Abstract

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.

Main

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).

Figure 1: Survey-sensitivity diagram.
Figure 1

Blue contours, expected number of detections from our survey if all lens stars have exactly one planet with orbit size a and mass M. Red points, all microlensing planet detections in the time span 2002–07, with error bars (s.d.) reported from the literature. White points, planets consistent with PLANET observing strategy. Red letters, planets of our Solar System, marked for comparison: E, Earth; J, Jupiter; S, Saturn; U,Uranus; N, Neptune. This diagram shows that the sensitivity of our survey extends roughly from 0.5 au to 10 au for planetary orbits, and from 5 M to 10 MJ. The majority of all detected planets have masses below that of Saturn, although the sensitivity of the survey is much lower for such planets than for more massive, Jupiter-like planets. Low-mass planets are thus found to be much more common than giant planets.

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) = eEEk/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).

Figure 2: Cool-planet mass function.
Figure 2

a, The cool-planet mass function, f, for the orbital range 0.5−10 au as derived by microlensing. Red solid line, best fit for this study, based on combining the results from PLANET 2002–07 and previous microlensing estimates18,25 for slope (blue line; error, light-blue shaded area, s.d.) and normalization (blue point; error bars, s.d.). We find dN/(dloga dlogM) = 10−0.62 ± 0.22 (M/MSat) 0.73 ± 0.17, where N is the average number of planets per star, a the semi-major axis and M the planet mass. The pivot point of the power-law mass function is at the mass of Saturn (MSat = 95 M). The grey shaded area is the 68% confidence interval around the median (dash-dotted black line). For comparison, the constraint from Doppler measurements27 (green line and point; error, green shaded area, s.d.) is also displayed. Differences can arise because the Doppler technique focuses mostly on solar-like stars, whereas microlensing a priori probes all types of host stars. Moreover, microlensing planets are located further away from their stars and are cooler than Doppler planets. These two populations of planets may then follow a rather different mass function. b, PLANET 2002–07 sensitivity, S: the expected number of detections if all stars had exactly one planet, regardless of its orbit. c, PLANET 2002–07 detections, k. Thin black curves, distribution probabilities of the mass for the three detections contained in the PLANET sample; red line, the sum of these distributions.

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.

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Acknowledgements

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.

Author information

Affiliations

  1. Probing Lensing Anomalies Network (PLANET) Collaboration, Institut d’Astrophysique de Paris, Université Pierre & Marie Curie, UMR7095 UPMC–CNRS 98 bis boulevard Arago, 75014 Paris, France

    • A. Cassan
    • , D. Kubas
    • , J.-P. Beaulieu
    • , M. Dominik
    • , K. Horne
    • , J. Greenhill
    • , J. Wambsganss
    • , J. Menzies
    • , A. Williams
    • , U. G. Jørgensen
    • , D. P. Bennett
    • , M. D. Albrow
    • , V. Batista
    • , S. Brillant
    • , J. A. R. Caldwell
    • , A. Cole
    • , Ch. Coutures
    • , K. H. Cook
    • , S. Dieters
    • , D. Dominis Prester
    • , J. Donatowicz
    • , P. Fouqué
    • , K. Hill
    • , N. Kains
    • , S. Kane
    • , J.-B. Marquette
    • , R. Martin
    • , K. R. Pollard
    • , K. C. Sahu
    • , C. Vinter
    • , D. Warren
    • , B. Watson
    •  & M. Zub
  2. Institut d’Astrophysique de Paris, Université Pierre & Marie Curie, UMR7095 UPMC–CNRS 98 bis boulevard Arago, 75014 Paris, France

    • A. Cassan
    • , D. Kubas
    • , J.-P. Beaulieu
    • , V. Batista
    • , Ch. Coutures
    •  & J.-B. Marquette
  3. Astronomischen Rechen-Instituts (ARI), Zentrum für Astronomie, Heidelberg University, Mönchhofstrasse 12–14, 69120 Heidelberg, Germany

    • A. Cassan
    • , J. Wambsganss
    •  & M. Zub
  4. European Southern Observatory, Alonso de Cordoba 3107, Vitacura, Casilla 19001, Santiago, Chile

    • D. Kubas
    •  & S. Brillant
  5. Scottish Universities Physics Alliance (SUPA), University of St Andrews, School of Physics & Astronomy, North Haugh, St Andrews, KY16 9SS, UK

    • M. Dominik
    •  & K. Horne
  6. University of Tasmania, School of Maths and Physics, Private bag 37, GPO Hobart, Tasmania 7001, Australia

