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Planet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 1990s, speculations about the occurrence of planets around other stars were informed only by planet formation theory, observations of circum- stellar disks, and the knowledge that at least one seemingly ordinary star — the Sun — had managed to make a variety of different planets. Since then, Doppler and transit surveys have exposed the population of planets around other Sun-like stars, especially those with orbital periods shorter than a few years. Over the last decade these surveys rose to a new level of perfection with Doppler spectrographs capable of 1 m s -1 precision, and space telescopes capable of detecting the transits of Earth- sized planets. This article is a brief introductory review of the knowledge of planet occurrence that has been gained from these surveys. Introduction If, in some cataclysm, all our knowledge of exoplanets were to be destroyed, and only one sentence passed on to the next generation of astronomers, what statement would contain the most helpful information? This challenge, a modified version of the one posed by Richard Feynman in his Lectures on Physics, is difficult to meet with words. Here is one possibility: Most Sun-like stars have planets, which display a wider range of properties — size, mass, orbital parameters — than the planets of the Solar System. The job would be easier if we could convey a mathematical function. The oc- currence rate density (or simply occurrence) is the expected number of planets per star that have certain properties. For example, transit observations reveal a planet’s radius R and orbital period P. We can summarize the findings of a transit survey with a function Princeton University, Department of Astrophysical Sciences, 4 Ivy Lane, Princeton, NJ 08544, USA, e-mail: [email protected] 1

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Page 1: Planet Occurrence Rates from Doppler and Transit · PDF filePlanet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 1990s, speculations about

Planet Occurrence Rates from Doppler andTransit Surveys

Joshua N. Winn

Abstract Prior to the 1990s, speculations about the occurrence of planets aroundother stars were informed only by planet formation theory, observations of circum-stellar disks, and the knowledge that at least one seemingly ordinary star — theSun — had managed to make a variety of different planets. Since then, Doppler andtransit surveys have exposed the population of planets around other Sun-like stars,especially those with orbital periods shorter than a few years. Over the last decadethese surveys rose to a new level of perfection with Doppler spectrographs capableof 1 m s−1 precision, and space telescopes capable of detecting the transits of Earth-sized planets. This article is a brief introductory review of the knowledge of planetoccurrence that has been gained from these surveys.

Introduction

If, in some cataclysm, all our knowledge of exoplanets were to be destroyed, andonly one sentence passed on to the next generation of astronomers, what statementwould contain the most helpful information? This challenge, a modified version ofthe one posed by Richard Feynman in his Lectures on Physics, is difficult to meetwith words. Here is one possibility: Most Sun-like stars have planets, which displaya wider range of properties — size, mass, orbital parameters — than the planets ofthe Solar System.

The job would be easier if we could convey a mathematical function. The oc-currence rate density (or simply occurrence) is the expected number of planets perstar that have certain properties. For example, transit observations reveal a planet’sradius R and orbital period P. We can summarize the findings of a transit surveywith a function

Princeton University, Department of Astrophysical Sciences, 4 Ivy Lane, Princeton, NJ 08544,USA, e-mail: [email protected]

1

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2 Joshua N. Winn

ΓR,P ≡ d2Nd logR d logP

, (1)

giving the mean number of planets per logarithmic interval of radius and period.A simple and accurate occurrence function would help our descendants design newinstruments to detect planets, and inspire theories for planet formation.

Occurrence depends on other planetary parameters, such as orbital eccentricity;and on the characteristics of the star, such as mass, metallicity, and age. Occurrenceis also likely to depend on the properties of any other planets that exist in a particularsystem. No analytic function could possibly represent all these parameters and theircorrelations. Ideally we would transmit a computer program that produces randomrealizations of planetary systems that are statistically consistent with everything wehave learned from planet surveys.

