Any negative completion or planet occurrence rates were set equal to zero.
In Petigura et. al. (2013), flux was not measured directly but calculated; given the published information, we were unable to determine the precise errors used for each component of the flux calculation, and hence we were unable to precisely propagate error.
First we assumed that the dominant source of error is the effective temperature, in which case we can approximate the propagated error of the occurrence rate calculation due to the fact that \(f_i \propto T_{eff}^2\). However, the produced errors were significantly lower than those published by Petigura, et. al. (2013). The second order source of error which we can account for is planetary flux, since \(f_i \propto F_P^{-1/2}\), and in turn \(F_P \propto \frac{L_*}{a^2}\). We have the necessary information to propagate the contribution of the error in stellar luminosity, but insufficient information to do so for \(a\) because it is derived from stellar mass estimates from color-color model fitting. However, the addition of \(\sigma_{L*}\) does raise our occurrence rate errors significantly, though they still remain smaller than those previously published. Upon further testing, Poisson errors are the dominant source of error in many of the bins. Overall, these effects are far more significant sources of error than planetary radius and far more significant error than period.
Our occurrence rate errors were significantly smaller (errors ranging from 0.03\(\%\)-0.51\(\%\)) than those found by Petigura et. al. (2013) (errors ranging from 0.1\(\%\)-2.1\(\%\)). The component missing in our analysis is error in semi-major axes, suggesting that error in semi-major axes is significant. Our conclusion is that we are accounting for the two dominant sources of error, but underestimating our errors overall due to higher order contributions we can’t account for given that some parameters are model-dependent.
Petigura et. al. (2013) use a completeness cutoff of 25\(\%\) per box (see Petigura et. al. Figure 2). We tested completeness cutoffs from 0\(\%\) up to 50\(\%\). These tests showed that completeness is high enough in the regions of interest, i.e. low radius and low flux, that completeness cutoffs make no significant impact until a cutoff of approximately 45\(\%\).
Similar to Group 1, we’ve stored our output in a pickled file called “occ rate bins.p”. It has four keys: “r bins”, the len(9) array with the edges of our bins in radius, “f p bins”, the len(11) array with the edges of our bins in \(F_P\), and the two 8x10 arrays of the occurrence rate and error for each bin. The indexing is as follows: