EXTRACTING PLANET MASS AND ECCENTRICITY FROM TTV DATA

The complex eccentricity of a planet is

$$\begin{equation} z = e e^{i \varpi }\,, \end{equation} \tag{ 11 }$$

where ϖ is the longitude of periapse. Near a first-order mean-motion resonance, this can be decomposed into free and forced parts,

$$\begin{equation} z = z_{\rm free} + z_{\rm forced}\,. \end{equation} \tag{ 12 }$$

The planet's forced eccentricity is forced by virtue of its companion's proximity to resonance. For the inner and outer planets,

$$\begin{eqnarray} &&\left(\begin{array}{@{} c @{}}z_{\rm forced} \\ z^{\prime }_{\rm forced}\end{array}\right)= -{1\over j\Delta }\left(\begin{array}{@{} c @{}}\mu ^{\prime }f(P/P^{\prime })^{1/3} \\ \mu g \end{array}\right)e^{i\lambda ^j} \ \end{eqnarray} \tag{ 13 }$$

(Equation (A15)). In the case of Δ > 0, the forced eccentricity of the inner planet is aligned with the longitude of conjunctions, and that of the outer planet is anti-aligned. The situation is reversed for Δ < 0.

The free eccentricities can take arbitrary values. They represent the degrees of freedom associated with the non-circularity of the orbits. Although there are four such degrees of freedom (corresponding to e, ϖ, e', ϖ'), they can only affect the TTV signal through the linear combination of Equation (10). The complex free eccentricities precess on a secular timescale, ∼P/μ, which is much longer than the super-period. Thus, during any short time-span TTV observation, they can be taken as constant. See Figure 1 for a schematic illustration of these concepts.

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Given two transiting planets, how much can be learned about their parameters from measuring their TTV? The analytical expressions allow for straightforward answers, sparing one the abstruseness of N-body simulations. If both transits are observed, then P, P', T, and T' are known, even in the absence of observed TTV. These yield Δ and λ^j. The TTV signals then allow one to measure four quantities: the real and imaginary parts of V and V' (via Equations (1) and (2)). Therefore, the four unknown parameters (μ, μ', and the real and imaginary parts of Z_free) could in principle be inferred by inverting Equations (8) and (9). However, as we discuss in the following, degeneracies often arise that prevent unique inversion.

The amplitudes of the complex TTV are

$$\begin{eqnarray} &&|V|\sim P{\mu ^{\prime }\over |\Delta |} \left(1+{|Z_{\rm free}|\over |\Delta |} \right) \ \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} &&|V^{\prime }|\sim P^{\prime }{\mu \over |\Delta |} \left(1+{|Z_{\rm free}|\over |\Delta |} \right), \end{eqnarray} \tag{ 15 }$$

after dropping order-unity coefficients; we also assume for the purposes of the present discussion that |Z_free/Δ| is either very large or very small (≫1 or ≪1). We infer that the TTV amplitude of the inner planet yields an upper limit on the mass of the outer planet, μ' ≲ |Δ||V|/P, and there is a corresponding limit on the inner planet's mass.⁴ But the value of μ' cannot be extracted from the TTV amplitude without knowing the value of |Z_free|: a smaller μ' can be compensated for by a higher |Z_free| without affecting the amplitude of V. Similarly, if both planets have measured TTVs, one can determine the ratio of their masses from the TTV amplitudes (within order-unity constants), but not the mass of either individually, without knowing |Z_free|.

Can the phases⁵ be used to determine mass and free eccentricity uniquely? In general this is impossible. From Equations (8) and (9), the two planets' TTV signals are exactly out of phase with each other (anti-correlated) in either the limit that |Z_free| ≪ |Δ| or |Z_free| ≫ |Δ|. This feature has been noted before and has been used to help confirm some Kepler planets (Steffen et al. 2012). In either limit, TTV signals only provide three independent quantities (|V|, |V'|, and one phase), and hence a degeneracy remains between planet mass and free eccentricity. To break it, additional information or assumptions are required, e.g., radial velocity measurements or the stability of the planetary system (Cochran et al. 2011; Fabrycky et al. 2012a). An alternative solution would be if TTV phases can be measured very accurately to discern the small deviation from anti-alignment.

Nonetheless, the phases contain important information, and in certain circumstances they can help break the degeneracy between mass and eccentricity. For ease of discussion, we first define

$$\begin{eqnarray} \phi _{\rm ttv}&\equiv & \angle (V\times {\rm sgn}\Delta) \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \, \, \, \phi _{\rm ttv}^{\prime }&\equiv &\angle (V^{\prime }\times {\rm sgn}\Delta), \end{eqnarray} \tag{ 17 }$$

where sgn Δ = 1 if Δ > 0, and −1 otherwise. With these definitions, ϕ_ttv = 0 and ϕ'_ttv = 180° when Z_free = 0, independent of the sign of Δ. Note that if ϕ_ttv = 0, then δt crosses zero from above to below whenever the longitude of conjunctions points along the line of sight (regardless of the sign of Δ); similarly, if ϕ'_ttv = 180°, δt' crosses zero from below when λ^j = 0. We consider the two possibilities.

