Time variation of Kepler transits induced by stellar rotating spots - a way to distinguish between prograde and retrograde motion I. Theory
aa r X i v : . [ a s t r o - ph . S R ] J a n Time variation of
Kepler transits induced by stellar rotating spots— a way to distinguish between prograde and retrograde motionI. Theory
Tsevi Mazeh , , Tomer Holczer , Avi Shporer , , Received ; accepted School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sci-ences, Tel Aviv University, Tel Aviv 69978, Israel The Jesus Serra Foundation Guest Program, Instituto de Astrofsica de Canarias, C. viaLactea S/N, 38205 L Laguna, Tenerife, Spain Division of Geological and Planetary Sciences, California Institute of Technology,Pasadena, CA 91125, USA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive,Pasadena, CA 91109, USA Sagan Fellow 2 –
ABSTRACT
Some transiting planets discovered by the
Kepler mission display transit tim-ing variations (TTVs) induced by stellar spots that rotate on the visible hemi-sphere of their parent stars. An induced TTV can be observed when a planetcrosses a spot and modifies the shape of the transit light curve, even if the timeresolution of the data does not allow to detect the crossing event itself. Wepresent an approach that can, in some cases, use the derived TTVs of a planet todistinguish between a prograde and a retrograde planetary motion with respectto the stellar rotation.Assuming a single spot darker than the stellar disc, spot crossing by the planetcan induce measured positive (negative) TTV, if the crossing occurs in the first(second) half of the transit. On the other hand, the motion of the spot towards(away from) the center of the stellar visible disc causes the stellar brightness todecrease (increase). Therefore, for a planet with prograde motion, the inducedTTV is positive when the local slope of the stellar flux at the time of transitis negative, and vice versa. Thus, we can expect to observe a negative (pos-itive) correlation between the TTVs and the photometric slopes for prograde(retrograde) motion. Using a simplistic analytical approximation, and also thepublicly available SOAP-T tool to produce light curves of transits with spot-crossing events, we show for some cases how the induced TTVs depend on thelocal stellar photometric slopes at the transit timings. Detecting this correlationin
Kepler transiting systems with high enough signal-to-noise ratio can allow usto distinguish between prograde and retrograde planetary motions. In comingpapers we present analyses of the KOIs and
Kepler eclipsing binaries, followingthe formalism developed here. 3 –
Subject headings: planetary systems — techniques: photometric — stars: spots —stars: rotation
1. Introduction
Formation and evolutionary processes of stellar and planetary systems are expected toleave their imprint on the present-day systems. One such imprint is the stellar obliquity, theangle between the stellar spin axis and the orbital angular momentum axis, also referred toas the spin-orbit angle. For star-planet systems the measurement of this angle is a matter ofintense study in recent years (e.g., Triaud et al. 2010; Moutou et al. 2011; Winn et al. 2011;Albrecht et al. 2012), primarily for hot Jupiters — gas-giant planets at short-period orbits.Some of the systems were found to be aligned, in a prograde orbit with spin-orbit angleclose to zero, while others were found to be misaligned, including systems in retrogrademotion where the spin-orbit angle is close to 180 ◦ (e.g., H´ebrard et al. 2011; Winn et al.2011).The growing sample and the wide range of spin-orbit angles measured for hot Jupiterscan be used for studying their orbital evolutionary history. For example, Winn et al. (2010)have noticed that hot stars, with an effective temperature above 6,250 K, tend to have awide obliquity range, while cool stars tend to have low obliquities, mostly consistent withwell aligned orbits. This was confirmed by a study of a larger sample by Albrecht et al.(2012) and is consistent with the results of Schlaufman (2010) and Hansen (2012) whoused different approaches. Those authors suggested that some mechanisms can cause theplanetary orbit to attain large obliquity (e.g., Fabrycky & Tremaine 2007; Naoz et al.2011; Batygin 2012). Then, tidal interaction with the host star (e.g., Winn et al. 2010) ormagnetic braking (e.g., Dawson 2014) act to realign the orbit. Since these processes areprobably inefficient for hot stars, those systems might still retain their wide obliquity range.So far spin-orbit alignment has been studied primarily through the Rossiter-McLaughlin(RM) effect (Holt 1893; Schlesinger 1910; Rossiter 1924; McLaughlin 1924), originallysuggested for stellar eclipsing binaries, and observed by monitoring the anomalous 5 –radial-velocity signal during eclipse, as the eclipsing star moves across the disc of theeclipsed star. The RM effect is sensitive to the sky-projected component of the spin-orbitangle, and was successfully measured for many transiting planet systems (e.g., Queloz et al.2000; Winn et al. 2006; Triaud et al. 2010), transiting brown dwarfs and low-mass starsystems (Triaud et al. 2013), and stellar binaries (Albrecht et al. 2007, 2009, 2011, 2014).The line-of-sight component of the spin-orbit angle can be measured usingasteroseismology (Gizon & Solanki 2003; Chaplin et al. 2013), or the observed rotationalbroadening of spectral lines, if the host star radius and rotation period are known withsufficient precision (Hirano et al. 2012, 2014, see also Schlaufman 2010). However, thesetwo methods require obtaining new data for each target, using valuable resources (e.g.,large telescopes or Kepler short-cadence data). Other methods have been presented, basedon stellar gravitational darkening (Barnes 2009; Szabo et al. 2011; Barnes et al. 2011), andthe beaming effect (Photometric RM — Shporer et al. 2012; Groot 2012).An interesting approach was taken by Nutzman et al. (2011) and Sanchis-Ojeda et al.(2011), who use the brief photometric signals during transit induced by the transiting objectmoving across spots located on the surface of the host object. This is based on the fact thatmany stars show photometric modulations resulting from the combination of stellar rotationand non-uniform longitudinal spots distribution (e.g., Irwin et al. 2009; Hartman et al.2011; McQuillan, Mazeh & Aigrain 2014). When such a star displays transits by an orbitingplanet, the transiting object might momentarily eclipse a stellar spot, inducing an increasein observed flux, if the surface brightness of the spot-covered area is lower than that ofthe non-spotted areas. The derivation of the stellar obliquity requires identification ofsuch ‘spot-crossing’ events within a few transits, and estimate the spot and the planetphases within their motion over the stellar disc. The method has since been applied toadditional systems using high-speed Kepler and CoRoT data (Sanchis-Ojeda & Winn 2011; 6 –D´esert et al. 2011; Deming et al. 2011; Sanchis-Ojeda et al. 2012, 2013).We present here another version of this approach that does not require suchhigh-speed photometry. Instead, we use the fact that a spot-crossing event can inducemeasurable transit time variation (TTV; e.g., Sanchis-Ojeda et al. 2011; Fabrycky et al.2012; Mazeh et al. 2013; Szab´o et al. 2013; Oshagh et al. 2013b), even for data that cannotresolve the event itself. Our approach relies on the expected correlation between the inducedTTV and the corresponding local photometric slope immediately outside the transit,presumably induced by the same spot. Detected correlation or anti-correlation between theTTVs and their local slope can in principle differentiate between prograde and retrograderotation of the primary star in stellar binaries or star-planet systems.We present here the basic concept and develop an analytical simplistic approximationfor the induced TTVs and the photometric slope. We also use the work of Boisse et al.(2012) and Oshagh et al. (2013a), who developed a numerical tool — SOAP-T , tosimulate a planetary transit light curve which includes a spot-crossing event. Oshagh et al.(2013b) used SOAP-T to derive detailed transit light curves, and then fitted them withtransit templates to obtain the expected TTVs, very similar to what is performed whenderiving the TTVs from the Kepler actual data (e.g., Mazeh et al. 2013). We show that ourapproximation yields TTVs with the same order of magnitude as the results of Oshagh et al.(2013b). Using our approximation and the SOAP-T tool we show that in some cases wecan expect a negative (positive) correlation between the TTVs induced by spot crossingand the local photometric slopes at the transit timings for prograde (retrograde) motionof the planet. We also discuss the limitations of this approach when applied to real data,showing that it can be applied only to a limited number of systems. Kepler planet candidates (Batalha et al. 2013).In that paper we show that indeed a few systems do show highly significant correlationbetween their derived TTVs and the local photometric derivatives, as predicted by thiswork. A forthcoming paper will present our analysis of the
Kepler eclipsing stellar binaries(Slawson et al. 2011).
2. The principle of the approach
To present our approach, we consider a transiting planet that crosses a stellar spotduring its apparent motion over the stellar disc. Let us assume, for the sake of simplicity,that only one spot is present on the stellar disc and that the stellar rotation and orbitalaxes are parallel to each other. This includes both prograde (complete alignment, withobliquity of 0 ◦ ) and retrograde (obliquity of 180 ◦ ) configurations. The sign of the inducedtime shift depends on whether the spot-crossing event occurs in the first (positive TTV) orsecond (negative TTV) half of the transit, which is determined by the location of the spoton the stellar disc at the time of transit. rotationAs depicted in Figures 1 & 2, the location of the spot over the stellar disc determines 8 –whether the star is becoming brighter or dimmer at the time of transit. When the spotis moving towards (away from) the center of the disc the stellar intensity is decreasing(increasing), because of the aspect effect, which changes the effective area of the spot onthe stellar visible disc — the projected area of the spot onto the sky plane. Therefore,when the spot is on the disc edge its effective area is minimal. On the other hand, theeffective area reaches its maximum when the spot is at its closest position to the centerof the visible disc, when the stellar surface is (almost) perpendicular to our line of sight.Another phenomenon, which also causes the star to become fainter when the spot movestowards the center of the disc is the limb-darkening effect, which is ignored at this point ofthe discussion.Now, when the stellar rotation and planetary motion have the same sense of rotation,the spot-crossing event in the first (second) half of the transit should always occur whenthe spot is moving towards (away from) the center of the disc. Therefore the signs ofthe induced TTV and the slope of the stellar brightness at the time of transit should beopposite. This is depicted in Figure 1 for prograde motion. For retrograde motion, positiveTTV should be associated with positive slope, as depicted in Figure 2.Therefore, we expect negative correlation between the derived TTVs and thecorresponding stellar photometric slopes for a system with planetary prograde motionand positive correlation for a system with retrograde motion. In the next sections wewill show that this is indeed the case for a limited number of cases by deriving analyticalapproximations for the TTVs and the photometric derivatives and by numerical simulationsfor the TTVs. 9 –Fig. 1.— Prograde motion — spot-crossing events during the first (left) and second (right)halves of the transit. The top panels display the stellar visible disc (yellow), the planet(black, small) and the spot (gray, large). The arrows represent the direction and speed ofthe planet and spot relative to the observed stellar disc. The middle panels show the lightcurve due to the spot passage over the stellar disc, spanning half a stellar rotation period.In the middle panels we also see the transits, occurring at phase 0.13 (left) and 0.37 (right)of the stellar rotation. The bottom panels show the light curves again, now zooming on thetransits, where the small ‘bumps’ are caused by the spot-crossing events. We consider onlya single spot, so the flux is equal to unity when the spot is on the stellar hemisphere hiddenfrom the observer’s view. Left (right) — the spot is at the first (second) half of its crossingover the stellar disc, and therefore the local photometric slope is negative (positive). Theplanet is at the first (second) half of the transit and therefore the derived transit timing shift(while the spot-crossing is unresolved) is positive (negative). 10 –Fig. 2.— Retrograde motion — see Figure 1 for details.
