On detecting terrestrial planets with timing of giant planet transits

Mon. Not. R. Astron. Soc. 59, 567 579 (005) doi:10.1111/j.165-966.005.089.x On detecting terrestrial planets with timing of giant planet transits Eric Agol, 1 Jason Steffen, 1 Re em Sari and Will Clarkson 1 Astronomy Department, University of Washington, Box 51580, Seattle, WA 98195, USA Theoretical Astrophysics, MS 10-, Caltech, Pasadena, CA 9115, USA Department of Physics and Astronomy, The Open University, Milton Keynes MK7 6AA Accepted 005 February 11. Received 005 February 4; in original form 004 November 4 ABSTRACT The transits of a distant star by a planet on a Keplerian orbit occur at time intervals exactly equal to the orbital period. If a second planet orbits the same star, the orbits are not Keplerian and the transits are no longer exactly periodic. We compute the magnitude of the variation in the timing of the transits, δt. We investigate analytically several limiting cases: (i) interior perturbing planets with much smaller periods; (ii) exterior perturbing planets on eccentric orbits with much larger periods; (iii) both planets on circular orbits with arbitrary period ratio but not in resonance; (iv) planets on initially circular orbits locked in resonance. Using subscripts out and in for the exterior and interior planets, µ for planet-to-star mass ratio and the standard notation for orbital elements, our findings in these cases are as follows. (i) Planet planet perturbations are negligible. The main effect is the wobble of the star due to the inner planet, and therefore δt µ in (a in /a out )P out. (ii) The exterior planet changes the period of the interior planet by µ out (a in /r out ) P in.asthe distance of the exterior planet changes due to its eccentricity, the inner planet s period changes. Deviations in its transit timing accumulate over the period of the outer planet, and therefore δt µ out e out (a in /a out ) P out. (iii) Halfway between resonances the perturbations are small, of the order of µ out a in /(a in a out ) P in for the inner planet (switch out and in for the outer planet). This increases as one gets closer to a resonance. (iv) This is perhaps the most interesting case because some systems are known to be in resonances and the perturbations are the largest. As long as the perturber is more massive than the transiting planet, the timing variations would be of the order of the period regardless of the perturber mass. For lighter perturbers, we show that the timing variations are smaller than the period by the perturber-to-transiting-planet mass ratio. An earth-mass planet in : 1 resonance with a three-dimensional period transiting planet (e.g. HD 09458b) would cause timing variations of the order of min, which would be accumulated over a year. This signal of a terrestrial planet is easily detectable with current ground-based measurements. For the case in which both planets are on eccentric orbits, we compute numerically the transit timing variations for several known multiplanet systems, assuming they are edge-on. Transit timing measurements may be used to constrain the masses, radii and orbital elements of planetary systems, and, when combined with radial velocity measurements, provide a new means of measuring the mass and radius of the host star. Keywords: eclipses planetary systems. Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018 1 INTRODUCTION The recent discovery of planets orbiting other stars ( exoplanets ) has opened a new field of astronomy with the potential to address fundamental questions about our own Solar system, which we can now compare with other planetary systems. The primary mode for the discovery of exoplanets has been the measurement of the stellar E-mail: agol@astro.washington.edu radial velocities via the Doppler effect. Currently, the small reflex motion of the star due to the orbiting planet can only be detected for m planet 10 m (Butler et al. 004; McArthur et al. 004; Santos et al. 004). More recently, a large number of planetary transit searches are being carried out, which are starting to yield a handful of giant planets (Charbonneau et al. 000; Konacki et al. 00; Alonso et al. 004; Bouchy et al. 004; Konacki et al. 004; Pont et al. 004), and many more planned searches should reap a large harvest of transiting planets in the near future (Horne 00). Despite these successes, the discovery of terrestrial exoplanets, C 005 RAS

568 E. Agol et al. similar in size to the Earth, awaits the development of several other techniques, such as astrometry, space-based transit searches, microlensing or direct imaging (Perryman 000; Borucki et al. 00; Charbonneau 00; Ford & Tremaine 00; Gould, Gaudi & Han 004). The first transiting planetary system, HD 09458b, was discovered with Doppler motions of the primary star (Charbonneau et al. 000). Because the mass of the planet is degenerate with orbital inclination, the planetary status of the companion was confirmed because the transits imply it is edge-on. Hubble Space Telescope (HST) observations yielded precision measurements of the transit light curve (Brown et al. 001), which made this the surest planetary candidate around a main-sequence star (other than our own). The ratio of the planetary radius to the stellar radius can be measured with extreme precision (Mandel & Agol 00). However, the absolute radii are uncertain due to a degeneracy between the radius and the mass of the host star (Seager & Mallén-Ornelas 00): an increase in the mass and radius of the star can yield an identical light curve and period. This can be broken with a photospheric measurement of the star s mass and radius, which has a precision limited to 5 per cent. The planets being discovered with the transit technique require the Doppler technique to confirm that they are not brown dwarfs or blended eclipsing binaries. About 10 per cent of stars with known planetary companions have more than one planet, while possibly as many as 50 per cent of them show a trend in radial velocity indicative of additional planets (Fischer et al. 001). If one or both of the planets is transiting, dynamical interactions between the planets will alter the timing of the transits (Dobrovolskis & Borucki 1996; Charbonneau et al. 000; Miralda-Escudé 00). In principle, timing transits of a giant planet could be used to detect a terrestrial planet companion. In the case that a transiting planet is terrestrial, radial velocity measurements may not be precise enough to measure its mass, while transit timing of a giant companion is