Transit Light Curves with Finite Integration Time: Fisher Information Analysis
aa r X i v : . [ a s t r o - ph . E P ] A ug Submitted to The Astrophysical Journal (ApJ)
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TRANSIT LIGHT CURVES WITH FINITE INTEGRATION TIME: FISHER INFORMATION ANALYSIS
Ellen M. Price
California Institute of Technology1200 East California Boulevard, Pasadena, CA 91125, USA
Leslie A. Rogers California Institute of Technology, MC249-171200 East California Boulevard, Pasadena, CA 91125, USA
Submitted to The Astrophysical Journal (ApJ)
ABSTRACT
Kepler has revolutionized the study of transiting planets with its unprecedented photometric preci-sion on more than 150,000 target stars. Most of the transiting planet candidates detected by
Kepler have been observed as long-cadence targets with 30 minute integration times, and the upcoming Tran-siting Exoplanet Survey Satellite (TESS) will record full frame images with a similar integration time.Integrations of 30 minutes affect the transit shape, particularly for small planets and in cases of lowsignal-to-noise. Using the Fisher information matrix technique, we derive analytic approximations forthe variances and covariances on the transit parameters obtained from fitting light curve photometrycollected with a finite integration time. We find that binning the light curve can significantly increasethe uncertainties and covariances on the inferred parameters when comparing scenarios with constanttotal signal-to-noise (constant total integration time in the absence of read noise). Uncertainties onthe transit ingress/egress time increase by a factor of 34 for Earth-size planets and 3.4 for Jupiter-sizeplanets around Sun-like stars for integration times of 30 minutes compared to instantaneously-sampledlight curves. Similarly, uncertainties on the mid-transit time for Earth and Jupiter-size planets in-crease by factors of 3.9 and 1.4. Uncertainties on the transit depth are largely unaffected by finiteintegration times. While correlations among the transit depth, ingress duration, and transit durationall increase in magnitude with longer integration times, the mid-transit time remains uncorrelatedwith the other parameters. We provide code in Python and
Mathematica for predicting the variancesand covariances at . Subject headings: methods: analytical — occultations — planetary systems — planets and satellites:general — techniques: photometric INTRODUCTIONThe
Kepler mission has discovered thousands of tran-siting planet candidates, ushering in a new era of exo-planet discovery and statistical analysis. The light curveproduced by the transit of a planet across the disk ofits star can provide insights into the planet inclination;eccentricity; stellar density; multiplicity, using transit-timing variations (TTVs); and planet atmosphere, usingtransmission spectroscopy. As the analysis of
Kepler datapushes toward Earth-size planets on Earth-like orbits, itis imperative to account for and to understand the un-certainties and covariances in the parameters that can beinferred from a transit light curve.Carter et al. (2008, hereafter C08) performed a Fisherinformation analysis on a simplified trapezoidal tran-sit light curve model to derive analytic approximationsfor transit parameters as well as their uncertainties andcovariances. These analytic approximations are usefulwhen planning observations (e.g. assessing how manytransits are needed for a given signal-to-noise on the de-rived planet properties), optimizing transit data analysis(e.g. by choosing uncorrelated combinations of parame-ters), and estimating the observability of subtle transiteffects. However, C08 assumed that the light curves were Hubble Fellow instantaneously sampled, and as a result did not accountfor the effect of finite integration times.Most
Kepler planets are observed with long-cadence,30-minute integration times. A finite integration time(temporal binning) induces morphological distortions inthe transit light curve. Kipping (2010) studied these dis-tortions and their effect on the measured light curve pa-rameters. The main effect of finite integration time is tosmear out the transit light curve into a broader shape,with the apparent ingress/egress duration increased byan integration time, and the apparent duration of the flatbottom of totality is decreased by an integration time. Asa consequence, the retrieved impact parameter may beoverestimated, while the retrieved stellar density is un-derestimated. Kipping (2010) provides approximate an-alytic expressions for the effect of integration on the lightcurves and discusses numerical integration techniques tocompensate for these effects, but his purpose was not toundertake a full Fisher analysis or to study the covari-ances between various parameters induced by the finiteintegration time.In this paper, we extend the analysis of Carter et al.(2008) to account for the effects of a finite integrationtime. We apply a Fisher information analysis to a time-integrated trapezoidal light curve to derive analytic ex-pressions for the uncertainties and covariances of model Price & Rogersparameters derived from fitting the light curve (Sec-tion 3 and Appendix A). We verify these expressions withMarkov chain Monte Carlo fits to synthesized
Kepler longcadence data (Section 4). Our analytic expressions canreadily be substituted for those of Carter et al. (2008)(e.g. their Equation 31) when calculating the variancesof transit parameters for any integration time. We pro-vide code online at thatcalculates the estimated variances and covariances of thetrapezoidal light curve parameters for any set of systemparameters. We discuss and conclude in Sections 7 and8. LINEAR APPROXIMATION TO BINNEDTRANSIT LIGHT CURVEA transit light curve represents the flux, as a functionof time, received from a star as a planet eclipses it. Ingeneral, modeling the transit light curve involves threemain ingredients. First, there is some model or parame-terization of spatial variations in the surface brightness ofthe star (due to limb darkening and/or star spots). Sec-ond, the stellar flux received is calculated as a functionof the planet-star center-to-center sky-projected distance(Mandel & Agol 2002; Seager & Mallén-Ornelas 2003).Third, the planet-star center-to-center sky-projected dis-tance must be evaluated as a function of time, eitherusing two-body Keplerian motion, or through N -bodysimulations if there are multiple dynamically interactingplanets.Following C08, we consider a simplified model for thelight curve of a dark spherical planet of radius R p tran-siting in front of a spherical star of radius R ∗ . We neglectlimb darkening and assume that the star has a uniformsurface brightness f . We assume that the orbital periodof the planet is long compared to the transit duration, sothat the motion of the planet can be approximated by aconstant velocity across the stellar disk. We then adoptthe C08 light curve model that approximates the transitlight curve as a piecewise linear function in time (Equa-tion 1), where δ is the transit depth, T is the full-widthhalf-max transit duration, and τ is the ingress/egress du-ration, as shown in Figure 1; see Winn (2011) for a morecomplete description of these parameters. F l ( t ; t c , δ, τ, T, f ) = f − δ, | t − t c | ≤ T − τ f − δ + δτ × (cid:18) | t − t c | − T τ (cid:19) , T − τ < | t − t c | < T + τ f , | t − t c | ≥ T + τ (1)As in C08, the parameters of the linear trapezoidallight curve model are related to the physical properties ofthe system (semi-major axis a , inclination i , eccentricity t c f f −δ T τ Fig. 1.—
Comparing the linear trapezoidal transit model (timevs. flux) of C08 (dashed, gray line) to the binned trapezoidal modelof this work (solid, black line), for an illustrative set of parameters.For both models, T is the full-width half-max duration of the tran-sit event, τ is the duration of ingress/egress, δ is the transit depth, f is the out-of-transit flux level, and t c is the time of transit center. e , longitude of periastron ω , and mean motion n ) by, δ = f r = f (cid:18) R p R ∗ (cid:19) (2) T = 2 τ p − b (3) τ = 2 τ r √ − b , (4)where b ≡ a cos iR ∗ (cid:18) − e e sin ω (cid:19) (5) τ ≡ R ∗ an √ − e e sin ω ! . (6)Here, b is the impact parameter, and τ is the timescalefor the planet to move one stellar radius (projected onthe sky).We integrate the C08 linear transit light curve in time,to account for a finite integration time, I . We denoteby F lb ( t ) the average received flux (in the linear model)over a time interval I centered on time t . We restrictour consideration to scenarios with I < T − τ , becauseotherwise the measurement of the transit depth duringtotality will be completely washed out by the integrationtime.ransit Light Curves with Finite Integration Time 3 F lb ( t ; t c , δ, τ, T, f , I ) = f − δ, if | t − t c | ≤ T − τ − I f − δ + δ τ I (cid:0) | t − t c | + I − T + τ (cid:1) , if T − τ − I < | t − t c | ≤ T − (cid:12)(cid:12) τ − I (cid:12)(cid:12) f − δ + δ max( τ, I ) (cid:16) | t − t c | − T + max( τ, I )2 (cid:17) , if T − (cid:12)(cid:12) τ − I (cid:12)(cid:12) < | t − t c | < T + (cid:12)(cid:12) τ − I (cid:12)(cid:12) f − δ τ I (cid:0) T + τ + I − | t − t c | (cid:1) , if T + (cid:12)(cid:12) τ − I (cid:12)(cid:12) ≤ | t − t c | < T + τ + I f , if | t − t c | ≥ T + τ + I (7)Equation 7 gives the the binned light curve model forcases where I < T − τ . Moving forward, we will differen-tiate between scenarios where the integration times lessthan the ingress/egress time, I < τ (case 1), and andscenarios where I > τ (case 2). FISHER INFORMATION ANALYSISWhen fitting a model to observed data, one is evaluat-ing the likelihood of the observed data, { y } , conditionedon a hypothesis, typically given in the form of a para-metric model f ( { a } ) . In this scenario, one is asking thequestion, “What is the probability of each of my datagiven the set of parameters { a } ?” In addition to seekingthe parameters that maximize the likelihood of observingthe data, one is often also interested in the sensitivity ofthe data to the model parameters, with the aim of plac-ing confidence intervals on the “best-fitting” parameters.In this case one is asking, “What is the sensitivity of mydata to small changes in the model parameters?”The Fisher information formalism provides a means ofaddressing this question. The diagonal elements of theFisher information matrix encode the variance of eachparameter, and the off-diagonal elements give the covari-ances of the parameters. The magnitudes of the variancesand covariances are a function of both the nature of themodel and the uncertainties in the data.In the special case that the observed data are nor-mally, identically, and independently distributed aboutthe model with constant total uncertainty σ , and as-suming flat priors on each of the model parameters, theFisher information matrix is simply the inverse of the co-variance matrix that is often a byproduct of a traditionalleast-squares analysis. The full details of the Fisher in-formation matrix derivation is given in Appendix A.We apply Fisher information formalism to the inte-grated trapezoidal light curve model to derive two dif-ferent sets of covariance matrices. First, we use the { t c , τ, T, δ, f } parametrization in both the τ > I and τ < I regimes; then, we transform these matrices to asecond, more physical parametrization adopted by C08, { t c , b , τ , r, f } . We consider this parameterization tobe more physical because its parameters are more closelyrelated to the properties of the system, and they are ofmore astronomical interest when characterizing planetsystems. All of the variances and covariances we derive in Ap-pendix A are scaled by the common factor σ / Γ , where Γ is the sampling rate of the light curve. Formally, theanalysis assumes a single transit light curve sampled atrate Γ . For phase-folded data spanning several transits,we can let T tot = P , where T tot is the length of the ob-servation baseline and P is the orbital period; then wemay substitute an “effective” sampling rate, Γ eff , whichcan be at most N Γ if N transits were observed. Practi-cally, we can define Γ eff as the reciprocal of the averagetime between consecutive phase-folded time points (seeSection 7.3 for a discussion of the effects of phase sam-pling on Γ eff ). Γ eff can be expressed independently of theintegration time I and can be substituted directly for Γ when appropriate. The Fisher information analysis as-sumes that the orbital period P is known with absolutecertainty when applied to phase-folded data, so the un-certainty on the transit midtime t c is not representativeof that for individual transits if Γ eff is used.In some cases the out-of-transit flux level, f , is knownto high enough precision that it can be fixed in the fittingprocess. In the following sections, we assume this is thecase and look at the implications of Equations A8, A9,A15, and A16 for the precision of the transit parametersderived from fitting flux-normalized transits ( f = 1) .It turns out that, under the assumption of a multivari-ate normal distribution of the parameters, marginalizingover f is equivalent to removing the row and columnthat contain the variance of f and covariances of f withthe other parameters and substituting the mean value of f (here assumed to be f = 1 ) in the remaining matrix(e.g., Coe 2009). NUMERICAL EXPERIMENTSThe covariance expressions (Equations A8, A9, A15,and A16) predict the uncertainties in the model parame-ters for a given dataset; we investigated their applicabil-ity with numerical experiments, in which we generatedsimulated transit photometry data and then fit the lightcurves to retrieve the transit parameters, uncertainties,and covariances.We synthesized light curves by numerically integratingthe C08 linear flux model over a 30-minute integrationtime, with time steps evenly spaced at 3-minute intervals.This choice corresponds to an effective sampling rate Γ eff = 10Γ , with Γ the 30-minute Kepler long-cadencerate. We assumed that ten transits were measured atsampling rate Γ and then phase-folded over one orbitalperiod (see Section 7.3 for a discussion on the effects ofsampling rate and phase). We then added white noiseusing a pseudo-random number generator to the relativephotometry at a level of σ i = 5 × − per long-cadencesample.To retrieve the variances (uncertainties) and covari-ances of the transit parameters, we fit a binned trape-zoidal light curve model (Equation 7) to our simu-lated, phase-folded photometry. We used the known,“true” light curve parameters as a starting point tosample the joint four-dimensional likelihood distribu-tion with emcee , an affine-invariant ensemble sampler forMarkov chain Monte Carlo (MCMC) implemented in thePython programming language (Foreman-Mackey et al.2013, proposed by Goodman & Weare 2010). The burn-in of the MCMC was sufficiently long that the starting Price & Rogersparameters should not have impacted the resulting pos-teriors. Using × MCMC chain samples, we esti-mated the covariance matrix with the Python numpy.cov method.For our numerical experiments, we considered nomi-nal planet-star parameters corresponding to a Solar-twin( R ⋆ = R ⊙ , M ⋆ = M ⊙ ) transited by a Jupiter-sizedplanet ( R p /R ⋆ = 0 . ) on an eccentric orbit ( e = 0 . )transiting at periastron ( ω = π/ ) with P = 9 .