• 1.  
  If the observed TTVs have a phase shift with respect to λ^j, that directly implies that free eccentricities are present. A large phase shift implies |Z_free| ≳ |Δ|.
• 2.  
  If there is no phase shift (i.e., ϕ_ttv = 0 and ϕ'_ttv = 180°), that does not necessarily imply that |Z_free| = 0. Instead, |Z_free| could be large but the phase of Z_free vanishes. However, such a coincidence is unlikely. Even if the phase of Z_free vanished initially, secular precession would operate on the timescale ∼P/μ to randomize the phase. So if the free eccentricity is large (|Z_free| ≫ |Δ|), TTV phases should be randomly distributed between 0 and 2π. Conversely, if |Z_free| ≲ |Δ|, phase shifts should be small. If many of the systems observed by Kepler have zero or near-zero phase shifts, one could argue that most of them have small free eccentricities (|Z_free| ≲ |Δ|). Such a result, if found true (see Section 3), has interesting implications for the dynamical history of these planets. Moreover, for the task at hand, it would break the degeneracy in the mass determination: when |Z_free/Δ| is negligible in Equations (8) and (9), μ and μ' are directly determined by the two TTV amplitudes.

Our above discussion is greatly aided by the analytical expressions. Using N-body simulations alone, it is difficult to elucidate the degeneracy between mass and free eccentricity. Moreover, the fact that TTV phases contain important information has hitherto been overlooked, only becoming transparent with the analytical formulae.

We conclude this subsection by comparing our results with previous work. Agol et al. (2005) estimate the TTV amplitudes for two near-resonant planets, assuming that the planets' initial orbits are circular. In that case, z_free = −z_forced initially, and hence Z_free ∼ μ/Δ (Equation (13)). Equations (14) and (15) then imply |V| ∼ P(μ/|Δ|)(1 + μ/|Δ|²), in agreement with Equations (29)–(31) of Agol et al. (2005) in the two corresponding limits (|Δ|²/μ ≪ 1 and ≫1). We note, though, that there is little reason why the initial orbits should be circular. If the planets suffer weak damping (e.g., by tides or a disk), that would tend to damp away the free eccentricities, but the forced eccentricities would remain intact as long as proximity to resonance is maintained. Agol et al. (2005) also analytically derive an expression for the TTV that is valid when Z_free = 0 (see their Appendix). Nesvorný & Morbidelli (2008) and Nesvorný (2009) derive a general analytic expression for the TTV, but their expressions have not been applied to Kepler planets due to their algebraic complexity, which arises partly because they do not distinguish free from forced eccentricities.

We test our TTV expressions with N-body simulations, choosing cases that illustrate our above discussion. Figure 2 shows TTVs from two separate N-body simulations for a pair of planets that resemble Kepler 18c/d, showing agreement with the predictions of Equations (1)–(10). In the top panel the planets have zero free eccentricity and hence TTVs are in phase with λ^j, while in the bottom panel the inner planet has a free eccentricity and hence the signals are not in phase. Numerically, we reach the state of zero free eccentricity by first weakly damping the planets' velocities to the local circular speed over a few million orbits. Many other forms of weak damping will also remove free eccentricities.

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Figure 3 illustrates the degeneracies inherent in extracting planet parameters from TTV, and how these can be partially removed with the phase information, as discussed in Section 2.2. In the top panel, we perform a simulation similar to the one in the top panel of Figure 2, with the same periods for the two planets, but in this case the masses of the two planets are reduced by factors of 0.2 and 0.37. Furthermore, after the free eccentricities are damped away the two planets' complex eccentricities are increased by z_free = z'_free = 0.05. The resulting TTVs are almost identical to those in the top panel of Figure 2, despite the vastly different parameters. Hence if one observed such TTVs, one could not determine the planets' masses. Nonetheless, the fact that the TTVs are in phase with λ^j in the top panel of Figure 3 is due to our judicious choice of ∠z_free = ∠z'_free = 0. Even though they are initially in phase, this cannot remain true for long: the black points in the bottom panel show the TTVs in the same simulation, but 10⁶ days later, by which time the two planets' free longitudes of periapse have precessed by ∼70°. As a result, the TTVs are no longer in phase with λ^j. By contrast, the TTVs of pairs with zero free eccentricity would always remain in phase with the longitude of conjunction (faint green line in the bottom panel of Figure 3). This example shows that if many systems are observed to have small TTV phase shifts, one could argue that most planetary systems likely have small free eccentricities, although one cannot be certain for any particular system. Furthermore, if the free eccentricities are small, one could determine planet masses from TTV data (see Figure 10).