Left (right) — the spot is at the first(second) half of its crossing over the stellar disc, and therefore the local photometric slope isnegative (positive). The planet is at the second (first) half of the transit and therefore thederived transit timing shift (while the spot-crossing is unresolved) is negative (positive). 11 –
3. Analytical approximation for the TTV induced by the spot-crossing event3.1. Center-of-light approximation
To present the concept behind our method in a more quantitative way, Figure 3 showsa simplified schematic diagram of a transit light curve with a single spot-crossing event. Weneglect the transit ingress and egress finite duration of both the transit and the spot-crossingevent. We also assume that at the time of each transit there is only one circular spot on thestellar disc. In the figure we also neglect the limb-darkening photometric modulation, andwill consider this effect later.In the figure, δ tr and δ sc are the depth of the transit and the amplitude of thephotometric increase inside the transit due to the spot-crossing event, respectively, ∆ tr and∆ sc are the duration of the transit and the spot-crossing event, respectively, and t sc is thetiming of the spot-crossing event relative to mid-transit time.From the figure one can see, using ‘center-of-light’ formulation, that we expect theTTV induced by the spot-crossing event to be: T T V sc ≃ − t sc δ sc ∆ sc δ tr ∆ tr − δ sc ∆ sc . (1)This result is similar to Equation (3) of Sanchis-Ojeda et al. (2011) after neglecting δ sc ∆ sc in the denominator. We will adopt this approximation below. 12 – TTV sc t sc δ tr δ sc ∆ sc ∆ tr Fig. 3.— Schematic diagram of a transit light curve with a spot-crossing event. The transitdepth is δ tr , while δ sc is the flux increase due to the spot-crossing event, ∆ tr and ∆ sc are thetransit and spot-crossing durations, and t sc is the timing of the spot-crossing event relative tothe mid transit. The vertical dashed black line represents the expected mid-transit timing,without any spot-crossing event, while the red dash-dot line represents the new mid-transitmeasurement, due to the shift induced by the spot crossing. The difference between the twolines, T T V sc , is the induced TTV. Approximately, T T V sc ≃ − t sc × δ sc ∆ sc / ( δ tr ∆ tr − δ sc ∆ sc ). 13 – To derive the analytical simplistic model we first consider a case for which • the impact parameter of the spot and the planet are both equal to zero, namely thatthey both cross the center of the stellar disc, and • there is no limb darkening.We lessen these two assumptions below.We denote the location of the spot on the stellar disc by the angle ψ , which is the anglebetween the observer and the spot, as seen from the stellar center. If the motion of the spotis equatorial, then ψ is the longitude of the spot on the stellar visible hemisphere: ψ ( t ) = ω ∗ t , (2)where ω ∗ is the stellar angular velocity. When the spot is on the stellar limb entering thevisible hemisphere, ψ gets the value of − π/
2, and when the location of the spot is in themiddle of its visible chord ψ = 0.We denote the angle corresponding to the spot crossing by ψ sc . The sky-projecteddistance of the spot from the stellar center, as seen by the observer, is d sc = R ∗ sin ψ sc ,where R ∗ is the stellar radius. The timing of the spot-crossing event, measured relative tothe middle of the transit, is therefore t sc = ∆ tr ψ sc . (3)In order to estimate the induced TTV, let us consider two extreme cases: a small spot,for which R spot ≪ R pl ≪ R ∗ , (4) 14 –and a large spot, for which R pl ≪ R spot ≪ R ∗ , (5)where R pl and R spot are the radii of the planet and the spot, respectively.For both cases we introduce a darkness parameter, 0 < α <
1, which measures thesurface brightness of the spot relative to the surface brightness of the star immediatelyoutside the spot. A completely dark spot would have α = 0, while α close to unity meansthe spot is almost as bright as the unspotted stellar area.For the small-spot approximation we can assume that the spot is completely coveredby the planet during the spot-crossing event and thereforeSmall spot : δ sc ≃ (1 − α ) (cid:18) R spot R ∗ (cid:19) cos ψ sc , ∆ sc ≃ ∆ tr (cid:18) R pl R ∗ (cid:19) . (6)As noted above, the factor cos ψ sc comes from the fact that the effective area of the spot isreduced by the aspect ratio, which is a function of the spot position on the visible stellardisc.For the large-spot approximation the planet is fully contained in the spotted areaduring the spot-crossing event, and therefore we getLarge spot : δ sc ≃ (1 − α ) (cid:18) R pl R ∗ (cid:19) , ∆ sc ≃ ∆ tr (cid:18) R spot R ∗ (cid:19) cos ψ sc . (7)Here the factor cos ψ sc comes from the fact that the time to cross the spot by the relativelysmall planet is reduced by the same aspect ratio.We now approximate the TTV to be T T V sc ≃ − t sc δ sc δ tr ∆ sc ∆ tr , (8) 15 –and the transit depth δ tr to be on the order of ( R pl /R ∗ ) . We therefore get for the small-spotapproximationSmall spot : T T V sc ≃ − t sc (1 − α ) (cid:18) R spot R ∗ (cid:19) R pl R ∗ (cid:18) R pl R ∗ (cid:19) − cos ψ sc = − t sc (1 − α ) R spot R pl R ∗ cos ψ sc , (9)and for the large-spot approximationLarge spot : T T V sc ≃ − t sc (1 − α ) R spot R ∗ cos ψ sc . (10)Note that when R spot → R pl , Equation (9) → Equation (10). To ease the discussion wedefine R as: R = R spot R pl R ∗ for small spot R spot R ∗ for large spot . (11)Using Equation (3) we get: T T V sc ≃ − (1 − α ) R ∆ tr ψ sc sin ψ sc , (12)which is valid both for the small- and large-spot approximations. The maximum observedTTV induced by the spot crossing ismax { T T V sc } ≃ (1 − α )4 R ∆ tr . (13) To include the stellar limb darkening effect in our model, we consider a quadraticlimb-darkening law of S = 1 − g (1 − cos ψ ) − g (1 − cos ψ ) , where S is the scaledstellar surface brightness and g and g are the two limb-darkening coefficients, such that g + g <
1. 16 –The induced TTV is proportional to δ sc , the increase of the stellar brightness duringthe spot crossing, which depends linearly on the stellar surface brightness S , which is nowa function of ψ . Therefore we get T T V sc = − (1 − α ) R ∆ tr ψ ( t ) sin ψ ( t ) (cid:8) − g (1 − cos ψ ) − g (1 − cos ψ ) (cid:9) = − (1 − α ) R ∆ tr (cid:8) (1 − g − g ) sin ψ ( t ) + ( g + 2 g ) sin ψ ( t ) cos ψ ( t ) − g sin ψ ( t ) cos ψ ( t ) (cid:9) cos ψ ( t ) . (14)Note that because of the limb darkening the transit light curve does not have arectangle shape, so our Equation (1) should be modified. Nevertheless, as this analyticalapproach is aimed only to understand the features of the TTVs as a function of thespot-crossing phase, we neglect this effect that will affect all phases alike. Another extension of our simplistic model accounts for a non-zero impact parameter, b = cos θ . Note that the stellar rotation is, as before, orthogonal to our line of sight. Inthis extension of the simplistic model, both planet and spot still have the same impactparameter, namely both move along the same chord on the stellar disc, a chord that doesnot go through the center of the disc. Therefore, the spot moves at a colatitude θ spot = θ ,with an impact parameter b spot = cos θ spot . In such a case, the angle ψ fulfill the relationcos ψ = sin θ cos φ , (15)where now φ is the longitude of the planet, and φ = 0 is when the planet crosses theprojection of the stellar rotational axis. The range of ψ is now different: π/ − θ ≤ | ψ | ≤ π/ t sc = ∆ btr φ sc , (16) 17 –where ∆ btr is the transit duration when b = 0. A good approximation would be∆ btr = ∆ tr sin θ .We now separate the discussion for the small and large spot approximations. For smallspot, the duration of the spot-crossing event, ∆ sc , is still the same as for b = 0, but thetransit duration is shorter by a factor of sin θ . The flux increase depends on cos ψ , as for b = 0. We can therefore writeSmall spot : δ sc ≃ (1 − α ) (cid:18) R spot R ∗ (cid:19) sin θ cos φ sc , ∆ sc ≃ ∆ btr (cid:18) R pl R ∗ (cid:19) θ . (17)Combining these expressions we getSmall spot : T T V sc ≃ − (1 − α ) R ∆ btr φ sc sin φ sc , (18)For the large spot case, the duration of the spot-crossing event, ∆ sc , is now different, asthe planet is crossing a spot which forms an ellipse on the stellar disc, whose axes are R spot and R spot cos ψ . One can show that the length of the planet’s path on the spotted area is R spot p cos θ + sin θ cos φ sc . We therefore getLarge spot : δ sc ≃ (1 − α ) (cid:18) R pl R ∗ (cid:19) , ∆ sc ≃ ∆ btr (cid:18) R spot R ∗ (cid:19) p cot θ + cos φ sc , (19)and thus Large spot : T T V sc ≃ − (1 − α ) R ∆ btr p cot θ + cos φ sc sin φ sc , (20) 18 –We can see that for a non-vanishing impact parameter there is a difference between thelarge and small planet cases, unlike in the basic model. The difference is due to the cot θ term under the square sign in Equation (20). Note that the approximation of the large spotis not valid for | φ | ≃ π/
2, where the projected area of the spot is small. Hence, we insertedinto the calculation of the large-spot case a correction factor that turns the TTV expressionto be similar to the small-spot one when | φ | → π/
2. This was done by multiplying thecot θ term with a Fermi function that is approximately unity, except for | φ | → π/
2, whenthe correction factor goes to zero.