sensitive to the terrestrial planet s mass. Given the dual motivations of detecting terrestrial planets and measuring their masses, we derive analytical and numerical results for transit timing variations (TTVs) due to the presence of a second planet. We begin our discussion by introducing the three-body system in Section. The signal from non-interacting planets is calculated first (Section ) and then we compute the effects of an eccentric exterior perturbing planet with a large period in Section 4. A derivation of the general transit timing differences for two planets with circular, coplanar orbits is presented in Section 5. The case of exact mean-motion resonance is analysed in Section 6. The case of two eccentric planets is considered in Section 7, along with numerical simulations of several known multiplanet systems (these are not transiting). We show how measurements of the dispersion of transit timings can be used to detect a secondary planet in the system (Section 8.1), we compare with other planet-search techniques (Section 8.), and we discuss how transit timing may be used to measure the absolute mass and radius of the star and planets (Section 8.). Finally, we discuss other effects that we have ignored, which an observer should be conscious of (Section 9). Throughout the rest of the paper we characterize the strength of TTVs as follows. For a series of transit times, t j,wefitthe times assuming a constant period, P.We compute the standard deviation, σ, of the difference between the nominal and actual times. Mathematically, σ = 1 N N 1 ] 1/ (t j t 0 + Pj), (1) j=0 where P and t 0 are chosen to minimize σ.ifthe variations are strictly periodic, then the amplitude of the timing deviation is simply times larger than σ. During the preparation of this paper, a proceedings contribution has appeared by Jean Schneider which considers several of the effects discussed here (Schneider 00); however, we find that Schneider s results are incorrect as he does not consider the differential force between the star and the transiting planet. In addition, calculations similar to those presented here have recently been published by Holman & Murray (005). EQUATIONS OF MOTION We are studying the three-body system in which the three bodies have labels 0, 1, and positions R i, i = 0, 1, (with an arbitrary origin). The exact Newtonian equations of motion are given by R j R i R i = Gm j R j R i. () j i Multiplying the equations for each particle by its mass and adding together, we find m 0 R 0 + m 1 R 1 + m R = 0. () This is simply a statement that the centre of mass of the system has no external forces. Because light travel time and parallax effects are negligible (Section 9.1), the transit problem is unaffected by the total velocity or position of the centre of mass, so we set i=0 R cm m i R i m = 0. (4) i=0 i This reduces the differential equations of motion to two, which we take to be that of the two planets, R 1 and R (for the two planetary masses). We use this system of equations for numerically solving the equations of motion. However, for analytical consideration it is more convenient to write the problem in Jacobian coordinates, which we discuss next. The Jacobian coordinate system is commonly used in perturbation theory for many bodies (see, for example, Malhotra 199a,b; Murray & Dermott 1999). For the three-body problem, the Jacobian coordinates amount to three new coordinates, which describe (i) the centre of mass of the system, (ii) the relative position of inner planet and the star (the inner binary ) and (iii) the relative position of the outer planet and the barycentre of the inner binary (the outer binary ). To distinguish from the body coordinates, we denote the Jacobian coordinates with a lower case r i. The Jacobian coordinates are r 0 = R cm = 0, r 1 = R 1 R 0, r = R 1 m 0 +m 1 m 0 R 0 + m 1 R 1 ]. (5) Using µ i = m i /M m i /m 0, where M = i=0 m i, the equations of motion may be rewritten in Jacobian coordinates r 1 = Gm 0 r 1 r 1 r 1 GMµ 1 µ 1 r1 r 1 r 1 GMµ r 1 r1 r = Gm 0 r 1 r 1 r 1 GMµ 1 µ r1 1 r 1 r 1, (6) where r 1 = µ 1 r 1 + r = R R 0. C 005 RAS, MNRAS 59, 567 579, Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018

NON-INTERACTING PLANETS: PERTURBATIONS DUE TO INTERIOR PLANET ON A SMALL ORBIT Throughout the rest of the paper we make the approximations that (i) the orbits of both planets are aligned in the same plane and (ii) the system is exactly edge-on (i.e. the inclination angle is 90 ). We also approximate the planet and star as spherical so that the transit is symmetric with a well-defined mid-point. If we take the limit as µ 1, µ 0inequation (6), then the orbits of the planets follow Keplerian trajectories with the equations of motion r r 1 = Gm 1 0, r 1 r r = Gm 0. (7) r This approximation requires that the periapse of the outer planet be much larger than the apoapse of the inner planet, (1 e )a (1 + e 1 )a 1 where a 1 and a are the semimajor axes of the inner and outer binaries and e 1 and e are the eccentricities. In this case, the inner binary orbits about its barycentre, which in turn orbits about the barycentre of the outer binary, but there is no perturbation to the relative motion of the inner binary due to gravitational interactions. Timing variations that arise are simply due to the reflex motion of the star (as shown in Fig. 1). The simplest case to consider is that in which both the inner and outer binaries are on approximately circular orbits. The transit occurs when the outer planet is nearly aligned with the barycentre of the inner binary and its motion during the transit is essentially transverse to the line of sight. The inner planet displaces the star from the barycentre of the inner binary by an amount x 0 = a 1 µ 1 sin π(t t 0 )/P 1 ], (8) where the inner binary undergoes a transit at time t 0 and P 1 is the orbital period of the inner binary. Thus, the timing deviation of the mth transit of the outer planet is δt x 0 P a 1 µ 1 sin π(mp t 0 )/P 1 ], (9) v v 0 πa where v i is the velocity of the ith body with respect to the line of sight. Typically v 0 v,sowehave neglected v 0 in the second expression in the previous equation. Computing the standard deviation of timing variations over many orbits gives σ = (δt ) 1/ = P a 1 µ 1 / πa. (10) Figure 1. Schematic diagram showing changes in the timing of transit due to a perturbing planet interior to the orbit of a transiting planet. C 005 RAS, MNRAS 59, 567 579 Transit timing variations 569 Note that if the periods have a ratio P : P 1 of the form q :1for some