transit timing variations for eccentric and inclined ... - IOPscience

The Astrophysical Journal, 701:1116–1122, 2009 August 20 C 2009.

doi:10.1088/0004-637X/701/2/1116

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

TRANSIT TIMING VARIATIONS FOR ECCENTRIC AND INCLINED EXOPLANETS David Nesvorny´ Department of Space Studies, Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, Colorado 80302, USA Received 2009 April 22; accepted 2009 June 22; published 2009 July 29

ABSTRACT The Transit Timing Variation (TTV) method relies on monitoring changes in timing of transits of known exoplanets. Nontransiting planets in the system can be inferred from TTVs by their gravitational interactions with the transiting planet. The TTV method is sensitive to low-mass planets that cannot be detected by other means. Inferring the orbital elements and mass of the nontransiting planets from TTVs, however, is more challenging than for other planet detection schemes. It is a difficult inverse problem. Here, we extended the new inversion method proposed by Nesvorný & Morbidelli to eccentric transiting planets and inclined orbits. We found that the TTV signal can be significantly amplified for hierarchical planetary systems with substantial orbital inclinations and/or for an eccentric transiting planet with anti-aligned orbit of the planetary companion. Thus, a fortuitous orbital setup of an exoplanetary system may significantly enhance our chances of TTV detection. We also showed that the detailed shape of the TTV signal is sensitive to the orbital inclination of the nontransiting planetary companion. The TTV detection method may thus provide important constraints on the orbital inclination of exoplanets and be used to test theories of planetary formation and evolution. Key words: planetary systems – celestial mechanics

planets are probably the easiest to detect via TTVs (Agol et al. 2005; Holman & Murray 2005). The long-term effects such as the apsidal precession produced by the perturbing planet are more difficult to detect if transit observations span only a few years (Miralda-Escudé 2002; Heyl & Gladman 2007). In addition, these effects can be masked by contributions to the apsidal precession from the oblateness of the central star, relativity and tides. The power of TTVs in finding and characterizing planetary companions of the transiting exoplanets requires the solution of a difficult inverse problem as the TTV signal can be controlled in nonintuitive ways by any combination of orbital (and resonant) frequencies of the interacting planets. The use of Fourier techniques for TTVs is also complicated by the sparse temporal sampling of orbital changes by transit observations. Therefore, the frequency peaks in the Fourier spectrum may (or may not) be aliases of unresolved frequencies. Several researchers have already used the TTV method to rule out Earth-mass and more massive planets in specific orbital locations near the known transiting planets (e.g., Steffen & Agol 2005; Agol & Steffen 2007; Alonso et al. 2008; Bean & Seifahrt 2008; Hrudková et al. 2009). They used numerical simulations to highlight parameter space where the expected TTV amplitude exceeds limits posed by transit observations. This method, however, has rather extreme CPU requirements, and so could only consider coplanar orbits. In Nesvorný & Morbidelli (2008; hereafter NM08) we developed and preliminarily tested a new inversion method that could be used to detect and characterize planets from TTVs. This method is based on perturbation theory (see Section 2). With this new method, the determination of δt(j ) for a planetary system can be achieved much faster than using direct numerical integrations. This drastically reduces the computing needs and makes the inversion problem feasible. The TTV inversion method was developed in NM08 assuming a circular orbit for the transiting planet. Most hot Jupiters are expected to have orbits circularized by tides, though it is difficult to rule out small eccentricity observationally (e.g.,

1. INTRODUCTION If the viewing geometry from Earth is favorable, an exoplanet orbiting its host star will be seen to transit in front of (primary transit) and/or behind (occultation or secondary transit) the stars’ disk. These events will happen once during the planet’s orbital period. Ignoring limb darkening, the four basic transit light curve parameters are the mid-transit time tc , transit depth Δ, total duration T, and ingress/egress duration τ . The transit parameters would be constant over the course of observations, except for variations produced by stellar activity and instrumental noise, if the transiting planet moved on a strictly Keplerian orbit. Several effects, however, can produce deviations from the Keplerian case and lead to Transit Lightcurve Variations (TLVs), where the values of transit parameters tc , Δ, T, and τ vary from transit to transit. For example, the tidal interaction between hot Jupiters (orbital periods P 3 d) and their host stars can lead to an extremely fast precession of apses that should be detectable from TLVs, given that some residual orbital eccentricity is maintained (Ragozzine & Wolf 2009). Here we study another source of TLVs, namely the one produced by the gravitational interaction between planets (MiraldaEscudé 2002). Specifically, we examine the effect of the perturbing mass on tc known as the Transit Timing Variations (TTVs). The TTV signal is a series of mid-transit times, tc (j ), with 1 j N, where N is the total number of observed transits. The orbital period of the transiting planet, P, can be estimated from this data by linear regression. When P is extracted from tc (j ), we end up with variation δt(j ) = tc (j ) − C(j )P (integer C(j ) denotes the transit cycle) that describes the difference of tc (j ) from a strictly periodic (i.e., single-frequency) signal. This variation can be produced by additional planets in the system that gravitationally perturb the orbit of the transiting planet and advance or delay individual transits. The dynamics of interacting planets can be complex. The short-period and (near-)resonant oscillations of the transiting planet’s orbit produced by gravitational perturbations from other 1116