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Given a sequence of transit times for two planets in a system, we obtain the parameters P, P', T, T', V, V' in Equations (1)–(4) as follows. We write the transit times for the inner planet as (Equation (1))

$$\begin{equation} t_{\rm trans}=T+Pi_{\rm trans}+{\rm Real}(V)\sin \lambda ^j+{\rm Imag}(V)\cos \lambda ^j, \end{equation} \tag{ 18 }$$

where i_trans = 0, 1, ... is the transit number. Since |V| ≪ P, we proceed in two steps. First, we fit the inner planet's t_trans versus i_trans with a straight line, thereby extracting P and T, and also do the same for the outer planet, extracting P', T'. Our fits are done by linear least squares (e.g., Press et al. 1992). Second, we fit for the four parameters explicit in Equation (18) (i.e., T, P, Real(V), Imag(V)) using a second least-squares fit for the inner planet's transit times; in this fit, λ^j is determined by Equations (3) and (4) at times t_trans, where the parameters T, P, T',  and P' are the ones obtained from the first fits. This refitting for P helps to remove the small linear trend that remains after the first fit. We repeat this procedure for the outer planet. This procedure implicitly assumes that the period of the TTV curve is the super-period (Equation (5)). That this assumption is valid for the six Kepler pairs of interest can be seen by comparing the fit to the data (Figures 4–6). The results of the fits are listed in Table 1.

To obtain error estimates for the best-fit values, we use the covariance matrix provided by the second least-squares fit and assume that the quoted 1σ errors in the transit time data are independent and Gaussian. The errors in periods (P, P') and fiducial transit times (T, T') are negligibly small (fractional error ≪10⁻⁴). All of our errors are quoted at 68% confidence.

Figure 7 depicts the inferred values of V and V' for the 12 planets. Figure 8, which plots just the phases, summarizes the main result of our analysis: three of the systems are consistent with having zero phase and three are not, but all six systems have |ϕ_ttv| ≲ 90°. This strongly suggests that |Z_free| ≲ |Δ| ∼ 0.01; otherwise, the six systems would have random phases between −180° and 180°.

Table 2 lists the masses that would be inferred by setting Z_free = 0 in Equations (8) and (9), i.e.,

$$\begin{eqnarray} m_{\rm nominal}&\equiv & M_* \left|{V^{\prime }\Delta \over P^{\prime }g}\right|\pi j \quad \, \, \quad \qquad \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} m^{\prime }_{\rm nominal}&\equiv & M_* \left|{V\Delta \over Pf}\right|\pi j^{2/3}(j-1)^{1/3} . \end{eqnarray} \tag{ 20 }$$

Figure 9 displays these nominal masses versus planet radii. The true mass is related to the nominal mass by

$$\begin{equation} {m\over m_{\rm nominal}}={1\over \left|1-Z_{\rm free}/(2g\Delta /3) \right| }, \end{equation} \tag{ 21 }$$

and similarly for the outer planet after replacing g → −f. Although Z_free is not known, the inference that |Z_free| ≲ |Δ| implies that the nominal masses are typically close to, but a little bigger than, the planets' true masses (Section 3.4).

From Figures 4 and 5, we see that Kepler 18, 24, and 25 all have very small phase shifts: δt crosses through zero from above at the times when λ^j = 0, and δt' crosses from below. This can also be seen in Figures 7 and 8.

Kepler 18. This system has currently one of the best measured TTVs, and its TTV phase lies very close to zero. If we assume that this system has zero free eccentricity, then we may deduce the masses of the two planets with small error bars. These values lie close to those obtained using an N-body fit (Table 2).

But to obtain these mass estimates and small error bars, we must assume that the free eccentricities vanish (or |Z_free| ≪ |Δ|). Without this assumption, the masses of the planets are degenerate with the free eccentricity: one could choose an infinite sequence of Z_free with ∠Z_free ≈ 0 or π to reproduce Kepler 18's TTV signals, each corresponding to a different set of masses. However, the secular precession of Z_free implies that a specially aligned Z_free would be unlikely. It is far more likely that the free eccentricities are very small, |Z_free| ≲ |Δ| ∼ 0.03.