What follows is an brief introductory review of the progress toward this goalthat has come from Doppler and transit surveys. The cited works were chosen to beuseful entry points to the literature, not to provide a comprehensive bibliography.The basics of the Doppler and transit methods themselves are left for other reviews,such as those by Lovis and Fischer (2010), Winn (2010), Wright (this volume) andBakos (this volume). The next section describes methods for occurrence calcula-tions. Surveys have shown major differences in occurrence between giant planetsand small planets, with a dividing line of about 6 R⊕ or 30 M⊕. Thus the results forgiants and small planets are presented separately, in two sections. After that comesa review of what is known about other types of stars, followed by a discussion offuture prospects.

Methods

Life would be simple if planets came in only one type, and we could detect themunerringly. We would search N stars, detect Ndet planets, and conclude that the in-tegrated occurrence is Γ ≈ Ndet/N. But detection is not assured, because small sig-nals can be lost in the noise. If the detection probability were pdet in all cases, thenwe would have effectively searched only pdetN stars, and the estimated occurrencewould be Ndet/(pdetN).

In reality, pdet depends strongly on the characteristics of the star and planet, asillustrated in Figure 1. Detection is easier for brighter stars, larger planets (relative tothe star), and shorter orbital periods. For this reason we need to group the detectedplanets according to orbital period and other salient characteristics for detection:radius R, for transit surveys; and m ≡ M sin I for Doppler surveys, where M is themass and I is the orbital inclination. Then our estimate becomes

Γi ≈ Nd

N

∑j=1

p−1det,i j, (2)

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Planet Occurrence Rates 3

1 10 100 1000 10000orbital period [days]

1

10

100

1000

mas

s [M

⊕]

Mmin~P1/3

Mmin~P3/2

Surveyduration

Fig. 1 Idealized Doppler survey of 104 identical Sun-like stars. Each star has one planet on arandomly-oriented circular orbit, with a mass and period drawn from log-uniform distributionsbetween the plotted limits. Each star is observed 50 times over one year with 1 m s−1 precision.The small dots are all the planets; the large blue dots are those detected with 5σ confidence. ForP shorter than the survey duration, the threshold mass is proportional to P1/3, corresponding to aconstant Doppler semi-amplitude K ≈ 1 m s−1. For longer P, the threshold mass increases morerapidly, with an exponent depending on the desired confidence level (Cumming 2004).

where the indices i and j specify the type of planet and star, respectively. Transitsurveys have the additional problem that there is no signal at all unless cos I <R?/ac,where ac is the orbital distance at the time of inferior conjunction. Thus we mustalso divide by the probability for this condition to be met, which is equal to R?/afor randomly-oriented circular orbits.

This conceptually simple method has been the basis of many investigations. Theresults of Doppler surveys are often presented as a matrix of Γ values for rectangularregions in the space of m ≡ M sin I and P; for transit surveys the regions are in thespace of R and P. Ideally, each region should be large enough to contain manydetected planets, and yet small enough that the detection probability does not varytoo much from one side to the other.

In practice these conditions are rarely achieved, and more complex methods arepreferable. One approach is to posit a functional form for the occurrence rate density,such as a power law

Γm,P =d2N

d lnm d lnP=Cmα Pβ , (3)

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4 Joshua N. Winn

1 10 100 1000 10000orbital period [days]

1

2

3

456789

radi

us [R

⊕]

Rmin ~ P1/6

2 x surveyduration

Fig. 2 Idealized transit survey of 104 identical Sun-like stars. Each star has one planet on arandomly-oriented circular orbit, with a radius and period drawn from log-uniform distributionsbetween the plotted limits. Each star is observed continuously for one year with a photon-limitedphotometric precision corresponding to 100 parts per million over 6 hours. The small dots are allthe planets; the large blue dots are those detected with 10σ confidence based on at least two transitdetections. Compared to the Doppler survey, the transit survey finds fewer planets and is morestrongly biased toward short periods. This is because the geometric transit probability is low and isproportional to P−2/3. The threshold radius varies as P1/6 out to twice the survey duration (Pepperet al. 2003).

and use this function to construct a likelihood function for the outcome of a survey.The likelihood function must take into account the detection probability, the proper-ties of the detected systems, and the properties of the stars for which no planets weredetected. Then the values of the adjustable parameters C, α , and β are determinedby maximizing the likelihood. Details of this method are provided by Tabachnikand Tremaine (2002) or Cumming et al. (2008) for Doppler surveys, and Youdin(2011) for transit surveys. Foreman-Mackey et al. (2014) generalized this techniqueto cope with uncertainties in the planet properties such as R and P. They also castthe problem in the form of Bayesian hierarchical inference.

