\documentclass{article}
\usepackage[affil-it]{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
\usepackage{url}
\usepackage{hyperref}
\hypersetup{colorlinks=false,pdfborder={0 0 0}}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[ngerman,english]{babel}
\newcommand{\truncateit}[1]{\truncate{0.8\textwidth}{#1}}
\newcommand{\scititle}[1]{\title[\truncateit{#1}]{#1}}
\begin{document}
\title{Transit Light Curves with Finite Integration Time: Fisher Information Analysis (Fix Title)}
\author{Leslie Rogers}
\affil{Affiliation not available}
\author{Ellen Price}
\affil{Affiliation not available}
\author{John Johnson}
\affil{Affiliation not available}
\date{\today}
\maketitle
\selectlanguage{english}
\begin{abstract}
\textit{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 \textit{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 \url{www.its.caltech.edu/~eprice}.
%
%****
%
%This analysis is relevant to Kepler long cadence data (which is integrated over 30 minutes) and will be relevant to TESS full frames images (which also have tentatively exposures of 30 minutes).
%
%
%We already know how to estimate covariances and variances for the transit parameters for non-binned data.
%
%You need to take binning into account, because that changes the shape of the light curve.
%
%And that's what we did.
%
%We found that binning the light curve can significantly increase the uncertainty and covariances on the inferred parameters, depending on the physical configuration of the planet-star system.
%
%For planet observers, the transit depth and mid-transit time are of prime importance due to their relation to measuring the planet radius and transit timing variations.
%
\end{abstract}%
\section{Introduction}
%Paragraph: Intro \\
%-Exoplanets are awesome and exciting \\
%-Transiting Planets are awesome \\
%-Kepler is awesome \\
%The transit of a planet across the disk of its star Info from transits: atmosphere, inclination, eccentricity, stellar density
The {\it 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 {\it 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.
%Paragraph: Benefit of Analytic Fisher analysis\\
\citet{CarterEt2008ApJ} 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.
%Paragraph: Effect of Finite integration Time\\
Most {\it 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. \citet{Kipping2010MNRAS} 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.
%Paragraph: What we do in this paper\\
In this paper, we extend the analysis of \citet{CarterEt2008ApJ} 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 {\it Kepler} long cadence data (Section~\ref{sec:MCMC}). Our analytic expressions can readily replace those of \citet{CarterEt2008ApJ} (e.g. their Equation 31) when calculating the variances of transit parameters. We provide code online at \url{www.its.caltech.edu/~eprice}. We discuss and conclude in Sections~\ref{sec:dis} and \ref{sec:con}.
\section{Linear Approximation to Binned Transit Light Curve}
%Describe Full transit light curve
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 \citep{Mandel&Agol2002ApJ,SeagerMO2003ApJ}. 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 {\it N}-body simulations if there are multiple dynamically interacting planets.
%Describe linear approximation to the light curve
Following \citet[][hereafter C08]{CarterEt2008ApJ}, 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{multline}
F_l \left( t; t_c, \delta, \tau, T, f_0 \right) \\ =
\begin{cases}
f_0 - \delta, &\selectlanguage{ngerman}
\left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} \\
f_0 - \delta + \frac{\delta}{\tau} \left( \left| t - t_c \right| -\selectlanguage{ngerman}
\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{multline}
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{eqnarray}
\delta &=& f_0r^2 = f_0\left(\frac{R_p}{R_*}\right)^2\\
T &=& 2\tau_0\sqrt{1-b^2}\\ %\label{eq:T}
\tau &=& 2\tau_0\frac{r}{\sqrt{1-b^2}},\label{eq:tau}
\end{eqnarray}
\noindent where
\begin{eqnarray}
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{eqnarray}
Here, $b$ is the impact parameter, and $\tau_0$ is the timescale for the planet to move one stellar radius (projected on the sky).
%Describe binned light curve.
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 flux (in the linear model) over a time interval $t_{exp}$ centered on time $t$. Equations~\ref{eqn:lcbinned1} and \ref{eqn:lcbinned2} give the the binned lightcurve model for exposure times less than the ingress/egress time, $t_{exp} < \tau$ (case 1), and $t_{exp} > \tau$ (case 2), respectively.
%Binned lightcurve model for $t_{exp} < \tau$.
\begin{multline}
F_{lb1} \left( t; t_c, \delta, \tau, T, f_0, t_{exp} \right) \\ =
\begin{cases}
f_0 - \delta, &\selectlanguage{ngerman}
\left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} \\
f_0 - \delta + \frac{\delta}{2\tau t_{exp}} \left( \left| t - t_c \right| +\selectlanguage{ngerman}
\frac{t_{exp}}{2} - \frac{T}{2} + \frac{\tau}{2} \right)^2, &
\frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} < \left| t - t_c \right| \le\selectlanguage{ngerman}
\frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} \\
f_0 - \delta + \frac{\delta}{\tau} \left( \left| t - t_c \right| -\selectlanguage{ngerman}
\frac{T}{2} + \frac{\tau}{2} \right), &
\frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} < \left| t - t_c \right| <
\frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} \\
f_0 - \frac{\delta}{2 \tau t_{exp}} \left( \frac{T}{2} + \frac{\tau}{2} +\selectlanguage{ngerman}
\frac{t_{exp}}{2} - \left| t - t_c \right| \right)^2, &
\frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} \le \left| t - t_c \right| <
\frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} \\
f_0, &
\left| t - t_c \right| \ge \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2}
\end{cases}
\label{eqn:lcbinned1}
\end{multline}
%Binned lightcurve model for $t_{exp} > \tau$.
\begin{multline}
F_{lb2} \left( t; t_c, \delta, \tau, T, f_0, t_{exp} \right) \\ =
\begin{cases}
f_0 - \delta, &
\left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} \\
f_0 - \delta + \frac{\delta}{2\tau t_{exp}} \left( \left| t - t_c \right| +
\frac{t_{exp}}{2} - \frac{T}{2} + \frac{\tau}{2} \right)^2, &
\frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} \left| t - t_c \right| \le
\frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} \\
f_0 - \delta + \frac{\tau \delta}{2 t_{exp}} + \frac{\delta}{t_{exp}} \left(
\left| t - t_c \right| + \frac{t_{exp}}{2} - \frac{T}{2} - \frac{\tau}{2} \right), &
\frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} < \left| t - t_c \right| <
\frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} \\
f_0 - \frac{\delta}{2 \tau t_{exp}} \left( \frac{T}{2} + \frac{\tau}{2} +
\frac{t_{exp}}{2} - \left| t - t_c \right| \right)^2, &
\frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} \le \left| t - t_c \right| <
\frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} \\
f_0, &
\left| t - t_c \right| \ge \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2}
\end{cases}
\label{eqn:lcbinned2}
\end{multline}
\noindent In both cases, we restrict our consideration to scenarios with $t_{exp} t_{exp}$ case.}}
\end{table}\selectlanguage{english}
\begin{table}
\begin{tabular}{l|ccccc}
& \textbf{Totality} & \textbf{Totality/ingress/egress} & \textbf{Ingress/egress} & \textbf{Ingress/egress/Out-of-transit} & \textbf{Out-of-transit} \\
$\partial F_{lb2} \big/ \partial t_c$ & $0$ & $-\frac{\delta (-T+t_{exp}+\tau +2 \left| t-\text{tc}\right| ) \text{sgn}(t-\text{tc})}{2 t_{exp} \tau }$ & $-\frac{\delta \text{sgn}(t-\text{tc})}{t_{exp}}$ & $-\frac{\delta (T+t_{exp}+\tau -2 \left| t-\text{tc}\right| ) \text{sgn}(t-\text{tc})}{2 t_{exp} \tau }$ & $0$ \\
$\partial F_{lb2} \big/ \partial \tau$ & $0$ & $-\frac{\delta (-T+t_{exp}-\tau +2 \left| t-\text{tc}\right| ) (-T+t_{exp}+\tau +2 \left| t-\text{tc}\right| )}{8 t_{exp} \tau ^2}$ & $0$ & $\frac{\delta (T+t_{exp}-\tau -2 \left| t-\text{tc}\right| ) (T+t_{exp}+\tau -2 \left| t-\text{tc}\right| )}{8 t_{exp} \tau ^2}$ & $0$ \\
$\partial F_{lb2} \big/ \partial T$ & $0$ & $-\frac{\delta (-T+t_{exp}+\tau +2 \left| t-\text{tc}\right| )}{4 t_{exp} \tau }$ & $-\frac{\delta }{2 t_{exp}}$ & $-\frac{\delta (T+t_{exp}+\tau -2 \left| t-\text{tc}\right| )}{4 t_{exp} \tau }$ & $0$ \\
$\partial F_{lb2} \big/ \partial \delta$ & $-1$ & $\frac{(-T+t_{exp}+\tau +2 \left| t-\text{tc}\right| )^2}{8 t_{exp} \tau }-1$ & $-\frac{T+t_{exp}-2 \left| t-\text{tc}\right| }{2 t_{exp}}$ & $-\frac{(T+t_{exp}+\tau -2 \left| t-\text{tc}\right| )^2}{8 t_{exp} \tau }$ & $0$ \\
$\partial F_{lb2} \big/ \partial f_0$ & $1$ & $1$ & $1$ & $1$ & $1$ \\
\end{tabular}
\label{tab:DerivativesF2}
\caption{{Partial derivatives of linear flux model for $\tau < t_{exp}$ case.}}
\end{table}
In the $\tau > t_{exp}$ case, we find the Fisher information matrix for the parameters $p = \{t_c, \tau, T, \delta, f_0\}$ to be
\begin{equation}
B_{lb1} = \frac{\Gamma}{\sigma^2} \left(
\begin{array}{ccccc}
-\frac{2 \delta ^2 (t_{exp}-3 \tau )}{3 \tau ^2} & 0 & 0 & 0 & 0 \\
0 & \frac{\delta ^2 \left(t_{exp}^3-5 \tau ^2 t_{exp}+5 \tau ^3\right)}{30 \tau ^4} & 0 & -\frac{\delta \left(2 t_{exp}^3-5 \tau t_{exp}^2+10 \tau ^3\right)}{60 \tau ^3} & 0 \\
0 & 0 & -\frac{\delta ^2 (t_{exp}-3 \tau )}{6 \tau ^2} & \frac{\delta }{2} & -\delta \\
0 & -\frac{\delta \left(2 t_{exp}^3-5 \tau t_{exp}^2+10 \tau ^3\right)}{60 \tau ^3} & \frac{\delta }{2} & T+\frac{t_{exp}^3-5 \tau t_{exp}^2-10 \tau ^3}{30 \tau ^2} & -T \\
0 & 0 & -\delta & -T & T_{tot} \\
\end{array}
\right).
\label{eqn:FisherMatrix1}
\end{equation}
In the $\tau < t_{exp}$ case, we find the Fisher information matrix to be
\begin{equation}
B_{lb2} = \frac{\Gamma}{\sigma^2} \left(
\begin{array}{ccccc}
\frac{2 \delta ^2 (3 t_{exp}-\tau )}{3 t_{exp}^2} & 0 & 0 & 0 & 0 \\
0 & \frac{\delta ^2 \tau }{30 t_{exp}^2} & 0 & \frac{\delta \tau (3 \tau -10 t_{exp})}{60 t_{exp}^2} & 0 \\
0 & 0 & \frac{\delta ^2 (3 t_{exp}-\tau )}{6 t_{exp}^2} & \frac{\delta }{2} & -\delta \\
0 & \frac{\delta \tau (3 \tau -10 t_{exp})}{60 t_{exp}^2} & \frac{\delta }{2} & T+\frac{-10 t_{exp}^3-5 \tau ^2 t_{exp}+\tau ^3}{30 t_{exp}^2} & -T \\
0 & 0 & -\delta & -T & T_{tot} \\
\end{array}
\right).
\label{eqn:FisherMatrix2}
\end{equation}\selectlanguage{english}
\begin{table}
\begin{tabular}{ll}
\textbf{Symbol} & \textbf{Expression} \\
$\alpha$ & $\left( 10 \tau ^3+2 t_{exp}^3-5 \tau t_{exp}^2 \right) / \tau^3$ \\
$\beta$ & $\left( 5 \tau ^3+t_{exp}^3-5 \tau ^2 t_{exp} \right) / \tau^3$ \\
$\gamma$ & $\left( 9 t_{exp}^5 T_{tot}-40 \tau ^3 t_{exp}^2 T_{tot}+120 \tau ^4 t_{exp} (3 T_{tot}-2 \tau ) \right) / \tau^6$ \\
$\epsilon$ & $\left( \gamma \tau ^5+t_{exp}^4 (54 \tau -35 T_{tot})-12 \tau t_{exp}^3 (4 \tau +T_{tot})+360 \tau ^4 (\tau -T_{tot}) \right) / \tau^5$ \\
$\zeta$ & $\left( \beta \left(24 T^2 (t_{exp}-3 \tau )-24 T T_{tot} (t_{exp}-3 \tau )\right)+\tau ^3 \epsilon \right) / \tau^3$ \\
$\eta$ & $\left( 3 \tau ^2+T (t_{exp}-3 \tau ) \right) / \tau^2$ \\
$\theta$ & $\left( -60 \tau ^4+12 \beta \tau ^3 T-9 t_{exp}^4+8 \tau t_{exp}^3+40 \tau ^3 t_{exp} \right) / \tau^4$ \\
$\kappa$ & $\left( 2 T-T_{tot} \right) / \tau$ \\
$\lambda$ & $\left( -3 \tau ^2 t_{exp} \left(-10 T^2+10 T T_{tot}+t_{exp} (2 t_{exp}+5 T_{tot})\right)-t_{exp}^4 T_{tot}+8 \tau t_{exp}^3 T_{tot} \right) / \tau^5$ \\
$\mu$ & $\left( \lambda \tau ^2+60 \tau ^2+10 \left(-9 T^2+9 T T_{tot}+t_{exp} (3 t_{exp}+T_{tot})\right)-75 \tau T_{tot} \right) / \tau^2$ \\
$\nu$ & $\left( t_{exp} T_{tot}-3 \tau (T_{tot}-2 \tau ) \right) / \tau^2$ \\
$\xi$ & $\left( -360 \tau ^5-24 \beta \tau ^3 T (t_{exp}-3 \tau )+9 t_{exp}^5-35 \tau t_{exp}^4-12 \tau ^2 t_{exp}^3-40 \tau ^3 t_{exp}^2+360 \tau ^4 t_{exp} \right) / \tau^5$ \\
$\omicron$ & $\left( -3 t_{exp}^3 \left(8 T^2-8 T T_{tot}+3 t_{exp} T_{tot} \right)+120 \tau ^2 T t_{exp} (T-T_{tot})+8 \tau t_{exp}^3 T_{tot} \right) / \tau^5$ \\
$\upsilon$ & $\left( o \tau ^2+40 \left(-3 T^2+3 T T_{tot}+t_{exp} T_{tot}\right)-60 \tau T_{tot} \right) / \tau^2$ \\
$\phi$ & $\left( 2 t_{exp}-6 \tau \right) / \tau$ \\
\end{tabular}
\label{tab:cov1vars}
\end{table}
We define the variables in Table~\ref{tab:cov1vars} to simplify the covariance matrix in the $\tau > t_{exp}$ case, given in Equation~\ref{eqn:cov1}.