integer q, then the perturbations disappear because the argument of the sine function is the same for each orbit. Another observable is the duration of the transit, which scales as t R (v v 0 ). (11) This leads to significant variations only if v 0 v,orµ 1 > a 1/a, which requires a very large axial ratio. More interesting variations occur if either or both planets are on eccentric orbits. Because both planets are following approximately Keplerian orbits, the TTVs and duration variations can be computed by solving the Kepler problem for each Jacobian coordinate. Because we are assuming that the planets are coplanar and edge-on, four coordinates each suffice to determine the planetary positions: e 1,, a 1,, ϖ 1, (longitude of pericentre measured from the sky plane) and f 1, (true anomaly). As in the circular case, the change in the transit timing is approximately δt x 0 /v. The position of the star with respect to the barycentre of the inner binary is x 0 = µ 1 r 1 cos f 1 + ϖ 1 ]. (1) If a 1 a, the outer planet is in nearly the same position at the time of each transit and its velocity perpendicular to the line of sight is v = πa (1 e sin ϖ ), (1) P 1 e where we have used the fact that f = ϖ π(m + 1/) at the timing of the transit. Thus, to first order in a 1 /a δt = P µ 1 r 1 cos f 1 + ϖ 1 ] 1 e. (14) πa (1 e sin ϖ ) The standard deviation of δt can be computed analytically as well. Over many transits by the outer planet, the inner binary s position populates all of its orbit provided the planets do not have a period ratio that is the ratio of two integers. Consequently, we find the mean transit deviation by averaging over the probability that the inner binary is at any position in its orbit, p( f 1 ) = n 1 / ḟ 1 (where n 1 = π/p 1 ), times the transit deviation at that point. This gives π δt = 1 d f 1 δt p( f 1 ) π 0 = µ a 1 1 e 1 cos ϖ 1. (15) v Because the star spends more time near apoapse, the mean timing grows as e 1. The symmetry of the orbit about ϖ = 0 and π explains the dependence on cos ϖ 1.Asimilar calculation gives δt and the resulting standard deviation is σ = ( δt δt ) 1/ = P a 1 µ 1 1 e / πa (1 e sin ϖ ) 1 e 1 ( 1 + sin ϖ 1 ) ] 1/. (16) This agrees with equation (10) in the limit e 1 0. Averaging again over ϖ 1 and ϖ gives ] 1/ σ ϖ1,ϖ = P a 1 µ 1 1 4 e 1 ( ) / πa 1 e 1/4. (17) Note that an eccentric inner orbit reduces σ because the inner binary spends more time near apoapse as the eccentricity increases, thus reducing the variation in position when averaged over time. As Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018

570 E. Agol et al. e 1 approaches unity for an orbit viewed along the major axis, σ reduces to zero because the minor axis approaches zero, leaving no variation in the x 0 position. 4 PERTURBATIONS DUE TO EXTERIOR PLANET ON A LARGE ECCENTRIC ORBIT In this section we include planet planet interactions and compute the timing variations due to the presence of a perturbing planet on an eccentric orbit with a semimajor axis much larger than that of a transiting planet on a nearly circular orbit. In this limit, resonances are not important and the ratio of the semimajor axes can be used as a small parameter for a perturbation expansion. A general formula for this case has been derived by Borkovits et al. (00). Here we present a shorter derivation which clarifies the primary physical effects for coplanar planets viewed edge-on. The equations describing the inner binary can be divided into a Keplerian equation (7) and a perturbing force proportional to m. The perturbing acceleration δr 1 on the inner binary due to the outer planet is given by r 1 r 1 δr 1 = GMµ r 1 r 1 GMµ. (18) r1 We expand this in a Legendre series and keep terms up to first order in the ratio of the radii of the inner and outer orbit: δr 1 = Gm r r 1 r 1 r ] 1 r r r + O(r 1 /r ). (19) To compute the perturbed orbital period we must find the change in the force on the inner binary due to the outer planet averaged over the orbital period of the inner binary. Because the inner binary is nearly circular, the angle of the inner binary is given by θ 1 = f 1 + ϖ 1 n 1 (t τ 1 ) + ϖ 1, where we have approximated e 1 0. Differentiating this with respect to time gives θ 1 = ṅ 1 (t τ 1 ) + n 1 n 1 τ 1 + ϖ 1. (0) Now, we write ṅ 1 = n 1 /(a 1 )ȧ 1, and express ȧ 1, ϖ 1 and τ 1 in terms of the radial, tangential and normal components of the force (see section.9 of Murray & Dermott 1999). Plugging these expressions into θ 1 gives a cancellation of most terms to lowest order in e 1, and after setting the normal force to zero leaves the remaining term ] a1 θ 1 = n 1 1 G(m 0 + m 1 ) R, (1) where R is the radial disturbing force per unit mass, R = (δ r 1 ˆr 1 ) = (1/)Gm a 1 /r. Thus, the presence of the second planet causes a change in the effective mass of the inner binary by an amount (1/) m (a 1 /r ) (this is a tidal force), which results in a slight increase in the period of the orbit. The increase in period would be constant if the second planet were on a circular orbit. However, for an eccentric orbit, the time variation of r induces a periodic change in the orbital frequency of the inner binary with period equal to P. Now, the time of the (N + 1)th transit occurs at t ec t 0 = = f0 +πn f 0 f0 +πn f 0 d f 1 θ 1 1 d f 1 n 1 1 1 + m m 0 + m 1 ( a1 r ) ], () where f 0 is the true anomaly of the inner binary at the time of the first transit. Following Borkovits et al. (00), we change the variable of integration from f 1 to f, the true anomaly of the outer planet, d f 1 = P r ( ) P 1 1 e 1/ d f. () a Because we are assuming that the orbit of the outer planet is eccentric, r = a (1 e )/(1 + e cos f ), which gives the transit time t ec t 0 = NP 1 + m ( ) 1 e / P1 ( f + e sin f ), (4) π(m 0 + m 1 ) P where f is the true anomaly of the outer binary at the timing of the (N 1) transit. The unperturbed f includes the mean motion, n (t τ ), which grows linearly with time. To find the deviation of the time of transits from a uniform period, we subtract off this mean motion as well as NP 1, which results in δt 1 = β ( 1 e ) / f n (t τ ) + e sin f ] m P1 β =. (5) π(m 0 + m 1 ) P This agrees with the expression of Borkovits et al. (00) in the limit I = 1 (i.e. coplanar orbits). The timing variations scale as a / over the period of the outer planet ( a / ), which is a much shallower scaling than the effect of secular perturbations, a, estimated by Miralda-Escudé (00). Numerical calculations of the three-body problem show that this approximation works extremely well in the limitr 1 r (see Fig. ). If P 1 P and the period ratio is non-rational, then over a long time the transits of the inner planet sample the entire phase of the outer planet. Thus, we can compute the standard deviation of the TTVs as in equation (16) σ 1 = δt 1 π 1/ 1 = d f δt 1 π p( f ), (6) 0 Figure. Contour plot of the logarithm of the dispersion of the normalized timing variations, log (σ 1 n 1 µ 1 ), for an inner circular planet and an outer eccentric planet (for example, at the contour an orbit lasting πyears with a perturbing planet of mass 10 M would have a transit timing standard deviation of 10 5 yr, or 5 min). The dotted line is the approximation given in equation (7). C 005 RAS, MNRAS 59, 567 579 Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018

because δt 1 =0, where p( f ) = n / ḟ. This integral turns out to be intractable analytically, but an expansion in e yields βe σ 1 = ( ) 1 e / 1 16 e 47 196 e4 41 ] 1/ 7648 e6, (7) which is accurate to better than per cent for all e. Fig. shows a comparison of this approximation with the exact numerical results averaged over ϖ (because there is a slight dependence on the value of ϖ ). This approximation breaks down for a (1 e ) 5a 1 because resonances and higher-order terms contribute strongly when the planets have a close approach. It also breaks down for e 0.05 because the perturbations to the semimajor axes caused by interactions of the planets contribute more strongly than the tidal terms, which become weaker with smaller eccentricity. 5 PERTURBATIONS FOR TWO NON-RESONANT PLANETS ON INITIALLY CIRCULAR ORBITS In this section we estimate the amplitude of timing variations for two planets on nearly circular orbits. The resonant forcing terms are most important in determining the amplitude, even for nonresonant planets. The planets interact most strongly at conjunction, so the perturbing planet causes a radial kick to the transiting planet, giving it eccentricity. Because the planets are not exactly on resonance, the longitude of conjunction will drift with time, causing the kicks to cancel after the longitude drifts by π in the inertial frame. Thus, the total amplitude of the eccentricity grows over a time equal to half of the period of circulation of the longitude of conjunction. The closer the planets are to a resonance, the longer the period of circulation and thus the larger the eccentricity grows. The change in eccentricity causes a change in the semimajor axis and mean motion. For two planets that are on circular orbits near a j : j + 1 resonance, conjunctions occur every P conj = π/(n 1 n ) jp (we take the limit of large j and we ignore factors of order unity). We define the fractional distance from resonance, ɛ = 1 (1 + j 1 )P 1 /P < 1, where ɛ = 0 indicates exact resonance. Then, because the planets are not exactly on resonance, the longitude of conjunction changes with successive conjunctions and the longitude of conjunction returns to its initial value over a period P cyc = Pj 1 ɛ 1. The number of conjunctions per cycle is N c = P cyc /P conj j ɛ 1. Each conjunction changes the eccentricity of the planets by e µ pert (a/ a) (using the perturbation equations for eccentricity and the impulse approximation, where µ pert is the planet star mass ratio of the perturbing planet). Over half a cycle, the eccentricities grow to about N c e µ pert (1 P 1 /P ) 1 ( jɛ) 1 µ pert ɛ 1. To find the change in the transit timing, we use the orbital elements to compute the variation in the instantaneous orbital frequency, θ. To first order in e n (1 + e cos f ) θ = n (1 e ) / 0 + δn + en 0 cos λ ϖ ], (8) where n 0 is the unperturbed mean motion. There are two terms which contribute to timing variations: fluctuations in the mean motion and fluctuations due to a non-zero eccentricity. In the first case, δn may be found by applying the Tisserand relation to the lighter planet (we now use subscripts light and heavy ), resulting in δn light /n light j ( e light ) jµ ɛ (where µ is µ heavy ). These changes to the period accumulate over an entire cycle, giving δt light µ ɛ P. (9) C 005 RAS, MNRAS 59, 567 579 Transit timing variations 571 By conservation of energy, the fractional change in the semimajor axis (or period) of the heavy planet is reduced by a factor of m light /m heavy,sothat δt heavy (m light /m heavy )µ ɛ P. (0) The eccentricity-dominated term gives a timing variation of δt µ pert ɛ 1 P. (1) So, for ɛ µ 1/ the perturbed eccentricity dominates, but closer to resonance for j 1/ µ / ɛ µ 1/ the perturbed mean motion dominates (this range is the same for both the light and heavy planets, except for factors of order unity). For smaller values of ɛ, the planets are trapped in mean-motion resonance, which is discussed in the next section. Halfway between resonances, ɛ j,sothe timing deviation become δt 0.7µ pert a/(a a 1 )] P. () A more precise derivation in the eccentricity-dominated case using perturbation theory is given in Appendix A. So far we have discussed the timing variations for planets nearby a first-order resonance. For larger period ratios, the eccentricity of the inner planet grows to e in µ out (P in /P out ),soδt in µ out (P in /P out ) P in.for an outer transiting planet, the motion of the star dominates over the perturbation due to the inner planet for P out > (π) /4 P in. Fig. shows a numerical calculation of the standard deviation of the TTVs. We have used small masses to avoid chaotic behaviour because resonant overlap occurs for j µ /7 (Wisdom 1980). Fig. 4 zooms in on the : 1 resonance. As predicted, the amplitude scales as ɛ 1 (equation 1), and then steepens to ɛ (equations 0 and 9) closer to resonance. Because the strength of the perturbation is independent of whether the perturbing planet is interior or exterior, the strengths of the resonances are similar and the shape of the standard deviation of the TTVs is symmetric about P in = P out. The dashed curve in Fig. shows the analytical approximation from equation (10), which agrees well for P pert < (π) /4 P trans. The numerical Figure. Transit timing standard deviation for two planets of mass m trans = 10 5 m 0 and m pert = 10 6 m 0 on initially circular orbits in units of the period of the transiting planet. The solid (dotted) line is the numerical calculation for the inner (outer) planet averaged over 100 orbits of the outer planet with the planets initially aligned with the observer. The dashed line is equation (10). The large dots are equations () on resonance and equation () halfway between resonances. Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018