55 days at impact parameter b = 0 . . We explored the effect ofvarying the parameters R p /R ⋆ , e , b , and P in Figures 2to 7. For the experiment varying P only, we fixed thebaseline of the observations to be days instead of as-suming that ten transits were observed. Our analyticexpressions for the variances and covariances of both theshape parameters and the physical parameters agree wellwith the results of the MCMC numerical experiments. RESULTSOne of the most important effects of a finite integra-tion time is to increase the magnitudes of the covariancesamong the parameters derived from fitting a transit lightcurve. A finite integration time also increases the vari-ances of both the shape and the physical parameters de-rived from a transit light curve. In some regimes (small R p /R ⋆ , short P , and low signal-to-noise ratio, S/N) thevariances on T and τ can increase by an order of magni-tude (Figure 2). The scaling of the variances with R p /R ⋆ and P is also affected. Finite integration time makes thescaling of the variances with R p /R ⋆ universally stronger,while the dependence on P becomes more complicatedthan a simple power law relation; the orbital period of aplanet influences whether it falls in the τ < I or I < τ regime, and the variance on τ increases substantially once τ < I .The variance on the transit depth, δ , which governsthe precision with with which the planet-to-star radiusratio can be inferred, is not strongly affected by a fi-nite integration time. Accounting for a finite integrationtime, the variance in δ increases by a factor 1.14 (as-suming the nominal orbit parameters described in theprevious section). This corresponds to a factor 1.06 in-crease on the uncertainty in the transit depth for Kepler ’s30-minute long-cadence data compared to the 1-minuteshort-cadence data. The transit depth shows strongercorrelations with T (negative correlation) and τ (posi-tive correlation), especially in cases with low S/N.The time of transit center is especially important formeasuring transit timing variations (TTVs). The preci-sion that can be obtained with the time of transit center t c scales with the signal-to-noise of the transit detectionin the light curve. From Equations A8 and A9, σ t c = Q q τT √ − I τ τ ≥ I Q q I T √ − τ I I > τ , where Q = √ Γ T δσ is the total signal-to-noise ratio of thetransit in the limit r → . We note that for TTVs, t c is measured for each individual transit; thus, the singletransit sampling rate, Γ , should be used to predict σ t c for an individual transit, and not Γ eff for a phase-foldedtransit. From C08, the expected uncertainty in the tran- sit time derived from an instantaneously sampled transitlight curve is lim I→ σ t c = 1 Q r τ T . (8)A finite integration time introduces a I /τ dependent cor-rection factor, and effectively substitutes I for τ in theformula for σ t c in the I > τ regime. Importantly, t c remains uncorrelated with the other parameters when afinite integration time is taken into account.The dependence of variances and covariances of thelight curve parameters on R p /R ⋆ are shown in Figures 2and 3, respectively. The MCMC variances and covari-ances start to deviate from the analytic predictions once R p /R ⋆ < . . This could be due to the fact that our in-tegral approximation to the finite sums is breaking downat that point. Indeed, we see that Γ eff τ < for thosesmall planet radii (see Section 7.3). Another possibility isthat the posterior distribution of τ is no longer Gaussianat this point (see Section 7.4).In Figure 4, we plot the predicted and measured uncer-tainties of the “physical” parameters r , b , and τ . Thedeviation of the relative uncertainty in τ at R p /R ⋆ < . seems to be caused by the corresponding deviationof the relative uncertainty of τ . We note a significantdeviation in the measured σ b , but it does not have suchan obvious explanation. Our results illustrate that b isthe most difficult of the physical parameters to constrain,particularly at small R p /R ⋆ . As an additional test, bysampling from these parameters, instead of the trape-zoidal parameters, in an MCMC fit to synthetic lightcurve data, we verified that the variances and covari-ances of the physical parameters obtained from fittingare consistent with these results.Figures 5, 6, and 7 show the predicted and measureduncertainties of the trapezoidal light curve parametersas functions of eccentricity e , impact parameter b , andorbital period P , respectively. Our MCMC measureduncertainties appear to agree with the predictions in allcases, even at large values of e and b (see Section 7.1for a discussion of the effects of grazing transits on ourapproximations). At large e , the relative uncertainty in τ increases by more than an order of magnitude fromthe C08 prediction, highlighting the importance of ac-counting for finite integration in such cases. Similarly,for small P , the relative uncertainty on τ increases sig-nificantly compared to previous predictions.Our analytic expressions for the covariances and vari-ances clearly agree better with the results from the sim-ulated Kepler long cadence data than the finite cadencecorrections from C08. The finite cadence correctionsfrom C08 do not account for the averaging of the planetlight curve over a finite integration time. In some caseswhere finite cadence corrections come to bear (e.g. inthe variances of T and τ as R p /R ⋆ gets smaller), finiteintegration time may actually improve the variances of T and τ compared to the predictions of C08 Equation 26.With finite integration time information on the ingressand egress is spread over any long exposure spanning theingress and egress. In contrast, if the light curve is in-stantaneously sampled at the same cadence, the ingressand egress may be completely missed.ransit Light Curves with Finite Integration Time 5 ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - R p ‘ R * Σ ∆ / ∆ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - - R p ‘ R * Σ T / T ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - R p ‘ R * Σ Τ / Τ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - R p ‘ R * Σ t c ( s ) C08, linear modelC08, finite cadencePrice & Rogers,binned model
Fig. 2.—
Relative uncertainties of the trapezoidal transit parameters derived from
Kepler long-cadence data, as a function of R p /R ⋆ .The fiducial planet and star properties assumed are: R ⋆ = R ⊙ , M ⋆ = M ⊙ , e = 0 . , P = 9 .