\begin{equation}
\text{Cov}\left( \{t_c, \tau, T, \delta, f_0\}, \{t_c, \tau, T, \delta, f_0\};~\tau > t_{exp} \right) = \\
\frac{\sigma ^2}{\Gamma } \left(
\begin{array}{ccccc}
-\frac{3 \tau }{\delta ^2 \phi } & 0 & 0 & 0 & 0 \\
0 & \frac{24 \tau \mu }{\delta ^2 \zeta } & \frac{36 \kappa \tau \alpha }{\delta ^2 \zeta } & -\frac{12 \nu \alpha }{\delta \zeta } & -\frac{12 \eta \alpha }{\delta \zeta } \\
0 & \frac{36 \kappa \tau \alpha }{\delta ^2 \zeta } & \frac{6 \tau \upsilon }{\delta ^2 \zeta } & \frac{72 \kappa \beta }{\delta \zeta } & \frac{6 \theta }{\delta \zeta } \\
0 & -\frac{12 \nu \alpha }{\delta \zeta } & \frac{72 \kappa \beta }{\delta \zeta } & -\frac{24 \nu \beta }{\tau \zeta } & -\frac{24 \eta \beta }{\tau \zeta } \\
0 & -\frac{12 \eta \alpha }{\delta \zeta } & \frac{6 \theta }{\delta \zeta } & -\frac{24 \eta \beta }{\tau \zeta } & \frac{\xi }{\tau \zeta } \\
\end{array}
\right)
\label{eqn:cov1}
\end{equation}\selectlanguage{english}
\begin{table}
\begin{tabular}{l|l}
\textbf{Symbol} & \textbf{Expression} \\
$a$ & $\left( 6 t_{exp}^2-3 t_{exp} T_{tot}+\tau T_{tot} \right) / t_{exp}^2$ \\
$b$ & $\left( \tau T+3 t_{exp} (t_{exp}-T) \right) / t_{exp}^2$ \\
$c$ & $\left( \tau ^3-12 T t_{exp}^2+8 t_{exp}^3+20 \tau t_{exp}^2-8 \tau ^2 t_{exp} \right) / t_{exp}^3$ \\
$d$ & $\left( 6 T^2-6 T T_{tot}+t_{exp} (5 T_{tot}-4 t_{exp}) \right) / t_{exp}^2$ \\
$e$ & $\left( 10 t_{exp} - 3 \tau \right) / t_{exp}$ \\
$f$ & $\left( 12 d t_{exp}^3+4 \tau \left(-6 T^2+6 T T_{tot}+t_{exp} (13 T_{tot}-30 t_{exp})\right) \right) / t_{exp}^3$ \\
$g$ & $\left( f t_{exp}^5+4 \tau ^2 t_{exp}^2 (12 t_{exp}-11 T_{tot})+\tau ^3 t_{exp} (11 T_{tot}-6 t_{exp})-\tau ^4 T_{tot} \right) / t_{exp}^5$ \\
$h$ & $\left( 3 T^2-3 T T_{tot}+t_{exp} T_{tot} \right) / t_{exp}^2$ \\
$k$ & $\left( 8 h t_{exp}^4+20 \tau t_{exp}^2 T_{tot}-8 \tau ^2 t_{exp} T_{tot}+\tau ^3 T_{tot} \right) / t_{exp}^4$ \\
$m$ & $\left( -\tau ^4+24 T t_{exp}^2 (\tau -3 t_{exp})+60 t_{exp}^4+52 \tau t_{exp}^3-44 \tau ^2 t_{exp}^2+11 \tau ^3 t_{exp} \right) / t_{exp}^4$ \\
$n$ & $\left( -15 d t_{exp}^3+10 h \tau t_{exp}^2+15 \tau ^2 (2 t_{exp}-T_{tot}) \right) / t_{exp}^3$ \\
$p$ & $\left( n t_{exp}^5+2 \tau ^3 t_{exp} (4 T_{tot}-3 t_{exp})-\tau ^4 T_{tot} \right) / t_{exp}^5$ \\
$q$ & $\left( T_{tot}-2 T \right) / t_{exp}$ \\
$s$ & $\left( 6 t_{exp}-2 \tau \right) / t_{exp}$ \\
\end{tabular}
\label{tab:cov2vars}
\end{table}
Similarly, we define the variables in Table~\ref{tab:cov2vars} to express the covariance matrix in the $\tau < t_{exp}$ case, given in Equation~\ref{eqn:cov2}.
\begin{equation}
\text{Cov} \left(\{t_c, \tau, T, \delta, f_0\}, \{t_c, \tau, T, \delta, f_0\};~\tau < t_{exp} \right) = \\
\frac{\sigma ^2}{\Gamma} \left(
\begin{array}{ccccc}
\frac{3 t_{exp}}{\delta ^2 s} & 0 & 0 & 0 & 0 \\
0 & -\frac{24 p t_{exp}^2}{\delta ^2 g \tau } & \frac{36 e q t_{exp}}{\delta ^2 g} & \frac{12 a e}{\delta g} & \frac{12 b e}{\delta g} \\
0 & \frac{36 e q t_{exp}}{\delta ^2 g} & \frac{6 k t_{exp}}{\delta ^2 g} & \frac{72 q}{\delta g} & \frac{6 c}{\delta g} \\
0 & \frac{12 a e}{\delta g} & \frac{72 q}{\delta g} & \frac{24 a}{g t_{exp}} & \frac{24 b}{g t_{exp}} \\
0 & \frac{12 b e}{\delta g} & \frac{6 c}{\delta g} & \frac{24 b}{g t_{exp}} & \frac{m}{g t_{exp}} \\
\end{array}
\right)
\label{eqn:cov2}
\end{equation}
Following C08, we transform the covariance matrices to a more physical parameter space, parameterized by the variables $t_c$, $b^2$, $\tau_0^2$, $r$, and $f_0$, given by the inverse mapping
\begin{equation}
r = \left( \frac{\delta}{f_0} \right)^{1/2}
\label{eqn:r}
\end{equation}
\begin{equation}
b^2 = 1 - \frac{r T}{\tau}
\label{eqn:bsq}
\end{equation}
\begin{equation}
\tau_0^2 = \frac{T \tau}{4 r}
\label{eqn:tau0sq}
\end{equation}
\noindent The covariance matrix of the physical parameters is then found by the transformation
\begin{equation}
\text{Cov}'(...) = J^T \text{Cov}(...) J
\end{equation}
\noindent with $J$ the Jacobian matrix
\begin{equation}
J = \frac{\partial (t_c, b^2, \tau_0^2, r, f_0)}{\partial (t_c, \tau, T, \delta, f_0)} =
\left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & \frac{T r}{\tau ^2} & \frac{T}{4 r} & 0 & 0 \\
0 & -\frac{r}{\tau } & \frac{\tau }{4 r} & 0 & 0 \\
0 & -\frac{T}{2 f_0 r \tau } & -\frac{T \tau }{8 f_0 r^3} & \frac{1}{2 f_0 r} & 0 \\
0 & \frac{T r}{2 f_0 \tau } & \frac{T \tau }{8 f_0 r} & -\frac{r}{2 f_0} & 1 \\
\end{array}
\right).
