Transit Light Curves with Finite Integration Time: Fisher Information Analysis (Fix Title)

Abstract

*Kepler* has revolutionized the study of transiting planets with its unprecedented photometric precision on more than 150,000 target stars. Most of the thousands of transiting planet candidates detected by *Kepler* have been observed as long-cadence targets with 30 minute exposure times, and the upcoming Transiting Exoplanet Survey Satellite (TESS) will record full frame images with a similar integration time. Analytic approximations for the variances and covariances on the transit parameters can be derived from fitting non-binned light curve photometry to a non-binned model. Integrations of 30 minutes affect the transit shape, particularly for small planets and in cases of low signal to noise. We derive light curve models in terms of the transit parameters and exposure time, and we used the Fisher information matrix technique to derive the variances and covariances among the parameters due to fitting these binned models to binned data. We found that binning the light curve can significantly increase the uncertainties and covariances on the inferred parameters. Uncertainties on the transit ingress/egress time can increase by a factor of 34 for Earth-size planets and 3.4 for Jupiter-size planets around Sun-like stars for exposure times of 30 minutes compared to instantaneously-sampled light curves. Similarly, uncertainties on the mid-transit time for Earth- and Jupiter-size planets increase by factors of 3.9 and 1.4, respectively. On the other hand, uncertainties on the transit depth are largely unaffected by finite exposure times (increasing by a factor of only 1.07 under the influence of 30 minute exposure times). While correlations among the transit depth, ingress duration, and transit duration all increase in magnitude with longer exposure times, the mid-transit time remains uncorrelated with the other parameters. We provide code for predicting the variances and covariances of any set of planet parameters and exposure times at www.its.caltech.edu/~eprice.

The *Kepler* mission has discovered thousands of transiting planet candidates, ushering in a new era of exoplanet discovery and statistical analysis. The lightcurve produced by the transit of a planet across the disk of its star can provide insights into the planet inclination; eccentricity; stellar density; multiplicity, using transit-timing variations (TTVs); and — in special cases — the planet atmosphere, using transmission spectroscopy. As the analysis of *Kepler* data pushes toward Earth-size planets on Earth-like orbits, it is imperative to account for and understand the uncertainties and covariances in the parameters that can be inferred from a transit light curve.

Carter et al. (2008) performed a Fisher information analysis on a simplified trapezoidal transit light curve model to derive analytic approximations for transit parameters as well as their uncertainties and covariances. These analytic approximations are useful when planning observations (e.g. assessing how many transits are needed for a given signal-to-noise on the derived planet properties), optimizing transit data analysis (e.g. by choosing uncorrelated combinations of parameters), and estimating the observability of subtle transit effects. However, Carter et al. assumed that the light curves were instantaneously sampled, and as a result did not account for the effect of finite integration times.

Most *Kepler* planets are observed with long-cadence, 30-minute exposure times. A finite integration time (temporal binning) induces morphological distortions in the transit light curve. Kipping (2010) studied these distortions and their effect on the measured light curve parameters. The main effect of finite exposure time is to smear out the transit light curve into a broader shape, with the apparent ingress/egress increased by an integration time, and the apparent duration of the flat bottom of totality is decreased by an integration time. As a consequence, the retrieved impact parameter may be overestimated, while the retrieved stellar density is underestimated. Though Kipping provides approximate analytic expressions for the effect of integration on the light curves and discusses numerical integration techniques to compensate for these effects, he does not undertake a full Fisher analysis or study the covariances between various parameters induced by the finite integration time.

In this paper, we extend the analysis of Carter et al. (2008) to account for the effects of a finite integration time. We apply a Fisher information analysis to a time-integrated trapezoidal light curve to derive analytic expressions for the uncertainties and covariances of model parameters derived from fitting the light curve (Section \ref{sec:FI}). We verify these expressions with Markov chain Monte Carlo fits to *Kepler* long cadence data (Section \ref{sec:MCMC}). Our analytic expressions can readily replace those of Carter et al. (2008) (e.g. their Equation 31) when calculating the variances of transit parameters. We provide code online at www.its.caltech.edu/~eprice. We discuss and conclude in Sections \ref{sec:dis} and \ref{sec:con}.

A transit light curve represents the flux, as a function of time, received from a star as a planet eclipses its host star. In general, modeling the transit light curve involves three main ingredients. First, there is some model or parameterization of spatial variations in the surface brightness of the star (due to limb darkening and/or star spots). Second, the stellar flux received is calculated as a function of the planet-star center-to-center sky-projected distance (Mandel et al., 2002; Seager et al., 2003). Third, the planet-star center-to-center sky-projected distance must be evaluated as a function of time, either using two-body Keplerian motion, or through *N*-body simulations if there are multiple dynamically interacting planets.

Following Carter et al. (2008), hereafter C08, we consider a simplified model for the light curve of a dark spherical planet of radius \(R_p\) transiting in front of a spherical star of radius \(R_*\). We neglect limb darkening and assume that the star has a uniform surface brightness \(f_0\). We assume that the orbital period of the planet is long compared to the transit duration, so that the motion of the planet can be approximated by a constant velocity across the stellar disk. We then adopt the C08 light curve model that approximates the transit light curve as a piece-wise linear function in time (Equation \ref{eqn:lclinear}).

\[\begin{gathered} F_l \left( t; t_c, \delta, \tau, T, f_0 \right) \\ = \begin{cases} f_0 - \delta, & \left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} \\ f_0 - \delta + \frac{\delta}{\tau} \left( \left| t - t_c \right| - \frac{T}{2} + \frac{\tau}{2} \right), & \frac{T}{2} - \frac{\tau}{2} < \left| t - t_c \right| < \frac{T}{2} + \frac{\tau}{2} \\ f_0, & \left| t - t_c \right| \ge \frac{T}{2} + \frac{\tau}{2} \end{cases} \label{eqn:lclinear}\end{gathered}\]

As in C08, the parameters of the linear trapezoidal light curve model are related to the physical properties of the system (semi-major axis \(a\), inclination \(i\), eccentricity \(e\), longitude of periastron \(\omega\), and mean motion \(n\)) by, \[\begin{aligned} \delta &=& f_0r^2 = f_0\left(\frac{R_p}{R_*}\right)^2\\ T &=& 2\tau_0\sqrt{1-b^2}\\ \tau &=& 2\tau_0\frac{r}{\sqrt{1-b^2}},\label{eq:tau}\end{aligned}\]

where

\[\begin{aligned} b &\equiv& \frac{a\cos{i}}{R_*}\left(\frac{1-e^2}{1+e\sin\omega}\right)\\ \tau_0 &\equiv& \frac{R_*}{an}\left(\frac{\sqrt{1-e^2}}{1+e\sin\omega}\right).\end{aligned}\]

Here, \(b\) is the impact parameter, and \(\tau_0\) is the timescale for the planet to move one stellar radius (projected on the sky).

We integrate the C08 linear transit light curve in time, to account for an finite exposure time \(t_{exp}\). We denote by \(F_{lb}(t)\) the average received

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