Without explicitly assuming that the free eccentricity is small, Cochran et al. (2011) obtain mass estimates and errors similar to ours by fitting the TTV signal (alone) with N-body simulations (Table 2). We suggest that this result is fortuitous: their N-body simulations have not exhaustively searched all possible mass–eccentricity combinations. Nearly the same TTV signals can be produced by much lower planet masses, as can be seen by comparing the top panels of Figures 2 and 3. In truth, the TTVs in those two panels differ slightly. Such "chopping" (Holman et al. 2010; Fabrycky et al. 2012a) or other non-sinusoidal behavior might be used to break the degeneracy between mass and eccentricity. But it is not clear if the data are of high enough quality to distinguish the difference between those two panels. By contrast, in Kepler 36, for example, which lies close to a high j resonance (7:6), the TTV signals are significantly non-sinusoidal, and that likely allowed the N-body fit to break the degeneracy (Carter et al. 2012).

In the case of Kepler 18, additional non-TTV information such as radial velocity measurements and theoretical expectations for planet density allow one to eliminate most of the solutions. But that would not be possible for most systems. This example accentuates the importance of an analytical understanding: the degeneracy between mass and eccentricity is explicit in the formula.

Kepler 25. The TTV phases of Kepler 25b/c are also consistent with zero, a strong indication that the planets' free eccentricities are very small. From our fits for |V| and |V'| listed in Table 1, we derive nominal masses for the planets as listed in Table 2. These are likely close to the true planet masses. They are consistent with those from N-body fits (Table 2). However, as for the Kepler 18 case, the N-body fit without any assumption on the value of the free eccentricity is incomplete. An alternative solution that matches the observations is, e.g., inner and outer masses of 0.59 M_⊕, 3.0 M_⊕, and Z_free = −0.05.

Kepler 24. Similar arguments apply to Kepler 24, for which the phases are also consistent with zero. We derive masses of 28.4  ±  5.9 M_⊕ and 50.4  ±  7.9 M_⊕ for the inner and outer planets when we assume that the free eccentricity vanishes. Fabrycky et al. (2012a) list masses from N-body fits of 56.1 ± 15.8 M_⊕ and 102.8 ± 21.4 M_⊕, and large eccentricities, of order 0.4 and 0.3 for the inner and outer planets. However, the TTV signal suggests smaller eccentricities.⁶

With planet radii of 2.4 and 2.8 R_⊕ (Ford et al. 2012; Batalha et al. 2012), our nominal masses imply surprisingly high densities of ∼12 g  cm⁻³, comparable to the density of lead (Figure 9). However, the true masses and densities could be smaller than these nominal values by a factor of ∼2 if the free eccentricities were ∼0.01, even though in that case it would have to be a coincidence that the phase nearly vanishes.

Three out of six systems we analyze are inconsistent with zero free eccentricity: their TTV phases differ significantly from ϕ_ttv = 0° and from ϕ'_ttv = 180°. All are near 3:2 resonances. Planet masses in these cases cannot be reliably determined. However, as is apparent from Figure 8, there is an absence of systems with large phase shifts.

Kepler 23. This pair has the largest phase shift, ∼70°. The nominal masses correspond to high planet densities ${\sim } 10\,\rm g\,\, \rm cm^{-3}$, comparable to silver. Equation (21) shows that if one takes |Z_free| = 4Δ = 0.03, for instance, then the real masses would be three times lower than the nominal ones.

Kepler 28 and 32. The nominal masses in these two cases yield planet densities ${\sim } 3\,\rm g\,\, \rm cm^{-3}$. The true masses are likely smaller (Equation (21)). However, if the collective behavior of these TTV pairs indeed implies that |Z_free| ≈ |Δ|, then the true masses lie close to the nominal ones.

Given the measured TTV phases and nominal masses (Figures 8 and 9), what can be concluded about the planets' eccentricities and masses? As emphasized above, the mass–eccentricity degeneracy can only be broken in a statistical way. Because of the small number of systems analyzed in this paper, we leave a proper statistical study to future work. Instead, here we model our results by assuming that the free eccentricities of all planets are randomly drawn from a Rayleigh distribution (Binney & Tremaine 2008) with random phases, i.e., that the real and imaginary parts of the complex free eccentricities (z_free ≡ e_x + ie_y) are drawn from independent Gaussians

$$\begin{equation} P(e_x) = {1\over {\sqrt{2 \pi }}\sigma } \exp \left(- {{e_x^2}\over {2\sigma ^2}} \right), \end{equation} \tag{ 22 }$$

and e_y similarly. The top panel of Figure 10 shows the resulting phase distribution for three values $\sigma _Z = \sigma \sqrt{f^2 + g^2}$ (assuming that fΔ is fixed). From the fact that half of the systems have phases |ϕ_ttv| ≲ 10°, we infer that σ_Z ∼ 0.002–0.008. Therefore, the free eccentricities of the planets are quite small, ≲ 0.006.

