\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
\[\frac{P_{orb}}{t_{exp}} = n \pm \Delta\]
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
\[\begin{aligned} \frac{t_{exp}}{P_{orb}} & = & \frac{1}{n \pm \Delta} \\ & = & \frac{1}{n} \left( 1 \mp \frac{\Delta}{n} + \frac{\Delta^2}{n^2} \mp \cdots \right) \\ & \approx & \frac{1}{n} \left( 1 \mp \frac{\Delta}{n} \right).\end{aligned}\]
Now, let \(\delta \phi \equiv {^\Delta/_n}\), so
\[\frac{n t_{exp}}{P_{orb}} \approx 1 \mp \delta \phi\]
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
\[N = \frac{t_{exp}}{P_{orb} \times \delta \phi} = \frac{t_{exp} n}{P_{orb} \Delta}.\]
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.