No abstract

We present the first study to synthesize results from five different exoplanet surveys using three independent detection methods: microlensing, radial velocity, and direct imaging. The constraints derived herein represent the most comprehensive picture of the demographics of large-separation ( 2 AU) planets orbiting the most common stars in our Galaxy that has been constructed to date. We assume a simple, joint power-law planet distribution function of the form d 2 N pl /(d log m p d log a) = A(m p /M Sat ) α (a/2.5 AU) β with an outer cutoff radius of the separation distribution function of a out . Generating populations of planets from these models and mapping them into the relevant observables for each survey, we use actual or estimated detection sensitivities to determine the expected observations for each survey. Comparing with the reported results, we derive constraints on the parameters {α, β, A, a out } that describe a single population of planets that is simultaneously consistent with the results of microlensing, RV, and direct imaging surveys. We find median and 68% confindence intervals of α = −0.86 +0.21 −0.19 (−0.85 +0.21 −0.19 ), β = 1.1 +1.9 −1.4 (1.1 +1.9 −1.3 ), A = 0.21 +0.20 −0.15 dex −2 (0.21 +0.20 −0.15 dex −2 ), and a out = 10 +26 −4.7 AU (12 +50 −6.2 AU) assuming "hot-start" ("cold-start") planet evolutionary models. These values are consistent with all current knowledge of planets on orbits beyond ∼ 2 AU around M dwarfs.

Motivated by the order-of-magnitude difference in the frequency of giant planets orbiting M dwarfs inferred by microlensing and radial velocity (RV) surveys, we present a method for comparing the statistical constraints on exoplanet demographics inferred from these methods. We first derive the mapping from the observable parameters of a microlensing-detected planet to those of an analogous planet orbiting an RV-monitored star. Using this mapping, we predict the distribution of RV observables for the planet population inferred from microlensing surveys, taking care to adopt reasonable priors for, and properly marginalize over, the unknown physical parameters of microlensing-detected systems. Finally, we use simple estimates of the detection limits for a fiducial RV survey to predict the number and properties of analogs of the microlensing planet population such an RV survey should detect. We find that RV and microlensing surveys have some overlap, specifically for super-Jupiter mass planets (m p 1 M Jup ) with periods between ∼ 3 − 10 years. However, the steeply falling planetary mass function inferred from microlensing implies that, in this region of overlap, RV surveys should infer a much smaller frequency than the overall giant planet frequency (m p 0.1 M Jup ) inferred by microlensing. Our analysis demonstrates that it is possible to statistically compare and synthesize data sets from multiple exoplanet detection techniques in order to infer exoplanet demographics over wider regions of parameter space than are accessible to individual methods. In a companion paper, we apply our methodology to several representative microlensing and RV surveys to derive the frequency of planets around M dwarfs with orbits of 30 years.

In contrast to radial velocity surveys, results from microlensing surveys indicate that giant planets with masses greater than the critical mass for core accretion (∼ 0.1 M Jup ) are relatively common around low-mass stars. Using the methodology developed in the first paper, we predict the sensitivity of M-dwarf radial velocity (RV) surveys to analogs of the population of planets inferred by microlensing. We find that RV surveys should detect a handful of super-Jovian (> M Jup ) planets at the longest periods being probed. These planets are indeed found by RV surveys, implying that the demographic constraints inferred from these two methods are consistent. We show that if total RV measurement uncertainties can be reduced by a factor of a few, it is possible to detect the large reservoir of giant planets (0.1 − 1 M Jup ) comprising the bulk of the population inferred by microlensing. We predict that these planets will likely be found around stars that are less metal-rich than the stars which host super-Jovian planets. Finally, we combine the results from both methods to estimate planet frequencies spanning wide regions of parameter space. We find that the frequency of Jupiters and super-Jupiters (1 m p sin i/M Jup 13) with periods 1 ≤ P/days ≤ 10 4 is f J = 0.029 +0.013 −0.015 , a median factor of 4.3 (1.5 − 14 at 95% confidence) smaller than the inferred frequency of such planets around FGK stars of 0.11 ± 0.02. However, we find the frequency of all giant planets with 30 m p sin i/M ⊕ 10 4 and 1 ≤ P/days ≤ 10 4 to be f G = 0.15 +0.06 −0.07 , only a median factor of 2.2 (0.73 − 5.9 at 95% confidence) smaller than the inferred frequency of such planets orbiting FGK stars of 0.31 ± 0.07. For a more conservative definition of giant planets (50 m p sin i/M ⊕ 10 4 ), we find f G ′ = 0.11 ± 0.05, a median factor of 2.2 (0.73 − 6.7 at 95% confidence) smaller than that inferred for FGK stars of 0.25 ± 0.05. Finally, we find the frequency of all planets with 1 ≤ m p sin i/M ⊕ ≤ 10 4 and 1 ≤ P/days ≤ 10 4 to be f p = 1.9 ± 0.5.