The TRENDS High-Contrast Imaging Survey. IV. The Occurrence Rate of Giant Planets around M-Dwarfs

Doppler-based planet surveys have discovered numerous giant planets but are incomplete beyond several AU. At larger star-planet separations, direct planet detection through high-contrast imaging has proven successful, but this technique is sensitive only to young planets and characterization relies upon theoretical evolution models. Here we demonstrate that radial velocity measurements and high-contrast imaging can be combined to overcome these issues. The presence of widely separated companions can be deduced by identifying an acceleration (long-term trend) in the radial velocity of a star. By obtaining high spatial resolution follow-up imaging observations, we rule out scenarios in which such accelerations are caused by stellar binary companions with high statistical confidence. We report results from an analysis of Doppler measurements of a sample of 111 M-dwarf stars with a median of 29 radial velocity observations over a median time baseline of 11.8 yr. By targeting stars that exhibit a radial velocity acceleration ("trend") with adaptive optics imaging, we determine that 6.5% +/- 3.0% of M-dwarf stars host one or more massive companions with 1 < m/M_Jupiter < 13 and 0 < a < 20 AU. These results are lower than analyses of the planet occurrence rate around higher-mass stars. We find the giant planet occurrence rate is described by a double power law in stellar mass M and metallicity F = [Fe/H] such that f(M,F) = 0.039(+0.056,-0.028) M^(0.8(+1.1,-0.9)) 10^((3.8 +/- 1.2)F). Our results are consistent with gravitational microlensing measurements of the planet occurrence rate; this study represents the first model-independent comparison with microlensing observations.