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Laughlin et al. 2005). Planets with wider orbits and/or smaller masses are not as effected by circularization and can have substantial orbital eccentricities (e.g., the transiting planets GJ 436b with e = 0.15 and HD 147506b with e = 0.5). With surveys like Kepler (Borucki et al. 2003, 2007) that were designed to detect planets in distant orbits, and given the wide eccentricity distribution of planets discovered by radial velocity (RV) measurements, the fraction of detected transiting planets with significant eccentricities is expected to grow. It is, therefore, important to extend the NM08 method to include the case of eccentric transiting planets. Doppler RV measurements can be used to determine Mp sin i, where Mp is the mass of the planet and i is the unknown orbital inclination with respect to the plane perpendicular to the line of sight. Hence, RV measurements cannot determine the mutual inclinations of orbits, since the planetary masses are not known a priori. Indeed, there are currently no important constraints on mutual inclinations in exoplanetary systems, except those required for dynamical stability.1 This limits our ability to understand the formation and evolution of planetary systems in general. For example, several theories have been proposed to explain the wide distribution of exoplanet eccentricities, including models of (1) planet–planet scattering (Rasio & Ford 1996; Weidenschilling & Marzari 1996; Lin & Ida 1997; Moorhead & Adams 2005; Chatterjee et al. 2008; Jurić & Tremaine 2008; Ford & Rasio 2008) and (2) torques from the gaseous disk (Goldreich & Sari 2003). These theories have different implications for orbital inclinations. A wide inclination distribution is expected in (1), while (2) should lead to small inclinations. One way to test these theories is therefore to look into the inclination distribution of eccentric planets in multi-planet systems. Here, we show that TTVs are sensitive to the orbital inclination of planetary companions of the transiting planets. This method can be applied to cases where transits for at least one planet are detected. If the planetary orbits are not coplanar, the other planets may not be transiting. Yet, TTVs may provide important constraints on their orbital inclinations (and other parameters). In section Section 2, we describe modifications of the NM08 method for the case of eccentric transiting planet and/or inclined planetary orbits. The new method is tested in Section 3. In Section 4, we show how the amplitude and shape of the TTV signal changes as a function of eccentricities and inclinations, and point out examples of potential degeneracies that may complicate an unique determination of the nontransiting planet’s parameters. In a companion paper (D. Nesvorný et al. 2009, in preparation), we use the method developed here and efficient χ 2 minimization techniques to solve the TTV inversion problem for an arbitrary planetary system. 2. METHOD Due to planet–planet interactions, the true longitude of the transiting planet, θ1 , can slightly lead or trail that of the unperturbed Keplerian orbit at the time of transit, thereby producing the timing variation. For an elliptic orbit, θ1 can be 1

The angle between a planet’s orbital axis and the spin axis of its parent star can be measured via the Rossiter–McLaughlin effect. Based on observations of the Rossiter–McLaughlin effect, the XO-3 system was found to be strongly misaligned while the other 10 systems for which data are currently available are consistent with perfect alignment. This may imply two distinct modes of planet migration (see Fabrycky & Winn 2009 and the references therein).

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written as a function of the mean longitude, λ1 , as θ1 = λ1 +

∞

Hk (e1 ) sin k(λ1 − 1 ),

(1)

k=1

where e1 and 1 are the eccentricity and pericenter longitude of the transiting planet, and Hk (e1 ) are polynomials in e1 with the lowest power of Hk in e1 being k. Therefore, the TTV signal may arise from perturbations of λ1 , e1 and 1 . Since e1 can be very small we introduce regularized variables h1 = e1 sin 1 and k1 = e1 cos 1 in Equation (1). Moreover, we isolate the short period variations of orbital elements, δλ1 , δh1 , and δk1 , by writing λ1 = λ¯ 1 + δλ1 , h1 = h¯ 1 + δh1 , and k1 = k¯1 + δk1 , where h¯ 1 and k¯1 are constant (or varying slowly), λ¯ 1 = n1 t + λ(0) 1 with n1 being the mean orbital frequency of the transiting planet and λ(0) 1 its initial phase. We also denote ¯ ¯ θ1 = θ1 +δθ1 , where θ1 stands for the true longitude of a fixed (or slowly varying) elliptic orbit and δθ1 is the short-term variation produced by δλ1 , δh1 , and δk1 . From now on, we will assume that t = 0 at a selected transit. Since we are interested √ in the behavior of tc only, the impact parameter2 b ≈ 1− Δ(T /τ ) will be set to zero. The coordinate system Oxyz is then chosen so that its origin O coincides with that of the star, the x-axis points toward the observer and y-axis lies in the orbital plane of the transiting planet. Thus, the orbital inclination of the transiting planet, i1 , is zero in the reference frame used here. Note that this definition differs from that used by observers who measure the orbital inclination with respect to the plane perpendicular to the line of sight. Without perturbations, the primary transits would occur when θ¯1 = 0. It follows that λ¯ 1 = 2h¯ 1 + O(e12 ) at transits. Since δλ1 1, δh1 1 and δk1 1, Equation (1) can be linearized. Using λ¯ 1 = 2h¯ 1 , we obtain 3¯ 5¯ ¯ δθ1 = (1 + 2k1 )δλ1 + h1 δk1 − 2 + k1 δh1 + O(e12 ) . (2) 2 2 The transit timing variations δt are related to δθ1 via 3/2 1 − e12 δθ1 δt = . 2 (1 + e1 cos 1 ) n1

(3)

Therefore, up to O(e1 ), Equations (2) and (3) lead to 3 n1 δt = δλ1 − 2δh1 + (k¯1 δh1 + h¯ 1 δk1 ) . 2

(4)

For conciseness, we do not explicitly list terms O(e12 ) in the above equation. These and higher order terms can be taken into account if e1 were very large (we show in Section 3 that the linear approximation is satisfactory for e1 0.5). Note that n1 δt = δλ1 − 2δh1 for e1 = 0 in agreement with Equation (3) in NM08. In the following, we will assume that m1 , m2 m0 , where m0 , m1 , and m2 are masses of the star, inner (transiting) and outer planets, respectively, and calculate the short-period variations δλ1 , δh1 , and δk1 using linear perturbation theory (hereafter PT; Hori 1966; Deprit 1969). See NM08 for a detailed discussion of 2

The impact parameter is equal to 0 and 1 for central and grazing transits, respectively, and ranges linearly with distance between these two extremes.

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¯ j , λ¯ j denoting the usual mean ¯ j,Ω the method. With a¯ j , e¯j , i¯j , orbital elements and i1 = 0, we have ∂χ1 β ∂χ1 ∂χ1 1 2a¯ 1 , x1 δλ1 = + − y1 L1 ∂ a¯ 1 D ∂ h¯ ∂ k¯1 1 β ∂χ1 x1 ∂χ1 , δh1 = − + L1 ∂ k¯1 D ∂ λ¯ 1 β ∂χ1 y1 ∂χ1 , (5) δk1 = + L1 ∂ h¯ 1 D ∂ λ¯ 1 √ where L√ Gm0 a¯ 1 , G being the gravitational constant, 1 = √ x1 = − 2P1 sin ¯ 1 , y1 = 2P1 cos ¯ 1 , P1 = L1 (1 − β), √ 2 D = 2L1 (1 + β), and β = 1 − e¯1 . In Equation (5), function χ1 can be written as Gm2 χ1 = a¯ 2

¯ j1 ¯ j2 l,j ¯ Ck (α) i1 i2 l1 l2 sin ι e¯1 e¯2 sin k n + k2 n2 2 2 |k1|+|k2|=0 1 1

¯ 1 + k6 Ω ¯ 2 ) exp ι(k1 λ¯ 1 + k2 λ¯ 2 ) (6) × exp ι(k3 ¯ 1 + k4 ¯ 2 + k5 Ω √ l,j l,j with ι = −1, Ck (α) = C−k (α), α = a¯ 1 /a¯ 2 < 1, and multiindices l = (l1 , l2 ), j = (j1 , j2 ), and k = (k1 , k2 , k3 , k4 , k5 , k6 ).

The general properties of Equation (6) are such that 6n=1 kn = 0, and that j1 + j2 , l1 − |k3 |, l2 − |k4 |, j1 − |k5 |, and j2 − |k6 | are non-negative even integers. In the i1 = 0 case considered here it follows that j1 = k5 = 0 and j2 is an even integer. The lowest combined power of eccentricities and inclinations that appears in front of each Fourier term in Equation (6) is then l1 + l2 + j2 = |k3 | + |k4 | + |k6 | 0. l,j Coefficients Ck can be obtained: (1) in terms of Laplace (i) (α) (e.g., Brouwer & Clemence 1961; Ellis & coefficients bs/2 Murray 2000); and (2) in power series of α (e.g., Kaula 1962). These expressions are equivalent as Laplace coefficients can be expanded in power series of α. The domain of convergence of these series is given by the Sundman criterion (Sundman 1916), which limits the validity of Equation (6) to a certain range of eccentricity values. See NM08 for a discussion. In Equation (5), the derivatives of χ1 with respect to a¯ 1 , λ¯ 1 , h¯ 1 = e¯1 sin ¯ 1 , and k¯1 = e¯1 cos ¯ 1 were calculated analytically from Equation (6); they are not listed here for brevity. To summarize, transit timing variations δt can be determined from Equation (4) with δλ1 , δk1 , and δh1 given by Equation (5). This method has the same strengths as those discussed in NM08 and is also valid e1 = 0 and i2 = 0. ˇ We used a computer code (Sidlichovsk´ y 1990) to tabulate l,j coefficients Ck (and their derivatives with respect to α) for 50 values of α between 0.1 and 1. Since the coefficients vary smoothly as a function of α, their values for any specific values of α in the 0.1–1 range were obtained by interpolation from the tabulated values. To assure that the PT method is valid for moderate to large values of e2 , we used terms e2l2 in Equation (6) up to l2 = 15. We also used terms sinj2 i22 up to j2 = 6. We verified that increasing the truncation order does not significantly improve the results. Our full model uses 17,731 terms in Equation (6) with different values of k.3 If e1 = 0 or i2 = 0, the number of terms drops to 12,666 or 3385, respectively. If both e1 = 0 and i2 = 0, the number of terms is 2293. Also, a significant speed-up of the algorithm can be achieved when the maximum combined power l1 + l2 + j2 is limited to 10 (8864 terms) or 5 (1708 terms). The later option, however, would compromise the precision of the PT theory for large eccentricities.

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Figure 1. Validity domain of the PT method as a function of the perturbing planet’s inclination (i2 ) and eccentricity (e2 ). Contours show 5%, 10%, 15%, and 20% values of the QPT factor and express the fractional precision of the PT method as a function of i2 and e2 . The dashed region denotes planetary parameters for which the precision of the PT method is better than 10%. Different panels correspond to different values of a2 : (a) a2 = 0.15 AU, (b) a2 = 0.2 AU, (c) a2 = 0.32 AU, and (d) a2 = 0.49 AU. We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, and e1 = i1 = 0.

Our second code, known as ptmet.f,4 uses the tabulated l,j l,j values of Ck and dCk /dα, and calculates δt according to the PT method described above. This code first determines amplitudes of different Fourier terms for the required a1 , a2 , e1 , e2 , and i2 values. Once these amplitudes are known, the code efficiently calculates TTVs for any combination of the 1 , 2 , Ω2 , and λ(0) 2 . As implemented, this later evaluation has minimal CPU requirements. Also, due to the linear dependence of TTVs on m2 , the mass of the planetary companion can be obtained by a linear least-squares fit to transit data. 3. TESTS We performed the following experiments to test the precision of the PT method. In each test, we fixed masses m0 , m1 , and m2 and conducted a large number of exact N-body integrations with individual runs starting from different initial orbits of the two planets. The inner planet was assumed to be transiting in these tests. The orbital evolution was followed for a fixed time span, 0 < t < Tint , with the Bulirsch–Stoer integrator (Press et al. 1992). During this time span we interpolated for and recorded all transit times of the inner planet. The same initial orbits were then used to determine the transit times from the PT method. We typically used Tint = 1000 d in these tests. To start with, we considered a case with m0 = MSun and a1 = 0.1 AU. This set the orbital period of the transiting planet to P1 = 11.6 days. With Tint = 1000 days, we thus have 87 primary transits. Parameters e1 , a2 , e2 , and i2 were varied to test the precision of the perturbation method in different situations 4

Available at http://www.boulder.swri.edu/∼davidn/TTVs/ptmet.f.

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Figure 2. Sensitivity of TTVs to the inclination of the planetary companion, i2 . The solid lines denote TTVs determined numerically and plus symbols denote TTVs calculated from the PT method. There is a good agreement between the two methods. Panels correspond to different planetary systems: (a) a2 = 0.32 AU, e2 = 0 and two values of i2 (0 and 40◦ ); (b) The same as (a) but for e2 = 0.1; (c) The same as (a) but for e2 = 0.2; and (d) a2 = 0.2 AU, e2 = 0.1 and two values of i2 (0 and 40◦ ). In (c) and (d), TTVs with larger amplitude correspond to i2 = 40◦ ; the ones with smaller amplitude correspond to i2 = 0. We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU and e1 = i1 = 0. The TTV values in (a) and (b) were shifted by ±5 s for clarity.

(i1 = 0 in all these tests). The initial value of λ1 was set so that θ1 = 0 for t = 0 (i.e., λ(0) 1 ≈ 2h1 ; see Section 2). The remaining orbital angles, λ2 , 1 , 2 , and Ω2 , were all set to zero at t = 0. We found that the shape and amplitude of the TTV signal is nearly invariant to changes of λ(0) 2 . Also, the PT method precision is not sensitive to the choice of 1 , 2 , and Ω2 (we discuss the effect of 1 , 2 , and Ω2 on the shape and amplitude of TTVs in Section 4). In addition, the amplitude of the short-periodic TTV signal is independent of m1 and scales (nearly) linearly with m2 because of the linear dependence on m2 in Equation (6). We, therefore, do not need to extensively test different mass values. To quantify the precision of the PT method we defined a quality parameter, QPT, as a ratio between the rms of the TTV difference between the N-body and PT methods and the amplitude of the TTV signal. Figure 1 shows QPT as a function of i2 and e2 for four different values of a2 . These values correspond to the three nonresonant cases discussed in NM08. One new case was selected to illustrate TTVs for the planetary system with large radial separation of orbits (1/α = a2 /a1 = 4.9). We used e1 = 0. Figure 1(d) shows the result for 1/α = 4.9. The precision of the PT method is excellent in this case. For i2 = 0, QPT < 10% for e2 < 0.5. The method gradually looses precision for e2 > 0.5 because of the effects of the nearby 11:1 mean motion resonance between planets. The strength of this resonance increases with e2 so that for a2 = 0.49 AU and e2 > 0.5 the resonant variations of tc start to be important. As discussed in NM08, the resonant modes are not taken into account in the PT method; they must be removed from χ1 to avoid the divergence of series in Equation (6). The short-periodic oscillations in the underlying signal can be correctly modeled by the PT method even in the resonant case.

Figure 3. Validity domain of the PT method as a function of eccentricities of the transiting (e1 ) and perturbing planets (e2 ). Contours show 5%, 10%, 15%, and 20% values of the QPT factor and express the fractional precision of the PT method as a function of e1 and e2 . The dashed region denotes planet parameters for which the precision of the PT method is better than 10%. Different panels correspond to different values of a2 : (a) a2 = 0.15 AU, (b) a2 = 0.2 AU, (c) a2 = 0.32 AU, and (d) a2 = 0.49 AU. We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, and i1 = i2 = 0.

In Figure 1(d), QPT < 10% for e2 < 0.3 and the i2 values up to 60◦ . Interactions of planets with even very large orbital

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Figure 6. Same as Figure 5(c) but for different values of 1 : (a) 1 = 90◦ , and (b) 1 = 180◦ . We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, a2 = 0.32 AU, and i1 = i2 = 2 = 0.

Figure 4. Dependence of TTVs on the eccentricity of the transiting planet, e1 . The solid lines denote TTVs determined numerically and plus symbols denote TTVs calculated from the PT method. There is a good agreement between the two methods. Panels correspond to different planetary systems: (a) a2 = 0.2 AU, e2 = 0 and two values of e1 (0 and 0.4); and (b) a2 = 0.32 AU, e2 = 0.3 and two values of e1 (0 and 0.3). TTVs with larger amplitude correspond to larger values of e1 . We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, and i1 = i2 = 0.

Figure 7. Dependence of the TTV amplitude on the inclination of the planetary companion, i2 . The solid lines in each panel denote the TTV amplitude for four values of e2 . Different panels correspond to different values of a2 : (a) a2 = 0.15 AU, (b) a2 = 0.2 AU, (c) a2 = 0.32 AU, and (d) a2 = 0.49 AU. We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, and e1 = i1 = 0.

Figure 5. Dependence of the TTV amplitude on the eccentricity of the transiting planet, e1 . The solid lines in each panel denote the TTV amplitude for four values of e2 . Different panels correspond to different values of a2 : (a) a2 = 0.15 AU, (b) a2 = 0.2 AU, (c) a2 = 0.32 AU, and (d) a2 = 0.49 AU. We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, and i1 = i2 = 1 = 2 = 0.

inclinations can thus be properly described by the PT method. Figure 2 illustrates the effect of inclination on the shape and amplitude of the TTV signal.

Parameter space where the PT method works well shrinks for more compact planetary systems (1/α = 3.2 in Figure 1(c), 1/α ≈ 2 in Figure 1(b) and 1/α ≈ 1.5 in Figure 1(a)). As for the large e2 values, the degraded precision stems from the poor convergence of χ1 in Equation (6). For example, the expansion is not convergent for a2 = 0.32 AU and e2 0.6 (Sundman 1916; NM08). Fortunately, however, the divergence domain of χ1 largely overlaps with the orbital domain where planets are generally not stable. These cases should not therefore correspond to actual planetary systems. The convergence/ stability criteria have been discussed in NM08 for e1 = i2 = 0. The limits of validity of the PT method for large i2 values are less well understood. They may be related to orbital resonances. For example, the strength of mean motion resonances grows with i2 so that different (inclination) resonances are expected to be very important for large i2 , where the PT method fails (e.g., for i2 25◦ in Figure 1(a)). On the other hand, these highly inclined orbits may also not be stable. For example, we find that

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4. TTVS FOR e1 = 0 AND/OR i2 = 0

Figure 8. Same as Figure 7(c) but for different values of Ω2 : (a) Ω2 = 0, (b) Ω2 = 45◦ , (c) Ω2 = 90◦ , and (d) Ω2 = 135◦ . We used m0 = MSun , m2 = 10−4 m0 , a1 = 0.1 AU, a2 = 0.32 AU, and e1 = i1 = 0.

orbits with i2 30◦ are violently unstable in the most compact planetary system discussed here (1/α = 1.5; Figure 1(a)). The instability is probably related to the resonant overlap (Wisdom 1980) and/or destabilizing Kozai oscillations (Kozai 1962). A detailed analysis of the stability of inclined planetary systems is beyond the scope of this paper. Figure 3 helps us to understand the extent of the orbital eccentricity domain where the PT method performs well and where it does not. For 1/α = 3.2 (panel c) and 1/α = 4.9 (d), the results are valid up to very large values of e1 . This was not expected a priori because we used the expansion of χ1 in e1 up to l1 = 5 only and because terms O(e12 ) were neglected in Equation (4). It shows that TTVs are less sensitive to the expansion order in e1 than they are to that of e2 . Using l1 < 5, however, would be problematic and lead to a degraded precision for e1 0.2. The validity domain of the PT method shrinks both in e1 and e2 for compact planetary systems (1/α = 1.5 in Figure 3(a) and 1/α = 2 in Figure 3(b)). As we discussed above, these trends stem from the poor convergence of the χ1 expansion in cases where the apocenter distance of the inner planet, Q1 = a1 (1 + e1 ), approaches the pericenter distance of the outer planet, q2 = a2 (1 − e2 ). This limitation of the PT method, however, is not as critical as it might seem because planets with q2 ∼ Q1 are generally not stable (Gladman 1993; see NM08 for a discussion). Figure 4 shows examples of TTVs for different values of e1 . The results discussed above were obtained with Ω2 = 1 = 2 = 0. This assumption implies that the planetary orbits are aligned and do not (immediately) cross each other even for very large values of e1 and e2 . We also tested other orientations of planetary orbits and found that the precision of the PT method is nearly independent of the actual Ω2 , 1 and 2 values. Thus, the results shown in Figures 1 and 3 are representative for a wide range of planetary systems.

Here, we describe how the amplitude and shape of the TTV signal depends on the eccentricity of the transiting planet, e1 , and on the orbital inclination of the planetary companion, i2 . We start by discussing the former case. For aligned orbits (i.e., 1 = 2 ), the TTV signal amplitude usually drops by a factor of few by slightly increasing e1 from e1 = 0 (Figure 5). After reaching a minimum for e1 ≈ 0.03–0.1, the amplitude starts increasing with e1 with a gradient mainly depending on the α and e2 values. For example, it ramps up faster with e1 for small α (i.e., widely separated orbits) and large e2 (Figure 5(c) and (d)). The opposite happens for more compact planetary systems (1/α < 2) where the amplitude grows more significantly with e1 for e2 = 0 (Figure 5(a)). Overall, the amplitude for e1 = 0.3 can be up to ∼5–10 times larger than the one for e1 = 0. The TTV signal for planetary orbits that are not aligned (i.e., 1 = 2 ) shows a more monotonic dependence on eccentricities than for the aligned orbits (Figure 6). Specifically, the amplitude always grows with e1 (and e2 ). The growth rate is the strongest for 2 − 1 = 180◦ (Figure 6(b)). Cases with 0 < |2 − 1 | < 180◦ (Figure 6(a)) are intermediate between those with 2 − 1 = 0 (Figure 5(c)) and 2 − 1 = 180◦ (Figure 6(b)). These results show that the most relaxed orbits with the lowest TTV amplitude are those with aligned periapses. Also, we find that it is not overly important for the TTV amplitude whether the transits of the inner planet occur near the pericenter (1 = 0) or elsewhere along the orbit. Interestingly, the detailed shape of the TTV signal is very sensitive to the value of e1 . Figure 4 illustrated this result. Moreover, we find that the shape of the TTV signal is also sensitive to 1 and 2 . It may, therefore, be possible to obtain important constraints on the mutual orientation of orbits from a careful analysis of TTVs. We show how this can be achieved in detail in D. Nesvorný et al. (2009, in preparation). Figure 7 shows how the amplitude of the TTV signal depends on i2 for Ω2 = 0 (orbital node in the line of sight). The amplitude is generally larger than in the case with i2 = 0. For hierarchical planetary systems with 1/α 3, the amplitude grows monotonically with i2 (Figure 7(d)). For compact planetary systems, however, the amplitude may achieve a maximum for moderate values of i2 (e.g., i2 ≈ 27◦ for 1/α = 1.5 and e2 = 0.15; Figure 7(a)) and then drop for larger values of i2 . The overall amplitude change for 0 < i2 < 60◦ can be ∼50% to more than a factor of ∼5 depending on individual case. We also tested how the TTV amplitude scales with Ω2 . Given the symmetry of gravitational interactions between planets for e1 = 2 = 0, which assures that the TTV amplitude with Ω2 + 180◦ is the same as the one with Ω2 , we only tested the Ω2 values between 0 and 180◦ (Figure 8). For Ω2 = 90◦ (Figure 8(c)), the functional dependence of the TTV amplitude on i2 is significantly different from that obtained for Ω2 = 0 (Figure 8(a)). It shows a dip with the minimum at i2 ∼ 20◦ –30◦ . Cases with Ω2 = 45◦ and Ω2 = 135◦ are intermediate between those with Ω2 = 0 and Ω2 = 90◦ . While the TTV amplitude may or may not change much with i2 depending on other orbital parameters of a planetary system, we find that the shape of the TTV signal is usually diagnostic of i2 (and Ω2 ) and might thus be used to determine the mutual inclination of orbits. We showed several examples of TTVs in Figure 2, where it is clear that the shape TTV signal significantly

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changes with i2 (see panels a, b, and c for a2 = 0.32 AU). The determination of i2 should be possible in these cases from the TTV observations alone. For planetary systems with more closely packed planets (1/α 2), the effect of i2 on TTVs seems to be degenerate with the mass of the planetary companion, m2 (Figure 2(d)), so that larger mutual inclinations between orbits may be compensated by smaller values of m2 . Note that this degeneracy is different from that of the Doppler RV method. Thus, the parallel use of the TTV and RV methods for a compact planetary system where at least one planet is transiting may help to remove the degeneracy and lead to the unique determination of the orbital inclination (and mass). We leave a detailed analysis of this problem for elsewhere. 5. CONCLUSIONS In NM08, we developed a method that can be used to determine planetary parameters from the transit timing observations. This method can now be applied also in cases when the transiting planet has an eccentric orbit and/or planetary companion of the transiting planet has a significant orbital inclination. In D. Nesvorný et al. (2009, in preparation), the new method is used to solve the TTV inversion problem for an arbitrary planetary system. Here we discussed how the amplitude and shape of the TTV signal depend on various parameters of the planetary system, including planetary eccentricities, inclinations, and orientation of orbits in space. We found that the TTV amplitude can vary by up to an order of magnitude in function of these parameters. Specifically, the TTV signal is significantly amplified for a hierarchical planetary system with substantial orbital inclinations and/or in the case of eccentric transiting planet with anti-aligned orbit of the planetary companion (i.e., 2 − 1 = 180◦ ). Thus, a fortuitous orbital setup of an exoplanetary system may significantly enhance our chances of the TTV detection. Perhaps the most interesting result that comes out of this work is that the shape of the TTV signal is generally sensitive to the orbital inclination of the nontransiting planetary companion. Thus, the TTV method can provide means of determining mutual inclinations in systems in which at least one planet is transiting. This parameter cannot be determined by any other existing method. For compact planetary systems (1/α 2), for which the effect of inclinations on TTVs is degenerate with that of the mass of the planetary companion, the orbital inclination and mass can probably still be uniquely determined in most cases by combining constraints from the TTV and RV observations. The method developed here can be used to analyze TTVs found for any of the potentially hundreds of planets expected to be discovered by Kepler (Beatty & Gaudi 2008). Kepler should be able to detect transit timing variations of only a few seconds (Holman & Murray 2005; Ragozzine & Wolf 2009), which