To further extend our simplistic model, we consider now a case for non-zero impactparameter and quadratic limb darkening together. As before, we divide the discussionbetween the cases of small and large spot. Following Equation (18), but now multiplying itby the limb darkening brightness factor, we get for the small spot case:Small spot :
T T V sc ≃ − (1 − α ) R ∆ btr (cid:8) (1 − g − g ) sin φ ( t )++ ( g + 2 g ) sin θ sin φ ( t ) cos φ ( t ) −− g sin θ sin φ ( t ) cos φ ( t ) (cid:9) cos φ ( t ) , (21)while for the large spot case, following Equation (20), we get:Large spot : T T V sc ≃ − (1 − α ) R ∆ btr (cid:8) (1 − g − g ) sin φ ( t )++ ( g + 2 g ) sin θ sin φ ( t ) cos φ ( t ) −− g sin θ sin φ ( t ) cos φ ( t ) (cid:9)q cot θ + cos φ ( t ) . (22) 19 – The last case we consider is when the apparent planetary chord along the stellar discgoes through the center ( b pl = 0), but is inclined with the angle η relative to the stellarequator. We nevertheless assume that in some transits spot-crossing events happen, withspots that have different latitudes. In such cases, t sc is proportional to the distance ofthe spot-crossing event from the center of the disc, as in the basic model (Equation (3)).Similar considerations show that here also we get, as in Equation (12): T T V sc ≃ − (1 − α ) R ∆ tr ψ sc sin ψ sc , which is true for small and large spot cases alike. The extension for limb darkening alsoholds in this case. To visualize the expected TTVs derived by our analytical approximation for non-vanishing impact parameter cases, we plotted in Figure 4 the calculated TTVs for differentvalues of the impact parameter, with the large-spot approximation, using R spot /R ∗ = 0 . R pl /R ∗ = 0 .
05 values. We chose a typical parameters for a transiting system — a planetorbiting a star with solar radius in a 3 d orbit. The duration of the transit (mid-ingress tomid-egress) is about 2.62 hours, a value on which we based our estimations.One can see in the figure that the amplitude of the induced TTV is about 5 min. Thederived TTVs display almost linear slope as a function of the spot-crossing position, up toa maximum at distance of 0.6–0.85 stellar radii from the center of the stellar disc, and thena sharp drop to zero at the edge of the stellar disc. 20 – TT V ( s e c ) Position of spot (stellar radius) b = 0b = 0.3b = 0.6b = 0.9
Fig. 4.— The analytic approximation for the induced TTV as a function of the spot-crossingposition on the stellar disc for different values of the impact parameter, using the large-spotexpression of Equation (22). The position of the spot-crossing event is measured relative tothe center of the stellar disc, in units of the stellar radius. The graphs are for a Jupiter-size planet that orbits a star with solar radius in a 3-d orbit. The duration of the transit(mid-ingress to mid-egress) is about 2.62 hours, a value on which we based our estimations.The spot and planet radii were chosen as R spot /R ∗ = 0 .
15 and R pl /R ∗ = 0 .
05. The limbdarkening coefficients used are [g , g ] = [0.29,0.34]. 21 –
4. Comparison with numerical simulations
As noted in the introduction, Boisse et al. (2012) and Oshagh et al. (2013a) developeda numerical tool — SOAP-T , to simulate stellar photometric modulations induced by arotating spot, including a planetary transit light curve which includes a spot-crossing event.Oshagh et al. (2013b) used SOAP-T to derive detailed transit light curves, and then fittedthem with transit templates to obtain the expected TTVs, very similar to what is performedwhen deriving the TTVs from the Kepler actual data (e.g., Mazeh et al. 2013). This ismuch more accurate derivation than that of the previous section, where we estimated theTTVs by the center-of-light approach. It is therefore useful to compare the TTVs obtainedby our analytical approximation with the ones derived with the SOAP-T numerical codeand the transit fitting.To do that we perform in this section two comparisons. First, we used ourselves thepublicly available SOAP-T tool to produce transit light curves with spot-crossing eventsand fitted them with the Mazeh et al. (2013) codes to produce TTVs for a few cases andcompare them with the analytical approximations. Second, we derive with our analyticalcenter-of-light approach some TTVs for the cases derived by Oshagh et al. (2013b), andcompare the results.In Figure 5 we plotted our analytical approximation for the same system as before— a 3 d transiting planet orbiting a solar-like star. We used limb darkening of g = 0 . g = 0 . R pl /R ∗ = 0 . R spot /R ∗ = 0 .
1, a dark spot, with α = 0, and impactparameter of zero. We can see from the figure that the maximum expected TTV based onour approximation is similar to the one obtained when simulating the spot-crossing event.The obvious difference is the phase dependence — while the analytical approximation has Distance from center in stellar radius TT V [ s e c ] Fig. 5.— Comparison of the analytic approximation for the induced TTV with numericalsimulations, as a function of the spot-crossing phase. The approximated TTV (red) wasderived by Equation (14), while the light curves obtained by the SOAP-T tool (blue) wereanalyzed to derive the TTV. The error bars was derived from the Mazeh et al. (2013) codes. R pl /R ∗ = 0 . R spot /R ∗ = 0 .
