Periastron Precession Measurements in Transiting Extrasolar Planetary

arXiv:0806.0629v1 [astro-ph] 4 Jun 2008

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András Pál General of Level the Relativity at Systems Planetary Transiting Extrasolar in Measurements Precession Periastron cetd.. eevd..;i rgnlfr ... form original in ; ... Received ..., Accepted c 2 1 3 eateto srnm,Loránd st Eötvös Astronomy, Garden Pá of 60 University, Department Astrophysics, for Center Harvard-Smithsonian eateto tmcPyis nttt fPyis Lorán Physics, of Institute Physics, Atomic of Department -al [email protected] E-mail: e tp/eolnte o pt aeinformation date to up for http://exoplanet.eu See -al [email protected] E-mail: 08RAS 2008 1 , 2 ⋆ n ec Kocsis Bence and 000 – 20)Pitd2 oebr21 M L (MN 2018 November 25 Printed (2008) 1–9 , inl–tcnqe:photometric techniques: – tional rcso eleceigtelvlpeitdb eea relativity. general words: by Key predicted level the exceeding well precision h rsneo te lnt ntesse.Ti eua precess secular relativ This system. general the in to change in due long-term planets the precesses other through slowly of ellipse presence the the of periastron The estvt yodr fmgiueoe h T esrmn.W fi We measurement. TTV the over a pr magnitude over the of measurements improves orders measurement e by using TDV sensitivity 17156b) HD The and instruments. 436b eccent GJ known space-borne 147506b, the HD (XO-3b, of systems observations tary future prec repeated corresponding for the estimate precision We TDV). variations, duration sit h otsa.I rniigpae so necnrcobt h du the orbit, eccentric an on is bot planet for transiting information a of If spectrum T star. widest host the provide the they amon since important tems surpassingly are systems exoplanetary Transiting ABSTRACT h rcsino h ri eut nasseai aito nbt t succes both between in period variation observed systematic the a in and events results orbit transit the individual of precession The D ssniiet h retto fteobtlelperltv oteln o line the to relative ellipse orbital the of orientation the to sensitive is ∼ 1 10 hs tran- These . iais cisn lntr ytm eaiiy–mtos obser methods: – relativity – systems planetary – eclipsing binaries: − 6 1 , 3 10 – † ma yP t /,Bdps -17 Hungary H-1117, Budapest 1/A, st. zmány P. − ¨t¨ sUiest,PamayP t /,Bdps H-111 Budapest Eötvös 1/A, Pázmányd st. University, P. ys- on ∼ 7 et abig,M,018 USA 02138, MA, Cambridge, reet, t ; , erpro a yial eettepeeso aet a to rate precession the detect typically can period year 4 oicto fteosre iepro ewe succes- between period time transits observed sive the of modification cor- dependent axis semimajor a rection. tran- that requires verify and law to Kepler’s o amplitude order third correction velocity in simultaneously, radial gravitoelectric periodicity the the siting measuring test by also GR principle observations light in TEP anisotropic that can shown the has 200 to (2006) Fabrycky due Iorio Furthermore, see life Yarkovski-effect, stellar predicte (a.k.a. of also redistribution are scale timing) time transit the the on semimajor in the therefore be in (and variations can Secular axis 2007). or 2002 Gladman (Miralda-Escude & star, (GR) Heyl in- relativity host precessi general periastron provide the by prograde could predicted additional of the system, detect oblateness to the used the satel- in on or 2007) co-orbita 2007), formation al. et 2008), Holman (Simon & al. Ford et lites see Miller-Ricci Murray (Trojans, 2005; & companions Holman al. et e.g. Agol (see 2005; companions planetary other of P obs napoersuy iad-sue(02 eie the derived (2002) Miralda-Escude study, pioneer a In tasttmn aitos T)o in or TTV) variations, timing (transit P A obs T E tl l v2.2) file style X aldtasttmn aitos(TTVs), variations timing transit called , i rniigexoplane- transiting ric o a edetected be can ion sinmeasurement ession h lntr sys- planetary the g ietransits, sive csindetection ecession itn rplanned or xisting aino transits of ration h lntand planet the h t n possibly and ity edrto of duration he dta TDV that nd ,Hungary 7, T D sight. f (tran- P obs va- the 8). on d f . l ; 2 A. Pál and B. Kocsis caused by the standard periastron precession due to GR (e.g. Y (north) Misner, Thorne, & Wheeler 1973) and the perturbations of other planets if present. Recently, Heyl & Gladman (2007) have extended these studies and estimated the precision of precession rate measurements for long–term mock obser- Z vations of eccentric transiting extrasolar planets (ETEPs). Both studies restricted to small eccentricities. At that time, X the existence of close eccentric planets was known only (west) through radial velocity measurements, and no ETEPs had been observed. Since their publication, four transiting extrasolar planets have been discovered with significant eccentricity: XO-3b (Johns-Krull et al. 2007), HD 147506b (a.k.a. ω HAT-P-2, Bakos et al. 2007), GJ 436b (Gillon et al. 2007; ξ η Butler et al. 2004), and HD 17156b (Fischer et al. 2007). (observer, Earth) Therefore it is now possible, for the first time, to make specific predictions for future, long–term measurements of Figure 1. The geometry of the orbit of the transiting planet. The periastron precession effects for real exoplanetary systems. plane of the sky is defined by the X+ (west) and Y + (north) axes In this paper we determine the precision by which re- while the Z+ axis points away from the Earth. The orbital plane peated long–term future ETEP observations will be able to is defined by the axes ξ+ and η+, where ξ+ is anti-parallel with detect the periastron precession rate for existing systems. X+ axis, η+ is in the plane of (Y +,Z+) and the angle between Y + and η+ is the inclination, nearly 90◦. The major axis of the In addition to TTVs, i.e. the slow modulation of Pobs con- sidered previously (Miralda-Escude 2002; Heyl & Gladman orbit is marked by the dotted line. 2007), the periastron precession also changes the time durations TD of individual transits. We examine whether these the instance2 of the transit. From the definitions of (ξ, η), transit duration variations (TDVs) can be used to improve observing from Earth is equivalent to setting ϕ0 = π/2. the sensitivity of periastron precession measurements. We Now let us denote the mean longitude of the planet at estimate the precession rate measurement precision for long the instance of the transit by λ. For a circular orbit, λ = term repeated observations of Pobs and TD for the known ϕ0. From its standard definition in celestial mechanics, it is ETEPs. Since several of the observed ETEPs have large ec- straightforward to derive the mean longitude for arbitrary centricities, we derive expressions for both TTVs and TDVs eccentricities (see Appendix A). The result is which are applicable for arbitrary eccentricities. We estimate whether future observations of currently known ETEPs will λ λ(ϕ0,k,h) λ(ϕ0,e cos ω,e sin ω)= ≡ ≡ be able to reach the sensitivity necessary to test the pre- he⊥ ke⊥ = arg k + cos ϕ0 + , h + sin ϕ0 diction of GR, using existing or planned space-borne instru- 2 ℓ − 2 ℓ − ments. We refer the reader to a recent independent study „ − − « e⊥(1 ℓ) by Jordan & Bakos (2008), of precession rates in eccentric − , (1) − 1+ e transiting extrasolar planets. k The next section of this paper introduces the geometri- where e = k cos ϕ0 + h sin ϕ0, e⊥ = k sin ϕ0 h cos ϕ0 are k − cal description which is the basis of our calculations, and de- the components of the eccentricity vector relative to the line- rives the expected transit timings and durations for planets of-sight, ℓ = 1 √1 e2 is the oblateness of the orbit, and − − orbiting a star with an arbitrarily large eccentricity. In 3, arg(x,y) = arctan(y/x) if x > 0 and π + arctan(y/x) other- § we utilize our results for the confirmed four ETEP systems, wise. Plugging in the observed values of e and ω, equation (1) and give predictions for future observations of periastron provides a simple way of calculating the mean longitude of precession with space-borne observations. Our conclusions the orbit for an arbitrary transit observation. Note that this are discussed in 4. formalism omits the direct usage of the mean anomaly, true § anomaly and eccentric anomaly which have no meanings for e 0. All of our derived formulae are based on the well– → behaved parameters λ, k, and h, and therefore are valid for arbitrary eccentricities. 2 TRANSIT TIMINGS AND DURATIONS FOR In the following subsections, we derive the expressions ECCENTRIC ORBITS describing TTVs and TDVs, discuss the corresponding ob- The reference frame used for the description of exoplane- servational implications and estimate the precision of peri- tary systems as well as for binary/multiple stellar systems astron precession observations. is fixed to the sky: the plane of the sky is defined by the (X+,Y +) while Y + points towards to north. For planetary transit observations, the line-of-sight lies close to the orbital 2.1 Modulation of the transit period plane, i.e. perpendicular to the plane of the sky. The orbit is The period between successive transits Pobs is modified by a given by Cartesian coordinates (ξ, η), where ξ+ is parallel to possible slow precession of the orbital elements. These mod- X and η+ oriented toward the observer (see also Fig. 1). − ulations are referred to as transit timing variations (TTVs). The Lagrange vector or eccentricity vector is given in these coordinates as (k, h)=(e cos ω,e sin ω). Let us define the 2 angle ϕ0 as the angle relative to ξ+ in the orbital plane at i.e. at the center of a transit