\end{equation}\selectlanguage{english}
\begin{table}
\begin{tabular}{ll}
\textbf{Symbol} & \textbf{Expression} \\
$\Delta$ & $T^2 \left(\delta ^2 \xi +24 f_0^2 (2 \alpha \nu -\beta \nu +4 \mu )+48 \delta f_0 \eta (\beta -\alpha )\right) / \left( \tau^2 f_0^2 \right)$ \\
$\Theta$ & $\left( 24 f_0^2 (6 \beta \kappa \tau +\nu T (\alpha -\beta ))-12 \delta f_0 (\theta \tau +2 \eta T (\alpha -2 \beta ))+\delta ^2 \xi T \right) / \left( \tau f_0^2 \right)$ \\
$\Lambda$ & $\left( 24 f_0^2 (\nu T (\alpha -\beta )-6 \beta \kappa \tau )+12 \delta f_0 (\theta \tau -2 \eta T (\alpha -2 \beta ))+\delta ^2 \xi T \right) / \left( \tau f_0^2 \right)$ \\
$\Pi$ & $\left( \delta \xi T-12 f_0 (\theta \tau +2 \eta T (\alpha -\beta )) \right) / \left( \tau f_0 \right)$ \\
$\Sigma$ & $\left( 12 f_0 (\theta \tau +2 \eta T (\beta -\alpha ))+\delta \xi T \right) / \left( \tau f_0 \right)$ \\
$\Upsilon$ & $\left( 288 f_0^2 \kappa \tau T (\beta -\alpha )-24 \delta f_0 \theta \tau T \right) / \left( \tau^2 f_0^2 \right)$ \\
$\Phi$ & $24 \upsilon$ \\
$\Psi$ & $\left( \delta \xi +24 \beta f_0 \eta \right) / f_0$ \\
$\Omega$ & $\left( \delta ^2 \xi -24 \beta f_0 (f_0 \nu -2 \delta \eta ) \right) / f_0^2$ \\
\end{tabular}
\label{tab:cov3vars}
\end{table}
\noindent As before, we define several variables so that we can write the covariance matrix compactly. For $\tau > t_{exp}$, they are given in Table~\ref{tab:cov3vars}. With these definitions, the transformed covariance matrix in the $\tau > t_{exp}$ case becomes
\begin{equation}
\text{Cov}(\{t_c,b^2,\tau_0^2,r,f_0\},\{t_c,b^2,\tau_0^2,r,f_0\};~\tau > t_{exp}) = \\
\frac{\sigma ^2}{\Gamma} \left(
\begin{array}{ccccc}
-\frac{3 \tau }{f_0^2 r^4 \phi } & 0 & 0 & 0 & 0 \\
0 & \frac{\Delta +\Upsilon +\Phi }{4 f_0^2 r^2 \zeta \tau } & \frac{\tau (\Delta -\Phi )}{16 f_0^2 r^4 \zeta } & -\frac{\Theta }{4 f_0^2 r^2 \zeta \tau } & \frac{\Pi }{2 f_0 r \zeta \tau } \\
0 & \frac{\tau (\Delta -\Phi )}{16 f_0^2 r^4 \zeta } & \frac{\tau ^3 (\Delta -\Upsilon +\Phi )}{64 f_0^2 r^6 \zeta } & -\frac{\tau \Lambda }{16 f_0^2 r^4 \zeta } & \frac{\tau \Sigma }{8 f_0 r^3 \zeta } \\
0 & -\frac{\Theta }{4 f_0^2 r^2 \zeta \tau } & -\frac{\tau \Lambda }{16 f_0^2 r^4 \zeta } & \frac{\Omega }{4 f_0^2 r^2 \zeta \tau } & -\frac{\Psi }{2 f_0 r \zeta \tau } \\
0 & \frac{\Pi }{2 f_0 r \zeta \tau } & \frac{\tau \Sigma }{8 f_0 r^3 \zeta } & -\frac{\Psi }{2 f_0 r \zeta \tau } & \frac{\xi }{\zeta \tau } \\
\end{array}
\right).
\label{eqn:cov3}
\end{equation}\selectlanguage{english}
\begin{table}
\begin{tabular}{ll}
\textbf{Symbol} & \textbf{Expression} \\
$A$ & $\left( -24 a f_0^2 \tau ^2 T^2 (2 e t_{exp}-\tau )+48 b \delta f_0 \tau ^2 T^2 (e t_{exp}-\tau )-96 f_0^2 p T^2 t_{exp}^3+\delta ^2 m \tau ^3 T^2 \right) / \left( \tau^5 f_0^2 \right)$ \\
$B$ & $\left( 24 b f_0 T (e t_{exp}-\tau )-12 c f_0 \tau t_{exp}+\delta m \tau T \right) / \left( \tau^2 f_0 \right)$ \\
$F$ & $\left( 24 b f_0 T (e t_{exp}-\tau )+12 c f_0 \tau t_{exp}+\delta m \tau T \right) / \left( \tau^2 f_0 \right)$ \\
$G$ & $\left( 24 b f_0-\delta m \right) / f_0$ \\
$H$ & $24 k t_{exp}^2 / \tau^2$ \\
$J$ & $\left( 288 f_0^2 q T t_{exp} (\tau -e t_{exp})-24 c \delta f_0 \tau T t_{exp} \right) / \left( \tau^3 f_0^2 \right)$ \\
$K$ & $\left( 24 e f_0 T t_{exp} (a f_0-b \delta )-\tau \left(24 f_0^2 (a T+6 q t_{exp})-12 \delta f_0 (4 b T+c t_{exp})+\delta ^2 m T\right) \right) / \left( \tau^2 f_0^2 \right)$ \\
$L$ & $\left( 24 e f_0 T t_{exp} (a f_0-b \delta )-\tau \left(24 f_0^2 (a T-6 q t_{exp})+12 \delta f_0 (c t_{exp}-4 b T)+\delta ^2 m T\right) \right) / \left( \tau^2 f_0^2 \right)$ \\
$M$ & $\left( 24 a f_0^2-48 b \delta f_0+\delta ^2 m \right) / f_0^2$ \\
\end{tabular}
\label{tab:cov4vars}
\end{table}
Similarly, in the $\tau < t_{exp}$ case, we define the variables in Table~\ref{tab:cov4vars} such that the physical parameter covariance matrix is
\begin{equation}
\text{Cov}(\{t_c,b^2,\tau_0^2,r,f_0\},\{t_c,b^2,\tau_0^2,r,f_0\};~\tau < t_{exp}) = \\
\frac{\sigma^2}{\Gamma} \left(
\begin{array}{ccccc}
\frac{3 t_{exp}}{s f_0^2 r^4} & 0 & 0 & 0 & 0 \\
0 & \frac{A+H+J}{4 f_0^2 g t_{exp} r^2} & \frac{\tau ^2 (A-H)}{16 f_0^2 g t_{exp} r^4} & \frac{K}{4 f_0^2 g t_{exp} r^2} & \frac{B}{2 f_0 g t_{exp} r} \\
0 & \frac{\tau ^2 (A-H)}{16 f_0^2 g t_{exp} r^4} & \frac{\tau ^4 (A+H-J)}{64 f_0^2 g t_{exp} r^6} & \frac{\tau ^2 L}{16 f_0^2 g t_{exp} r^4} & \frac{\tau ^2 F}{8 f_0 g t_{exp} r^3} \\
0 & \frac{K}{4 f_0^2 g t_{exp} r^2} & \frac{\tau ^2 L}{16 f_0^2 g t_{exp} r^4} & \frac{M}{4 f_0^2 g t_{exp} r^2} & \frac{G}{2 f_0 g t_{exp} r} \\
0 & \frac{B}{2 f_0 g t_{exp} r} & \frac{\tau ^2 F}{8 f_0 g t_{exp} r^3} & \frac{G}{2 f_0 g t_{exp} r} & \frac{m}{g t_{exp}} \\
\end{array}
\right).
\label{eqn:cov4}
\end{equation}
In some cases the out-of-transit flux level, $f_0$, is known to high enough precision that it can be fixed in the fitting process. In the following sections, we assume this is the case and look at the implications of Equations~\ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3}, and \ref{eqn:cov4} for the precision of the transit parameters derived from fitting flux-normalized transits $\left(f_0=1\right)$.
It turns out that, under the assumption of a multivariate gaussian distribution of the parameters, marginalizing over $f_0$ is equivalent to removing the row and column that contain the variance of $f_0$ and covariances of $f_0$ with the other parameters and substituting the mean value of $f_0$ (here assumed to be $f_0=1$) in the remaining matrix \citep[e.g.,][]{Coe2009arXiv}.
%Under the assumption of a multivariate gaussian distribution of the parameters Given the full covariance matrices, one can easily marginalize over $f_0$ by removing the row and column that contain the variance of $f_0$ and covariances of $f_0$ with the other parameters.
\section{Validating Covariance Expressions}
\label{sec:MCMC}
%Describe approach
We validated the covariance expressions (Equations~\ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3}, and \ref{eqn:cov4}) with numerical experiments, in which we generated simulated transit photometry data and then fit the light curves to retrieve the transit parameters, uncertainties and covariances. We generated simulated transit light curves by numerically integrating, with fixed-order Gaussian quadrature, the C08 linear flux model at time steps evenly spaced at 3-minute intervals. This choice corresponds to an effective sampling rate $\Gamma_{eff} = 10\Gamma$, with $\Gamma$ the 30-minute {\emph Kepler} long-cadence rate; we assume that ten transits were measured at sampling rate $\Gamma$ and then phase-folded over one orbital period (see Section~\ref{sec:PhaseSampling} for a discussion on the effects of sampling rate and phase). We then added white noise using a pseudo-random number generator to the relative photometry at a level of $\sigma_i = 5 \times 10^{-5}$ per long-cadence sample.
