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Spatial distribution of exoplanet candidates based on Kepler and $Gaia$ data

The spatial distribution of exoplanets in the Galaxy is important for our understanding of planet formation and evolution. We aim to determine the spatial gradients of exoplanet occurrence in the Solar neighbourhood and in the vicinity of open clusters. We combined Kepler and Gaia DR2 data for this purpose, splitting the volume sampled by the Kepler mission into certain spatial bins. We determined an uncorrected and bias-corrected exoplanet frequency and metallicity for each bin. There is a clear drop in the uncorrected exoplanet frequency with distance for F-type stars, a decline with increasing distance along the Galactic longitude l=90 deg, and a drop with height above the Galactic plane. We find that the metallicity behaviour cannot be the reason for the drop of the exoplanet frequency around F stars with increasing distance. We argue that the above-mentioned gradients of uncorrected exoplanet frequency result from a single bias of undetected smaller planets around fainter stars. When we correct for observational biases, most of these gradients in exoplanet frequency become statistically insignificant. Only a slight decline of the planet occurrence with distance for F stars remains significant at the 3 sigma level. Apart from that, the spatial distribution of exoplanets in the Kepler field of view is compatible with a homogeneous one. At the same time, we do not find a significant change in the exoplanet frequency with increasing distance from open clusters. As a byproduct, we identified six exoplanet host star candidates that are members of open clusters. Four of them are in the NGC 6811 (KIC 9655005, KIC 9533489, Kepler-66, Kepler-67) and two belong to NGC 6866 (KIC 8396288, KIC 8331612). Two out of the six had already been known to be cluster members.