55 days , and b = 0 . . The solid redline gives R p /R ⋆ , corresponding to the analytic predictions from Carter et al. (2008) (their Equation 20), the dashed red curve gives theanalytic predictions from Carter et al. (2008) including a finite-cadence correction (their Equation 26), and the solid black curve presentsthe analytic predictions accounting for a finite integration time from this work (Equations A8 and A9). The uncertainties derived from anMCMC analysis of simulated long-cadence Kepler data (blue crosses) agree well with the predictions of this work; we plot the measureduncertainty scaled by the true value of the parameter (where appropriate), so this plot does not reflect any systematic error in parametermeasurement. ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - - - R p ‘ R * C o v ( T , ∆ )( - s ) ‰‰ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - R p ‘ R * C o v ( Τ , ∆ )( - s ) ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - R p ‘ R * C o v ( Τ , T )( s ) C08, linear model C08, finite cadence Price & Rogers, binned model
Fig. 3.—
Covariances in the trapezoidal approximation transit parameters derived from simulated
Kepler long-cadence data, as a functionof R p /R ⋆ , corresponding to the same scenarios presented in Figure 2. SURVEY PLANNINGThere are several space- and ground-based transit sur-veys on the horizon, including TESS, K2, and PLATO.Equations A8, A9, A15 and A16 can be used to helpchoose an optimal integration time for photometric sur-veys for transiting planets when combined with modelsfor the frame rate and photometric measurement uncer-tainty of the particular instrument.In the Equations A8, A9, A15 and A16, the photo-metric precision σ and integration time I are separateparameters. In practice, the uncertainty on a given pho-tometric point will depend on the integration time cho-sen. For photon-noise, σ/f ∝ I − / . We have kept ourequations explicitly in terms of σ (instead of directly sub- stituting in the assumption of photon-noise) so that theycan be more flexibly applied to cases where additionalwhite noise sources add to the photometric measurementuncertainty.The integration time also affects the effective phasesampling of the light curve. For continuous photomet-ric observations over a time baseline, T tot , the effectivesampling of the phase-folded light curve can be up to Γ eff = T tot P ( I + t read ) . (9)We denote by t read the time needed for the photometer toread out; ( I + t read ) − is the CCD frame rate. The fac-tor T tot /P accounts for the number of transits detected Price & Rogers ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - R p ‘ R * Σ b / b ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - R p ‘ R * Σ Τ / Τ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - R p ‘ R * Σ r (cid:144) r ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - R p ‘ R * Σ t c ( s ) C08, linear modelC08, finite cadencePrice & Rogers,binned model
Fig. 4.—
Relative uncertainties in the physical transit parameters derived from simulated
Kepler long-cadence data, as a function of R p /R ⋆ , corresponding to the same scenarios presented in Figure 2. We scale by the true value of the parameter to isolate the uncertaintyfrom systematic error in parameter measurement. ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - e Σ ∆ / ∆ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - e Σ T / T ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - e Σ Τ / Τ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ e Σ t c ( s ) C08, linear modelC08, finite cadencePrice & Rogers,binned model
Fig. 5.—
Relative uncertainties of the trapezoidal transit parameters derived from
Kepler long-cadence data, as functions of eccentricity e . We have assumed nominal planet parameters R ⋆ = R ⊙ , M ⋆ = M ⊙ , P = 9 . days, b = 0 . , and r = 0 . . To isolate systematic error inparameter measurement from parameter uncertainty, the relative uncertainties are scaled by the true value of the parameter. ransit Light Curves with Finite Integration Time 7 ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - b Σ ∆ / ∆ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - b Σ T / T ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - b Σ Τ / Τ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ b Σ t c ( s ) C08, linear modelC08, finite cadencePrice & Rogers,binned model
Fig. 6.—
Relative uncertainties of the trapezoidal transit parameters derived from
Kepler long-cadence data, as functions of impactparameter b . We have assumed nominal planet parameters R ⋆ = R ⊙ , M ⋆ = M ⊙ , P = 9 . days, e = 0 . , and r = 0 . . To isolatesystematic error in parameter measurement from parameter uncertainty, the relative uncertainties are scaled by the true value of theparameter. ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ P (days) Σ ∆ / ∆ ( - ) ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ P (days) Σ T / T ( - ) ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ P (days) Σ Τ / Τ ( - ) ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ P (days) Σ t c ( s ) C08, linear modelC08, finite cadencePrice & Rogers,binned model
Fig. 7.—
Relative uncertainties of the trapezoidal transit parameters derived from
Kepler long-cadence data, as functions of impactparameter P ; for this experiment only, we fix the baseline of the observations to be days, instead of assuming that ten transits wereobserved. We have assumed nominal planet parameters R ⋆ = R ⊙ , M ⋆ = M ⊙ , b = 0 . , e = 0 . , and r = 0 . . To isolate systematic errorin parameter measurement from parameter uncertainty, the relative uncertainties are scaled by the true value of the parameter. Price & Rogersover the span of the observations. This maximal effec-tive sampling rate is achieved in the case where thereis no clustering of photometric points at specific orbitalphases.In the case of a photon-noise limited survey with negli-gible read out time ( t read = 0) , the I dependence cancelsin the σ / Γ eff prefactor that scales all the covariance ma-trices. In these limits, the integration time dependencecomes solely from the body of the covariance matrix el-ements in Equations A8, A9, A15 and A16. We plot inFigure 8 how the uncertainties on the transit parameterspredicted for a Jupiter transiting a Sun-twin in a photon-noise limited survey with negligible read-out time dependon I , T tot , and P . We assumed a nominal photometricprecision of σ/f = 5 × − for a 30 minute exposure;choosing a different value for σ would simply amount torescaling the vertical axis on the figure. In Figure 9, wemake the same plot for a R ⊕ planet orbiting a Sun-twin, assuming the same nominal orbit parameters.Lower integration times mean better precision, but af-ter a certain point there is a plateau regime in whichshorter integration times do not improve the relative pre-cision of the transit parameters derived from the lightcurve (see Figures 8 and 9). In planning a transit survey,choosing an integration time near the “knee” would beoptimal to minimize both the data rate and the relativeuncertainties on the derived planet properties. The crit-ical integration time depends on both the planet orbitalperiod and R p /R ∗ , but is not significantly affected by thesurvey duration, T tot . Planets with shorter P and smaller R p /R ∗ have smaller critical integration times, and theircharacterization would benefit more greatly from shortcadence observations.The critical integration time delimiting the beginningof the plateau regime is different for different transit pa-rameters of interest. The critical integration time for τ is the shortest. In planning a survey, one would want toconsider the smallest critical integration time among theparameters of interest. Exposure times of 3, 10, and 30minutes are optimal for sampling the transits of Jupiter-sized planets orbiting Sun twins on 1-day, 10-day, and100-day orbits, respectively (Figure 8). In contrast, forEarth-sized planets with R p /R ∗ = 10 − the plateau inthe relative uncertainty in τ occurs at I < (Figure 9), regardless of the orbital period. DISCUSSIONIn this section we revisit some of the approximationsinvolved in deriving our covariance matrices, exploringtheir effects and quantifying the limitations they imposeon the applicability of Equations A8, A9, A15, and A16.7.1.
