Most known extrasolar planets (exoplanets) have been discovered using the radial velocity1,2 or transit3 methods. Both are biased towards planets that are relatively close to their parent stars, and studies find that around 17–30% (refs 4, 5) of solar-like stars host a planet. Gravitational microlensing6,7,8,9, on the other hand, probes planets that are further away from their stars. Recently, a population of planets that are unbound or very far from their stars was discovered by microlensing10. These planets are at least as numerous as the stars in the Milky Way10. Here we report a statistical analysis of microlensing data (gathered in 2002–07) that reveals the fraction of bound planets 0.5–10 au (Sun–Earth distance) from their stars. We find that of stars host Jupiter-mass planets (0.3–10 MJ, where MJ = 318 M⊕ and M⊕ is Earth’s mass). Cool Neptunes (10–30 M⊕) and super-Earths (5–10 M⊕) are even more common: their respective abundances per star are and . We conclude that stars are orbited by planets as a rule, rather than the exception.
Gravitational microlensing is very rare: fewer than one star per million undergoes a microlensing effect at any time. Until now, the planet-search strategy7 has been mainly split into two levels. First, wide-field survey campaigns such as the Optical Gravitational Lensing Experiment (OGLE; ref. 11) and Microlensing Observations in Astrophysics (MOA; ref. 12) cover millions of stars every clear night to identify and alert the community to newly discovered stellar microlensing events as early as possible. Then, follow-up collaborations such as the Probing Lensing Anomalies Network (PLANET; ref. 13) and the Microlensing Follow-Up Network (μFUN; refs 14, 15) monitor selected candidates at a very high rate to search for very short-lived light curve anomalies, using global networks of telescopes.
To ease the detection-efficiency calculation, the observing strategy should remain homogeneous for the time span considered in the analysis. As detailed in the Supplementary Information, this condition is fulfilled for microlensing events identified by OGLE and followed up by PLANET in the six-year time span 2002−07. Although a number of microlensing planets were detected by the various collaborations between 2002 and 2007 (Fig. 1), only a subset of them are consistent with the PLANET 2002–07 strategy. This leaves us with three compatible detections: OGLE 2005-BLG-071Lb (refs 16, 17) a Jupiter-like planet of mass M ≈ 3.8 MJ and semi-major axis a ≈ 3.6 au; OGLE 2007-BLG-349Lb (ref. 18), a Neptune-like planet (M ≈ 0.2 MJ, a ≈ 3 au); and the super-Earth planet OGLE 2005-BLG-390Lb (refs 19, 20; M ≈ 5.5 M⊕, a ≈ 2.6 au).
To compute the detection efficiency for the 2002–07 PLANET seasons, we selected a catalogue of unperturbed (that is, single-lens-like) microlensing events using a standard procedure21, as explained in the Supplementary Information. For each light curve, we defined the planet-detection efficiency ε(logd,logq) as the probability that a detectable planet signal would arise if the lens star had one companion planet, with mass ratio q and projected orbital separation d (in Einstein-ring radius units; ref. 22). The efficiency was then transformed23 to ε(loga,logM). The survey sensitivity S(loga,logM) was obtained by summing the detection efficiencies over all individual microlensing events. It provided the number of planets that our survey would expect to detect if all lens stars had exactly one planet of mass M and semi-major axis a.
We used 2004 as a representative season from the PLANET survey. Among the 98 events monitored, 43 met our quality-control criteria and were processed24. Most of the efficiency comes from the 26 most densely covered light curves, which provide a representative and reliable sub-sample of events. We then computed the survey sensitivity for the whole time span 2002–07 by weighting each observing season relative to 2004, according to the number of events observed by PLANET for different ranges of peak magnification. This is described in the Supplementary Information, and illustrated in Supplementary Fig. 2. The resulting planet sensitivity is plotted in blue in Fig. 1, where the labelled contours show the corresponding expected number of detections. The figure shows that the core sensitivity covers 0.5−10 au for masses between those of Uranus/Neptune and ten times the mass of Jupiter, and extends (with limited sensitivity) down to about 5 M⊕. As inherent to the microlensing technique, our sample of event-host stars probes the natural mass distribution of stars in the Milky Way (K–M dwarfs), in the typical mass range of 0.14−1.0 M⊙ (see Supplementary Fig. 3).