c 2008 RAS, MNRAS 000, 1–9 Periastron Precession of TEPs 3

Table 1. Basic data of the four known eccentric transiting exoplanetary systems: the mass (M⋆) of the parent star, period (P , in days), eccentricity (e) and the argument of pericenter (ω), the half-duration of a transit event (H, in days) and the impact parameter (b). In the last two columns, we provide the calculated values of the secular period caused by the GR periastron precession, and the minimum mass – semimajor axis ratio for a hypothetical exterior perturber at a2 & 3a that would lead to the same magnitude of periastron precession. .

m2/M⊕ System M /M P (d) e ω (degrees) H (d) b P (years) 3 ⋆ ⊙ sec (a2/3a) HD 147506b 1.298 ± 0.07 5.63341 0.520 ± 0.010 179.3 ± 3.6 0.083 0 19790 ± 740 12.2 XO-3b 1.41 ± 0.08 3.19154 0.260 ± 0.017 344.6 ± 6.6 0.050 0.8 9280 ± 410 15.9 GJ 436b 0.452 ± 0.013 2.64385 0.150 ± 0.012 351.0 ± 1.2 0.065 0.85 ± 0.02 15180 ± 400 2.6 HD 17156b 1.2 ± 0.1 21.21725 0.6717 ± 0.0027 121.23 ± 0.40 0.098 0.50 ± 0.12 143000 ± 7900 5.9

