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 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 (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.