Effect of grazing transits
We have thus far limited our discussion to cases inwhich the integration time does not exceed the time be-tween second and third contact, when the planet disk iscontained completely within the disk of the star. Oncethe integration time exceeds T − τ , the maximum ap-parent depth of the transit light curve starts to decreaseas all exposures taken during totality are diluted by fluxduring ingress, egress, and/or out-of-transit. The ap-parent maximum depth of the transit light curve can bereduced by as much as T / I ; this maximum value corre-sponds to I > T + τ . By focusing on cases with I < T − τ , we are effectivelylimiting the range of orbital inclinations (or impact pa-rameters, b ) considered. We are neglecting grazing tran-sits, with impact parameters b in excess of a maximumvalue, b max , given by combining Equations 3 and 4, b max = " − r x r + s(cid:18) x r (cid:19) (cid:18) x r + 2 (cid:19)! / , (10)where x = I τ . In the case of long orbital periods( x ≪ r ) b max approaches − r , the limiting value forthe disk of the planet to be circumscribed by the stellardisk. For shorter periods ( . days) the constraints on b for which our results apply can be significantly more re-strictive (Figure 10). Limb-darkening could have an im-portant effect on these grazing transits, and may lead tothe breakdown of the C08 trapezoidal light curve approx-imations anyway, in the excluded regime of b > b max .Though we have not provided analytic equations forthe covariance matrix in the case where I > T − τ , thesecan be readily derived following a similar approach asin Appendix A, below. There are in fact, three moreregimes to be considered (in addition to cases 1 and 2given in Equation 7): T − τ < I < T + τ and I < τ (case3); T − τ < I < T + τ and I > τ (case 4); and I > T + τ (case 5). 7.2. Effect of limb darkening
So far, we have neglected the effect of limb-darkening(following C08), and have considered a planet transit-ing a star with uniform surface brightness. To ex-plore the effects of limb darkening, we generated syn-thetic transit data with a Python implementation ofthe Eastman et al. (2013) EXOFAST occultquad rou-tine, which generates a Mandel & Agol (2002) quadrat-ically limb-darkened light curve. We chose the limbdarkening parameters for HAT-P-2 as our test case, ob-taining the parameters with the Eastman et al. (2013)limb darkening parameter applet , which interpolates theClaret & Bloemen (2011) quadratic limb darkening ta-bles.We fit each synthetic data set with three different mod-els (the trapezoidal model described in this paper, a Man-del & Agol model with fixed limb darkening parameters,or a Mandel & Agol model with limb darkening coeffi-cients as free parameters). We used a procedure similarto that described in Section 4 but with . × sam-ples. To minimize the covariances between model param-eters, we parametrized all models in terms of δ , T , τ , and t c . The limb darkened model also included the Kipping(2013) limb darkening parameters q and q , which mapdirectly to the quadratic coefficients u and u , as freeparameters. Since fitting eccentricity e and argument ofperiastron ω is difficult and computationally intensive,we have restricted this test to e = 0 cases only.We show the results of this analysis in Figures 11 and12. In the case of the trapezoidal model, our EquationsA8, A9, A15, and A16 do well to predict the uncertaintieson δ , T , and t c , but we overpredict the uncertainty in τ . http://astroutils.astronomy.ohio-state.edu/exofast/limbdark.shtml ransit Light Curves with Finite Integration Time 9 - - - I H min L Σ ∆ / ∆ - - - I H min L Σ T / T - - - I H min L Σ Τ / Τ - I H min L Σ t c ( s ) P = P =
10 days P =
100 days T tot = T tot = T tot = T tot = P Fig. 8.—
Uncertainties of trapezoidal transit parameters for representative observing cases as functions of integration time I , as predictedby the binned trapezoidal light curve model, for Jupiter-size planets orbiting Sun twins. We show cases for orbits of 1 day (red curves), 10days (blue curves), and 100 days (black curves) and total survey lengths of 1 month (TESS-like; solid curves), 3 months (dashed curves),4 years ( Kepler -like; dotted curves), and one orbital period (dot-dashed curves). We assume nominal parameter values b = 0 . , e = 0 . ,and r = 0 . , to remain consistent with the figures above; we scale the photometric uncertainty such that a Kepler σ = 5 × − , and we assume photon-noise is the only noise source. Γ eff is given by Equation 9 when T tot > P ; otherwise, Γ eff = 1 / I . The special case T tot = P corresponds to the precision on the parameters derived from fitting a single transit light curve. T becomes more correlated with δ and with τ when limbdarkening is taken into account. When the synthetic dataare fit with a Mandel & Agol model with fixed limb dark-ening, the uncertainty in the transit depth δ increases.Our predictions apply well to T and t c in this case, whilethe uncertainty in τ is still overpredicted. Finally, whenlimb darkening coefficients are added as free parameters,the uncertainty in δ increases further, and we underpre-dict the uncertainty in T significantly, as expected. Toaccurately predict the uncertainties in this case, q and q should be included as parameters in the Fisher infor-mation analysis. The uncertainty in τ is overpredicted inthe small R p /R ⋆ regime and underpredicted in the large R p /R ⋆ regime. When a Mandel & Agol model is used,we find that our equations are not as useful for predictingthe parameter covariances.7.3. Effect of finite phase sampling
Our analysis makes the approximation that the data issampled at a uniform rate. The Γ eff that we have definedfor phase-folded data (Equation 9) is the maximum pos-sible sampling rate for a continuous photometric timeseries. The approximation of a constant effective sam-pling rate for phase-folded data may break down if theplanet orbital period is an integer (or rational number)multiple of the sampling cadence. In these scenarios,the photometric observations of different transits clus-ter at specific phases within the planet orbit and transitlight curve, and the equal weighting of different phases inEquation A5 is no longer fully valid. This will produce scatter about our analytic expressions for the covariancematrix in the idealized sampling scenario.Another obstacle to applying our variance and covari-ance approximations arises if too few transits have beenobserved to sufficiently cover the full range of planetphases during transit. In these cases, the integral ap-proximation of equation A4 breaks down and the finitesums (Equation A3) must be evaluated numerically.7.4. Effect of non-Gaussian posteriors
The posterior distribution of the trapezoidal parame-ters obtained from our MCMC fits to simulated
Kepler long-cadence light curves are well approximated by Gaus-sians in high S/N scenarios. In cases of low S/N, however,the “normally-distributed parameters” assumption uponwhich the Fisher information matrix analysis relies canbreak down. The ingress/egress duration is physicallyconstrained to be τ > . When the uncertainty σ τ be-comes comparable to the magnitude of τ itself, the trun-cation at τ = 0 induces non-Gaussian posteriors. Thismay account for some of the deviations at small R p /R ⋆ in Figures 2 and 3. Solving numerically for the valueof R p /R ⋆ where the value of τ becomes comparable to σ τ for nominal values of the other parameters, we findthat τ is equal to σ τ at R p /R ⋆ ≈ . ; it is equal to σ τ when R p /R ⋆ ≈ . and σ τ when R p /R ⋆ ≈ . .As R p /R ⋆ → , we expect the posterior to approach aGaussian centered at and truncated at ; τ = 3 σ τ is theapproximate lower limit of τ down to which truncationshould not be apparent in the posterior distribution. The0 Price & Rogers - - I H min L Σ ∆ / ∆ - - I H min L Σ T / T I H min L Σ Τ / Τ I H min L Σ t c ( s ) P = P =
10 days P =
100 days T tot = T tot = T tot = T tot = P Fig. 9.—
Uncertainties of trapezoidal transit parameters for representative observing cases as functions of integration time I , as predictedby the binned trapezoidal light curve model, for Earth-size planets orbiting Sun twins. We show cases for orbits of 1 day, 10 days, and100 days and total survey lengths of 1 month, 3 months, 4 years, and one orbital period. We assume nominal parameter values b = 0 . , e = 0 . , and r = 0 . and scale the photometric uncertainty such that a Kepler σ = 5 × − , and weassume photon-noise is the only noise source. Γ eff is given by Equation 9 when T tot > P ; otherwise, Γ eff = 1 / I . The line styles and colorsare identical to those in Figure 8. - P (days) b m a x Fig. 10.—