We derive the modulation of Pobs due to periastron preces- Miralda-Escude 2002; Heyl & Gladman 2007) is very im- sion in two steps. First we demonstrate that the change in precise for moderate to large eccentricites. Depending on the period between successive transits is simply related to the actual value of ω, equation (7) can result even 6 8 2 2 − the change in the mean longitude ∆λ. Then relating the times smaller or larger values for ∂ λ/∂ω as its first order mean longitude shift to the periastron precession rate we approximation for eccentricities 0.5 0.7. − derive the expected TTV rate. We have to mention here that the observed period and Let P0 be the orbital period, n = 2π/P0 be the mean therefore the individual transit timings are also affected by angular velocity. The mean longitude of the planet increases the light time effect (LTE). Since the precession of an ellip- steadily in time, λ = nt+λ0. The variation in the mean lon- tical orbit causes the distance between the host star and the gitude at the transit center would result in a variation in the planet at the transit instances to vary, the light time delay transit cadence. During a transit at time t1, the mean longi- will change for transit event to transit event. The magnitude is λtr(t1)= nt1 + λ0, and after an observed revolution, tude of this effect can be derived as follows. The distance at the instance t2 = t1 + Pobs it is λtr(t2) + 2π = nt2 + λ0. between the star and the planet at transit time (i.e. when Therefore the observed period between transits is λ = ϕ0 = π/2) is 2π +∆λ ∆λ a(1 e2) a(1 e2) a(1 e2) Pobs = = P0 1+ (2) r = − = − = − . (8) n 2π 1+ e cos v 1+ e cos(ϕ0 ω) 1+ e sin ω „ « − where ∆λ = λtr(t2) λtr(t1). This difference in the distance implies an additional r(1 − − − In the following we assume that the shift in the mean µ)/c time shift, respective to the barycentric reference frame longitude is caused by the perihelion shift ∆ω = P0ω˙ per of the star–planet system. Here, µ = Mp/(Mp + M⋆) 1 ≪ period, (i.e. we assume a constant eccentricity), then from is the mass parameter, a is the semimajor axis of the orbit the chain rule and c is the speed of light. Therefore, the correction in the ∂λ ∂λ ∂k ∂λ ∂h ∆λ = ∆ω = ∆ω + ∆ω. (3) observed period is ∂ω ∂k ∂ω ∂h ∂ω −1 +LTE a 2 ∂(1 + e sin ω) Here ∂k/∂ω = e sin ω and ∂h/∂ω = e cos ω, from definition P = Pobs (1 e ) P0ω˙ = − obs − c − ∂ω (see above), and the partial derivatives ∂λ/∂k and ∂λ/∂h a (1 e2)e cos ω P can be found from equation (1) and are given explicitly in 0 = Pobs + 2π − 2 , (9) the Appendix (A8–A9). At transit, we get c (1 + e sin ω) Psec ∂λ e2 e2 +(2+ e sin ω)e sin ω where we neglected the barycentric correction. Thus, the = + 1 e2 . (4) variation in this period (corrected for LTE) due to the vari- ∂ω 1+ √1 e2 − (1 + e sin ω)2 − p ation in ω is Combining equations (2)-(4) and defining the secular period 2 ˙ +LTE ˙ 2 2 e(e + e cos ω + sin ω) a P0 of the periastron precession as Psec = (2π)/ω˙ , we get Pobs = Pobs 4π (1 e ) 3 2 .(10) − − (1 + e sin ω) c Psec 2 2 P0 ∂λ ∂λ P0 Pobs = P0 + ω˙ = P0 + . (5) Comparing equation (6) and (10), and assuming that the 2π ∂ω ∂ω Psec motion of the planet is non-relativistic, a/c P0, we find +LTE ≪ Since ∂λ/∂ω itself is not constant due to periastron pre- that P˙ P˙obs P˙obs . We conclude that the period | obs − | ≪ | | cession, the observed period between transits slowly changes. variation due to the LTE is negligible compared to the period Differentiating equation (5) with respect to time, we get variation caused by the changing geometry. 2 2 2 3/2 2 In summary, equations (5)-(6), along with equation (4), ˙ ∂ λ P0 4π(1 e ) e cos ω P0 Pobs = 2π 2 2 = − 3 2 , (6) give the modulation of the actual observable period between ∂ω Psec (1 + e sin ω) Psec transit events. These results are valid for arbitrary eccentric- since the partial derivative of equation (4) with respect to ities and are independent of the physical mechanism causing ω is the secular precession of the periastron. We calculate the ∂2λ 2(1 e2)3/2e cos ω secular precession period caused by general relativity (see = − . (7) ∂ω2 (1 + e sin ω)3 e.g., Wald 1984) using 2 2 Note that this equation clearly shows that the small ec- c (1 e ) 5/3 2 2 Psec = − P , (11) centricity approximation ∂ λ/∂ω e cos ω (e.g. used by 3(2πGM )2/3 0 ≈ ⋆ c 2008 RAS, MNRAS 000, 1–9 4 A. Pál and B. Kocsis where M⋆ is the mass of the parent star, G is Newton’s to Kepler’s Second Law, the apparent tangential velocity of gravitational constant. This secular period is of order 104– the transiting object will increase (resulting a shorter tran- 105 years for the known ETEP systems (see Table 1 and sit duration). Therefore at a certain value of the inclination Section 3 for more details). We note that if other planets and/or the impact parameter, the two effects cancel each are also present in these systems, they might also cause ad- other yielding no TDV effect. Equation 16 clearly shows that ditional periastron precession of a larger magnitude. The it occurs when the impact parameter is b = 1/√2 0.707. ≈ Yarkovski-effect (Fabrycky 2008) and the tidal circulariza- The long-term variation in the duration of the transit is then tion (see Johns-Krull et al. 2007, and the references therein) √ 2 2 lead to negligible modifications for our purposes, as these ef- ˙ ∂H 1 e e cos ω 1 2b R⋆ P0 H = ω˙ = − 2 − (17) ∂ω (1 + e sin ω) √1 b2 a Psec fects are relevant on timescales exceeding 0.1 Gyr for these − systems. In the following, we compare the precession mea- Comparing equation (6) and equation (17) the TDV effect surement sensitivities with the general relativistic rate Psec relates to the TTV effect as given by equation (11). 2 H˙ 1+ e sin ω 1 2b R⋆ = − (18) 2 P˙obs 6π √1 b RSch 2.2 Modulation of the transit duration − 2 where RSch = 2GM/c is the Schwarzschild radius of the 5 Here we investigate how the periastron precession affects the star. As an example, note that R⋆/RSch = 2.5 10 for × duration of a transit. Let us denote the half duration of the the Sun. Therefore, equation (18) shows that the TDV ef- transit by H = TD/2, which we define as half the time be- fect is always much larger than the TTV effect. In partic- tween the instances when the center of the planetary disk ular, the ratio is larger for increasing b. In the limit b 1 → intersects the limb of the star, i.e. between the center of the equation (17) breaks down because the periastron precession ingress and egress. Note that this is not the time between shifts the orbit out of the transiting region. the first and last contact. This is important because the instances of the center of the ingress and egress can be measured more accurately than their beginning or end. Since we 2.3 Observational implications are interested in estimating the variations of the duration of the transit to leading order, we perform a first-order calcu- Now, using the results for the TDV and TTV effects, lation, i.e. assuming a constant apparent tangential velocity equation (6) and (17), we can estimate how these timing for the planet and neglecting the changes in the impact pa- and transit duration variations might be observed on long rameter due to the elliptical orbit and/or the curvature of timescales. In the following we discuss these observational the projection of the orbit due to the inclination. From Ke- implications. pler’s Second Law and equation (8), one can calculate the tangential distance ∆x traveled by the planet during time H, 2.3.1 Transit Timing Variations 1+ e sin ω Equation (6) shows that the observed period between suc- ∆x = vtanH = an H (12) √1 e2 cessive transits increases or decreases at a practically con- − stant rate during the observations, P˙obs. The time of the (see Appendix B for the derivation of vtan). This can be related to the impact parameter b and the radius of the star mth transit from an arbitrary epoch T0 can be found from adding up the contributions of the observed m number of R⋆ for the geometry of the transit as periods ∆x = 1 b2. (13) 2 R⋆ − Tm = T0 + Pobsm + Dm , (19) p Thus, to leading order, where Pobs P0 denotes the time of the first observed orbit, ≈ D is the transit timing variation factor, √ 2 3 P0 R⋆ 1 e 2 R⋆ H = − 1 b + . (14) ˙ 2 2π a 1+ e sin ω − O a P0Pobs P0 ( "„ « #) D = = 2πGTV(e,ω)P0 , (20) p 2 Psec Note that equation (14) depends on ω through the (1 + „ « e sin ω)−1 term and also implicitly through b, where we have introduced the geometrical factor