57 E. Agol et al. Figure 4. Amplitude near the : 1 resonance versus the difference in period from exact resonance for two systems: one with m 1 = 10 5 m 0 (top solid) and m = 10 (lower solid), and the other with m 1 = 10 6 m 0 (top dotted) and m = 10 5 (lower dotted). The large dots are equation (). The dashed line is equation (1), while the dash-dotted line is equation (0). results match the perturbation calculation (equations A7 and A8), except for near resonance where the change in mean motion dominates (we have not bothered to overplot the perturbation calculation because it is indistinguishable from the numerical results). There is a dip in σ near P out =.5P in, which occurs because the amplitude of the timing differences due to the orbit of the star about the barycentre (equation 10) are opposite in sign and comparable in amplitude to the differences due to the perturbation of the outer planet by the inner planet (equation A8). The analysis in this section breaks down near each resonance because we have not considered changes to the orbital elements of the perturbing planet. In the next section we consider what happens to planets trapped in resonance. 6 PERTURBATIONS FOR TWO PLANETS IN MEAN-MOTION RESONANCE The analysis in the previous section assumes that the perturbation to the orbit of each planet is small, so that the interaction can be calculated using the unperturbed orbits (linear perturbation theory). This is clearly not the case near a mean-motion resonance. We investigate the case of low, initially zero, eccentricity where we found the standard analyses of this case (e.g. Murray & Dermott 1999) to be incorrect. Here we provide a physically motivated, order of magnitude, derivation of the perturbations and the TTVs for two planets in a first-order mean-motion resonance. A rigorous derivation is left for elsewhere, but we verify our findings with numerical simulations. Consider a first order, j: j + 1, resonance where the lighter planet is a test particle. Qualitatively, the physics of low-eccentricity resonance is as follows. On the nominal resonance, the two planets have successive conjunctions at exactly the same longitude in inertial space. The strong interactions that occur at conjunctions build up the eccentricity of the test particle and cause a change in semimajor axis and period. The change in period of the test particle causes the longitude of conjunction to drift. Once the longitude of conjunction shifts by about π relative to the original direction, the eccentricity begins to decrease, making a libration cycle. The libration of the semimajor axes causes the timing of the transits to change. This qualitative discussion leads directly to an estimate of the drifts in transit times. Within each libration cycle, the longitude of conjunction shifts by about half an orbit, mostly due to the period change of the lighter planet. Because conjunctions occur only once every j orbits, the largest transit time deviation of the lighter planet during the period of libration is P/j (in this order of magnitude derivation we ignore factors of order unity, and take the limit of large j so that j j + 1 and P P 1 ). The observationally more interesting case is probably that in which the heavier planet is the transiting one. Then, conservation of energy for the orbiting planets implies that the change in periods is inversely proportional to the masses; therefore, the timing variations are given by (m light /m heavy )P/ j.we find an excellent fit to the data for δt max P m pert. () 4.5 j m pert + m trans The calculations shown in Fig. verify this analytical scaling with j. Calculating the libration period is slightly more complicated, but still straightforward. Suppose the period of the test particle deviates from the nominal resonance by a small fraction ɛ. Then, consecutive conjunctions drift in longitude by about π j ɛ. The number of conjunctions, N c, before a drift of order π in the longitude of conjunctions accumulates is N c j ɛ 1.Wenow estimate ɛ indirectly. The test particle gains an eccentricity of order j µ in each conjunction due to the radial force from the massive planet (this can be computed from the impulse approximation and the perturbation equation for eccentricity). The eccentricity given in N c conjunctions is then of order e µ ɛ 1. Using the Tisserand relation, the fractional change in the semimajor axis associated with this change in eccentricity is jµ ɛ. Because this is also the fractional change in the period, we have ɛ j 1/ µ / and a libration period of P lib 0.5 j 1 ɛ 1 P 0.5 j 4/ µ / P. (4) We numerically computed the amplitude and period of the TTVs at the : 1 resonance. Fig. 4 shows a plot of the amplitude of the Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018 C 005 RAS, MNRAS 59, 567 579

timing variations versus the mass ratio of the perturbing planet to the transiting planet. As predicted, the amplitude is of the order of the period of the transiting planet when the transiting planet is lighter, and varies as the mass ratio when the transiting planet is heavier. The libration period measured from the numerical simulations shows the predicted behaviour, scaling precisely as µ / for the more massive planet (with a coefficient of 0.7 for j = 1 and 0.5 for j > 1inequation 4). We have compared the numerical values of the amplitude and period of libration on resonance as a function of j. Despite the fact that the above scalings were derived in the large-j limit, the agreement is better than 10 per cent for j, and accurate to about 40 per cent for j = 1. Fig. 4 shows the more detailed behaviour of the amplitude near the : 1 resonance. The amplitude is maximum slightly below resonance at the location of the cusp. This may be understood as follows. Because the simulations are started with e 1 = e = 0, after conjunction the eccentricity grows and the outer planet moves outwards, while the inner planet moves inward. This causes the planets to move closer to resonance, causing a longer time between conjunctions, leading to a larger change in eccentricity and semimajor axis. The cusp is the location where the planets reach exact resonance at the turning point of libration, at which point δt is maximum. To the right of the cusp, the libration causes the planets to overshoot the resonance, so the change in eccentricity and semimajor axis is Transit timing variations 57 somewhat smaller, and hence the amplitude is smaller. Fig. 4 shows that the width of the resonance scales as µ / (the