%Double check Gamma_eff equation is included in section 3
%Analysis Approach
To retrieve the variances (uncertainties) and covariances of the transit parameters we fitted a binned trapezoidal light curve model (Equations~\ref{eqn:lcbinned1} and \ref{eqn:lcbinned2}) to our simulated, phase-folded photometry. We used the known, ``true'' light curve parameters as a starting point to sample the joint five-dimensional likelihood distribution with the code {\tt emcee}, an affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) implemented in the Python programming language \citep[][, proposed by Goodman & Weare (2010)]{ForemanMackeyEt2013PASP}. The burn-in of the MCMC was sufficiently long that the starting parameters should not have impacted the resulting posteriors. We measured the covariance matrix from $3\times10^5$ MCMC chain samples, estimating the covariance using the Python \texttt{numpy.cov} method.
%Describe planet Case studies
For our numerical experiments we considered nominal planet-star parameters corresponding to a Solar-twin ($R_\star = R_{\odot}$, $M_\star = M_{\odot}$) transited by a Jupiter-sized planet ($R_p / R_\star = 0.1$) on an eccentric orbit ($e = 0.16$) transiting at periastron ($\omega = {^\pi/_2}$) with $P_{orb} = 9.55~\rm{days}$ at impact parameter $b = 0.2$. We explored the effect of varying the parameters $R_p / R_\star$, $e$, and $b$ in Figures \ref{fig:varRpTrapz} to \ref{fig:varImpactTrapz}. Our analytic expressions for the variances and covariances of both the shape parameters and the physical parameters are validated by their good agreement with the results of the MCMC numerical experiments.
\section{Results}
%Describe main effect of finite integration time on Variances
A finite integration time increases the variances of both the shape and the physical parameters derived from a transit lightcurve. In some regimes (small $R_p/R_\star$, short $P$, and low signal-to-noise ratio, S/N) the variances on $T$ and $\tau$ can increase by multiple orders of magnitude (Figure~\ref{fig:varRpTrapz}). The scaling of the variances with $R_p/R_\star$ and $P$ is also affected. Finite integration time makes the scaling of the variances with $R_p/R_\star$ universally stronger, while the dependence on $P_{orb}$ becomes more complicated than a simple power law relation; the orbital period of a planet influences whether it falls in the $\tau \tau
% \end{array}
% \right.
%\end{displaymath}
\begin{displaymath}
\sigma_{t_c} = \left\{
\begin{array}{lc}
\frac{1}{Q}\sqrt{\frac{\tau T}{2}} \frac{1}{\sqrt{1-\frac{t_{exp}}{3\tau}}} & \tau\geq t_{exp}\\
\frac{1}{Q}\sqrt{\frac{t_{exp} T}{2}} \frac{1}{\sqrt{1-\frac{\tau}{3t_{exp}}}} & t_{exp}>\tau
\end{array}
\right.,
\end{displaymath}
\noindent where $Q = \sqrt{\Gamma T}\frac{\delta}{\sigma}$ is the total signal to noise ratio of the transit in the limit $r\to0$. We note that for transit timing variations, $t_c$ is measured for each individual transit; $\Gamma$ is the sampling rate for an individual transit, and not $\Gamma_{eff}$ for a phase-folded transit. From \citet{CarterEt2008ApJ}, the expected uncertainty on the transit time derived from an instantaneously sampled transit light curve is $\sigma_{t_c}=\frac{1}{Q}\sqrt{\frac{\tau T}{2}}$.
% $\sigma_{t_c} = \frac{1}{Q}\frac{\tau T}{2} \frac{t_{exp}}{\tau \sqrt{1-\frac{t_{exp}}{3\tau}}}$
A finite exposure time introduces a $t_{exp}/\tau$ dependent correction factor, and effectively substitutes $t_{exp}$ for $\tau$ in the formula for $\sigma_{t_c}$ in the $t_{exp}>\tau$ regime. Importantly, $t_c$ remains uncorrelated to the other parameters when a finite integration time is taken into account. %DOUBLE CHECK definition of $\sigma$ with Ellen.
We emphasize that
%Describe the behavior of the uncertainties as a function of orbital period, radius, b, and e
The dependence of variances and covariances of the light curve parameters on $R_p/R_\star$ are shown in Figures~\ref{fig:varRpTrapz} and \ref{fig:covRpTrapz}. The MCMC variances and covariances start to deviate from the analytic predictions once $R_p/R_\star < 0.04$. This could be due to the fact that our integral approximation to the finite sums is breaking down at that point. Indeed, we see that $\Gamma\tau < 3$ for those small planet radii. Another possibility is that the posterior distribution of $\tau$ is no longer Gaussian at this point (see Section~\ref{sec:NonGaussian}). %DOUBLE CHECK.
In Figure~\ref{fig:varRpPhys}, we plot the predicted and measured uncertainties of the ``physical'' parameters of Equations~\ref{eqn:r}, \ref{eqn:bsq}, and \ref{eqn:tau0sq}. The deviation of the relative uncertainty on $\tau_0^2$ seems to be caused by the corresponding deviation of the relative uncertainty of $\tau$. We note a significant deviation in the measured $\sigma_{b^2}$, however, does not have such an obvious explanation. Our results seem to support that $b^2$ is the most difficult of the physical parameters to constrain, particularly at small $R_p/R_\star$.
Finally, Figures~\ref{fig:varEccTrapz} and \ref{fig:varImpactTrapz} show the predicted and measured uncertainties of the trapezoidal light curve parameters as functions of eccentricity $e$ and impact parameter $b$, respectively. Our MCMC measured uncertainties appear to agree with the predictions in all cases, even at large values of $e$ and $b$ (see Section~\ref{sec:grazing} for a discussion of the effects of grazing transits on our approximations). At large $e$, the relative uncertainty in $\tau$ increases by more than an order of magnitude from the C08 prediction, highlighting the importance of finite integration in such cases.
%Comment on the Carter et al. 2008 finite cadence correction, and how it does not account for the time-integraton of the light curve.
Our analytic expressions for the covariances and variances clearly agree better with the results from the simulated Kepler long cadence data than the finite cadence corrections from C08. The finite cadence corrections from C08 do not account for the averaging of the planet light curve over a finite exposure time.
In some cases where finite cadence corrections come to bear (e.g. in the variances of $T$ and $\tau$ as $R_p/R_\star$ gets smaller), finite integration time actually may improve the variances of $T$ and $\tau$ compared to the predictions of C08 Equation 26. With finite exposure time information on the ingress and egress is spread over any long exposure spanning the ingress and egress, while if the lightcurve is instantaneously sampled at the same cadence the ingress and egress may be completely missed. %NEED TO INVESTIGATE FURTHER.
%NOTE: The covariance on $T$ and $\tau$ from C08 Equation 26 looks like it may be off (possible sign error?); it predicts a positive correlation of $T$ and $\tau$, while Carter et al. Equation 20, our analytic expression and the MCMC results all predict a negative correlation. LOOK INTO
%Compare to analytic Expressions from \citet{Kipping2010MNRAS}. What additional insights do we gain from this Fisher Analysis that was not apparent in the Kipping et al paper?\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/radiusratio-trapz-var/var}
\caption{{\label{fig:varRpTrapz}
Relative uncertainties of the trapezoidal transit parameters derived from \textit{Kepler} long-cadence data, as a function of $R_p/R_\star$. The fiducial planet and star properties assumed are: $R_\star = R_{\odot}$, $M_\star = M_{\odot}$, $e = 0.16$, $P_{orb}=9.55~\rm{days}$, and $b=0.2$. The solid red line gives $R_p/R_\star$, corresponding to the analytic predictions from \protect\citet{CarterEt2008ApJ} (their Equation 20), the dashed red curve gives the analytic predictions from \protect\citet{CarterEt2008ApJ} including a finite-cadence correction (their Equation 26), and the solid black curve presents the analytic predictions accounting for a finite exposure time from this work (Equations \ref{eqn:cov1} and \ref{eqn:cov2}). The uncertainties derived from an MCMC analysis of simulated long-cadence Kepler data (blue crosses) agree well with the predictions of this work; we plot the measured uncertainty scaled by the true value of the parameter (where appropriate), so this plot does not reflect any systematic error in parameter measurement.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/radiusratio-trapz-cov/cov}
\caption{{\label{fig:covRpTrapz}
Covariances in the trapezoidal approximation transit parameters derived from simulated \textit{Kepler} long-cadence data, as a function of $R_p/R_\star$, corresponding to the same scenarios presented in Figure~\ref{fig:varRpTrapz}.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/radiusratio-phys-var/var}
\caption{{\label{fig:varRpPhys}
Relative uncertainties in the physical transit parameters derived from simulated \textit{Kepler} long-cadence data, as a function of $R_p/R_\star$, corresponding to the same scenarios presented in Figure~\ref{fig:varRpTrapz}. We scale by the true value of the parameter to isolate the uncertainty from systematic error in parameter measurement.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/eccentricity-trapz-var/var}
\caption{{\label{fig:varEccTrapz}
Relative uncertainties of the trapezoidal transit parameters derived from \textit{Kepler} long-cadence data, as functions of eccentricity $e$. We have assumed nominal planet parameters $R_\star = R_\odot$, $M_\star = M_\odot$, $P_{orb} = 9.55~\text{days}$, $b = 0.2$, and $r = 0.1$. To isolate systematic error in parameter measurement from parameter uncertainty, the relative uncertainties are scaled by the true value of the parameter.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/impactparam-trapz-var/var}
\caption{{\label{fig:varImpactTrapz}
Relative uncertainties of the trapezoidal transit parameters derived from \textit{Kepler} long-cadence data, as functions of impact parameter $b$. We have assumed nominal planet parameters $R_\star = R_\odot$, $M_\star = M_\odot$, $P_{orb} = 9.55~\text{days}$, $e = 0.16$, and $r = 0.1$. To isolate systematic error in parameter measurement from parameter uncertainty, the relative uncertainties are scaled by the true value of the parameter.%
}}
\end{center}
\end{figure}
%\subsection{Properties of Variance/Covariance Figures to explain \& look into}
%
%Rough Notes to be incorporated into other sections
%
%Overall, the our analytic expressions for the variances and covariances of the shape parameters agree well with the results of the MCMC numerical experiments.
%
%The Price \& Rogers analytic expressions for the variance of $b^2$ and its covariances with the physical parameters are systematically larger than those obtained from the MCMC experiments. We need to track down the reason for this. Here are some possibilities.
%\begin{enumerate}
%\item The posterior of $b^2$ may be non-Gaussian (actually, it is necessarily non-gaussian because it is truncated [0, ~1]), which may account for this discrepancy.
%\item Our expressions for the binned light curve do not properly treat grazing transits (see discussion below), although this could be readily extended. If the MCMC is sampling configurations with $b$ outside the formal range of applicability of our equations, that could lead to trouble.
%\item Does the Jacobian transformation of the shape parameters to the physical parameters somehow amplify the small errors on the covariance matrix in the shape parameters, producing larger discrepancies in the covariances and variance of $b^2$?
%\end{enumerate}
%
%The Covariance of $T$ and $\tau$ shows significant scatter about (but is still centered on) our analytic %prediction. The covariances and variances as a function of orbital period also show a fair bit of scatter. Both of %these effects may be due to the phasing of the light curve points. To test this, Ellen is regenerating the plots %with simulated light curves with a random time offset of the starting time. We can also rerun the MCMC experiments %on $\sim10$ simulated light curves (instead of just 1), to get a measure of the distribution of each %variance/covariance obtained from the MCMC (just due to differences in white noise, and time-point phasing).
%
%The MCMC variances and covariances start to deviate from the analytic predictions once $R_p/R_\star<0.04$. This could be due to the fact that our integral approximation to the finite sums is breaking down at that point. Indeed, we see that $\Gamma\tau<3$ for those small planet radii.
%
%The finite cadence corrections from \citet{CarterEt2008ApJ} do not account for the averaging of the planet light curve over a finite exposure time. Our Price \& Rogers analytic expressions for the covariances and variances clearly agree better with the results from the simulated Kepler long cadence data. The covariance on T and tau from Carter et al. Equation 26 looks like it may be off (possible sign error?); it predicts a positive correlation of T and tau, while Carter et al. Equation 20, our analytic expression and the MCMC results all predict a negative correlation.
%
%Overall, a finite integration time increases the variance and the magnitude of the covariances betweeen shape parameters compared to an instantaneously sampled transit light curve. In some cases where finite cadence corrections come to bear (e.g. in the variances of $T$ and $\tau$ as $R_p/R_\star$ gets smaller), finite integration time actually may improve the variances of $T$ and $\tau$ compared to the predictions of Carter et al. 2008 equation 26. With finite exposure time information on the ingress and egress is spread over any long exposure spanning the ingress and egress, while if the lightcurve is instantaneously sampled at the same cadence the ingress and egress may be completely missed. Need to investigate further.
%
%In the plots that Ellen has made (keeping all parameters of the system fixed but one), the variance of $\delta$ increases with $e$, increases with $b$, decreases with $P$, and increases with $R_p/R_\star$.
%The variance of $T$ behaves non-monotonically with $e$, increases with $b$, increases with $P$, and decreases with $R_p/R_\star$.
%The variance of $\tau$ increases with $e$, behaves non-monotonically with $b$, decreases with $P$, and decreases with $R_p/R_\star$.
%The variance of $t_c$ decreases slightly with $e$, increases with $b$, increases with $P$, and decreases with $R_p/R_\star$.
%Ask Ellen what value of $\omega$ she used, to understand the behavior of the $e$ dependence.
%
%Elaborate on effect of finite integration time on $\tau$ variance and covariance. Reverse dependence of $sigma_{\tau}$ on $P$, $b$.
\section{Discussion}
\label{sec:dis}
In this section we revisit some of the approximations that went into deriving our covariance matrix, exploring their effects and quantifying the limitations they impose on the applicability of Equations \ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3}, and \ref{eqn:cov4}.
\subsection{Application to Survey Planning}
Equations~\ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3} and \ref{eqn:cov4} can be used to help chose an optimal exposure time for photometric surveys for transiting planets, when combined with models for the frame rate and photometric measurement uncertainty of the particular instrument.
In the Equations~\ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3} and \ref{eqn:cov4}, the photometric precision $\sigma$ and exposure time $t_{exp}$ are separate parameters. In practice, the uncertainty on a given photometric point will depend on the exposure time chosen. For photon-noise, $\sigma/f_0 \propto t_{exp}^{-1/2}$. We keep our equations explicitly in terms of $\sigma$ (instead of directly substituting in the assumption of photon-noise assumption) so that they can be more flexibly applied to cases where additional white noise sources add to the photometric measurement uncertainty.
The exposure time also affects the effective phase sampling of the light curve. For continuous photometry observations over a time baseline, $T_{tot}$, the effective sampling of the phase-folded light curve is,
\begin{equation}
\Gamma_{eff} = \frac{T_{tot}}{P_{orb}\left(t_{exp}+t_{read}\right)}.
\label{eqn:gamma_eff}
\end{equation}
\noindent We've denoted by $t_{read}$ the time needed for photometer to read out; $\left(t_{exp}+t_{read}\right)^{-1}$ is the CCD frame rate. The factor $\frac{T_{tot}}{P_{orb}}$ accounts for the number of transits detected over the span of the observations.