Most studies report the probability for a star to have a planet with certain prop-erties, regardless of any other planets in the system. Accounting for multiple-planetsystems is more difficult. For Doppler surveys, the main problem is that the staris pulled by all the planets simultaneously. As a result, the detectability of a givensignal to depend on the properties of any other detectable planets — especially theirperiods — and on the timespan and spacings between the data points. This makes itdifficult to calculate the detection probability.

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Planet Occurrence Rates 5

For transit surveys, the different planetary signals do not overlap very much. In-stead, the problem is the degeneracy between multiplicity and inclination dispersion(Tremaine and Dong 2012). A star with only one detected planet could lack addi-tional planets, or it could have a system of many planets only one of which has aninclination close enough to 90◦ to exhibit transits. In principle this degeneracy canbe broken by combining the results of Doppler and transit surveys and solving forboth the occurrence rate density as well as the planetary mass-radius relationship.

Doppler surveys have uncovered a total of about 500 planets. The most infor-mative surveys for planet occurrence were based on observations with the HighResolution Echelle Spectrometer (HIRES) on the Keck I 10-meter telescope (Cum-ming et al. 2008; Howard et al. 2010) and the High Accuracy Radial-velocity PlanetSearcher (HARPS) on the La Silla 3.6-meter telescope (Mayor et al. 2011). Bothinstruments were used to monitor ∼103 stars for about a decade, with a precision ofa few meters per second. Additional information about wide-orbiting giant planetscomes from a few lower-precision and longer-duration surveys.

For transits, the ground-based surveys have discovered about 200 planets, but arenot well suited to occurrence calculations because both the sample of searched starsand the detection probabilities are poorly characterized. Instead our most importantsource is the NASA Kepler mission, which used a 1-meter space telescope to mea-sure the brightness of 150,000 stars every 30 minutes for 4 years (Borucki 2016).The typical photometric precision over a 6-hour time interval was of order 100 partsper million (ppm). This was sufficient to detect several thousand planets.

Giant planets

Overall occurrence For giant planets the key references are Cumming et al. (2008)and Mayor et al. (2011) for Doppler surveys, and Santerne et al. (2016) for Kepler.These studies agree that giant planets with periods shorter than a few years are foundaround ≈10% of Sun-like stars (see Fig. 3). In particular Cumming et al. (2008)studied planets with a minimum mass m in the range from 0.3-10 MJup and P from2-2000 days. They found the best-fitting power-law of the form given by Eqn. (3) tohave α = −0.31±0.20 and β = 0.26±0.10, and a normalization such that 10.5%of Sun-like stars have such a planet. They also found the data to be equally welldescribed by a distribution uniform in logP from 2-300 days (i.e., β = 0), followedby a sharp increase by a factor of 4-5 for longer periods. The Kepler data are alsoconsistent with this description (Santerne et al. 2016).

Hot Jupiters Hot Jupiters are the easiest type of planet to detect. However, they arealso intrinsically rare, with an occurrence of 0.5-1% for periods 1-10 days. For giantplanets with even shorter periods, dN/d lnP is lower by at least an order of magni-tude (Howard et al. 2012; Sanchis-Ojeda et al. 2014). There is a growing literatureaddressing the apparent 2-3σ discrepancy between the Hot Jupiter occurrence of0.8-1.2% measured in Doppler surveys (Wright et al. 2012; Mayor et al. 2011) and

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6 Joshua N. Winn

the lower figure of 0.6% measured using Kepler data (Howard et al. 2012; Petiguraet al. 2017b). Two aspects of the Kepler sample of stars that might be contributingto the difference are its slightly lower mean metallicity (Guo et al. 2017) and higherbinary fraction (Wang et al. 2015) than the Doppler sample.

