Our discovery of a hot Jupiter in the Hyades bolsters the statistics of short period giant planets in open clusters (of which three are now known). These discoveries, along with a careful treatment of survey completeness and the properties of the stellar sample, can now be used to refine the calculation of the frequency of hot Jupiters in clusters.
4.3.1 Metal-Rich Clusters
Together, members of the Hyades and Praesepe open clusters constitute a large sample of metal-rich stars. Because of the enhanced hot Jupiter fraction for metal-rich field stars, it is possible that the architectures of metal-rich planetary systems are significantly different from those of metal-poor systems, and these differences may significantly influence the efficiency of migration in open clusters, where close stellar encounters are expected to be more frequent. For example, simulations by Shara et al. (2016) suggest that close stellar fly-bys in clusters may result in an increase in perturbations to planetary orbits, which lead to a higher incidence of planet-planet scattering events. In such a scenario, the hot Jupiter frequency may be affected, particularly if planet-planet scattering is an important contributor to hot Jupiter migration. As such, we separately examine the hot Jupiter occurrence rate in metal-rich
clusters before calculating the occurrence rate for our entire open cluster sample.
Including the Paulson et al. (2004b) null result in the Hyades (0 planets among 74 FGK stars), our Hyades sample (1 in 25; Quinn et al. 2014), and our Praesepe sample (2 in 53; Quinn et al. 2012), a total of 3 out of 154 stars host a hot Jupiter. To be able to make a better comparison to field star surveys, we have excluded from these totals stars with RV variations that appear to indicate the presence of a close stellar companion that would be expected to prevent the formation or survival of planets.
While the raw planet counts can provide an estimate of the occurrence rates, survey completeness must be considered in order to make an accurate calculation. To assess the sensitivity of our survey (and that of Paulson et al. 2004b), we simulate planetary orbits and sample them according to the actual observing cadences and measurement errors for each
star. Because we cannot know, a priori, the planetary mass and orbital period distributions
in open clusters, we draw these quantities from a smoothed empirical distribution of hot
Jupiters orbiting field stars with periods less than 10 days and masses less than 13MJup.
While we do expect that environmental factors will create some differences between cluster and field star distributions, we do not expect them to be so different that drawing from the incorrect distribution will significantly impact our calculation of completeness. We also note that a characterization of the joint distribution of giant planet masses and periods can be found in the literature (e.g., Cumming et al. 2008), but we do not use this distribution because it does not account for the shape of the distribution in small ranges of period (such as in the local short-period peak corresponding the hot Jupiters). Each orbit is given a random
time of inferior conjunction and an orbital inclination drawn from an isotropic distribution (i.e., prob(i)∝sini?). We assume circular orbits for periods less than 7 days, and for longer
periods, we draw eccentricities following a Rayleigh distribution with Rayleigh parameter
σ = 0.3 (a good match to the eccentricities of giant planets according to Juri´c & Tremaine
2008), with randomly oriented longitudes of periastron.
For each synthetic RV data set, we ask whether we would have detected the planet
according to our P(χ2) threshold. If the simulated planet meets this criterion, we call it a
detectable planet, because in our actual survey we would have followed it up and detected it. We happily point out that adopting a clear, quantitative criterion for follow-up, as we have done in our survey, makes completeness calculations straightforward. For the Paulson Hyades survey, we make the assumption that they, too, would have detected planets with
P(χ2)<0.001, even though they did not adopt this exact criterion in their survey. We argue that this is a valid assumption because they did follow up their low-amplitude variables, and none of the remaining stars in their survey exhibits P(χ2)<0.001.
After correcting for completeness and calculating Poisson errors following the prescription in Gehrels (1986), we find a hot Jupiter frequency of 1.97+1−1..9207% in the metal-rich Praesepe and Hyades open clusters. However, giant planet occurrence scales with metallicity approx- imately as 102[Fe/H] (Fischer & Valenti 2005). If we take [Fe/H] ≈ +0.15 as representative of the combined Praesepe and Hyades sample, the solar-metallicity-adjusted hot Jupiter frequency in clusters is 0.99+0−0..9654%. Although more discoveries are needed to reduce the un-
et al. 2012), and improves the evidence that planet frequency is the same in clusters and the field.
4.3.2 600 Myr Clusters
While it is important to bear in mind the effect that stellar environment may have on the formation and evolution of planetary systems, it can also be illustrative to examine the characteristics of a larger, more diverse population. Here, we treat our entire sample of adolescent (500-625 Myr) clusters (once again including the null result from Paulson et al. 2004b) as one population and calculate the hot Jupiter occurrence rate. In Coma Berenices and Ursa Major, we detect no hot Jupiters among 31 and 16 non-binary Sun-like members,
respectively. This brings our total sample to 201 stars, of which 3 host a hot Jupiter.
We again adjust for the metallicity of the stellar samples and for the survey completeness. The final derived solar-metallicity-equivalent hot Jupiter occurrence rate in these adolescent clusters is 0.93+0−0..9050%.
EVIDENCE FOR HIGH ECCENTRICITY MIGRATION