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The bottom panel of Figure 10 shows the distribution of the ratio of true to nominal mass (Equation (21)). For the inferred σ_Z, we see that the true masses are comparable to the nominal masses, within around a factor of two. We emphasize, however, that it is not necessarily the case that the free eccentricities in these six systems are drawn from the same distribution. Instead, it could be that the three systems consistent with zero phase all have precisely Z_free = 0. In that case, the nominal masses for those systems would be their true masses. A better assessment of probable values awaits future data analysis.

We have derived simple analytical expressions for the TTV from two planets near a first-order mean-motion resonance (Equations (1)–(10)). These show that the amplitude and phase of the TTV depend on both planet mass and free eccentricity. There is an inherent degeneracy between mass and free eccentricity which in general prevents either from being determined independently of the other. This degeneracy, however, may be (partially) broken under certain circumstances, based on probability arguments.

There is a special moment in time when the longitude of conjunction points along the line of sight. If the phase of TTV is zero relative to this time, then it is likely that the free eccentricity in the system is zero. Moreover, if the free eccentricity is zero, the TTV amplitudes can be used to uniquely determine planet masses.

When applying this technique to six published systems, we find that three of them are consistent with zero TTV phase, while the other three deviate by less than a radian. This clustering around zero phase can be most naturally explained if all systems have free eccentricity of order a percent or less—which is comparable to the typical distance to resonance in these systems. Furthermore, because the free eccentricities are small, the nominal masses determined by TTV are likely close to the true masses, within a factor of ≲ 2.

Without the analytical TTV expressions and relying only on N-body simulations, it would be hard to reach these conclusions.

The very small free eccentricities suggest that these planets have experienced damping, as suggested also by the resonant repulsion theory (Lithwick & Wu 2012; see also Batygin & Morbidelli 2012). In that work, we found that if Kepler planets have experienced substantial energy dissipation, but substantially less angular momentum damping, the two planets will be repelled from each other. This would explain the observed pile-up of planets just wide of resonances. A corollary of this theory is that low-mass planets in the Kepler sample should have little if any free eccentricity.

Although this appears to be confirmed by three of the six systems we analyze, we are puzzled by the small but finite free eccentricities in the other three systems. Assuming that resonant repulsion indeed occurred, the planets' eccentricities would have been subsequently excited, perhaps by interplanetary interactions. Such a scenario would argue against tides as the mechanism of dissipation causing resonant repulsion, because tides would have to act over very long times to be effective. Instead, dissipation by a disk of gas or planetesimals is a more plausible damping agent.

TTV data have also been reported for Kepler 9b/c and Kepler 30b/c (Holman et al. 2010; Fabrycky et al. 2012a). These pairs contain one or two giant planets. We find preliminary evidence that these systems have large TTV phases. This, if true, will indicate the presence of large free eccentricities in systems of giant planets, in contrast to the lower mass planets discussed here. We are currently analyzing Kepler public light curve data to distill more TTV systems. One interesting issue to pursue is whether planet pairs at much shorter or much longer orbital periods have the same characteristics as those analyzed here.

Many planet pairs are also near resonance with a third planet. This may bring further complications to our TTV analysis but is not considered here.

All six systems we analyze have orbital periods between 6 and 15 days, and planet radii ranging from 2 to 7 R_⊕. We confirm the mass estimates of Cochran et al. (2011) for Kepler 18c/d, and the mass estimates of Fabrycky et al. (2012a) for Kepler 25b/c, under the assumption that these systems have |Z_free| ≲ |Δ|, consistent with their small phase. Densities of these planets range between 0.3 and $2\,\rm g\,\, \rm cm^{-3}$. For Kepler 28b/c, 32b/c, we obtain nominal mass upper limits that lead to density upper limits of ${\sim } 3\,\rm g\,\, \rm cm^{-3}$. We also argue that these upper limits are likely not too different from the real masses. For Kepler 24b/c and 23b/c, the nominal densities are ${\sim } 10\,\rm g\,\, \rm cm^{-3}$.

We derive the TTVs for two planets near a first-order (j: j − 1) mean-motion resonance, assuming that the planets are coplanar with each other and with the line of sight. We solve perturbatively, taking the following quantities to be small: the eccentricities (e, e'), the mass ratios of the planets to the star (μ, μ'), and the fractional distance to resonance (Δ). Typical values for Kepler planets are very roughly e ≲ 0.1, μ ≲ 10⁻⁴, and |Δ| ≲ 0.05. Section A.2 provides further restrictions on these parameters for our perturbative treatment to be valid.

We first give a brief overview of the calculation that follows. When the equations of motion are solved perturbatively (where the unperturbed solution is circular), the leading-order solution for the complex eccentricity is a sum of the forced and free terms (Equation (A15)); the leading-order solution for the semimajor axis is determined by the product of the free and forced eccentricities (Equation (A19)); and the perturbed solution for the longitudes is determined by the perturbed semimajor axis (divided by Δ; Equation (A22)). These solutions determine the angular positions of the planets as functions of time (via Equation (A23)). Finally, the angular positions are trivially inverted to obtain the times of transit, and hence the TTV.