Jupiter analogs A perennial question is whether the Solar System is typical orunusual in some sense. Several groups have used Doppler surveys to try and answerthis question by measuring the occurrence of Jupiter-like planets. The idea is thatJupiter would be the most easily detected planet around the Sun in an extraterrestrialDoppler survey. The answer depends on the definition of “Jupiter analog”, a termwithout a precise meaning. (The same problem arises when trying to measure theoccurrence of “Earth-like” planets.) Wittenmyer et al. (2016) presented the latesteffort, finding the occurrence to be 6.2+2.8

−1.6% for planets of mass 0.3-13 MJup withorbital distances from 3-7 AU and eccentricities smaller than 0.3.

Long-period giants Regarding the more general topic of wide-orbiting giant plan-ets, Foreman-Mackey et al. (2016) measured the occurrence of “cold Jupiters”through an automated search of the Kepler data for stars showing only one tran-sit over 4 years. They concluded

ΓR,P =d2N

d lnR d lnP= 0.024±0.007 (4)

over the range of periods from 2-25 years and planetary radius R = 0.1-1 RJup.Early work by Wright et al. (2009) suggested that when one giant planet is de-

tected in a Doppler survey, the odds of detecting a wider-orbiting companion withina few years are at least 30%. More recently Bryan et al. (2016) used high-resolutionimaging and long-term Doppler monitoring to search for wide-orbiting companionsto 123 giant planets with orbital distances ranging from 0.01 to 5 AU. They foundthat for the outer companions, dN/d lnP is a declining function of period, unlike themore uniform distribution observed for the inner planets. They measured an occur-rence of (53± 5)% for outer companions of mass 1-20 MJup and orbital distancesof 5-20 AU. This is about twice has high as the occurrence that was measured byForeman-Mackey et al. (2016) without regard to multiplicity, suggesting a linkagebetween the formation of inner and outer giant planets.

Metallicity The earliest Doppler surveys revealed that the occurrence of giant plan-ets with periods shorter than a few years is a steeply rising function of the host star’smetallicity (Santos et al. 2003; Fischer and Valenti 2005). This is widely interpretedas support for the theory of giant-planet formation via core accretion. The logic isthat the rapid assembly of a massive solid core — an essential step in the theory —is easier to arrange in a metal-rich protoplanetary disk. Most recently, Petigura et al.(2017b) used Kepler data to determine the best-fitting parameters of

ΓP,z =d2N

d lnP d lnz∝ CPα zβ , (5)

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Planet Occurrence Rates 7

where z is the iron-to-hydrogen abundance relative to the solar value. For planetslarger than Neptune they found β ≈ 4, a remarkably strong dependence. For smallerplanets the metallicity dependence was found to be weaker, in agreement with pre-vious studies (see, e.g., Mayor et al. 2011; Buchhave et al. 2012). Nevertheless, forperiods shorter than about 10 days, even planets smaller than Neptune are associatedwith slightly elevated metallicity (Mulders et al. 2016). This trend is not as easy tounderstand in terms of core accretion theory.

Other properties The giant-planet population is distinguished by other featuresthat are beyond the scope of this review, but are mentioned here for completeness.They have a broad range of orbital eccentricities (see, e.g., Udry and Santos 2007).Their occurrence seems to fall precipitously for masses above ≈10 MJup, a phe-nomenon that is often called the “brown dwarf desert” (Grether and Lineweaver2006; Sahlmann et al. 2011; Triaud et al. 2017). There is a significant population oftwo-planet systems in mean-motion resonances (Wright et al. 2011). The rotation ofthe star can be grossly misaligned with the orbit of the planet, especially if the staris more massive than about 1.2 M� (Triaud, this volume). These and other topicswere reviewed recently by Winn and Fabrycky (2015) and Santerne (this volume).

1 10 100 1000orbital period [days]

0.0001

0.0010

0.0100

occu

rrenc

e, ∆

N / ∆

log

P

Santerne et al. (2016)Mayor et al. (2011)Cumming et al. (2008)

Fig. 3 Occurrence of giant planets ( >∼ 0.1 MJup) as a function of orbital period, from two inde-pendent Doppler surveys (Cumming et al. 2008; Mayor et al. 2011) and the Kepler transit survey(Santerne et al. 2016).