horizontal axis has been scaled with µ / so that the curves overlap), so for larger mass planets the resonant variations have a wider range of influence than the non-resonant variations discussed in the previous section. The curves in Fig. 4 demonstrate that on-resonance the amplitude scales as min(1, µ pert /µ trans )/ j, while off-resonance the amplitude scales as µ pert. 7 NON-ZERO ECCENTRICITIES When either eccentricity is large enough, higher-order resonances become important. In particular, the resonances that are 1:m begin to dominate as the ratio of the semimajor axes becomes large; as the eccentricity of the outer planet approaches unity, these resonances become as strong as first-order resonances (Pan & Sari 004). Fig. 5 shows the results of a numerical calculation where the transiting planet HD 09458b, with a mass of approximately 0.67 Jupiter masses, is perturbed by a 1-M planet with various eccentricities (we have taken HD 09458b to have a circular orbit). Near the mean-motion resonances, the signal is large enough that an earth-mass planet would be detectable with current technology. The amplitude increases everywhere with eccentricity. This graph can be applied to systems with other masses and periods as the timing Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018 Figure 5. Dispersion, σ trans,oftiming variations of HD 09458b due to perturbations induced by an exterior earth-mass planet with various eccentricities and periods. The colour bar has σ trans for a planet of mass m and for a transiting period of.5 d. Near the period ratio of 4 : the system becomes chaotic. The increase in the number of resonance peaks, particularly those beyond the : 1 resonance, are from higher-order terms in the expansion of the Hamiltonian, which were truncated for the near-circular case. C 005 RAS, MNRAS 59, 567 579

574 E. Agol et al. variation scales as δt P trans m pert (except for planets trapped in resonance). When both planets have non-zero eccentricity, the parameter space becomes quite large: the four phase-space coordinates for each planet, assuming both are edge-on, and two mass ratios give 10 free parameters. On resonance, the analysis remains similar to the circular case. The libration amplitude will still be of order Pj 1 for the lighter planet and Pj 1 (µ light /µ heavy ) for the heavier planet. However, the period of libration will decrease significantly as the eccentricity increases because P lib e 1/ µ 1/. On the secular time-scale, the precession of the orbits will lead to a significant variation in the transit timing (Miralda-Escudé 00). The period of precession, P ν, may be driven by other planets, by general relativistic effects, or by a non-spherical stellar potential, but leads to a magnitude of transit timing deviation which just depends on the eccentricity for P ν P. Miralda-Escudé (00) has shown that the maximum deviation for e 1isgivenby δt = ep (5) π and the timing variations vary with a period that is equal to the period of precession. Thus, for large period ratios, this effect dominates over perturbations due to an exterior eccentric perturber for e 1 e (m /m 0 )(P 1 /P ) (for e 1 and e small but non-zero); however, the secular perturbations accumulate over the much longer secular time-scale, which may be of the order of centuries. For arbitrary eccentricity, the maximum deviation is δt = P ( sin 1 y + sin 1 z + x x π x 4), (6) where x = (1 e ) 1/4, y = (1 x)/e and z = (1 x )/e (this is derived from the Keplerian solution with a slowly varying ϖ ). This approaches P/ as e 1. Typically the eccentricity will vary on the secular time-scale, so these expressions only hold as long as the variation in e is much smaller than its mean value. Rather than systematically studying the entire parameter space, we now investigate several specific cases of known extrasolar planets to demonstrate that detection of this effect should be possible once a transiting multiplanet system is found. Most of these systems have non-zero eccentricities and several are in resonance, causing a significant signal. We summarize the amplitude of the signals of most known multiplanet systems, if they were seen edge-on, in Table 1 (in some cases other planets are present, which would cause additional perturbations). The extrasolar planetary system Gliese 876 contains two Jupitermass planets on modestly eccentric orbits which are near the : 1 mean-motion resonance, P 1 = 0.1 d and P = 61.0 d (Marcy et al. 001). Due to the small size of the M4 host star, the inner planet has a 1.5 per cent probability of transiting for an observer at ar- Table 1. Timing variations for known multiplanet systems. System P in (d) P out /P in σ 1 σ 55 Cnc e, b.81 5.1 10.5 s.68 s 55 Cnc b, c 14.7.0 1.61 h 14.7 h Ups And b, c 4.6 5. 1.0 s 1.61 min Gliese 876 0.1.07 1.87 d 14.6 h HD 74156 51.6 9. 4.98 min 4.4 min HD 16844 58.1 9.9 1.9 min.6 h HD 714 15 9.81.4 d 11. d HD 894.00 4.9 d 0.7 d PSR 157+1 b, c 66.5 1.48 15. min. min Earth/Jupiter 65 11.9 1.4 min 4.1 s Figure 6. (a) Transit timing variations for Gliese 876 B and (b) Gliese 876 C. The vertical axis is in units of h, while the horizontal axis is in units of the period of the transiting planet (given in parentheses for reference). bitrary inclination. The orbital motion involves both mean-motion resonance as well as a secular resonance in which the planets librate about their apsidal alignment. The apsidal alignment is in turn precessing at a rate of 41 yr 1 (Laughlin & Chambers 001; Rivera & Lissauer 001; Nauenberg 00; Ji et al. 00; Laughlin et al. 004). Fig. 6 shows the predicted timing variations if this system were seen edge-on and if the planets are coplanar using the orbital elements from Laughlin et al. (004). The two most prominent periodicities in Fig. 6 are those associated with the : 1 libration, with a period of roughly 600 d (0 orbits of the inner planet; Laughlin & Chambers 001), and the long-term precession of the apsidal angle with a period of about 00 d (110 orbits of the inner planet, corresponding to 41 yr 1 ). Evaluating equation (5) gives a peak amplitude of 1.4 d for the inner planet and 18 h for the outer planet, which both compare well with the numerical results given that the eccentricities are not constant. The extrasolar planetary system 55 Cancri contains a set of planets, b and c, near the : 1 resonance having 15- and 45-d periods. There is some evidence for another planet, d, in an extremely long orbit, and recently a fourth low-mass planet, e, was found with a.8-d period (McArthur et al. 004). The planets e, b and c have transit probabilities of 1, 4 and per cent, respectively, for an observer at arbitrary inclination. The orbit of planet b is approximately circular while planet c is somewhat eccentric (Marcy et al. 00). Table 1 gives the amplitude of the variations for the planets. We have C 005 RAS, MNRAS 59, 567 579 Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018