The total energy, or Hamiltonian, is (e.g., Murray & Dermott 2000)

$$\begin{eqnarray} &&H=-{GM_*m\over 2a}-{GM_*m^{\prime }\over 2a^{\prime }} - {Gmm^{\prime }\over a^{\prime }}R^j, \end{eqnarray} \tag{ A1 }$$

where M_* is the stellar mass, and the disturbing function due to the j: j − 1 resonance is

$$\begin{eqnarray} &&R^j= f e\cos (\lambda ^j-\varpi) +g e^{\prime }\cos (\lambda ^j-\varpi ^{\prime }), \end{eqnarray} \tag{ A2 }$$

for

$$\begin{eqnarray} &&\lambda ^j\equiv j\lambda ^{\prime }-(j-1)\lambda . \end{eqnarray} \tag{ A3 }$$

In the above, {m, a, e, λ, ϖ, m', a', e', λ', ϖ'} are the mass and standard orbital elements for the inner (unprimed) and outer (primed) planets, following the notation of Murray & Dermott (2000). The coefficients f and g are order unity, and are functions of j and

$$\begin{equation} \alpha \equiv {a/a^{\prime }}. \end{equation} \tag{ A4 }$$

These are tabulated for a few resonances in Table 3.

Table 3. Coefficients of the Disturbing Function, Defined via Equations (A1) and (A2)

	j: j − 1a	2:1	3:2	4:3	5:4
f	$-jb_{1/2}^j-{\alpha \over 2}Db_{1/2}^j$	−1.190 + 2.20Δ	−2.025 + 6.21Δ	−2.840 + 12.20Δ	−3.650 + 20.15Δ
g	$(j-{1\over 2}) b_{1/2}^{j-1}+{\alpha \over 2} Db_{1/2}^{j-1}+{\rm indirect}_{2:1}$	0.4284-3.69Δb	2.484-5.99Δ	3.283-11.9Δ	4.084-19.86Δ

Notes. Numerical values expanded to first order in Δ, the relative distance to resonance (Equation (A16)). ^ab^j_s ≡ b_s^j(α) are Laplace coefficients and D ≡ d/dα (Murray & Dermott 2000). ^bThe 2:1 value of g contains the indirect term for an internal perturber (=−1/(2α²)). One should use this in Equations (9) and (10) for the 2:1; however, in Equation (8) one should use Z_free = fz_free + g_extz'_free, where g_ext = 0.4284-1.17Δ, appropriate for an external perturber. Nonetheless, this distinction is unlikely to be of practical importance unless one is interested in the O(Δ) corrections to the free eccentricities.

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The equations of motion are Hamilton's equations, after expressing the orbital elements in the above Hamiltonian in terms of canonical Poincaré variables (e.g., Murray & Dermott 2000). The resulting equation for the inner planet's longitude is

$$\begin{eqnarray} \quad {d\lambda \over dt}&=&{1\over \sqrt{GM_*}}{2\sqrt{a}\over m}{\partial H\over \partial a} \end{eqnarray} \tag{ A5 }$$

$$\begin{eqnarray} &\approx & \sqrt{{GM_*\over a^3}}, \quad \, \, \end{eqnarray} \tag{ A6 }$$

and similarly for λ'. We drop the derivative of the disturbing function in the above and justify this below. Variations in the semimajor axes are second order in eccentricity (see below); hence

$$\begin{equation} \left(\begin{array}{@{} c @{}}\lambda \\ \lambda ^{\prime }\end{array}\right)= \left(\begin{array}{@{} c @{}}\frac {2\pi}{P} (t-T)\\ \ms {2\pi \over P^{\prime }}(t-T^{\prime }) \end{array}\right) +\left(\begin{array}{@{} c @{}}\delta \lambda \\ \delta \lambda ^{\prime }\end{array}\right), \end{equation} \tag{ A7 }$$

where the periods are

$$\begin{eqnarray} &&P\equiv 2\pi \sqrt{a^3\over GM_*},\ P^{\prime }\equiv 2\pi \sqrt{a^{\prime 3}\over GM_*}. \end{eqnarray} \tag{ A8 }$$

The periods will henceforth be treated as constants (i.e., considered to be functions of the unperturbed semimajor axes), T and T' are constant reference times, and the variations in the longitudes δλ, δλ' = O(e²) remain to be determined.