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8 Joshua N. Winn

Smaller planets

Overall occurrence About half of Sun-like stars have at least one planet with anorbital period shorter than a year, and asize in between those of Earth and Neptune.Planet formation theories generally did not predict this profusion of planets. Indeedsome of the most detailed theories predicted that such close-orbiting “super-Earth”or “sub-Neptune” planets would be especially rare (Ida and Lin 2008). Their sur-prisingly high abundance has led to new theories in which small planets can formin very short-period orbits, rather than forming farther away from the star and thenmigrating inward (see, e.g., Hansen and Murray 2012; Chiang and Laughlin 2013).

Doppler surveys provided our first glimpse at this population of planets, andthen Kepler revealed them in vivid detail. For planets with periods shorter than 50days and minimum masses 3-30 M⊕, two independent Doppler surveys found theoccurrence to be (15±5)% (Howard et al. 2010) and (27±5)% (Mayor et al. 2011).For this same period range, and planets with a radius between 2-4 R⊕, the Keplersurvey found the occurrence to be (13.0±0.8)% (Howard et al. 2012). The resultsof the two surveys are compatible, given reasonable guesses for the relation betweenplanetary mass and radius (Howard et al. 2012; Figueira et al. 2012).

Size, mass, and period The surveys also agree that within this range of periodsand planet sizes, the occurrence rate is higher for the smallest planets (Howard et al.2010, 2012):

dNd lnm

∝ m−0.5,dN

d lnR∝ R−2. (6)

For even smaller or longer-period planets, Kepler provides almost all the availableinformation (Fressin et al. 2013). Figure 4 shows some of the latest results. Theperiod distribution dN/d lnP rises as ∼P2 between 1-10 days, before leveling offto a nearly constant value between 10-300 days.

Multiple-planet systems Small planets occur frequently in closely-spaced systems(Fabrycky et al. 2014). The most fecund Kepler systems have 7 transiting planets,all with periods shorter than a year. The period ratios tend to be in the neighborhoodof 1.5-3. In units of the mutual Hill radius,

aH ≡(

Min +Mout

3M?

)1/3(ain +aout

2

), (7)

which is more directly relevant to orbital stability, the typical spacing is 10-30 (Fangand Margot 2013). A few percent of these systems exhibit mean-motion resonances,suggesting that the orbits have been sculpted by planet-disk gravitational interac-tions. These resonant systems also offer the gift of transit-timing variations (Agol& Fabrycky, this volume). These observable manifestations of planet-planet gravi-tational interactions sometimes enable measurements of planetary masses as well asorbital eccentricities and inclinations. Such studies and some other lines of evidenceshow that the compact multiple-planet systems tend to have orbits that are circular

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Planet Occurrence Rates 9

low detection probability

1 10 100orbital period [days]

1

2

4

8

16

radi

us [R

⊕]

M dwarfs (Dressing et al. 2015)

1

2

4

8

16

radi

us [R

⊕]

hot Neptune desert

FGK dwarfs (Petigura et al. 2018)

low detection probability

Fig. 4 Planet occurrence around FGK dwarfs (top) and M dwarfs (bottom) based on Kepler data.The blue dots are a random sample of 103 planets drawn from the occurrence rate densities derivedby (Petigura et al. 2017b) and Dressing and Charbonneau (2015). Compared to FGK stars, the Mstars have a higher occurrence of small planets and a lower occurrence of giant planets, at least forperiods shorter than ∼100 days. For the M dwarfs, occurrence rates for planets larger than 4 R⊕were not reported because only four such planets were detected.

(Hadden and Lithwick 2014; Xie et al. 2016; Van Eylen and Albrecht 2015) andcoplanar (Fabrycky et al. 2014).

