Figure S11 in Petigura et al. (2013) provides a discrete (binned) representation of the completeness grid. We decided to instead transform the completeness grid into a continuous function rather than rely only on the binned completeness grid. Our motivation for this comes from the fact that the completeness drops off drastically exactly where the inner HZ begins. For example, the 2x2 square of bins at \(R=1-2{R_\oplus}\) and \(P=100-200\) days ranges from 53% completeness in the top left box to just 5% in the bottom right box, a very steep gradient. Any planet on the edge of such a 2D bin is not accurately described by the completeness value at the center of the bin.
Therefore, we fit a bivariate, polynomial spline to the discrete completeness grid, which creates a 2D interpolation of the grid that allows us to more accurately calculate the completeness by creating a function that can guess the completeness value at any given period and radius, rather than just the bin center. The difference will obviously be the most pronounced near the edges and corner of each bin.
We attempted to use this fit to calculate the completeness values required for this project but we found that our interpolation was systematically lower in all grid points. Thus, we used the grid provided by Petigura and a 2D, third order polynomial fit using the SciPy:RectBivariateSpline fitting function. Our results are presented in Fig. \ref{fig_comp}.
The occurrence rate per planet is calculated as such (derived from the caption of Figure 2):
\(f = \frac{1}{n_{*}} \frac{a}{R_{*}} \frac{1}{C}\)
\(L_{*}=4 \pi R_{*}^{2} \sigma T_{eff}^{4}\)
\(F_{P}=\frac{L_{*}}{4\pi a^{2}} = \frac{4\pi R_{*}^{2} a T_{eff}^{4}}{4\pi a^{2}}\)
\(\frac{a}{R_{*}} = \left(\frac{\sigma T_{eff}^{4}}{F_{P}}\right)^{\frac{1}{2}}\)
This binning of our occurrence rate per planet will be the subject of Group 2’s work. Note that the occurrence rate is only a useful number after binning. The first significant difference from Petigura et al. (2013) is our desire to bin the occurrence rate by planet radius and incident flux rather than by planet radius and orbital period. The advantage of using period is obvious in that, for transits, the period is known exactly, so there is no associated error bar. However, habitable zones requires knowing the incident flux, so this advantage disappears when calculating potential habitability. On the other hand, binning by incident flux allows us to show the equivalent of Figure 4 from Petigura et al. (2013) in a more useful way. The Kopparapu et al. (2013) HZ boundaries change as a function of stellar \(T_{eff}\), which is not captured in any way by Figure 4. Instead, we think Figure 4 would best be replaced by a plot of stellar \(T_{eff}\) vs. \(F\), with symbol size representing planet radius, in which case the HZ estimates from Kopparapu et al. (2013) can be directly plotted onto the figure, providing an easy way to demonstrate the final measurement: the occurrence rate of planets in the HZ.