ignored planet d; however, it is at a large enough semimajor axis to produce a -s variation due to light travel time as the barycentre of the inner binary orbits the barycentre of the triple system were the inner planets transiting. The double planet system Upsilon Andromedae has a semimajor axial ratio of 14, which is not in a mean-motion resonance (Butler et al. 1999; Marcy et al. 001). The inner planet has a short period of 4.6 d, and thus a significant probability of transiting of about 1 per cent, but it has variations that are too small to currently be detected from the ground or space. The outer planet has much larger TTVs due to its smaller velocity, but a much smaller probability of transiting. The planetary system HD 714 has two planets with a period ratio of 10 and a period of the inner planet of 41 d (Vogt et al. 000). The outer planet is highly eccentric, e = 0.69, and so its periapse passage produces a large and rapid change in the transit timing of the inner planet. If this system were transiting, the variations would be large enough to be detected from the ground. HD 894 is in a : 1 resonance giving variations of the order of the periods of the planets. The pulsar planets are near a : resonance, which would cause large TTVs were they seen to transit the pulsar progenitor star. Finally, alien civilizations observing transits of the Sun by Jupiter would have to have 10-s accuracy to detect the effect of the Earth. 8 APPLICATIONS 8.1 Detection of terrestrial planets The possibility of detecting terrestrial planets using the transit timing technique clearly depends strongly on (i) the period of the transiting planet, (ii) the nearness to resonance of the two planets and (iii) the eccentricities of the planets. The detectability of such planets also depends on the measurement error, the intrinsic noise due to stellar variability, and the number of transit timing measurements. One requirement for the case of an external perturbing planet is that observations should be made over a time longer than the period of the timing variations, which can be longer than the period of the perturbing planet. Ignoring these complications, a rough estimate of detectability can be obtained from comparing the standard deviation of the transit timing with the measurement error. It is worthwhile to provide a numerical example for the case of a hot Jupiter with a -d period that is perturbed by a lighter, exterior planet on a circular orbit in exact : 1 resonance. The timing deviation amplitude is about 0 per cent of the period ( d) times the mass ratio (00) or about min (equation ): ( ) mpert δt = min. (7) m These variations accumulate over a time-scale of the order of the period ( d) times the planet to star mass ratio to the power of /, which for a transiting planet of the order of a Jupiter mass is about five months (equation 4): ( ) / mtrans t cycle = 150 d. (8) m J Such a large signal should easily be detectable from the ground. With a relative photometric precision of 10 5 from space or from future ground-based experiments, less massive objects or objects further away from resonance could be detected. The observations could be scheduled in advance and require a modest amount of C 005 RAS, MNRAS 59, 567 579 Transit timing variations 575 Figure 7. Mass sensitivity of various planet detection techniques to secondary planets in HD 09458. The vertical axis is the perturbing planet s mass in units of M. The horizontal axis is the period ratio of the planets. The solid line is for the transit timing technique, the dashed line is astrometric, and the dotted line is the radial velocity technique. observing time with the possible payoff of being able to detect a terrestrial-sized planet. 8. Comparison with other terrestrial planet search techniques To attempt a comparison with other transit timing techniques, we have estimated the mass of a planet which may be detected at an amplitude of 10 times the noise for a given technique. We compare the current (and estimated) abilities for measuring the mass of planets with three techniques: (i) radial velocity variations of the star; (ii) astrometric measurements; (iii) TTVs. We assume that radial velocity measurements have a limit of 0.5 m s 1 (giving a detectable planet signal of 5 m s 1 ), which is about the highest accuracy that has been achieved from the ground, and may be at the limit imposed by stellar variability (Butler et al. 004). We assume that astrometric measurements have an accuracy of 1 µarcsec, which is the accuracy that is projected to be achieved by the Space Interferometry Mission (Ford & Tremaine 00; Sozzetti et al. 00). Finally, we assume that the transit timing can be measured to an accuracy of 10 s, which is the highest accuracy of transit timing measurements of HD 09458 (Brown et al. 001); it is possible to increase the signal-to-noise above that estimated here with the observations of many transits, as pointed out by Miralda-Escudé (00). We concentrate on HD 09458 because it is the best studied transiting planet. This system is at a distance of 46 pc and has a period of.5 d. Fig. 7 shows a comparison of the 10σ sensitivity of these three techniques. The solid curve is computed for m trans = 6.7 10 4 M and m pert = 10 7 M and both planets on circular orbits. The amplitude of the timing variations scales as m pert /m 0,sowe scale the results to the mass of the perturber to compute where the timing variations are 100 s this determines the sensitivity. The TTV technique is more sensitive than the astrometric technique at semimajor axial ratios smaller than about. Off-resonance, radial velocity measurements are the technique of choice for this system, while on-resonance the TTV is sensitive to much smaller planet masses. Note that in Fig. 7 the TTV and astrometric techniques have the same slope at small P pert /P trans. This is because the transit timing technique is measuring the reflex motion of the host star due Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018