For the eccentricity equations, we introduce the complex eccentricities (e.g., Ogilvie 2007)

$$\begin{eqnarray} z&\equiv & ee^{i\varpi } \end{eqnarray} \tag{ A9 }$$

$$\begin{eqnarray} z^{\prime }&\equiv & e^{\prime }e^{i\varpi ^{\prime }}, \end{eqnarray} \tag{ A10 }$$

in terms of which the disturbing function may be written as

$$\begin{eqnarray} &&R^j={1\over 2}(fz^*+gz^{\prime *})e^{i\lambda ^j} + {\rm c.c.}, \end{eqnarray} \tag{ A11 }$$

where c.c. denotes the complex conjugate of the preceding term. The eccentricity equation for the inner planet is

$$\begin{eqnarray} {dz\over dt}&=&{-}{1\over \sqrt{GM_*}}{2i\over m\sqrt{a}}{\partial H\over \partial z^*} \end{eqnarray} \tag{ A12 }$$

to leading order in eccentricity, and similarly for the outer planet, i.e.,

$$\begin{eqnarray} &&{d\over dt}\left(\begin{array}{@{} c @{}}z \\ z^{\prime }\end{array}\right)= i{2\pi \over P^{\prime }}\left(\begin{array}{@{} c @{}}\mu ^{\prime } f/\sqrt{\alpha } \\ \mu g \end{array}\right)e^{i\lambda ^j}, \end{eqnarray} \tag{ A13 }$$

where the mass ratios are

$$\begin{equation} \mu \equiv m/M_*, \ \mu ^{\prime }\equiv m^{\prime }/M_*. \end{equation} \tag{ A14 }$$

Solving the eccentricity equations to first order yields a sum of free and forced terms:

$$\begin{eqnarray} &&\left(\begin{array}{@{} c @{}}z \\ z^{\prime }\end{array}\right)= \left(\begin{array}{@{} c @{}}z_{\rm free} \\ z^{\prime }_{\rm free}\end{array}\right) -{1\over j\Delta }\left(\begin{array}{@{} c @{}}\mu ^{\prime }f/\sqrt{\alpha } \\ \mu g \end{array}\right)e^{i\lambda ^j}, \end{eqnarray} \tag{ A15 }$$

where the free terms are constant and the normalized distance to resonance is

$$\begin{equation} \Delta \equiv {j-1\over j}{P^{\prime }\over P}-1, \end{equation} \tag{ A16 }$$

assumed to satisfy |Δ| ≪ 1. In the above, we have used the relation:

$$\begin{equation} {d\over dt}\lambda ^j=-(j\Delta) {2\pi \over P^{\prime }} + O(e^2). \end{equation} \tag{ A17 }$$

We proceed to determine the O(e²) changes to a, a', and thereby to obtain δλ, δλ' to this order. To do so, we note that the resonant Hamiltonian (Equation (A1)) has two constants of motion in addition to the energy: K = Λ + (j − 1)(Γ + Γ') and K' = Λ' − j(Γ + Γ'), where the Λ and Γ are the usual Poincaré momenta (Murray & Dermott 2000). From the constancy of K, the variation in a over the course of the planets' orbits (i.e., δa) satisfies

$$\begin{eqnarray} &&m\sqrt{a}{\delta a\over 2a}+(j-1) \left(m\sqrt{a}{e^2\over 2}+ m^{\prime }\sqrt{a^{\prime }}{e^{\prime 2}\over 2} \right)={\rm const.}, \nonumber\\ \end{eqnarray} \tag{ A18 }$$

discarding terms of higher order in δa, δa', e²,  and e'². Inserting the eccentricities from Equation (A15), only the cross terms between the free and forced eccentricities yield time-varying components to δa, implying

$$\begin{eqnarray} &&\left(\begin{array}{@{} c @{}}\delta a/a\\ \delta a^{\prime }/a^{\prime } \end{array}\right) = \left(\begin{array}{@{} c @{}}\mu ^{\prime } \frac{j-1}{j} {\alpha }^{-1/2} \\ -\mu \end{array}\right){Z_{\rm free}^*\over \Delta }e^{i\lambda ^j}+c.c., \end{eqnarray} \tag{ A19 }$$

where

$$\begin{equation} Z_{\rm free}\equiv fz_{\rm free}+gz^{\prime }_{\rm free} \end{equation} \tag{ A20 }$$

is a weighted sum of the two planets' free eccentricities.

