We present the first measurement of the planet frequency beyond the "snow line," for the planet-to-star mass-ratio interval −4.5 < log q < −2, corresponding to the range of ice giants to gas giants. We find d 2 N pl d log q d log s = (0.36 ± 0.15) dex −2 at the mean mass ratio q = 5 × 10 −4 with no discernible deviation from a flat (Öpik's law) distribution in logprojected separation s. The determination is based on a sample of six planets detected from intensive follow-up observations of high-magnification (A > 200) microlensing events during 2005-2008. The sampled host stars have a typical mass M host ∼ 0.5 M , and detection is sensitive to planets over a range of planet-star-projected separations (s −1 max R E , s max R E), where R E ∼ 3.5 AU (M host /M) 1/2 is the Einstein radius and s max ∼ (q/10 −4.3) 1/3. This corresponds to deprojected separations roughly three times the "snow line." We show that the observations of these events have the properties of a "controlled experiment," which is what permits measurement of absolute planet frequency. High-magnification events are rare, but the survey-plus-follow-up high-magnification channel is very efficient: half of all high-mag events were successfully monitored and half of these yielded planet detections. The extremely high sensitivity of high-mag events leads to a policy of monitoring them as intensively as possible, independent of whether they show evidence of planets. This is what allows us to construct an unbiased sample. The planet frequency derived from microlensing is a factor 8 larger than the one derived from Doppler studies at factor ∼25 smaller star-planet separations (i.e., periods 2-2000 days). However, this difference is basically consistent with the gradient derived from Doppler studies (when extrapolated well beyond the separations from which it is measured). This suggests a universal separation distribution across 2 dex in planet-star separation, 2 dex in mass ratio, and 0.3 dex in host mass. Finally, if all planetary systems were "analogs" of the solar system, our sample would have yielded 18.2 planets (11.4 "Jupiters," 6.4 "Saturns," 0.3 "Uranuses," 0.2 "Neptunes") including 6.1 systems with two or more planet detections. This compares to six planets including one twoplanet system in the actual sample, implying a first estimate of 1/6 for the frequency of solar-like systems.

We report the first unambiguous detection and mass measurement of an isolated stellar-mass black hole (BH). We used the Hubble Space Telescope (HST) to carry out precise astrometry of the source star of the long-duration (t E ≃ 270 days), high-magnification microlensing event MOA-2011-BLG-191/OGLE-2011-BLG-0462 (hereafter designated as MOA-11-191/OGLE-11-462), in the direction of the Galactic bulge. HST imaging, conducted at eight epochs over an interval of 6 yr, reveals a clear relativistic astrometric deflection of the background star’s apparent position. Ground-based photometry of MOA-11-191/OGLE-11-462 shows a parallactic signature of the effect of Earth’s motion on the microlensing light curve. Combining the HST astrometry with the ground-based light curve and the derived parallax, we obtain a lens mass of 7.1 ± 1.3 M ⊙ and a distance of 1.58 ± 0.18 kpc. We show that the lens emits no detectable light, which, along with having a mass higher than is possible for a white dwarf or neutron star, confirms its BH nature. Our analysis also provides an absolute proper motion for the BH. The proper motion is offset from the mean motion of Galactic disk stars at similar distances by an amount corresponding to a transverse space velocity of ∼45 km s−1, suggesting that the BH received a “natal kick” from its supernova explosion. Previous mass determinations for stellar-mass BHs have come from radial velocity measurements of Galactic X-ray binaries and from gravitational radiation emitted by merging BHs in binary systems in external galaxies. Our mass measurement is the first for an isolated stellar-mass BH using any technique.

Aims. We report the discovery of a planet with a high planet-to-star mass ratio in the microlensing event MOA-2009-BLG-387, which exhibited pronounced deviations over a 12-day interval, one of the longest for any planetary event. The host is an M dwarf, with a mass in the range 0.07 M < M host < 0.49 M at 90% confidence. The planet-star mass ratio q = 0.0132 ± 0.003 has been measured extremely well, so at the best-estimated host mass, the planet mass is m p = 2.6 Jupiter masses for the median host mass, M = 0.19 M . Methods. The host mass is determined from two "higher order" microlensing parameters. One of these, the angular Einstein radius θ E = 0.31 ± 0.03 mas has been accurately measured, but the other (the microlens parallax π E , which is due to the Earth's orbital motion) is highly degenerate with the orbital motion of the planet. We statistically resolve the degeneracy between Earth and planet orbital effects by imposing priors from a Galactic model that specifies the positions and velocities of lenses and sources and a Kepler model of orbits. Results. The 90% confidence intervals for the distance, semi-major axis, and period of the planet are 3.5 kpc < D L < 7.9 kpc, 1.1 AU < a < 2.7 AU, and 3.8 yr < P < 7.6 yr, respectively.

Bennett et al.

We present a general framework for matching the point-spread function (PSF), photometric scaling, and sky background between two images, a subject which is commonly referred to as difference image analysis (DIA). We introduce the new concept of a spatially varying photometric scale factor which will be important for DIA applied to wide-field imaging data in order to adapt to transparency and airmass variations across the field-of-view. Furthermore, we demonstrate how to separately control the degree of spatial variation of each kernel basis function, the photometric scale factor, and the differential sky background. We discuss the common choices for kernel basis functions within our framework, and we introduce the mixed-resolution delta basis functions to address the problem of the size of the least-squares problem to be solved when using delta basis functions. We validate and demonstrate our algorithm on simulated and real data. We also describe a number of useful optimisations that may be capitalised on during the construction of the least-squares matrix and which have not been reported previously. We pay special attention to presenting a clear notation for the DIA equations which are set out in a way that will hopefully encourage developers to tackle the implementation of DIA software.