576 E. Agol et al. to the inner planet, which is also being measured by astrometry. The solid curve is an upper limit to the minimum mass detectable in HD 09458 because a non-zero eccentricity will lead to larger timing variations (Fig. 5) and thus a smaller detectable mass. 8. Measuring the mass radius relation In addition to the detection and characterization of planets, transit timing could be used to measure the absolute mass and absolute radii of the host stars. Transit timing provides an extra observable to break the mass radius degeneracy, discussed by Seager & Mallén- Ornelas (00), without using a theoretical mass radius relation (a similar degeneracy exists for single-lined spectroscopic binaries; only for double-lined spectroscopic binaries can the mass and radius of the stars be measured). This technique could be used as a check on photospheric measurements of the stellar mass and radius, and in turn would yield the absolute mass and radii of the planets. From observations of the depth and duration of the transit, one can measure the density of the star (Seager & Mallén-Ornelas 00). For anedge-on, circular system with no limb-darkening ρ = P (9) π GtT for either planet, where t T is the duration of the transit from midingress to mid-egress (this differs slightly from the expression in Seager & Mallén-Ornelas 00, because we define the transit duration from mid-in/egress). The density of the planet can be derived in a similar manner. In the case of two transiting planets, both on circular orbits, and in which the timing deviations of the outer planet are dominated by the reflex motion of the star (equation 10), the ratio of the mass to the radius of the star is m 0 = 1 P 7/ 1 P / K1, (40) R 0 8π G t T1 σ where K 1 is the Doppler amplitude of the inner planet and σ is the standard deviation of the transit timing deviations of the outer planet. Combined with the measurement of the density, this gives the absolute mass and radius of the star. Because the ratio of the planet mass to star mass and the ratio of the planet radius to star radius can also be measured from the transit depth and duration and Doppler amplitude, this technique allows an absolute measurement of the mass and radius of the transiting planets. Without the simple assumptions used here, a numerical treatment is required, but the same concept applies for breaking the degeneracy. 9 EFFECTS WE HAVE IGNORED We now discuss several physical effects that we have ignored, which ought to be kept in mind by observers measuring TTVs. is much smaller in planetary systems than in binary stars because their masses and semimajor axes are small, having an amplitude t star = a ( )( ) m p mp a 0.5s, (41) c m M J 1au where M J is the mass of Jupiter and a is the semimajor axis of the perturbing planet. This effect is present in the absence of deviations from a Keplerian orbit because the inner binary orbits about about the centre of mass. There can also be changes in the timing of the transit as the distance of the transiting planet from the star varies. In this case, the time of transit is delayed by the light travel time between the different locations where the planet intercepts the beam of light from the star. The amplitude of these variations is smaller than the σ we have calculated by a factor of v trans /c, where v trans is the velocity of the transiting planet. So, only very precise measurements will require taking into account light travel time effects, which should be borne in mind in future experiments (of course, the light travel time due to the motion of the observer in our solar system must be taken into account with current experiments). 9. Inclination We have assumed that the planets are strictly coplanar and exactly edge-on. The first assumption is based on the fact that the solar system is nearly coplanar and the theoretical prejudice that planets forming out of discs should be nearly coplanar; however, because some extrasolar planetary systems have been found with rather large eccentricities, it is entirely possible that non-coplanar systems will be found as well (as predicted by, for example, models of planetary scattering). If two planets in the same system were detected as transiting, then their mutual inclination can be measured. Small noncoplanar effects will change our results slightly (Miralda-Escudé 00), while large inclination effects would require a reworking of the theory. Systems in which two (or more) planets transit will be biased to have small relative inclinations. Transit timing in twoplanet systems in which only one planet transits, combined with Doppler measurements of the other planet, will allow a constraint on the relative inclination of the planets because the mass of the Doppler planet varies as 1/sin i (although this is partly offset by the reduction in the mutual force of the planets at conjunction caused by their increased separation due to mutual inclination). The assumption that the systems are edge-on is based on the fact that a transit can occur only for systems that are nearly edge-on. For small inclinations our formulae will only change slightly, but may result in interesting effects such as a change in the duration of a transit, or even the disappearance of transits due to the motion of the star about the barycentre of the system. On a much longer time-scale (centuries), the precession of an eccentric orbit might cause the disappearance of transits because the projected shape of the orbit on the sky can change. This possibility was mentioned by Laughlin et al. (004) for GJ 876. Downloaded from https://academic.oup.com/mnras/article-abstract/59//567/9875 by guest on 0 November 018 9.1 Light travel time Deeg et al. (000) carried out a search for perturbing planets in the eclipsing binary stellar system, CM Draconis, using the changes in the times of the eclipse due to the light travel time to measure a tentative signal consistent with a Jupiter-mass planet at 1 au(their technique would in principle be sensitive to a planet on an eccentric orbit as well; cf. equation 4). The Römer effect due to the change in light travel time caused by the reflex motion of the inner binary 9. Other sources of timing noise Aside from the long-term effects that have been ignored, there are several sources of timing noise that must be included in the analysis of observations of transiting systems. These sources of noise could come from the planet or the host star. If the planet has a moon or is a binary planet, then there is some wobble in its position causing a change in both the timing and duration of a transit (Sartoretti & Schneider 1999; Brown et al. 001). A moon or ring system C 005 RAS, MNRAS 59, 567 579