Radius gap The radius distribution of planets with periods shorter than 100 dayshas an interesting feature: a dip in occurrence between 1.5-2 R⊕ (Fulton et al. 2017;Van Eylen et al. 2017). Such a feature had been seen in theoretical calculations ofthe atmospheric erosion of low-mass planets due to the intense irradiation from thehost star (Owen and Wu 2013; Lopez and Fortney 2013). Thus the “radius gap”seems to be a precious example in exoplanetary science of a prediction fulfilled,with many implications for the structures and atmospheres of close-orbiting planets(Owen and Wu 2017).

Hot Neptunes As mentioned above, the occurrence dN/d lnP changes from a ris-ing function for P <∼ 10 days to a more constant value for P = 10-100 days. Thecritical period separating these regimes is longer for larger planets. The effect is to

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10 Joshua N. Winn

0.7 1.0 1.3 1.8 2.4 3.5 4.5 6.0 8.0 12.0 20.0Planet Size [Earth radii]

0.00

0.02

0.04

0.06

0.08

0.10

0.12Nu

mbe

r of P

lanet

s per

Sta

r(O

rbita

l per

iod <

100

day

s)typicaluncert.

Fig. 5 From Fulton et al. (2017). Occurrence as a function of radius based on Kepler data, forplanets with orbital periods shorter than 100 days. There is a “valley” in the occurrence rate densitybetween about 1.5-2 R⊕, possibly the result of the erosion of planetary atmospheres by high-energyradiation from the star.

create a diagonal boundary in the space of lnR and lnP, above which the occurrenceis very low (see Fig. 4). The same phenomenon is seen in Doppler data and hasbeen called the “hot Neptune desert” (Mazeh et al. 2016). The desert may be an-other consequence of atmospheric erosion, although other explanations have beenoffered. One clue is that those few hot Neptunes that do exist are strongly associatedwith metal-rich stars (Dong et al. 2017; Petigura et al. 2017b), making them similarto hot Jupiters and unlike small planets more generally (Buchhave et al. 2012). Thehot Neptunes are also similar to hot Jupiters in that they tend not to have planetarycompanions in closely-spaced coplanar orbits (Dong et al. 2017).

Earth-like planets The goal of exoplanetary science with the broadest appeal ismeasuring the occurrence of Earth-sized planets orbiting Sun-like stars within the“habitable zone”, the range of distances within which a rocky planet could plausiblyhave oceans of liquid water. The Kepler mission provided by far the best data thathas ever been available for this purpose. However, even Kepler was barely sensitiveto such planets. The number of detections is of order 10, depending critically on thedefinitions of “Earth-sized”, “Sun-like” and “habitable zone”. Thus the measure-ment of

Γ⊕ ≡ d2Nd lnP d lnR

∣∣∣∣P=1 yr, R=R⊕

(8)

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Planet Occurrence Rates 11

may require extrapolation from the measurements for shorter periods and largerplanets. The Kepler team has published a series of papers reporting steady advance-ments in efficiency of detection, elimination of false positives, and understanding ofinstrumental artifacts. Nevertheless the most recent effort to determine η⊕ found thedata to be compatible with any value from 0.04 to 11.5, given the range of possiblesystematic errors (Burke et al. 2015, see Figure 6). Since then the Kepler team andother groups have clarified the properties of the stars that were searched (Petiguraet al. 2017a; De Cat et al. 2015), and the most recent installments by Twicken et al.(2016) and Thompson et al. (2017) quantified the sensitivity of the algorithms forplanet detection and validation. These developments have brought us right to thethreshold of occurrence rate calculations for Earth-like planets.

Γ⊕ 0.01 0.03 0.10 0.32 1.00 3.16 10.00

Log10[Γ⊕

]

Prob

abili

ty D

ensi

ty

Youd

in

Petig

ura

Fore

man

-Mac

key

Don

g

-2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3Optimistic EfficiencyPessimistic EfficiencyOriginal KICDV RpLow ReliabilityHigh Reliability

Direct BaselineTrimmed Direct BaselineExtrapolated Baseline

Fig. 6 From Burke et al. (2015). Estimates for Γ⊕ based on Kepler data. The orange histogramis the posterior probability distribution considering only the uncertainties from counting statisticsand extrapolation. The other curves illustrate the effects of some systematic errors: uncertaintyin the detection efficiency, orbital eccentricities, stellar parameters, and reliability of weak planetcandidates. These systematic effects led to a range in Γ⊕ spanning an order of magnitude. Alsoshown are other estimates of Γ⊕ by Foreman-Mackey et al. (2014); Petigura et al. (2013); Dongand Zhu (2013) and Youdin (2011).