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should easily exist in many systems, extrapolating from the RV planets (Agol et al. 2005; Fabrycky 2009). ˇ We are thankful to Miloˇs Sidlichovsk´ y for making available to us his code that we used to tabulate the perturbing function coefficients. We are also grateful to Cristián Beaugé, Alessandro Morbidelli, and Darin Ragozzine for stimulating discussions. This work was supported by the NSF AAG program. REFERENCES Agol, E., & Steffen, J. H. 2007, MNRAS, 374, 941 Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567 Alonso, R., Barbieri, M., Rabus, M., Deeg, H. J., Belmonte, J. A., & Almenara, J. M. 2008, A&A, 487, L5 Bean, J. L., & Seifahrt, A. 2008, A&A, 487, L25 Beatty, T. G., & Gaudi, B. S. 2008, ApJ, 686, 1302 Borucki, W. J., et al. 2003, Kepler Mission: A Mission to Find Earth-Size Planets in the Habitable Zone Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, ed. M. Fridlund, T. Henning, & H. Lacoste (ESA-SP-539; Noordwijk: ESA), 69 Borucki, W. J., et al. 2007, in ASP Conf. Ser. 366, KEPLER Mission Status. Transiting Extrapolar Planets Workshop, ed. C. Afonso, D. Weldrake, & T. Henning (San Francisco, CA: ASP), 309 Brouwer, D., & Clemence, G. M. 1961, Methods of Celestial Mechanics (New York: Academic) Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008, ApJ, 686, 580 Deprit, A. 1969, Celest. Mech., 1, 12 Ellis, K. M., & Murray, C. D. 2000, Icarus, 147, 129 Fabrycky, D. C. 2009, in IAU Symp. 253, Transiting Planets, ed. F. Pont, D. Sasselov, & M. Holman (Cambridge: Cambridge Univ. Press), 173 Fabrycky, D. C., & Winn, J. N. 2009, ApJ, 696, 1230 Ford, E. B., & Rasio, F. A. 2008, ApJ, 686, 621 Goldreich, P., & Sari, R. 2003, ApJ, 585, 1024 Gladman, B. 1993, Icarus, 106, 247 Heyl, J. S., & Gladman, B. J. 2007, MNRAS, 377, 1511 Holman, M. J., & Murray, N. W. 2005, Science, 307, 1288 Hori, G. 1966, PASJ, 18, 287 Hrudková, M., et al. 2009, MNRAS, submitted Jurić, M., & Tremaine, S. 2008, ApJ, 686, 603 Kaula, W. M. 1962, AJ, 67, 300 Kozai, Y. 1962, AJ, 67, 591 Laughlin, G., Marcy, G. W., Vogt, S. S., Fischer, D. A., & Butler, R. P. 2005, ApJ, 629, L121 Lin, D. N. C., & Ida, S. 1997, ApJ, 477, 781 Miralda-Escudé, J. 2002, ApJ, 564, 1019 Moorhead, A. V., & Adams, F. C. 2005, Icarus, 178, 517 Nesvorný, D., & Morbidelli, A. 2008, ApJ, 688, 636 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge: Cambridge Univ. Press) Ragozzine, D., & Wolf, A. S. 2009, ApJ, 698, 1778 Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954 ˇ Sidlichovsk´ y, M. 1990, Bull. Astron. Inst. Czech. Acad. Sci., 42, 116 Steffen, J. H., & Agol, E. 2005, MNRAS, 364, L96 Sundman, K. F. 1916, Ofversigt Finska Vetenskamps-Soc. Forh. 58 A, 24 Weidenschilling, S. J., & Marzari, F. 1996, Nature, 384, 619 Wisdom, J. 1980, AJ, 85, 1122