%We keep our equations explicitly in terms of $\sigma$ and $\Gamma$, instead of substituting in the assumption of photon-noise, so that they can be more flexibly applied to cases with finite read-out time, and additional white noise sources that add to the photometric uncertainty of the
In the case of a photon-noise limited survey with negligible read out time $\left(t_{read}=0\right)$, the $t_{exp}$ dependence cancels in the $\sigma^2/\Gamma_{eff}$ prefactor that scales all the covariance matrices. In these limits, the exposure-time dependence comes solely from the body of the covariance matrix elements in Equations~\ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3} and \ref{eqn:cov4}. We plot in Figure~\ref{fig:survey} how the uncertainties on the transit parameters predicted for a Jupiter transiting a sun-twin in a photon-noise limited survey with negligible read-out time depend on $t_{exp}$, $T_{tot}$, and $P_{orb}$. We assumed a nominal photometric precision of $\sigma/f_0=5\times10^{-5}$ for a 30 minute exposure; choosing a different value for $\sigma$ would simply amount to rescaling the vertical axis on the figure.
%Comment on main takeaway point.
Lower exposure times mean better precision, but after a certain point there is a plateau regime in which shorter exposure times do not improve the relative precision of the transit parameters derived from the light curve. In planning a transit survey, choosing an exposure time near the ``knee'' would be optimal to minimize both the data rate and the relative uncertainties on the planet properties. The critical exposure time depends on both the planet orbital period and $R_p/R_*$, but is not significantly affected by the survey duration, $T_{tot}$. Planets with shorter $P_{orb}$ and smaller $R_p/R_*$ have smaller critical exposure times, and their characterization would benefit more greatly from short cadence observations.
The critical exposure time delimiting the beginning of the plateau regime is different for different transit parameter of interest. The critical exposure time for $\tau$ is the shortest. In planning a survey, one would want to consider the smallest critical exposure time among the parameters of interest. Exposure times of 3, 10, and 30 minutes are optimal for sampling the transits of Jupiter-sized planets orbiting sun twins on 1-day, 10-day, and 100-day orbits, respectively (Figure~\ref{fig:survey}). In contrast, for Earth-sized planets with $R_p/R_*=10^{-2}$ the plateau in the relative uncertainty on $\tau$ occur at $t_{exp}<1~\rm{minute}$.
%Our Fisher analysis assumes that the photometric measurement uncertainty is dominated by white noise.
%Our Fisher Information analysis relies on the assumption that the photometric measurement uncertainty is dominated by white noise. Real photometry data often has correlated (red) noise; which may cause our analytic approximations to the transit parameter uncertainties to break down.
%this may be especially true when $t_{exp}$ is short.
%One potential way to reduce the effect of red-noise is to bin the lightcurve data to an effective exposure time that is larger than the timescales. Our analytic expressions for the effect of a finite exposure time on the covariance of transit light-curve-derived parameters will be helpful to assess the optimal exposure time, balancing the effect of red-noise against the additional variances and covariances induced with longer exposure times. FIX and ELABORATE.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/survey-planning/survey-planning}
\caption{{\label{fig:survey}
Uncertainties of trapezoidal transit parameters for representative observing cases as functions of exposure time $t_{exp}$, as predicted by the binned trapezoidal light curve model. For planets on orbits of 1 day (red curves), 10 days (blue curves), and 100 days (black curves), we show the expected uncertainties for total survey lengths of 1 month (solid curves), 3 months (dashed curves), and 4 years (dotted curves). We assume nominal parameter values $b = 0.2$, $e = 0.16$, and $r = 0.1$, to remain consistent with the figures above; we scale the photometric uncertainty such that a \textit{Kepler} 30-minute exposure corresponds to $\sigma = 5 \times 10^{-5}$. $\Gamma_{eff}$ is given by Equation \ref{eqn:gamma_eff} when $T_{tot} > P_{orb}$; otherwise, $\Gamma_{eff} = 1/t_{exp}$.%
}}
\end{center}
\end{figure}
\subsection{Effect of grazing transits}
\label{sec:grazing}
We have thus far limited our discussion to cases in which the exposure time does not exceed the time between second and third contact, when the planet disk is contained completely within the disk of the star. Once the exposure time exceeds $T-\tau$, the maximum apparent depth of the transit light curve starts to be decrease as all exposures taken during totality are diluted by flux during ingress, egress, and/or out-of-transit. The apparent maximum depth of the transit light curve can be reduced by as much as $T/t_{\exp}$; this maximum value corresponds to $t_{exp}>T+\tau$).
By focussing on cases with $t_{exp}b_{max}$ .
%This paragraph may be deleted/changed
Though we have not provided analytic equations for the covariance matrix in the case where $t_{exp}>T-\tau$, these can be readily derived following a similar approach as in Section~\ref{sec:FI}, above. There are in fact, three more regimes to be considered (in addition to cases 1 and 2 given in Equations~\ref{eqn:lcbinned1} and \ref{eqn:lcbinned2}): $T-\tau\tau$ (case 4); and $t_{exp}>T+\tau$ (case 5).
%Did we want to include these equations after all (for completeness)? Or shall we just make sure the code we place online treats these grazing transits?\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/limiting-porb/limiting-porb}
\caption{{\label{fig:bmax}
Maximum impact parameter, $b_{max}$, as a function of orbital period for Jupiter-sized ($r=0.1$, solid line) and Earth-sized ($r=0.01$, dashed) planets on circular orbits around a Sun-twin star with {\it Kepler} long-cadence sampling ($t_{exp}=30~\rm{minutes}$).%
}}
\end{center}
\end{figure}
\subsection{Effect of limb darkening}
So far, we have neglected the effect of limb-darkening (following Carter et al. 2008), and have considered a planet transiting a star with uniform surface brightness. We repeated the MCMC procedure outlined in Section~\ref{sec:MCMC}, generating transit data with a Python implementation of the \citet{ClaretEt2011AA} EXOFAST \texttt{occultquad} routine, which generates a \citet{Mandel&Agol2002ApJ} quadratically limb-darkened light curve; we chose the limb-darkening parameters for HAT-P-2 as our test case, obtaining the parameters with the \citet{EastmanEt2013ASP} limb-darkening parameter applet (\url{http://astroutils.astronomy.ohio-state.edu/exofast/limbdark.shtml}), which interpolates the \citet{ClaretEt2011AA} quadratic limb darkening tables.
We show the results of this analysis in Figures \ref{fig:varRpTrapzLD} and \ref{fig:covRpTrapzLD}. Our Equations \ref{eqn:cov1}, \ref{eqn:cov2}, \ref{eqn:cov3}, and \ref{eqn:cov4} still do well to predict the uncertainties on $\delta$, $T$, and $t_c$, but we overpredict the uncertainty on $\tau$. $T$ becomes more correlated with $\delta$ and with $\tau$ when limb darkening is taken into account.
%How will limb darkening affect the uncertainties?
%How does limb darkening affect the parameter values and uncertainties obtained from a
%We fit a trapezoidal model to both a non-limb-darkened and limb-darkened transit she found that adding limb-darkening decreases $T$, increases $\tau$, and decreased $\delta$ of the ``best fit" trapezoidal model.
%Ellen to do: Repeat figures 1 and 2 adding points of another color, indicating the covariance matrix elements obtained by fitting a trapezoidal model to limb-darkened light curve.
%Simulated a light curve including limb darkening
%Some of our formulas are still pretty applicable. For some physical parameters ($b^2$ and $\tau_0^2$) the physical parameters, we underestimate the uncertainties.
%The parameters become more correlated when we add limb darkening ($T$ becomes more correlated with $\delta$ and with $\tau$).
%Still do well in predicting the uncertainties on $r$ and $t_c$. The mid-transit time is the time about which the transit is symmetric. So, it makes sense that the addition of limb-darkening would not affect our ability to measure that value.
%Similarly, $r$ is the square route of the maximum depth.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/limbdarkening-trapz-var/var}
\caption{{\label{fig:varRpTrapzLD}
Relative uncertainties of the trapezoidal transit parameters for a \protect\citet{Mandel&Agol2002ApJ} quadratically limb-darkened light curve, integrated with $t_{exp} = 30~\text{minutes}$, as a function of $R_p/R_\star$. The uncertainties were measured with an MCMC analysis as before, with the relative uncertainties being scaled by the true value of the parameter (purple crosses). Our predictions (solid black lines) are still applicable except in the case of $\tau$, which has a smaller relative uncertainty than predicted. The \protect\citet{CarterEt2008ApJ} prediction (solid red line) and finite cadence prediction (dashed red line) are shown for comparison. We let $R_\star = R_\odot$, $M_\star = M_\odot$, $e = 0.16$, $P_{orb} = 9.55~\text{days}$, and $b = 0.2$, as in Figure~\ref{fig:varRpTrapz}.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/limbdarkening-trapz-cov/cov}
\caption{{\label{fig:covRpTrapzLD}
Covariances in the trapezoidal approximation transit parameters derived from simulated \textit{Kepler} long-cadence data, as a function of $R_p/R_\star$, corresponding to the same scenarios presented in Figure~\ref{fig:varRpTrapzLD}. $T$ is significantly more correlated with $\delta$ and $\tau$ than we predict at smaller values of $R_p/R_*$.%
}}
\end{center}
\end{figure}
\subsection{Effect of finite phase sampling}
\label{sec:PhaseSampling}
The $\Gamma_{eff}$ effective time sampling approximation may break down if the planet orbital period is an integer multiple of the sampling cadence. Consider a single transit event sampled over one complete orbital period $P_{orb}$ with an exposure time equal to $t_{exp}$, with the condition that $t_{exp} < P_{orb}$. In this case, we may say
\begin{equation}
\frac{P_{orb}}{t_{exp}} = n \pm \Delta
\end{equation}
\noindent where $n$ is the largest integer possible such that $0 \le \Delta < 1$. Let us consider the case when $P_{orb}$ is nearly an integer multiple of the exposure time; by Taylor's theorem, we expand about ${^\Delta/_n} = 0$ to obtain
\begin{eqnarray}
\frac{t_{exp}}{P_{orb}} & = & \frac{1}{n \pm \Delta} \\
& = & \frac{1}{n} \left( 1 \mp \frac{\Delta}{n} + \frac{\Delta^2}{n^2} \mp \cdots \right) \\
& \approx & \frac{1}{n} \left( 1 \mp \frac{\Delta}{n} \right).
\end{eqnarray}
\noindent Now, let $\delta \phi \equiv {^\Delta/_n}$, so
\begin{equation}
\frac{n t_{exp}}{P_{orb}} \approx 1 \mp \delta \phi
\end{equation}
\noindent The number of transits $N$ needed to adequately sample this light curve, such that the data can be phase-folded and an effective sampling rate $\Gamma_{eff}$ can be used, is
\begin{equation}
N = \frac{t_{exp}}{P_{orb} \times \delta \phi} = \frac{t_{exp} n}{P_{orb} \Delta}.
\end{equation}
\noindent As $\Delta \rightarrow 0$, the number of observed transits needed to use our $\Gamma_{eff}$ approximation increases without bound; in these cases, Equation~\ref{eqn:FisherElementIntegral} still applies to the data which is not phase-folded.
Another obstacle to applying our variance and covariance approximations arises if too few transits have been observed to sufficiently cover the full range of planet phases during transit. In these cases, the integral approximation of equation~\ref{eqn:FisherElementIntegral} breaks down and the finite sums (Equation~\ref{eqn:FisherElementSum}) must be evaluated numerically.
%This caveat must be kept in mind when applying our covariance matrix expressions.
%We have found in our numerical expressions that our covariance expressions still provide useful general trends, even when the finite phase sampling starts to become an issue.
%The finite Cadence Correction from \citet{CarterEt2008ApJ} may translate over directly in this case. [Check]
%Our expression for $\Gamma_{eff}$ for phase-folded light curves assumes that the
%Elaborate on the values for $\Gamma_{eff}$ at which the integral approximation breaks down.
%Update Figure 3.
\subsection{Effect of Non-Gaussian Posteriors}
\label{sec:NonGaussian}
The posterior distribution of the trapezoidal parameters obtained from our MCMC fits to simulated {\it Kepler} long-cadence light curves are well approximated by Gaussians in high S/N scenarios. In cases of low S/N, however, the ``normally-distributed parameters'' assumption on which the Fisher information matrix analysis relies can break down. The ingress/egress duration is physically constrained to be $\tau > 0$. When the uncertainty $\sigma_{\tau}$ becomes comparable to the magnitude of $\tau$ itself, the truncation at $\tau = 0$ induces non-Gaussian posteriors. This may account for some of the deviations at small $R_p/R_\star$ in Figures \ref{fig:varRpTrapz} and \ref{fig:covRpTrapz}. Solving numerically for the value of $R_p/R_\star$ where the value of $\tau$ becomes comparable to $\sigma_\tau$ for nominal values of the other parameters, we find that $\tau$ is equal to $3\sigma_\tau$ at $R_p/R_\star \approx 0.042$; it is equal to $2\sigma_\tau$ when $R_p/R_\star \approx 0.037$ and $1 \sigma_\tau$ when $R_p/R_\star \approx 0.031$. As $R_p/R_\star \rightarrow 0$, we expect the posterior to approach a Gaussian centered at $0$ and truncated at $0$; $\tau = 3\sigma_\tau$ is the approximate lower limit of $\tau$ where truncation should not be apparent in the posterior distribution. The numerical results seem to coincide well with the value of $R_p/R_\star$ where the MCMC results begin to deviate from the analytic prediction.
\section{Conclusions}
\label{sec:con}
%Summarize main analysis we accomplished in this paper.
%Summarize main effect of finite integration time on the uncertainties and correlations of different light curve parameters.
%Emphasize applications of these results.
%You have to take finite exposure time into account when predicting the uncertainties on planet transit parameters, particularly in the limits of small planets, low signal-to-noise, and short orbital periods.
\citet{Kipping2010MNRAS} highlighted the necessity of fitting a binned light curve model to binned light curve data. We have updated the \citet{CarterEt2008ApJ} analytic expressions for the variances and covariances of parameters derived from fitting transit light curve data, to take finite exposure time into account.
With finite exposure time, the uncertainties on the transit parameters are strictly greater than what one could extract from an instantaneously sampled light curve. The magnitude of the correlations among transit ingress/egress duration, transit duration and transit depth all increase, while the mid-transit time (relevant for measuring TTVs) remains uncorrelated. For example, for a Hot Jupiter or close-in Earth-size planet on a three day orbit the variances on $\delta$, $t_c$, $\tau$, and $T$ are $1.2$, $2.5$, $24$, and $2.8$ times larger for 30-minute long-cadence data as compared to 1 minute short cadence data; the covariances can increase by as much as a factor of $30$. In contrast, for a transiting Earth-twin on a 1-year orbit, the variances themselves are larger in magnitude, but they do not change greatly with exposure time.
We provide Python and \textit{Mathematica} code for computing the predicted variances and covariances that could be measured using the binned light curve model.
Phasing, red noise, non-Gaussianities, and other effects can affect the actual uncertainties obtained from a full analysis.
Our analytic expressions are still helpful for target selection, observation planning, and rule of thumb intuition.
Today finite exposure time is relevant for Kepler long-cadence light curves, and will remain important in the analysis of data from upcoming transit missions K2 and TESS full frame images.
%Caveat on finite expose time always being worse, if the Carter et al. finite cadence correction turns out to be valid.
\section{Acknowledgements}
We would like to thank John Johnson of the Harvard-Smithsonian Center for Astrophysics for his valuable input on this project. We also thank him for establishing the Johnson Exolab as an environment where undergraduates and postdoctoral scholars can work together on projects like this one.
E. Price would like to thank Mr. and Mrs. Carl Larson for providing funding for her 2013 Carolyn Ash SURF Fellowship.
The LevelScheme \citep{Caprio2005CPC} scientific figure preparation system for \textit{Mathematica} was used in the preparation of this paper.
\selectlanguage{english}
\FloatBarrier
\bibliographystyle{plain}
\bibliography{bibliography/converted_to_latex.bib%
}
\end{document}