To derive the actual abundance of exoplanets from our survey, we proceeded as follows. Let the planetary mass function, f(loga,logM) ≡dN/(dloga × dlogM), where N is the average number of planets per star. We then integrate the product f(loga,logM) S(loga,logM) over loga and logM. This gives E(f), the number of detections we can expect from our survey. For k (fractional) detections, the model then predicts a Poisson probability distribution P(k|E) = e−EEk/k!. A Bayesian analysis assuming an uninformative uniform prior P(logf) ≡ 1 finally yields the probability distribution P(logf|k) that is used to constrain the planetary mass function.
Although our derived planet-detection sensitivity extends over almost three orders of magnitude of planet masses (roughly 5 M⊕ to 10 MJ), it covers fewer than 1.5 orders of magnitude in orbit sizes (0.5−10 au), thus providing little information about the dependence of f on a. Within these limits, however, we find that the mass function is approximately consistent with a flat distribution in loga (that is, f does not explicitly depend on a). The planet-detection sensitivity integrated over loga, or S(logM), is displayed in Fig. 2b. The distribution probabilities of the mass for the three detections (computed according to the mass-error bars reported in the literature) are plotted in Fig. 2c (black curves), as is their sum (red curve).
To study the dependence of f on mass, we assume that to the first order, f is well-approximated by a power-law model: f = f0 (M/M0)α, where f0 (the normalization factor) and α (the slope of the power-law) are the parameters to be derived and M0 a fiducial mass (in practice, the pivot point of the mass function). Previous works18,25,26,27 on planet frequency have demonstrated that a power law provides a fair description of the global behaviour of f with planetary mass. Apart from the constraint based on our PLANET data, we also made use in our analysis of the previous constraints obtained by microlensing: an estimate of the normalization18 f0 (0.36 ± 0.15) and an estimate of the slope25 α (−0.68 ± 0.2), displayed respectively as the blue point and the blue lines in Fig. 2. The new constraint presented here therefore relies on 10 planet detections. We obtained f = 10−0.62 ± 0.22 (M/M0)−0.73 ± 0.17 (red line in Fig. 2a) with a pivot point at M0 ≈ 95 M⊕; that is, at Saturn’s mass. The median of f and the 68% confidence interval around the median are marked by the dashed lines and the grey area.
Hence, microlensing delivers a determination of the full planetary mass function of cool planets in the separation range 0.5−10 au. Our measurements confirm that low-mass planets are very common, and that the number of planets increases with decreasing planet mass, in agreement with the predictions of the core-accretion theory of planet formation28. The first microlensing study of the abundances of cool gas giants21 found that fewer than 33% of M dwarfs have a Jupiter-like planet between 1.5−4 au, and even lower limits of 18% have been reported29,30. These limits are compatible with our measurement of for masses ranging from Saturn to 10 times Jupiter, in the same orbit range.
From our derived planetary mass function, we estimate that within 0.5−10 au (that is, for a wider range of orbital separations than previous studies), on average of stars host a ‘Jupiter’ (0.3−10 MJ) and of stars host Neptune-like planets (10−30 M⊕). Taking the full range of planets that our survey can detect (0.5−10 au, 5 M⊕ to 10 MJ), we find that on average every star has planets. This result is consistent with every star of the Milky Way hosting (on average) one planet or more in an orbital-distance range of 0.5–10 au. Planets around stars in our Galaxy thus seem to be the rule rather than the exception.
Support for the PLANET project was provided by the HOLMES grant from the French Agence Nationale de la Recherche (ANR), the French National Centre for Scientific Research (CNRS), NASA, the US National Science Foundation, the Lawrence Livermore National Laboratory/National Nuclear Security Administration/Department of Energy, the French National Programme of Planetology, the Program of International Cooperation in Science France–Australia, D. Warren, the German Research Foundation, the Instrument Center for Danish Astronomy and the Danish Natural Science Research Council. The OGLE collaboration is grateful for funding from the European Research Council Advanced Grants Program. K.Ho. acknowledges support from the Qatar National Research Fund. M.D. is a Royal Society University Research Fellow.
The file contains Supplementary Text and Data, Supplementary Figures 1-5 with legends, Supplementary Table 1 and additional references.
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Experimental Astronomy (2018)