    • J. Greenhill
    • , A. Cole
    • , S. Dieters
    • , K. Hill
    • , D. Warren
    •  & B. Watson
  7. South African Astronomical Observatory, PO Box 9 Observatory 7935, South Africa

    • J. Menzies
  8. Perth Observatory, Walnut Road, Bickley, Perth 6076, Australia

    • A. Williams
    •  & R. Martin
  9. Niels Bohr Institute and Centre for Star and Planet Formation, Juliane Mariesvej 30, 2100 Copenhagen, Denmark

    • U. G. Jørgensen
    •  & C. Vinter
  10. Optical Gravitational Lensing Experiment (OGLE) Collaboration, Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland

    • A. Udalski
    • , M. K. Szymański
    • , M. Kubiak
    • , R. Poleski
    • , I. Soszynski
    • , K. Ulaczyk
    • , G. Pietrzyński
    •  & Ł. Wyrzykowski
  11. Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland

    • A. Udalski
    • , M. K. Szymański
    • , M. Kubiak
    • , R. Poleski
    • , I. Soszynski
    • , K. Ulaczyk
    • , G. Pietrzyński
    •  & Ł. Wyrzykowski
  12. University of Notre Dame, Physics Department, 225 Nieuwland Science Hall, Notre Dame, Indiana 46530, USA

    • D. P. Bennett
  13. University of Canterbury, Department of Physics & Astronomy, Private Bag 4800, Christchurch 8140, New Zealand

    • M. D. Albrow
    •  & K. R. Pollard
  14. Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, Maryland 21218, USA

    • J. A. R. Caldwell
    •  & K. C. Sahu
  15. Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory, PO Box 808, California 94550, USA

    • K. H. Cook
  16. Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia

    • D. Dominis Prester
  17. Technical University of Vienna, Department of Computing, Wiedner Hauptstrasse 10, 1040 Vienna, Austria

    • J. Donatowicz
  18. Laboratoire Astrophysique de Toulouse (LATT), Université de Toulouse, CNRS, 31400 Toulouse, France

    • P. Fouqué
  19. European Southern Observatory Headquarters, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany

    • N. Kains
  20. NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue, Pasadena, California 91125, USA

    • S. Kane
  21. Microlensing Observations in Astrophysics (MOA) Collaboration, Department of Earth and Space Science, Osaka University, Osaka 560-0043, Japan

    • T. Sumi
  22. Department of Earth and Space Science, Osaka University, Osaka 560-0043, Japan

    • T. Sumi
  23. Universidad de Concepción, Departamento de Fisica, Casilla 160-C, Concepción, Chile

    • G. Pietrzyński
  24. Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

    • Ł. Wyrzykowski
  25. Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK

    • J.-P. Beaulieu

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Contributions

A.Ca. led the analysis and conducted the modelling and statistical analyses. A.Ca. and D.K. selected light curves from 2002–07 PLANET/OGLE microlensing seasons, analysed the data and wrote the Letter and Supplement. D.K. computed the magnification maps used for the detection-efficiency calculations. J.-P.B. and Ch.C. wrote the software for online data reduction at the telescopes. J.-P.B. led the PLANET collaboration, with M.D., J.G., J.M. and A.W.; P.F. and M.D.A. contributed to online and offline data reduction. M.D. contributed to the conversion of the detection efficiencies to physical parameter space and developed the PLANET real-time display system with A.W., M.D.A. and Ch.C.; K.Ho. and A.Ca. developed and tested the Bayesian formulation for fitting the two-parameter power-law mass function. J.G. edited the manuscript, conducted the main data cleaning and managed telescope operations at Mount Canopus (1 m) in Hobart. J.W. wrote the original magnification maps software, discussed the main implications and edited the manuscript. J.M., A.W. and U.G.J. respectively managed telescope operations in South Africa (South African Astronomical Observatory 1 m), Australia (Perth 0.61 m) and La Silla (Danish 1.54 m). A.U. led the OGLE campaign and provided the final OGLE photometry. D.P.B, V.B., S.B., J.A.R.C., A.Co., K.H.C., S.D., D.D.P., J.D., P.F., K.Hi., N.K., S.K., J.-B.M., R.M., K.R.P., K.C.S., C.V., D.W., B.W. and M.Z. were involved in the PLANET observing strategy and/or PLANET data acquisition, reduction, real-time analysis and/or commented on the manuscript. T.S. commented on the manuscript. M.K.S., M.K., R.P., I.S., K.U., G.P. and Ł.W. contributed to OGLE data.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to A. Cassan or A. Cole.

Supplementary information

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  1. 1.

    Supplementary Information

    The file contains Supplementary Text and Data, Supplementary Figures 1-5 with legends, Supplementary Table 1 and additional references.