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12 Joshua N. Winn

Other types of stars

Almost all the preceding results were based on studies of main-sequence stars withmasses 0.5-1.2 M�, i.e., spectral types from K to late F. Stars with masses 0.1-0.5 M�, the M dwarfs, are not as thoroughly explored. However they are very attrac-tive for planet surveys because their small masses and sizes lead to larger Dopplerand transit signals, and because planets in the habitable zone have conveniently shortorbital periods.

M dwarfs have a relatively low occurrence of giant planets, at least for periodsshorter than a few years. Cumming et al. (2008) showed that if planet occurrence ismodeled by the functional form of Eqn. (3), then M dwarfs have 3-10 times fewerplanets than FGK dwarfs, for masses exceeding 0.4 MJup and periods shorter than5.5 years. Likewise, Bonfils et al. (2013) found the occurrence of planets with m =0.3−3 MJup and P < 100 days to be 0.02+0.03

−0.01, which is 2-3 times smaller than theequivalent number for FGK stars.

On the other hand, M dwarfs exceed FGK dwarfs by a factor of 2-3 in the oc-currence of smaller planets over the same range of periods (Howard et al. 2012;Mulders et al. 2015). This is based on Kepler data, which remains our best source ofinformation on this topic despite the fact that only a few percent of the Kepler targetstars were M dwarfs. Dressing and Charbonneau (2015) performed the most com-prehensive analysis to date, finding the mean number of planets of radius 1-4 R� andP < 50 days to be 2.5±0.2. Since the habitable zones occur at periods of weeks tomonths, the implication is that the nearest habitable-zone planets are surely aroundM dwarfs, probably within just a few parsecs.

Some other comparisons with FGK dwarfs have been made. One similarity isthat M dwarfs often have compact systems of multiple planets (Ballard and Johnson2016). Another is that the “radius gap” between 1.5-2 R⊕ and the “hot Neptunedesert” appear to exist for M dwarfs (Hirano et al. 2017), although these featuresoccur for planets at lower levels of irradiation than those around higher-mass stars.Whether or not planet occurrence is associated with high metallicity for M dwarfsis unclear (see, e.g., Gaidos and Mann 2014).

Beyond the scope of this review, but nevertheless fascinating, are the occurrencerates that have been measured in surveys of other types of stars: evolved stars (John-son et al. 2010; Reffert et al. 2015), stars in open clusters (Mann et al. 2017) andglobular clusters (Gilliland et al. 2000; Masuda and Winn 2017), binary stars (Arm-strong et al. 2014), brown dwarfs (He et al. 2017), white dwarfs (Fulton et al. 2014),and neutron stars (Wolszczan 2012; Kerr et al. 2015).

Future Prospects

The volume of data produced by recent surveys will offer opportunities for progressfor at least another few years. The ongoing struggle to measure the occurrence ofEarth-like planets has already been mentioned. Another undeveloped area is the

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Planet Occurrence Rates 13

determination of joint and conditional probabilities; for example, given a planetwith radius R1 and period P1, what is the chance of finding another planet aroundthe same star with radius R2 and period P2? Conditional occurrence is probablymore useful than overall occurrence for testing theories of planet formation. Onlya few specific cases have been studied, such as the mutual radius distribution ofneighboring planets (Ciardi et al. 2013; Weiss et al. 2017), and the tendency for hotJupiters to have wider-orbiting companions (Huang et al. 2016; Bryan et al. 2016;Schlaufman and Winn 2016).

