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 N-body simulations if there are multiple dynamically interacting planets.
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}).
\[\begin{gathered} F_l \left( t; t_c, \delta, \tau, T, f_0 \right) \\ = \begin{cases} f_0 - \delta, & \left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} \\ f_0 - \delta + \frac{\delta}{\tau} \left( \left| t - t_c \right| - \frac{T}{2} + \frac{\tau}{2} \right), & \frac{T}{2} - \frac{\tau}{2} < \left| t - t_c \right| < \frac{T}{2} + \frac{\tau}{2} \\ f_0, & \left| t - t_c \right| \ge \frac{T}{2} + \frac{\tau}{2} \end{cases} \label{eqn:lclinear}\end{gathered}\]
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 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.
\[\begin{gathered} F_{lb1} \left( t; t_c, \delta, \tau, T, f_0, t_{exp} \right) \\ = \begin{cases} f_0 - \delta, & \left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} \\ f_0 - \delta + \frac{\delta}{2\tau t_{exp}} \left( \left| t - t_c \right| + \frac{t_{exp}}{2} - \frac{T}{2} + \frac{\tau}{2} \right)^2, & \frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} < \left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} \\ f_0 - \delta + \frac{\delta}{\tau} \left( \left| t - t_c \right| - \frac{T}{2} + \frac{\tau}{2} \right), & \frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} < \left| t - t_c \right| < \frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} \\ f_0 - \frac{\delta}{2 \tau t_{exp}} \left( \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} - \left| t - t_c \right| \right)^2, & \frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} \le \left| t - t_c \right| < \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} \\ f_0, & \left| t - t_c \right| \ge \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} \end{cases} \label{eqn:lcbinned1}\end{gathered}\]
\[\begin{gathered} F_{lb2} \left( t; t_c, \delta, \tau, T, f_0, t_{exp} \right) \\ = \begin{cases} f_0 - \delta, & \left| t - t_c \right| \le \frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} \\ f_0 - \delta + \frac{\delta}{2\tau t_{exp}} \left( \left| t - t_c \right| + \frac{t_{exp}}{2} - \frac{T}{2} + \frac{\tau}{2} \right)^2, & \frac{T}{2} - \frac{\tau}{2} - \frac{t_{exp}}{2} \left| t - t_c \right| \le \frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} \\ f_0 - \delta + \frac{\tau \delta}{2 t_{exp}} + \frac{\delta}{t_{exp}} \left( \left| t - t_c \right| + \frac{t_{exp}}{2} - \frac{T}{2} - \frac{\tau}{2} \right), & \frac{T}{2} + \frac{\tau}{2} - \frac{t_{exp}}{2} < \left| t - t_c \right| < \frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} \\ f_0 - \frac{\delta}{2 \tau t_{exp}} \left( \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} - \left| t - t_c \right| \right)^2, & \frac{T}{2} - \frac{\tau}{2} + \frac{t_{exp}}{2} \le \left| t - t_c \right| < \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} \\ f_0, & \left| t - t_c \right| \ge \frac{T}{2} + \frac{\tau}{2} + \frac{t_{exp}}{2} \end{cases} \label{eqn:lcbinned2}\end{gathered}\]
In both cases, we restrict our consideration to scenarios with \(t_{exp}<T-\tau\), because otherwise the measurement of the transit depth during totality will be completely washed out by the exposure time.