Maximum impact parameter, b max , as a function oforbital period for Jupiter-sized ( r = 0 . , solid line) and Earth-sized( r = 0 . , dashed) planets on circular orbits around a Sun-twin starwith Kepler long-cadence sampling ( I = 30 minutes ). numerical results seem to coincide well with the value of R p /R ⋆ where the MCMC results begin to deviate fromthe analytic predictions.7.5. Effect of other noise sources
Throughout this work, we have neglected the effectsof correlated (red) noise on the light curve parameteruncertainties. For light curve data with significant rednoise, we expect that our formulae will be less applicable,since we assume completely uncorrelated errors in the Fisher information analysis.We have also neglected the effects of read noise σ read ,noise that is intrinsic to the detector. Read noise addsin quadrature with photon-noise, thereby setting a mini-mum value of the overall uncertainty on each data pointin the light curve. For shorter integration times I andfainter stars, read noise may dominate the overall noise σ . In the σ → σ read limit, σ ∼ constant , Γ eff ∼ / I , and f ∼ I , so all of the covariance elements we derive in Ap-pendix A will be scaled by ( σ/f ) / Γ eff ∼ / I . Thus,in the read-noise dominated regime, the uncertainties ontransit parameters are expected to increase with shorterexposure times. In practice, the optimal integration timefor a given target (assuming white noise) will be eitherthe integration time at which read noise becomes thedominant noise source or the critical integration time as-suming photon noise (the “knee” in Figures 8 and 9),whichever is longer. CONCLUSIONSKipping (2010) highlighted the necessity of fitting abinned light curve model to binned light curve data. Wehave updated the Carter et al. (2008) analytic expres-sions for the variances and covariances of parameters de-rived from fitting transit light curve data, to take finiteintegration time into account.With finite integration time, the uncertainties on thetransit parameters are strictly greater than what onecould extract from an instantaneously sampled lightcurve. The magnitude of the correlations among tran-sit ingress/egress duration, transit duration and transitransit Light Curves with Finite Integration Time 11 ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò +++++++++++++++++++ è è è è è è è è è è è è è è è è è è - - - - R p ‘ R * Σ ∆ / ∆ ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò +++++++++++++++++++++++++ è è è è è è è è è è è è è è è è è è è è è è è è è - - - - - - R p ‘ R * Σ T / T ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò +++++++++++++++++++++++++ è è è è è è è è è è è è è è è è è è è è è è è è è - - - - - R p ‘ R * Σ Τ / Τ ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò +++++++++++++++++++++++++ è è è è è è è è è è è è è è è è è è è è è è è è è - - R p ‘ R * Σ t c ( s ) C08, linear modelC08, finite cadencePrice & Rogers,binned model
Fig. 11.—
Relative uncertainties of the trapezoidal transit parameters for a Mandel & Agol (2002) quadratically limb darkened lightcurve, integrated with I = 30 minutes, as a function of R p /R ⋆ . The uncertainties were measured with an MCMC analysis, with the relativeuncertainties being scaled by the true value of the parameter. We fit synthetic light curves with the trapezoidal model described in thispaper (purple triangles), a Mandel & Agol model with fixed limb darkening parameters (orange crosses), and a Mandel & Agol modelwith limb darkening coefficients as free parameters (green points). Our predictions (solid black lines) are applicable for the trapezoidalmodel, though the uncertainty in τ is overpredicted. The uncertainty in δ increases when limb darkening is taken into account. The C08prediction (solid red line) and finite cadence prediction (dashed red line) are shown for comparison. We let R ⋆ = R ⊙ , M ⋆ = M ⊙ , e = 0 , P = 9 . days, and b = 0 . . As before, the uncertainties are scaled by the true value of the parameter, so systematic offsets are not reflectedin this plot. òòòòòòòòòòòòòòòòò ++ ++++++++++++ - - - - - - R p ‘ R * C o v ( T , ∆ )( - s ) ò ò òòòòòòòòòòòòòòòòòòòò + +++++++ - - R p ‘ R * C o v ( Τ , ∆ )( - s ) òòòòòòòòòòòòòò ++ ++++++++++++ è è è è è - - - - - - R p ‘ R * C o v ( Τ , T )( s ) C08, linear model C08, finite cadence Price & Rogers, binned model
Fig. 12.—
Covariances in the trapezoidal approximation transit parameters derived from simulated
Kepler long-cadence data, as afunction of R p /R ⋆ , corresponding to the same scenarios presented in Figure 11. Fitting with a binned trapezoidal model yields values of T that are significantly more correlated with δ and τ than we predict at smaller values of R p /R ∗ . Our predictions are much less useful forpredicting the parameter covariances of the Mandel & Agol models. depth all increase, while the mid-transit time (relevantfor measuring TTVs) remains uncorrelated. For example,for a Hot Jupiter or close-in Earth-size planet on a threeday orbit the variances on δ , t c , τ , and T are . , . , ,and . times larger for 30-minute long-cadence data ascompared to 1 minute short cadence data, assuming thenominal orbit parameters we have used throughout thiswork; the covariances can increase by as much as a factorof . In contrast, for a transiting Earth-twin on a 1-yearorbit, the variances themselves are larger in magnitude,but they do not change greatly with integration time.We provide Python and Mathematica code for comput- ing the predicted variances and covariances that couldbe measured using the binned light curve model. Phas-ing, red noise, non-Gaussianities, and other effects canaffect the actual uncertainties obtained from a full anal-ysis. Our analytic expressions are still helpful for targetselection, observation planning, and rule of thumb intu-ition. Today, finite integration time is relevant for
Kepler long-cadence light curves, and will remain important inthe future analysis of data from K2 and from TESS fullframe images. ACKNOWLEDGEMENTS2 Price & RogersWe would like to thank John Johnson of the Harvard-Smithsonian Center for Astrophysics for his valuable in-put on this project and for establishing the Johnson Ex-olab as an environment where undergraduates and post-doctoral scholars can work together on projects like thisone. We would also like to thank the referee for provid-ing a very helpful and constructive review of this work.EMP acknowledges funding provided by Mr. and Mrs.Carl Larson for her 2013 Carolyn Ash SURF Fellowship. LAR acknowledges support provided by NASA throughHubble Fellowship grant
Mathematica was used in the preparation ofthis paper.APPENDIX
FULL DERIVATION OF FISHER INFORMATION AND COVARIANCE MATRICES
For N data points { y } and model points { y mod } which depend on a set of parameters { p } , and under the assumptionof uncorrelated errors of constant absolute magnitude σ , we begin with the likelihood function (see e.g., Gould 2003) L = 1 σ √ π exp " − σ − N X k =1 ( y k − y k, mod ) . (A1)The Fisher information matrix B is defined by (see e.g., Vallisneri 2008), B ij = h (cid:18) ∂∂p i log L (cid:19) (cid:18) ∂∂p j log L (cid:19) i = h σ − N X k =1 N X l =1 ( y k − y k, mod ) ( y l − y l, mod ) ∂y k, mod ∂p i ∂y l, mod ∂p j ! i = σ − N X k =1 (cid:18) ∂y k, mod ∂p i (cid:19) (cid:18) ∂y k, mod ∂p j (cid:19) (A2)where h x i denotes the expected value of x . For our application to transit light curves, the model function is y mod = F lb and p = { t c , τ, T, δ, f } , so B ij = σ − N X k =1 (cid:20) ∂∂p i F lb ( t k ; { p } ) (cid:21) (cid:20) ∂∂p j F lb ( t k ; { p } ) (cid:21) . (A3)Tables 5 and 6 give partial derivatives for the five regions of the binned light curve model for F lb and F lb , where F lb and F lb are equivalent to F lb limited to the τ > I and τ < I regimes, respectively.We assume that the data points are sampled uniformly with a uniform sampling rate Γ , beginning at time point t and for a total duration T tot . Like C08, we approximate the finite sums of Equation A3 by an integral over time,assuming that Γ is large enough to sufficiently sample the transit light curve: B ij = Γ σ t + T tot Z t (cid:20) ∂∂p i F ( t ; { p } ) (cid:21) (cid:20) ∂∂p j F ( t ; { p } ) (cid:21) dt. (A4)Substituting Γ eff for Γ and P for T tot , as described in Section 3 for phase-folded data, this equation becomes B ij = Γ eff σ t + P Z t (cid:20) ∂∂p i F ( t ; { p } ) (cid:21) (cid:20) ∂∂p j F ( t ; { p } ) (cid:21) dt. (A5)Evaluating Equation A4 with the partial derivatives given in Tables 5 and 6 yields the Fisher matrices in EquationsA6 and A7. In the τ > I case, we find the Fisher information matrix to be B lb = Γ σ − δ ( I− τ )3 τ δ ( I − τ I +5 τ ) τ − δ ( I − τ I +10 τ ) τ
00 0 − δ ( I− τ )6 τ δ − δ − δ ( I − τ I +10 τ ) τ δ T + I − τ I − τ τ − T − δ − T T tot . (A6)ransit Light Curves with Finite Integration Time 13 TABLE 1Variables defined for the simplification of the trapezoidal parameters covariance matrix of Equation A8.
