2 1 1 2 1 1 1 8 1 4 1 1 1 2 2 2 1 51 1 − e2 f (e )sin2g + 2K (e ,ω ,g ) + e2K (e ,ω ,g ) 2IC(2v + 2g ) 2 20 1 2 1 1 3 1 1 1 8 1 5 1 1 1 2 2 sin i cot i 2 3 1 + m 1 − 1 + e2 cos u + e2 cos(2g + u ) [1 + 2K (e ,ω )] I M− S(2v + 2g ) 1/2 1 m1 1 1 m1 1 1 1 2 2 1 − e2 5 2 2 1 1 2 3 + 1 + e2 sin u + e2 sin(2g + u ) [1 + 2K (e ,ω )] C(2v + 2g ) , (B.15) 2 5 2 1 m1 1 1 m1 1 1 1 2 2 where

3 1 5 3 7 K (e ,ω ) = ∓e sin ω + e2 f cos 2ω ± e3 f sin 3ω − e4 f cos 4ω ∓ e5 sin 5ω + e6 cos 6ω , (B.16) 1 1 1 1 1 4 1 3 1 2 1 6 1 16 1 8 1 16 1 1 64 1 1 3 1 5 K (e ,ω ,g ) = ∓e sin(ω − 2g ) + e2 f cos(2ω − 2g ) ± e3 f sin(3ω − 2g ) − e4 f cos(4ω − 2g ) 2 1 1 1 1 1 1 4 1 4 1 1 2 1 7 1 1 16 1 5 1 1 3 7 ∓ e5 sin(5ω − 2g ) + e6 cos(6ω − 2g ), (B.17) 16 1 1 1 64 1 1 1 3 1 5 K (e ,ω ,g ) = ∓e cos(ω − 2g ) − e2 f sin(2ω − 2g ) ± e3 f cos(3ω − 2g ) + e4 f sin(4ω − 2g ) 3 1 1 1 1 1 1 4 1 4 1 1 2 1 7 1 1 16 1 5 1 1 3 7 ∓ e5 cos(5ω − 2g ) − e6 sin(6ω − 2g ), (B.18) 16 1 1 1 64 1 1 1 3 K (e ,ω ,g ) = −e2 f cos(2ω + 2g ) ∓ e3 sin(3ω + 2g ) + e4 cos(4ω + 2g ), (B.19) 4 1 1 1 1 5 1 1 1 1 1 4 1 1 1 3 K (e ,ω ,g ) = −e2 f sin(2ω + 2g ) ± e3 cos(3ω + 2g ) + e4 sin(4ω + 2g ), (B.20) 5 1 1 1 1 5 1 1 1 1 1 4 1 1 1 A53, page 24 of 25 T. Borkovits et al.: Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. and 25 15 95 f = 1 + e2 + e4 + e6, (B.21) 1 8 1 8 1 64 1 31 23 f = 1 + e2 + e4, (B.22) 2 51 1 48 1 1 1 f = 1 + e2 + e4, (B.23) 3 6 1 16 1 1 1 f = 1 + e2 + e4, (B.24) 4 4 1 8 1 3 f = 1 + e2, (B.25) 5 4 1 3 f = 1 + e2, (B.26) 6 8 1 1 f = 1 + e2, (B.27) 7 2 1 3 f = 1 + e2. (B.28) 8 5 1

A53, page 25 of 25