2 2 3/2 e cos ω r cos i a 1 e GTV(e,ω)=(1 e ) . (21) b = = − cos i. (15) − (1 + e sin ω)3 R⋆ R⋆ 1+ e sin ω The variation in H caused by the variation in the periastron The chance of detecting the periastron precession in- can be found from equation (14) and (15), creases with the geometrical factor GTV(e,ω) which is related to the alignment of the orbital ellipse with the line ∂H e cos ω 1 2b2 of sight. The optimal value for detecting the precession is = H − . (16) ∂ω 1+ e sin ω 1 b2 ω = 0 or π for small eccentricities, i.e. the semimajor axis − Note that equation reflects the qualitative expections im- should be perpendicular to the line of sight. For arbitrary ec- plied by Kepler’s Second Law. Namely, if an eccentric orbit centricities, the optimal value for ω for the TTV observation advances, the distance between the planet and the star will is change. If this distance decreases, the impact parameter will best 3 6e ωTV = π arccos . (22) also decrease (resulting a longer transit duration) but due 2 ± √ 2 „1+ 1 + 24e « c 2008 RAS, MNRAS 000, 1–9 Periastron Precession of TEPs 5 which approaches 0◦ and 180◦ for small eccentricities as ex- Gaussian errors, the parameter estimation covariance matrix pected, and 270◦ for large eccentricities. In case of e = 0.5, can be found from the Fisher matrix method (Finn 1992): ◦ ◦ ωextr = 235.4 , 304.6 . The least favorable value for ω oc- −1 { } worst ◦ ◦ δpiδpj =( )ij (28) curs when GTV(e,ω) = 0, i.e. at ω = 90 , 270 ,. TV { } h i F Here is the Fisher matrix defined as F fid fid 2.3.2 Transit Duration Variations 1 ∂Tm ∂Tm ij = 2 (29) F σ (T ) ∂pi ∂pj Now let us turn to the TDV effect. The observed duration of m=0,d,...,X(N−1)d the mth transit can be calculated in the same way, namely fid where Tm is the fiducial value of Tm given by equation (19). Hm = H0 + F m, (23) The marginalized expected squared parameter estimation error is given by the diagonal elements of the covariance where F is the shift in the transit duration per orbit. This 2 −1 error matrix σ (pi)=( )ii. In particular, the resulting factor is F uncertainty of D becomes ˙ P0 F = P0H = 2πGDV(e,ω,b)H , (24) √180 σ(T ) Psec σ(D) = d2 N(N 2 1)(N 2 4) and − − 2 2 P 5 σ(T ) 1 2b e cos ω √180p 0 1+ . (30) GDV(e,ω,b)= − . (25) 2 1 b2 1+ e sin ω ≈ Ttot 2N √N − „ « „ « best Here, the first equality is valid for arbitrary N, while the sec- The optimal orientation ωDV for detecting the TDV ef- ond is its first order approximation for large N. The leading fect for fixed e, b, and P0 can be found by maximizing order approximation is verified against Press et al. (1992). HGDV(e,ω,b). The result is