Symbol Expression a (cid:0) τ + 2 I − τ I (cid:1) /τ a (cid:0) τ + I − τ I (cid:1) /τ a (cid:0) I T tot − τ I T tot + 120 τ I (3 T tot − τ ) (cid:1) /τ a (cid:0) a τ + I (54 τ − T tot ) − τ I (4 τ + T tot ) + 360 τ ( τ − T tot ) (cid:1) /τ a (cid:0) a (cid:0) T ( I − τ ) − T T tot ( I − τ ) (cid:1) + τ a (cid:1) /τ a (cid:0) τ + T ( I − τ ) (cid:1) /τ a (cid:0) − τ + 12 a τ T − I + 8 τ I + 40 τ I (cid:1) /τ a (2 T − T tot ) /τa (cid:0) − τ I (cid:0) − T + 10 T T tot + I (2 I + 5 T tot ) (cid:1) − I T tot + 8 τ I T tot (cid:1) /τ a (cid:0) a τ + 60 τ + 10 (cid:0) − T + 9 T T tot + I (3 I + T tot ) (cid:1) − τT tot (cid:1) /τ a ( I T tot − τ ( T tot − τ )) /τ a (cid:0) − τ − a τ T ( I − τ ) + 9 I − τ I − τ I − τ I + 360 τ I (cid:1) /τ a (cid:0) − I (cid:0) T − T T tot + 3 I T tot (cid:1) + 120 τ T I ( T − T tot ) + 8 τ I T tot (cid:1) /τ a (cid:0) a τ + 40 (cid:0) − T + 3 T T tot + I T tot (cid:1) − τT tot (cid:1) /τ a (2 I − τ ) /τ TABLE 2Variables defined for the simplification of the trapezoidal parameters covariance matrix of Equation A9.
Symbol Expression b (cid:0) I − I T tot + τT tot (cid:1) / I b ( τT + 3 I ( I − T )) / I b (cid:0) τ − T I + 8 I + 20 τ I − τ I (cid:1) / I b (cid:0) T − T T tot + I (5 T tot − I ) (cid:1) / I b (10 I − τ ) / I b (cid:0) b I + 4 τ (cid:0) − T + 6 T T tot + I (13 T tot − I ) (cid:1)(cid:1) / I b (cid:0) b I + 4 τ I (12 I − T tot ) + τ I (11 T tot − I ) − τ T tot (cid:1) / I b (cid:0) T − T T tot + I T tot (cid:1) / I b (cid:0) b I + 20 τ I T tot − τ I T tot + τ T tot (cid:1) / I b (cid:0) − τ + 24 T I ( τ − I ) + 60 I + 52 τ I − τ I + 11 τ I (cid:1) / I b (cid:0) − b I + 10 b τ I + 15 τ (2 I − T tot ) (cid:1) / I b (cid:0) b I + 2 τ I (4 T tot − I ) − τ T tot (cid:1) / I b ( T tot − T ) / I b (6 I − τ ) / I In the τ < I case, we find B lb = Γ σ δ (3 I− τ )3 I δ τ I δτ (3 τ − I )60 I
00 0 δ (3 I− τ )6 I δ − δ δτ (3 τ − I )60 I δ T + − I − τ I + τ I − T − δ − T T tot . (A7)The full covariance matrix for each model is found by taking the matrix inverse of the Fisher matrix. We define thevariables in Table 1 to simplify the covariance matrix in the τ > I case, given in Equation A8.Cov ( { t c , τ, T, δ, f } , { t c , τ, T, δ, f } ; τ > I ) = σ Γ − τδ a τa δ a a τa δ a − a a δa − a a δa a τa δ a τa δ a a a δa a δa − a a δa a a δa − a a τa − a a τa − a a δa a δa − a a τa a τa (A8)Similarly, we define the variables in Table 2 to express the covariance matrix in the τ < I case, given in Equation A9.4 Price & Rogers TABLE 3Variables defined for the simplification of the physical parameters covariance matrix of Equation A15.