1. The limb darkening coefficients that were used were [g ,g ] = [0.29,0.34].a smooth rise to the maximum, at phase of 0.65, the simulated light curves yielded TTVsthat are quite small for most phases, and rise sharply towards the maximum at phase 0.8.The reason for this difference comes from the different approaches of obtaining theTTV. The approach that fits a model to the simulated light curve ignores sometimes the‘bump’ in the light curve caused by the spot-crossing event, yielding a small TTV, whilethe center-of-light model is, in fact, integrating over the whole transit light curve. We willsee this difference again and again. Nevertheless, this difference does not change the resultof this paper — the negative (positive) correlation for prograde (retrograde) motion, as willbe shown below.Oshagh et al. (2013b) paper includes two figures that present their derived TTVsas a function of the orbital phase of the spot-crossing event. We applied our analytical 23 – TT V ( s e c ) Position of spot (stellar radius)
Rp/Rs=0.15, f=1%Rp/Rs=0.15, f=0.25%Rp/Rs=0.1, f=1%Rp/Rs=0.1, f=0.25%Rp/Rs=0.05, f=1%Rp/Rs=0.05, f=0.25%
Fig. 6.— The analytic approximation for the induced TTV as a function of the spot-crossingphase for different spot and planet sizes. Expected TTV were derived by using Equation (14).Rp/Rs is planet to star radius ratio and f is spot to star radius ratio squared. The limbdarkening coefficients used are [g , g ] = [0.29,0.34].approximation to all cases included in Oshagh et al. (2013b) figures, presented in the nexttwo figures. In Figure 6 we plot results of our analytical approximation that correspondingto the six cases of Oshagh et al. (2013b) Figure 3, where they have considered differentspot and planet relative sizes, keeping the same limb darkening parameters. We chosethe same ( R spot /R ∗ ) (what they call ’f’) values — 0.01 and 0.0025, and the same R pl /R ∗ values — 0.05, 0.1, and 0.15. We used the same limb darkening coefficients of [ g , g ] =[0.29,0.34], and assumed a completely dark spot ( α = 0 in our notation). As before, thetransit duration is set to be 2.62 hours. 24 –As in the previous figure, we see here that the maximum TTV is similar to the valuesobtained by Oshagh et al. (2013b), while the phase behavior of the two approaches isdifferent, as explained above.Another comparison was done by constructing Figure 7 and comparing it with Figure 6of Oshagh et al. (2013b), to study the effect of the limb darkening and spot darkness. Hereagain the amplitudes of the analytical approximation are similar to those of Oshagh et al.(2013b), while the phase dependence is different, like in our Figure 6.
5. Analytical approximation for the stellar photometric slopes
We turn now to approximate the local photometric slope at the time of the transit,assuming as before that the stellar brightness is modulated by a single circular spot.For no limb darkening and null impact parameter we approximate the stellar flux,modulated by the spot as F ∗ ( t ) ≃ − A cos ψ ( t ) , for − π/ ≤ ψ ≤ π/ , (23)where A is the observed amplitude of the photometric modulation. This is so because thespot area on the stellar disc is reduced by the aspect ratio cos ψ . The derivative of thestellar photometric brightness is therefore˙ F ∗ ( t ) ≃ ω ∗ A sin ψ ( t ) . (24)The amplitude of the observed stellar photometric modulation is a function of the spotradius and darkness. To express this relation we introduce the 0 < β < β measuresthe ratio of the area of the spot being crossed by the planet to the total neighboring 25 – TT V ( s e c ) Position of spot (stellar radius) case 1case 2case 3case 4
Fig. 7.— Expected TTV for different limb darkening parameters, using our analytical ap-proximation for R pl /R ∗ = R spot /R ∗ = 0 .
1. The limb darkening coefficients were in case 1[g , g ] = [0.29,0.34], in case 2 [0.38,0.37], in case 3 [0.6,0.16], and in case 4 [0.29,0.34]. InCase 4 the spot has half of the stellar brightness ( α = 0 . with the same phase . The total stellarmodulation due to the spots, relative to the maximum stellar brightness, is A ≃ − αβ (cid:18) R spot R ∗ (cid:19) . (25)In the case of limb darkening, the brightness of the spotted star takes the form F ∗ ( t ) ≃ − A cos ψ ( t ) (cid:8) − g (1 − cos ψ ( t )) − g (1 − cos ψ ( t )) (cid:9) , (26)as the photometry is modulated by the aspect ratio and the limb darkening at the spot’slocation. The photometric derivative is then:˙ F ∗ ( t ) = A ω ∗ (cid:8) (1 − g − g ) sin ψ ( t )+(2 g +4 g ) sin ψ ( t ) cos ψ ( t ) − g sin ψ ( t ) cos ψ ( t ) (cid:9) . (27)The stellar photometry for non-vanishing impact parameter is expressed like inEquation (23), but now cos ψ ( t ) = sin θ cos ω ∗ t , and therefore the stellar photometryderivative is ˙ F ∗ ( t ) ≃ ω ∗ A sin θ sin φ ( t ) , (28)where t is the time since the spot was in the middle of its trail, on the projection of thestellar spin (see below) on the stellar disc, and ω ∗ is the stellar rotation rate, as explainedin Section 3.3.4.For non-vanishing impact parameter and stellar limb darkening the stellar photometryis F ∗ ( t ) ≃ − A sin θ cos φ ( t ) (cid:8) − g (1 − sin θ cos φ ( t )) − g (1 − sin θ cos φ ( t )) (cid:9) . (29)and its derivative is˙ F ∗ ( t ) ≃ ω ∗ A (cid:8) (1 − g − g ) sin θ sin φ ( t ) + (2 g + 4 g ) sin θ sin φ ( t ) cos φ ( t ) − g sin θ sin φ ( t ) cos φ ( t ) (cid:9) . (30) 27 –When the obliquity of the system is non-vanishing, the spot moves on a chordorthogonal to the projection of the stellar rotational axis, at a colatitude θ spot , with b spot = cosθ spot . The spot chord is different from that of the planet, which we assume goesthrough the center of the stellar disc. Because of the inclination of the transit chord, at thetime of crossing sin θ spot sin ω ∗ t = sin ψ sc cos η (31)where t is the time since the spot was in the middle of its trail, on the projection of thestellar spin on the stellar disc, and ω ∗ is the stellar rotation rate. Therefore, the stellarphotometric derivative is like Equations (28) or (30), except for a sin θ spot factor. Note thatwhen η → π/ F ∗ ( t ) →
0, because the spot-crossing effect occurs near the photometricmaximum, and therefore the correlation with the TTVs becomes difficult to detect.