best 3 4e ωDV = π arccos . (26) 2 ± √ 2 „1+ 1 + 8e « 2.4.2 Transit Duration Variations worst ◦ ◦ and the worst orientation is at ωDV = 90 , 270 , just We can repeat the same calculations as above for the ob- best { best } like for the TTV case. Comparing ωTV and ωDV it is clear servation of the half transit duration Hm to measure the that the most favorable orientation in terms of the two ef- variation factor F . The merit function in this case fects are similar, hence the chance of detecting the periastron 2 Hm (H0 + Fm) motion through transit timing variations or transit duration χ2 = − , (31) DV σ(H) variations is correlated. Both effects go away if the eccen- m=0,d,...,(N−1)d » – tricity is oriented parallel to the line of sight. We also note X that for moderate values of e and small impact parameters, has to be minimized for the same set of observations. This minimization again leads to a linear set of equations in the GTV GDV which also implies that the most favorable | | ≈ | | geometry for detecting either TTVs or TDVs is similar. parameters H0 and F . The Fisher matrix method in this case gives the uncertainty in F as √12σ(H) 2.4 Error analysis σ(F ) = d N(N 2 1) ≈ Next we estimate the parameter measurement precision of − P 1 σ(H) the TTV and TDV effects for future observations. We con- √p 0 12 1+ 2 . (32) ≈ Ttot 2N √ sider the repeated observation of a particular transiting sys- „ « N tem over a total timespan Ttot, measuring the transit timing Figure 2 shows the detection significance of the TDV Tm and duration Hm for each transit with respective errors measurement F /σ(F ), if the precession rate in F is given | | σ(T ) and σ(H). (We discuss the specific values of σ(T ) and by the general relativistic formula, equation (24). Here each σ(H) for transit observations in Section 3.1). For simplic- transit is assumed to be measured (i.e. d = 1) with a ity, let us assume that these measurements are equidistant precision σ(H) = 5 sec for a total observation time of and in total N independent transits are observed, i.e. the Ttot = 4 years. These assumptions are realistic for the fu- mth transit is observed if m = 0,d,..., (N 1)d, where ture Kepler mission (see 3.1 below). Other parameters are − § d = Ttot/(NP0). M⋆ = M⊙, R⋆ = R⊙, b = 0, and we averaged over the possible orientations of ω. For other parameters,

1/3 2.4.1 Transit Timing Variations F /σ(F ) 1 1 R⋆ M⋆ | | = F0 /σ(F0) d √1 b2 R⊙ M⊙ · Using equation (19), we can fit a second-order polynomial | | „ «„ « − −1 3/2 to these observations with unknown coefficients T0, Pobs and σ(H) T tot , (33) D by minimizing the merit function · 5 sec 4 yr „ « „ « 2 2 Tm (T0 + Pobsm + Dm ) implying that the detection significance can be even better. χ2 = − . (27) TV σ(T ) Figure 2 clearly shows that the chances of detecting the m=0,d,...,X(N−1)d » – precession effects through the TDV effect is encouraging. The minimization of the above function results in a linear set The detection significance of the general relativistic preces- of equations in the parameters pi = T0, Pobs, D . Assuming sion of a transiting exoplanet with eccentricity e & 0.2 and { } c 2008 RAS, MNRAS 000, 1–9 6 A. Pál and B. Kocsis

1 Table 2. Transit timing variation factor (D, in seconds) and the number of transits (N20 y,3−σ,2 sec) what should be detected almost uniformly in a 20 year long timespan, each with an error 0.8 30 of 2 second to conﬁrm the precession within 3-σ. 10 0.6 3 System D (seconds) N20 y, 3−σ, 2 sec −7 0.4 HD 147506b −(5.9 ± 0.7) · 10 6600