The equation for the longitudes (Equation (A6)) becomes, to first order in δa,

$$\begin{equation} {d\over dt}\delta \lambda =-{3\over 2}{2\pi \over P}{\delta a\over a}, \end{equation} \tag{ A21 }$$

and similarly for the outer planet. The solutions are

$$\begin{eqnarray} &&\left(\begin{array}{@{} c @{}}\delta \lambda \\ \delta \lambda ^{\prime }\end{array}\right)= \left(\begin{array}{@{} c @{}}\mu ^{\prime }{j-1\over j}\alpha ^{-2} \\ -\mu \end{array}\right){3Z_{\rm free}^*\over 2i j\Delta ^2}e^{i\lambda ^j}+c.c.. \end{eqnarray} \tag{ A22 }$$

Now, to convert from λ to θ, we must add the following, valid to first order in eccentricity,

$$\begin{eqnarray} \theta -\lambda &=&2e\sin (\lambda -\varpi) \end{eqnarray} \tag{ A23 }$$

$$\begin{eqnarray} \qquad &=&{z^*\over i}e^{i\lambda }+c.c., \, \, \end{eqnarray} \tag{ A24 }$$

and similarly for primed quantities. Since we are ultimately interested in transit times, we need only insert for z the forced eccentricity (Equation (A15)), because the free eccentricity produces a term with the same period as the transits, and hence does produce variations from transit to transit. By the same logic, we may drop the e^iλ multiplying the forced eccentricity if we choose the observer to be at angular position 0. We then have

$$\begin{eqnarray} &&\left(\begin{array}{@{} c @{}}\theta -\lambda \\ \theta ^{\prime }-\lambda ^{\prime } \end{array}\right)= {1\over j\Delta } \left(\begin{array}{@{} c @{}}\mu ^{\prime } f/\sqrt{\alpha } \\ \mu g \end{array}\right){e^{i\lambda ^j}\over i}+c.c. \end{eqnarray} \tag{ A25 }$$

The final expression for θ, θ' is given by the sum of Equation (A7) (after inserting Equation (A22)) with Equation (A25), yielding

$$\begin{eqnarray} \theta &=& {2\pi \over P}(t-T)-{2\pi \over P} \left \lbrace {V\over 2i}e^{i\lambda ^j}+c.c. \right \rbrace \end{eqnarray} \tag{ A26 }$$

$$\begin{eqnarray} \theta ^{\prime } &=& {2\pi \over P^{\prime }}(t-T^{\prime })-{2\pi \over P^{\prime }} \left\lbrace {V^{\prime }\over 2i}e^{i\lambda ^j}+c.c. \right \rbrace, \end{eqnarray} \tag{ A27 }$$

where the amplitudes are

$$\begin{eqnarray} V&=&{P\over \pi }{\mu ^{\prime }\over j\Delta }\alpha ^{-1/2} \left({-f-{j-1\over j}\alpha ^{-3/2}{3Z_{\rm free}^*\over 2\Delta }} \right) \end{eqnarray} \tag{ A28 }$$

$$\begin{eqnarray} V^{\prime }&=&{P^{\prime }\over \pi }{\mu \over j\Delta }\left(- g+{3Z_{\rm free}^*\over 2\Delta } \right). \end{eqnarray} \tag{ A29 }$$

Note that α = (1 + Δ)^−2/3(1 − 1/j)^2/3, whence follows Equation (8) in the body of the paper after dropping O(Δ) corrections. (We somewhat inconsistently keep the O(Δ) corrections to f and g in Table 3 because these can be quite large, especially near the 4:3 and 5:4 resonances). Since we choose θ = 0 to point along the line of sight, transits of the inner planet occur whenever θ/2π = 0, 1, ..., and similarly for the outer planet. We conclude that the TTV signals for the inner and outer planets are as given in Equations (1)–(10).

• 1.  
  We have assumed that the system is not locked in resonance. Resonance locking occurs when δλ ∼ unity. From Equation (A22), this implies that our expressions are valid as long as

  $$\begin{equation*} e_{\rm free} \lesssim \Delta ^2/\mu. \nonumber \end{equation*}$$

  For typical Kepler systems, |Δ| ≳ 1% and μ ≲ 10⁻⁴, and hence this assumption is usually valid.
• 2.  
  In Equation (A6), we dropped the derivative of the disturbing function. This term is ∼μe, and hence is smaller than the term that is kept by ∼Δ (Equation (A21)).
• 3.  
  We have neglected secular effects. Secular precession occurs on the timescale ∼P/μ, which is much longer than the timescale of the resonant effects considered here, ∼P/Δ. Hence secular effects only lead to corrections to our formulae of order |μ/Δ| ≪ 1. Nonetheless, secular precession causes Z_free to precess on thousand-year timescales, which randomizes the orientation of Z_free. For that reason, we argue in this paper that if the data require ∠Z_free = 0 (or π) for many systems, it suggests that their Z_free nearly vanishes.
• 4.  
  The timescale for general relativistic precession is also too long to be important, specifically (dω/dt)⁻¹ ≈ Pac²/(6πGM_*) ≈ 4000 × P^5/3_5 dayyear radian⁻¹.
• 5.  
  We assume perfect coplanarity. The small inclination dispersion inferred for Kepler multi-planet systems justifies this assumption.