6. The correlation between
T T V sc and the stellar photometric slopes We are now ready to consider the expected correlation between the TTVs induced bythe spot-crossing events and the local slope of the stellar photometry at the time of thetransit.
Figure 8 displays our analytical approximation for the TTVs as a function of thephotometric derivatives for a few cases. The figure shows that the slope of the stellarbrightness at the time of each transit and the corresponding induced TTV have opposite signs for prograde motion, and therefore we expect negative correlation between the two.Obviously, the slope and the induced TTV have the same sign for retrograde motion,because of the argumentation presented in Section 2 and plotted in Figures 1 and 2 still 28 –holds, and therefore a positive correlation is expected in such a case.
Figure 8 portraits how the TTVs derived by our analytical approximation dependon the photometric slope, but it does not show the real expected TTV, nor includes anyobservational noise, associated with every derived TTV and photometric derivative series.To see how these two affect the expected correlation we added normally distributed noise toboth the simulated TTVs and the photometric derivatives, the results of which are plottedin Figure 9 for our analytically approximated TTVs, and in Figure 10 for TTVs derived bythe SOAP-T tool.In both figures we used the same fiducial system, but now with R spot /R ∗ = R pl /R ∗ = 0 . g , g ] = [0 . , . . −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.6−0.4−0.200.20.40.6 Flux derivative TT V BasicLimb Darkening, g =g =0.3Impact Parameter, Small Spot, b=0.5Impact Parameter, Large Spot, b=0.5 Fig. 8.— The induced TTV sc versus the photometric slope for prograde motion, usingarbitrary units on both axes. The blue line is the basic model, for b = 0 and no limbdarkening. The red line presents the limb-darkening, g = g = 0 .
3, model, the greenone is for b = 0 . b with the large spotapproximation. 30 – −3 −2 −1 0 1 2 3−600−400−2000200400600 Flux Derivative TT V [ s e c ] Fig. 9.— Simulation of TTV, derived by the analytical approximation, versus the cor-responding photometric slope for prograde motion, both with added normally distributedrandom noise. The noise r.m.s. equals to 50% of the maximum of the corresponding vari-able. The slope is scaled such that its maximum (before adding the noise) is unity. Thesimulation includes 500 phases selected at random. Correlation is − .
62. See text for details. 31 – −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−600−400−2000200400600800
Flux Derivative TT V [ s e c ] Fig. 10.— Simulation of TTV, derived by analyzing the transit light curves obtained by theSOAP-T tool, versus the corresponding photometric slope for prograde motion, both withadded normally distributed random noise. Correlation is − .
48. See Figure 9 and text fordetails. 32 –The two figures show similar results — there is a very clear anti-correlation betweenthe induced TTVs and the photometric slopes at the transit timings, even when some smallnoise is added. In fact, the noise covers up the fact that for some phases the dependence ofthe TTVs on the slope changes its sign, as we see in Figure 8.To estimate the expected effect of the noise on the measured correlation we ranextensive simulations, with different values of noise level and number of observed transits.For each choice of noise level, σ TTV , σ slope and number of transits, N , we chose N randomphases, derived their TTVs and photometric derivatives, added randomly distributednoise to both the TTVs and the stellar photometric slopes, and then derived the resulting(anti-)correlation. We repeated this simulation for 1000 times, with the same values ofnoise level and number of points. We then derived the median and scatter of the sample ofcorrelations obtained, which are plotted in Figure 11 as a function of the noise level and N .We chose five values for σ TTV and σ slope , each scaled as a fraction of the maximum ofits corresponding variable. The five noise-to-signal ratios we chose were [0, 0.15, 0.3, 0.51]. Each choice characterizes both the noise added to the TTVs and to the photometricslopes. For N we chose values of 50, 100, 500, and 1000. For short-period transiting planets Kepler light curves could have on the order of 1000 transits, but 200–400 was a more typicalnumber. All together we had 4 × σ spread of the correlation depends onthe noise level and the number of points. It goes from 0.13 for N = 50 and SNR of unitydown to 0.02 for N = 1000 and no noise. The figure suggests that we can easily detectthe correlation with SNR of unity, if we can measure on the order of 500 TTVs and theircorresponding photometric slopes. 33 – Number of measurements c o rr e l a t i on Noise =0%Noise =15%Noise =30%Noise =50%Noise =100%
Fig. 11.— The absolute value of the correlation of 1000 system samples of simulated inducedTTV with the stellar photometric slopes, for different noise levels and different number ofpoints. The points are the median of each sample and the error bars are the sample r.m.s.See text for details.
7. Discussion
We presented here a simple approach that can, in a few cases, use the derived TTVsof a transiting planet to distinguish between a prograde and a retrograde planetary motionwith respect to the stellar rotation, assuming the TTVs are induced by spot-crossingevents. Using a simplistic analytical approximation we showed that those TTVs mighthave negative (positive) correlation with the local stellar photometric slopes at the transittimings for prograde (retrograde) motion. We have shown that the correlation might bedetected for different stellar limb darkening and different impact parameters. Furthermore, 34 –we obtained similar correlated TTVs when we used the SOAP-T tool to simulate transitlight curves and derive the corresponding TTVs. We have shown also that even if weinclude certain amount of noise, the correlation is still detectable.Can such a correlation surface above the observational noise? The expected amplitudeof the TTV can be estimated by Equations (11) and (13). For example, a system with R spot ≃ R pl ≃ . R ∗ , (1 − α ) / ≃ .
25 and transit duration of 3 h should show an inducedTTV on the order of 5 min. So, we can expect to observe the (anti-)correlation between theTTVs and the photometric slopes only for systems with high enough signal-to-noise ratiothat allows timing precision of the order of 5 min or better. Note that for a 3 d transitingplanet in the
Kepler field we have at hand data for up to about 400 transits, enabling us todetect a correlation even if the noise is comparable with the signal.Obviously, the approximation and simulation presented here are quite simplistic. First,spotted stars probably have more than one spot. The spot eclipsed by the planet mightnot be the one dominating the stellar flux modulation, and hence the local photometricslope at the time of transit might be very different from the expressions we developed here.Note, however, that in our simulation we allowed an error of the photometric slope thatcan be as large as the slope itself, and showed that even in such a case the correlationstill can be detected. Second, spots have different stellar latitudes, so some transits mightnot have induced TTVs at all, contaminating the expected correlation. To deal with thisproblem one might consider the correlation of only the highly significant TTVs, which couldshow the signal better. Third, the system obliquity can be very different from 0 ◦ or 180 ◦ ,although most of the planets around cool stars, with a temperature below about 6000 K,apparently are aligned with the stellar rotation (Albrecht et al. 2012; Mazeh et al. 2015).We have shown that for systems with non-vanishing obliquity and null impact parameterthe shape of the dependence of the TTV on the photometric slope is the same, although 35 –the obliquity might decrease or even eliminate the correlation, because many transits mightnot include a spot-crossing event at all. Note, however, that even for significant obliquitythe correlation might still exist, assuming there will be enough induced TTVs, probablycaused by spots with different latitudes. Here again one might ignore the non-significantTTVs when searching for a correlation. Fourth, the observed transiting system might haveadditional planets that induce dynamical TTVs, completely shadowing the TTVs causedby spot crossing events.Despite all these obstacles, the correlation studied here might be solid enough to showup for a few KOIs. Although our method cannot give an accurate spin-orbit angle, butcan instead only indicate the sign of the orientation of the planetary motion, the methodmight be useful nevertheless, as it uses Kepler long-cadence data that is publicly availablefor all transiting planets. In the next paper (Holczer et al., in preparation) we report on asearch for correlation between the available TTVs and the corresponding local photometricslopes at the transit timings for all
Kepler
KOIs, and indeed find five convincing cases withsignificant correlations.The approach described here can in principle be applied to any eclipsing system,whether it is a transiting planet or a stellar binary. For a binary system, the inducedobserved minus calculated (O-C) eclipse timings can be estimated with the small-spotapproximation, for which the planetary radius is that of the secondary. We therefore expectthe TTVs to be on the same order of magnitude as for transiting planets. However, aseclipses in binaries are usually deeper and longer than the planetary transits, we expect theO-Cs in eclipsing binaries to be more precise.In fact, a negative correlation between the O-Cs and the local photometric slopeswas identified already for the stellar eclipsing binary in the Kepler-47 circumbinary planetsystem (Orosz et al. 2012). The authors detected O-C on the order of 1 min in the timing 36 –of the primary eclipse, and used the derived linear trend to correct the eclipse timings. Thedetection of a negative correlation for Kepler-47 is consistent with a more detailed analysisof the spot-crossing events, also done by Orosz et al. (2012), which indicates a progrademotion.The method presented here can be applied in the future to a large sample ofsystems monitored by current and future space missions, like K2 (Howell et al. 2014),TESS (Ricker et al. 2014), and PLATO (Rauer et al. 2014), helping discovering, withoutadditional observations, interesting systems that are worth following, and possibly find whatare the conditions for alignment or misalignment of stellar rotations and orbital motions ofplanets and stellar binaries.We are grateful to the referee for very helpful comments that helped us substantiallyimprove the paper. We are thankful to the authors of the SOAP-T tool that made it publiclyavailable. The research leading to these results has received funding from the EuropeanResearch Council under the EU’s Seventh Framework Programme (FP7/(2007-2013)/ ERCGrant Agreement No. 291352). T.M. also acknowledges support from the Israel ScienceFoundation (grant No. 1423/11) and the Israeli Centers of Research Excellence (I-CORE,grant No. 1829/12). T.M. is grateful to the Jesus Serra Foundation Guest Program and toHans Deeg and Rafaelo Rebolo, that enabled his visit to the Instituto de Astrof´ısica deCanarias, where the last stage of this research was completed. This work was performedin part at the Jet Propulsion Laboratory, under contract with the California Institute ofTechnology (Caltech) funded by NASA through the Sagan Fellowship Program executed bythe NASA Exoplanet Science Institute. 37 –
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