Eccentricity −7 1 XO-3b +(4.3 ± 0.6) · 10 1280 GJ 436b +(5.0 ± 0.5) · 10−8 44160 0.2 0.3 HD 17156b −(6.9 ± 0.8) · 10−8 97 · 106

0 0 2 4 6 8 10 Period (days) a2 & 3a and for non-resonant cases. Note that the minimum mass of the perturber scales with the third power of Figure 2. The significance of detecting the GR periastron preces- the semimajor axis ratio to cause a comparable precession sion |F0|/σ0(F ) through the TDV effect as a function of orbital rate as GR. The numbers show that the precession caused period and eccentricity. The transit duration is assumed to be by additional planets in the system, if present, can easily measured for 4 years each with a 5 sec error, for a Sun-like star cause a larger precession rate than GR. In case of orbital on a non-inclined orbit. Increasing the mass or radius of the star, or the impact parameter increases the detection significance (see resonances with exterior planets, the precession rate can be text). even larger (Holman & Murray 2005; Agol et al. 2005). To be conservative, we examine whether the precession rate can be measured to a precision better than that corresponding period P . 5 days, is typically over the 1-σ level. Gener- to GR. ally, equation (30) and equation (32) can be used directly To obtain a high significance, k-σ detection of the pe- to check what kind of observations are required to detect riastron precession using transit timing variations, we need kσ(D) D . Using equation (30), the total number of tran- the precession of the periastron through the TTV or TDV ≈ | | methods, respectively. sits necessary to measure D with this precision is 2 4 σ(T ) P0 NTV = 180 . (34) k D Ttot » | | – „ « 3 THE CASE OF XO-3b, HD 147506b, GJ436b Thus, the number of such required transit observations is AND HD 17156b extremely sensitive to the orbital period P0 and the ob- As of this writing, four TEPs are known with a non-zero servation timespan. Table 2 gives the corresponding val- eccentricity within 3-σ, namely XO-3b (Johns-Krull et al. ues of the transit timing variation factors for the known 2007), HD 147506b (Bakos et al. 2007), GJ 436bb ETEP systems and this number of observations, assuming a (Butler et al. 2004; Gillon et al. 2007), and HD 17156bb Ttot = 20 year long observational timespan, timing precision (Fischer et al. 2007; Barbieri et al. 2007). The planet TrES- of σ(T ) = 2 sec and 3-σ sensitivity of the GR periastron pre- 1 (Alonso et al. 2004) has also been reported as an object cession level. The table shows the recently discovered XO-3b with non-zero eccentricity, however, it is zero within 2-σ thus system is a promising candidate to detect the GR periastron we omit from our analysis. We note here that recently both precession through the TTV effect, while the other ETEP GJ 436 and HD 17156 have been suggested to have another systems require unrealistically many observations. We note planetary companions (see Ribas et al. 2008; Short et al. that if other perturbing planets are present in these systems 2008). The secular period of the periastron motion are de- and lead to a precession rate that is larger by a factor of 10 than the GR precession rate, then the number of detec- termined by the mass of the star M⋆, the orbital period P0, and the eccentricity e, while the timing variation constant tions (during the same Ttot timespan) is lower by a factor D is also affected by the actual argument of pericenter, ω. of 100. It is interesting that the best candidate (by far) is The transit duration variation factor F is affected indirectly XO-3b even though its eccentricity is not as large as that of HD 147506b or HD 17156b. by the geometrical ratio a/R⋆ and directly by the impact parameter b. These parameters are summarized in the first Let us now turn to the observational constraints for the seven columns of Table 1 for these four ETEP systems. The detection of transit duration variations. Using equation (32), the total number of required observations within Ttot is derived GR periastron precession period, Psec can be found in the 8th column of the table. 2 2 σ(H) P0 In addition to the inevitable periastron precession NDV = 12 . (35) k F Ttot caused by GR, there might be other sources of perturba- » | | – „ « tions causing periastron precession. The last column gives Note that NDV is not as sensitive to the P0/Ttot ratio as the minimum mass to semimajor axis ratio of a hypothet- NTV, and implies that a smaller number of observations is ical exterior perturber (e.g. a planet or an asteroid belt), typically necessary. in Earth mass units, which causes the same periastron pre- In Table 3 we present the values of the transit duration session rate as that caused by the general relativity. This variation factor, F , and the number of required observations estimate based on Price & Rush (1979), and is valid for to reach the same 3-σ confidence for detecting the GR peri-

c 2008 RAS, MNRAS 000, 1–9 Periastron Precession of TEPs 7

Table 3. Transit duration variation factor (F , in seconds) and Table 4. Uncertainties of the transit time and transit duration ′ the number of transits (N2 y, 3−σ, 2 sec) what should be detected measurements for the four known ETEPs, assuming Sloan z - almost uniformly in a 4 year long timespan, each with an error of band photometric data taken with a 1 sec cadence and 1 mmag 2 sec to conﬁrm the precession within 3-σ. photometric precision.