Symbol Expression A (cid:0) T (cid:0) f (2 a a + 4 a − a a ) + 48 a δf ( a − a ) + a δ (cid:1)(cid:1) / (cid:0) f τ (cid:1) A (cid:0) f ( a T ( a − a ) + 6 a a τ ) − δf (2 a T ( a − a ) + a τ ) + a δ T (cid:1) / (cid:0) f τ (cid:1) A (cid:0) f ( a T ( a − a ) − a a τ ) + 12 δf ( a τ − a T ( a − a )) + a δ T (cid:1) / (cid:0) f τ (cid:1) A ( a δT − f (2 a T ( a − a ) + a τ )) / ( f τ ) A (12 f (2 a T ( a − a ) + a τ ) + a δT ) / ( f τ ) A (cid:0) a f τT ( a − a ) − a δf τT (cid:1) / (cid:0) f τ (cid:1) A a A ( a δ + 24 a a f ) /f A (cid:0) a δ − a f ( a f − a δ ) (cid:1) /f Cov ( { t c , τ, T, δ, f } , { t c , τ, T, δ, f } ; τ < I ) = σ Γ I δ b − I b δ τb I b b δ b b b δb b b δb I b b δ b I b δ b b δb b δb b b δb b δb b I b b I b b b δb b δb b I b b I b (A9)Equation A8 reduces to the C08 covariance matrix (their Equation 20) in the limit that I → .Following C08, we transform the covariance matrices to a more physical parameter space, parameterized by thevariables t c , b , τ , r , and f , given by the inverse mapping r = (cid:18) δf (cid:19) / (A10) b = 1 − rTτ (A11) τ = T τ r . (A12)The covariance matrix of the physical parameters is then found by the transformationCov ′ ( ... ) = J T Cov ( ... ) J (A13)with J the Jacobian matrix J = ∂ ( t c , b , τ , r, f ) ∂ ( t c , τ, T, δ, f ) = T rτ T r − rτ τ r − T f rτ − T τ f r f r T r f τ T τ f r − r f . (A14)As before, we define several variables so that we can write the covariance matrix compactly. For τ > I , they are givenin Table 3. With these definitions, the transformed covariance matrix in the τ > I case becomesCov ( { t c , b , τ , r, f } , { t c , b , τ , r, f } ; τ > I ) = σ Γ − τf r a A + A + A f r a τ τ ( A − A )16 f r a − A f r a τ A f ra τ τ ( A − A )16 f r a τ ( A − A + A )64 f r a − τA f r a τA f r a − A f r a τ − τA f r a A f r a τ − A f ra τ A f ra τ τA f r a − A f ra τ a a τ . (A15)Similarly, in the τ < I case, we define the variables in Table 4 such that the physical parameter covariance matrix isransit Light Curves with Finite Integration Time 15 TABLE 4Variables defined for the simplification of the physical parameters covariance matrix of Equation A16.
Symbol Expression B (cid:0) − b f τ T (2 b I − τ ) + b δ τ T − b f T I + 48 b δf τ T ( b I − τ ) (cid:1) / (cid:0) f τ (cid:1) B ( b δτT + 24 b f T ( b I − τ ) − b f τ I ) / (cid:0) f τ (cid:1) B ( b δτT + 24 b f T ( b I − τ ) + 12 b f τ I ) / (cid:0) f τ (cid:1) B (24 b f − b δ ) /f B (cid:0) b I (cid:1) /τ B (cid:0) b f T I ( τ − b I ) − b δf τT I (cid:1) / (cid:0) f τ (cid:1) B (cid:0) b f T I ( b f − b δ ) − τ (cid:0) f ( b T + 6 b I ) + b δ T − δf (4 b T + b I ) (cid:1)(cid:1) / (cid:0) f τ (cid:1) B (cid:0) b f T I ( b f − b δ ) − τ (cid:0) f ( b T − b I ) + b δ T + 12 δf ( b I − b T ) (cid:1)(cid:1) / (cid:0) f τ (cid:1) B (cid:0) b f + b δ − b δf (cid:1) /f Cov ( { t c , b , τ , r, f } , { t c , b , τ , r, f } ; τ < I ) = σ Γ I b f r B + B + B f b I r τ ( B − B )16 f b I r B f b I r B f b I r τ ( B − B )16 f b I r τ ( B + B − B )64 f b I r τ B f b I r τ B f b I r B f b I r τ B f b I r B f b I r B f b I r B f b I r τ B f b I r B f b I r b b I . (A16) REFERENCESCaprio, M. 2005, Computer Physics Communications, 171, 107Carter, J. A., Yee, J. C., Eastman, J., Gaudi, B. S., & Winn,J. N. 2008, ApJ, 689, 499Claret, A., & Bloemen, S. 2011, A&A, 529, A75Coe, D. 2009, ArXiv e-prints, arXiv:0906.4123Eastman, J., Gaudi, B. S., & Agol, E. 2013, PASP, 125, 83Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J.2013, PASP, 125, 306 Goodman, J., & Weare, J. 2010, Communications in AppliedMathematics and Computational Science, 5, 65Gould, A. 2003, ArXiv Astrophysics e-prints, astro-ph/0310577Kipping, D. M. 2010, MNRAS, 408, 1758—. 2013, MNRAS, 435, 2152Mandel, K., & Agol, E. 2002, ApJ, 580, L171Seager, S., & Mallén-Ornelas, G. 2003, ApJ, 585, 1038Vallisneri, M. 2008, Phys. Rev. D, 77, 042001Winn, J. N. 2011, Exoplanet Transits and Occultations, ed.S. Seager, 55–77 P r i ce & R og e r s TABLE 5Partial derivatives of binned flux model the for τ > I case. Totality Totality/ingress/egress Ingress/egress Ingress/egress/out-of-transit Out-of-transit ∂F lb (cid:14) ∂t c − δ ( − T + I + τ + 2 | t − t c | ) sgn ( t − t c ) / (2 I τ ) − δ sgn ( t − t c ) /τ − δ ( T + I + τ − | t − t c | ) sgn ( t − t c ) / (2 I τ ) 0 ∂F lb (cid:14) ∂τ − δ ( − T + I − τ + 2 | t − t c | )( − T + I + τ + 2 | t − t c | ) / (8 I τ ) δ ( T − | t − t c | ) / (2 τ ) δ ( T + I − τ − | t − t c | )( T + I + τ − | t − t c | ) / (8 I τ ) 0 ∂F lb (cid:14) ∂T − δ ( − T + I + τ + 2 | t − t c | ) / (4 I τ ) − δ/ (2 τ ) − δ ( T + I + τ − | t − t c | ) / (4 I τ ) 0 ∂F lb (cid:14) ∂δ − − T + I + τ + 2 | t − t c | ) / (8 I τ ) − − ( T + τ − | t − t c | ) / (2 τ ) − ( T + I + τ − | t − t c | ) / (8 I τ ) 0 ∂F lb (cid:14) ∂f TABLE 6Partial derivatives of binned flux model for the τ < I case. Totality Totality/ingress/egress Ingress/egress Ingress/egress/out-of-transit Out-of-transit ∂F lb (cid:14) ∂t c − δ ( − T + I + τ + 2 | t − t c | ) sgn ( t − t c ) / (2 I τ ) − δ sgn ( t − t c ) / I − δ ( T + I + τ − | t − t c | ) sgn ( t − t c ) / (2 I τ ) 0 ∂F lb (cid:14) ∂τ − δ ( − T + I − τ + 2 | t − t c | )( − T + I + τ + 2 | t − t c | ) / (8 I τ ) 0 δ ( T + I − τ − | t − t c | )( T + I + τ − | t − t c | ) / (8 I τ ) 0 ∂F lb (cid:14) ∂T − δ ( − T + I + τ + 2 | t − t c | ) / (4 I τ ) − δ/ (2 I ) − δ ( T + I + τ − | t − t c | ) / (4 I τ ) 0 ∂F lb (cid:14) ∂δ − − T + I + τ + 2 | t − t c | ) / (8 I τ ) − − ( T + I − | t − t c | ) / (2 I ) − ( T + I + τ − | t − t c | ) / (8 I τ ) 0 ∂F lb (cid:14) ∂f0