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https://doi.org/10.1038/nature10684

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    • , Bertrand Bonfond
    • , Rosaria Bonito
    • , Aldo S. Bonomo
    • , John Robert Brucato
    • , Allan Sacha Brun
    • , Ian Bryson
    • , Waldemar Bujwan
    • , Sarah Casewell
    • , Bejamin Charnay
    • , Cesare Cecchi Pestellini
    • , Guo Chen
    • , Angela Ciaravella
    • , Riccardo Claudi
    • , Rodolphe Clédassou
    • , Mario Damasso
    • , Mario Damiano
    • , Camilla Danielski
    • , Pieter Deroo
    • , Anna Maria Di Giorgio
    • , Carsten Dominik
    • , Vanessa Doublier
    • , Simon Doyle
    • , René Doyon
    • , Benjamin Drummond
    • , Bastien Duong
    • , Stephen Eales
    • , Billy Edwards
    • , Maria Farina
    • , Ettore Flaccomio
    • , Leigh Fletcher
    • , François Forget
    • , Steve Fossey
    • , Markus Fränz
    • , Yuka Fujii
    • , Álvaro García-Piquer
    • , Walter Gear
    • , Hervé Geoffray
    • , Jean Claude Gérard
    • , Lluis Gesa
    • , H. Gomez
    • , Rafał Graczyk
    • , Caitlin Griffith
    • , Denis Grodent
    • , Mario Giuseppe Guarcello
    • , Jacques Gustin
    • , Keiko Hamano
    • , Peter Hargrave
    • , Yann Hello
    • , Kevin Heng
    • , Enrique Herrero
    • , Allan Hornstrup
    • , Benoit Hubert
    • , Shigeru Ida
    • , Masahiro Ikoma
    • , Nicolas Iro
    • , Patrick Irwin
    • , Christopher Jarchow
    • , Jean Jaubert
    • , Hugh Jones
    • , Queyrel Julien
    • , Shingo Kameda
    • , Franz Kerschbaum
    • , Pierre Kervella
    • , Tommi Koskinen
    • , Matthijs Krijger
    • , Norbert Krupp
    • , Marina Lafarga
    • , Federico Landini
    • , Emanuel Lellouch
    • , Giuseppe Leto
    • , A. Luntzer
    • , Theresa Rank-Lüftinger
    • , Antonio Maggio
    • , Jesus Maldonado
    • , Jean-Pierre Maillard
    • , Urs Mall
    • , Jean-Baptiste Marquette
    • , Stephane Mathis
    • , Pierre Maxted
    • , Taro Matsuo
    • , Alexander Medvedev
    • , Yamila Miguel
    • , Vincent Minier
    • , Giuseppe Morello
    • , Alessandro Mura
    • , Norio Narita
    • , Valerio Nascimbeni
    • , N. Nguyen Tong
    • , Vladimiro Noce
    • , Fabrizio Oliva
    • , Enric Palle
    • , Paul Palmer
    • , Maurizio Pancrazzi
    • , Andreas Papageorgiou
    • , Vivien Parmentier
    • , Manuel Perger
    • , Antonino Petralia
    • , Stefano Pezzuto
    • , Ray Pierrehumbert
    • , Ignazio Pillitteri
    • , Giampaolo Piotto
    • , Giampaolo Pisano
    • , Loredana Prisinzano
    • , Aikaterini Radioti
    • , Jean-Michel Réess
    • , Ladislav Rezac
    • , Marco Rocchetto
    • , Albert Rosich
    • , Nicoletta Sanna
    • , Alexandre Santerne
    • , Giorgio Savini
    • , Gaetano Scandariato
    • , Bruno Sicardy
    • , Carles Sierra
    • , Giuseppe Sindoni
    • , Konrad Skup
    • , Ignas Snellen
    • , Mateusz Sobiecki
    • , Lauriane Soret
    • , Alessandro Sozzetti
    • , A. Stiepen
    • , Antoine Strugarek
    • , Jake Taylor
    • , William Taylor
    • , Luca Terenzi
    • , Marcell Tessenyi
    • , Angelos Tsiaras
    • , C. Tucker
    • , Diana Valencia
    • , Gautam Vasisht
    • , Allona Vazan
    • , Francesc Vilardell
    • , Sabrine Vinatier
    • , Serena Viti
    • , Rens Waters
    • , Piotr Wawer
    • , Anna Wawrzaszek
    • , Anthony Whitworth
    • , Yuk L. Yung
    • , Sergey N. Yurchenko
    • , María Rosa Zapatero Osorio
    • , Robert Zellem
    • , Tiziano Zingales
    •  & Frans Zwart

    Experimental Astronomy (2018)

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