System F (seconds) N4 y, 3−σ, 2 sec System σ1,1(T ) (sec) σ1,1(H) (sec) HD 147506b −(1.8 ± 0.3) · 10−2 20 HD 147506b 5.3 4.8 XO-3b −(5.4 ± 0.6) · 10−3 70 XO-3b 6.9 4.7 GJ 436b −(4.1 ± 0.5) · 10−3 85 GJ 436b 6.7 8.4 HD 17156b −(3.2 ± 0.2) · 10−3 8970 HD 17156b 5.4 6.1

astron precession3. Here we assumed a shorter observation timespan, Ttot = 4 year (i.e. shorter compared to the 20 year σ(m) ∆τ σ(H) σ1,1(H) . (39) long timespan necessary for the detection of the TTV ef- ≈ 1 mmag 1 sec fect), the same timing precision of σ(H) = 2 sec and the r same level of detection, 3-σ. The best known ETEP system For comparison, note that the expected photometric preci- for TDV detection is therefore HD 147506b, but the number sion of the Kepler space telescope (see Borucki et al. 2007) of necessary observations is feasible for XO-3b and GJ 436b is 1 mmag for observing a light curve of a bright, Mv = 8.8 as well. star with a 1 sec sampling cadence. Since the star XO-3 has almost the same apparent magnitude, it is clear, the transit durations would be detected with an accuracy of 3.1 Photometric detection σ(H) 5 sec if this star was in the field of Kepler. There- ≈ The precision for measuring the transit timing and transit fore, equation (32) and Table 3 shows that the transit du- duration for a photometric observation can be estimated as ration variations would be detectable for HD 147506(b) or XO-3(b)–like systems due to the GR periastron precession, follows. Since the time of the ingress (TI) and the time of the within 3-σ confidence with a Kepler–type mission within ap- egress (TE) – i.e. when the center of the planet crosses the limb of the star inwards or outwards, respectively – defines proximately 3 or 4 years, respectively. both the time of the transit center and the half duration like 1 T = (T + T ), (36) 2 E I 1 H = (TE TI), (37) 2 − 4 SUMMARY moreover TI and TE can be treated as uncorrelated vari- The first four eccentric transiting exoplanetary systems have ables, therefore the uncertainties of the transit time and half been discovered during 2007. The precession of an eccentric duration would be nearly the same, i.e. σ(T ) σ(H). We ≈ orbit causes variations both in the transit timings and tran- have estimated these uncertainties for the four distinct plan- sit durations. We estimated the significance of measuring the ets using Monte-Carlo simulations by fitting transit light corresponding observable effects compared to the inevitable curves on mock data sets. We have used the observed plane- precession rate of general relativity. We applied these calcu- tary parameters as an input for these artificial light curves. lations to predict the significance of measuring the effect for The fit was performed assuming quadratic limb darkening the four known eccentric transiting planetary systems. Our ′ (see Mandel & Agol 2002) in the Sloan z band. The mock calculations show that a space-borne telescope is adequate light curves were sampled with ∆τ1 = 1 sec cadence and an to detect the change in the transit durations to a high signif- additional Gaussian noise of σ1(m) = 1 mmag was added. icance better than the GR periastron precession rate within The resulting uncertainties, σ1,1(T ) and σ1,1(H) for the four a 3 – 4 year timespan (in a continuous observing mode). The planets are presented in Table 4. Since the depth of the four same kind of instruments would need more than a decade transits are nearly the same (see the appropriate normalized to detect the corresponding transit time variations to this radii, p = Rp/R⋆, all between 0.068 . p . 0.085), the un- sensitivity even for the most optimistic known system. certainties σ1,1(T ) and σ1,1(H) are almost the same for the The CoRoT mission has already found two transiting four cases. Using these normalized values, one can easily es- planets (see Barge et al. 2008; Alonso et al. 2008) and there timate the uncertainties for arbitrary sampling cadence ∆τ are two known planets in the planned field-of-view of the and photometric precision σ(m) using Kepler mission (see O’Donovan et al. 2006; Pál et al. 2008). Our results suggest that if an eccentric transiting planet is σ(m) ∆τ σ(T ) σ (T ) , (38) found in the Kepler or CoRoT field, these missions will be 1,1 1 mmag 1 sec ≈ r able to measure the periastron precession rate to a very high significance within their mission lifetime or with the support 3 Note that since the measurement of the TDV effect relies on of ground-based observations on a longer time scale. This fitting 2 parameters, instead of 3 parameters for the TTV effect, will provide an independent test of the theory of general the 3-σ confidence corresponds to a higher confidence level for the relativity and will also be useful for testing for the presence TDV effect. of other planets in these systems.

c 2008 RAS, MNRAS 000, 1–9 8 A. P´al and B. Kocsis

ACKNOWLEDGMENTS λ = ω + E e sin E. The only thing what is to be done is − to calculate the eccentric anomaly E for the instance when The authors would like to thank the hospitality and support the orbiting body intersect the semi-line with the argument of the Harvard-Smithsonian Center for Astrophysics where angle ϕ . The latter means that the true anomaly v of the this work was partially carried out. We thank Andres Jordan 0 body is v = ϕ0 ω, by definition. The relation between the for useful comments on the manuscript. BK acknowledges − eccentric and true anomaly is support from OTKA grant No. 68228. E 1 e v tan = − tan , (A1) 2 1+ e 2 r REFERENCES which is equivalent with e + cos v Agol, E., Steffen, J., Sari, R., & Clarkson, W., 2005, MN- cos E = , (A2) RAS, 359, 567 1+ e cos v Alonso, R. et al., 2004, ApJ, 613, 153 √1 e2 sin v sin E = − . (A3) Alonso, R. et al., 2008, astro-ph:0803.3207 1+ e cos v Bakos, G. A.´ et al., 2007, ApJ, 670, 826 Using the addition theorem, the sine and cosine of the angle Barbieri, M. et al., 2007, A&A, 476, 13 ω + E can be written as: Barge, P. et al., 2008, astro-ph:0803.3202 Borucki, W. J. et al., 2007, ASP Conf. Ser., 366, 309 e + cos(ϕ0 ω) cos(ω + E) = cos ω − Brown, T. M. et al., 2001, ApJ, 552, 699 1+ e cos(ϕ0 ω) − − 2 Butler et al., 2004, ApJ, 617, 580 √1 e sin(ϕ0 ω) Charbonneau, D., Brown, T. M., Latham, D. W. & Major, sin ω − − , (A4) − 1+ e cos(ϕ0 ω) M., 2000, ApJ, 529, 45 − e + cos(ϕ0 ω) Fischer, D. A. et al., 2007, ApJ, 669, 1336 sin(ω + E) = sin ω − + 1+ e cos(ϕ0 ω) Fabrycky, D., 2008, astro-ph:0803.1839 − 2 Finn, L. S., 1992, Phys. Rev. D, 46, 5236 √1 e sin(ϕ0 ω) + cos ω − − . (A5) Ford, E. B. & Holman, M. J., 2007, ApJ, 664, 51 1+ e cos(ϕ0 ω) − Gillon, M. et al., 2007, A&A, 472, 13 Thus, the mean longitude itself is going to be Heyl, J. S. & Gladman, B. J., 2007, MNRAS, 377, 1511 Holman, M. J, & Murray, N. W., 2005, Science, 307, 1288 λ = ω + E e sin E = arg [cos(ω + E), sin(ω + E)] − − Iorio, L., 2006, NewA, 11, 490 √1 e2 sin v e − . (A6) Jordan,A. & Bakos, G, 2008, ApJ, in press − 1+ e cos v Johns-Krull, C .M. et al., 2007, astro-ph:0712.4283 If both arguments of the above arg[ , ] function is multiplied Mandel, K., Agol, E., 2002, ApJ, 580, 171 · · by the always positive common denominator 1 + e cos(ϕ0 Miller-Ricci, E. et al., 2008, astro-ph:0802.0718 − Miralda-Escude, J., 2002, ApJ, 564, 1019 ω), one gets after some simplification:

Misner, C. W., Thorne, K., & Wheeler, J. A., 1973, Grav- h(k sin ϕ0 h cos ϕ0) itation, Second Edition, W. H. Freeman and Company, ω + E = arg k + cos ϕ0 + − , 1+ √1 e2 San Francisco » − k(k sin ϕ0 h cos ϕ0) O’Donovan, F. T. et al., 2006, ApJ, 651, 61 h + sin ϕ0 − . (A7) − 1+ √1 e2 Pál, A. et al., 2008, astro-ph:0803.0746 − – Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flan- Putting all terms together and replacing the appropriate nery, B. P., 1992, Numerical Recipes in C: the art of sci- terms by e⊥ = k sin ϕ0 h cos ϕ0, e = k cos ϕ0 +h sin ϕ0 and − k entific computing, Second Edition, Cambridge University ℓ = 1 √1 e2, we get equation (1). The partial derivatives − − Press of equation (1) become Price, M. P., & Rush, W. E., 1979, Am. J. Phys., 47, 531 ∂λ h h +(2+ ek) sin ϕ0 Ribas, I., Font-Ribera, A. & Beaulieu, J., 2008, ApJ, 677, = (1 ℓ) , (A8) ∂k − 2 ℓ − − (1 + e )2 59 − k Short, D., Welsh, W. F., Orosz, J. A. & Windmiller G., ∂λ k k +(2+ ek) cos ϕ0 = + + (1 ℓ) 2 . (A9) 2008, astro-ph:0803.2935 ∂h 2 ℓ − (1 + ek) Simon, A., Szatmáry, K. & Szabó, Gy. M., 2007, A&A, 470, − 727 Steffen, J. H. & Agol, E., 2007, ASP Conf. Ser, 366, 158 Wald, R. M., 1984 General Relativity, The University of APPENDIX B: TANGENTIAL VELOCITY AND Chicago Press POSITION AT THE TRANSIT It is known from the theory of the two-body problem that the angular momentum of a body orbiting around a mass of GM = µ and having an orbit with the semimajor axis of a 2 APPENDIX A: MEAN LONGITUDE AT THE and eccentricity e is C = µa(1 e ). Since C = rvtan for − TRANSIT INSTANCES all points, the tangential velocity would be p The derivation of equation (1) goes as follows. According to C 2 1+ e cos(ϕ0 ω) vtan = = µa(1 e ) − (B1) Kepler’s equation, E e sin E = M = λ ω, one can write r a(1 e2) − − − p − c 2008 RAS, MNRAS 000, 1–9 Periastron Precession of TEPs 9

Using Kepler’s Third Law, i.e. µ = n2a3, the above equation can be reordered to

1+ e cos(ϕ0 ω) vtan = an − . (B2) √1 e2 − For ϕ0 = π/2, equation (B2) becomes 1+ e sin ω vtan = an . (B3) √1 e2 −

c 2008 RAS, MNRAS 000, 1–9