Petigura et al. (2013) calculated the frequency of Earth-like planets (\(R=1-2{R_\oplus}\)) in the habitable zone (HZ) of their host star to be \(22\pm8\%\), with their HZ defined to be \(0.25-4.0{F_\oplus}\), where \({F_\oplus}\) is the incident flux Earth receives from the Sun. The authors created an independent transit search algorithm and fitting routine called TERRA and applied it to the Kepler light curves of the “Best42k”, or the brightest (\(K_p=10-15\)) 42,557 Sun-like stars (\(T_{eff}=4100-6100K\), \(logg=4.0-4.9\)) exhibiting the lowest photometric noise, selecting those stars on the main sequence or just starting the subgiant phase, according to the initial stellar parameters. After false positive checks and other vetting procedures, 603 “eKOIs” remained, in analogy to the Kepler Objects of Interest (KOIs).
The key method that allows Petigura et al. (2013) to calculate the occurrence rate of Earth-like planets in the HZ is a thorough analysis of their completeness. They measured completeness by injecting 40,000 planets at different periods and planet radii into the light curves of the actual Kepler light curves of the Best42k stars. They then applied TERRA to check how many of their injected transits they could recover. They are then able to correct for transiting planets missed by their TERRA code by dividing the number of planets recovered by the completeness value. One can then correct this number by the geometric probability that a planet would transit: \(P_T=R_*/a\), where \(R_*\) is the stellar radius and \(a\) is the semi-major axis of the planet. They use these corrections to calculate the frequency of planets in the inner HZ, which they define as \(1-4{F_\oplus}\), where they claim a high enough completeness to get an accurate measurement of the occurrence rate of Earth-like planets. Their estimate of the occurrence rate for Earth-like planets in the inner HZ is \(11\pm4\%\).
For the outer HZ, which they define to be \(0.25-1.00{F_\oplus}\), they must rely on an extrapolation. Petigura et al. (2013) find a roughly constant occurrence rate in equally binned log(P), which translates into a roughly constant occurrence rate in equally binned log(F). Therefore, since \(1-4{F_\oplus}\) and \(0.25-1{F_\oplus}\) are equally spaced in log(F) (a factor of 4), the occurrence rate in the outer HZ is also \(11\pm4\%\). That makes the overall occurrence rate in the HZ \(22\pm8\%\).
We aim to use the data available in Petigura et al. (2013) to redo the analysis as best we can by attempting to calculate the occurrence rate ourselves with different assumptions. For example, a HZ extending to \(4{F_\oplus}\) is almost certainly too hot to be habitable for most terrestrial worlds, although particularly dry planets may be an exception. Another problem is that the completeness drops drastically at the exact point where the inner HZ begins. As such, Petigura et al. (2013) had to extrapolate to achieve an occurrence rate for their outer HZ, \(0.25-1{F_\oplus}\), which is actually where the bulk of the HZ is according to many HZ calculations, such as Kopparapu et al. (2013). We would like to explore different definitions of what is habitable, what is Earth-like, and the methods to calculate the occurrence rate in the full HZ.