New data are also forthcoming. The Transiting Exoplanet Survey Satellite (TESS)is scheduled for launch in 2018 (Ricker et al. 2015). This mission was not designedfor measuring planet occurrence. Rather, the main goal is to pluck low-hanging fruit:to discover short-period transiting planets around nearby and bright stars. Less wellappreciated is that TESS should be superior to Kepler in measuring the occurrence ofplanets larger than Neptune with periods shorter than 10 days. When TESS was con-ceived in 2006, it was expected that the limitations of data storage and transmissionwould limit the search to ∼ 105 pre-selected stars, similar to the Kepler mission. By2013, it became clear that entire TESS images could be stored and transmitted with30-minute time sampling. As a result, although ≈105 Sun-like stars will still be se-lected for special treatment (two-minute time sampling), it will be possible to searchmillions of stars for large, short-period planets. Indeed TESS will excel at findingvery rare, large-amplitude, short-period photometric phenomena of all kinds.

For smaller planets around Sun-like stars, it will be difficult to achieve an order-of-magnitude improvement over the existing data. There is more room for improve-ment in the study of low-mass stars, using new Doppler spectrographs operating atinfrared wavelengths, and ground-based transit surveys focusing exclusively on low-mass stars. Particularly encouraging was the discovery of TRAPPIST-1, a system ofseven Earth-sized planets orbiting an “ultra-cool dwarf” that barely qualifies as a star(Gillon et al. 2017). This system was found after searching ≈50 similar objects witha detection efficiency of around 60% (Burdanov et al., this volume; M. Gillon, pri-vate communication), and the transit probability for the innermost planet is 5%. Thissuggests the occurrence of such systems is approximately (50 ·0.6 ·0.05)−1 = 0.7.Thus, while TRAPPIST-1 seems extraordinary, it may represent a typical outcomeof planet formation around ultra-cool dwarfs.

In the decades to come, the domains of all the planet detection techniques —including astrometry, gravitational microlensing, and direct imaging — will beginoverlapping. Some efforts have already been made to construct occurrence modelsbased on data from several very different techniques (see, e.g. Clanton and Gaudi2016). We can look forward to a much more holistic view of the occurrence ofplanets around other stars, barring any civilization-destroying cataclysm.

Cross-References

• Populations of extrasolar giant planets from transit and radial velocity surveys

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14 Joshua N. Winn

• Multiplicity of brown dwarfs and brown dwarfs as companions to stars• Radial velocities as an exoplanet discovery method• Transit photometry as an exoplanet discovery method• Space Missions for Exoplanet Science: TESS• Space Missions for Exoplanet Science: Kepler / K2• Biases Affecting Population Studies• Planet Populations as a Function of Star Properties

Acknowledgements The author is grateful to Chris Burke, B.J. Fulton, Courtney Dressing, andErik Petigura for allowing their data and figures to be reproduced here.

References

Armstrong DJ, Osborn HP, Brown DJA et al. (2014) On the abundance of circumbinary planets.MNRAS444:1873–1883

Ballard S Johnson JA (2016) The Kepler Dichotomy among the M Dwarfs: Half of Systems Con-tain Five or More Coplanar Planets. ApJ816:66

Bonfils X, Delfosse X, Udry S et al. (2013) The HARPS search for southern extra-solar planets.XXXI. The M-dwarf sample. A&A549:A109

Borucki WJ (2016) KEPLER Mission: development and overview. Reports on Progress in Physics79(3):036901

Bryan ML, Knutson HA, Howard AW et al. (2016) Statistics of Long Period Gas Giant Planets inKnown Planetary Systems. ApJ821:89

Buchhave LA, Latham DW, Johansen A et al. (2012) An abundance of small exoplanets aroundstars with a wide range of metallicities. Nature486:375–377

Burke CJ, Christiansen JL, Mullally F et al. (2015) Terrestrial Planet Occurrence Rates for theKepler GK Dwarf Sample. ApJ809:8

Chiang E Laughlin G (2013) The minimum-mass extrasolar nebula: in situ formation of close-insuper-Earths. MNRAS431:3444–3455

Ciardi DR, Fabrycky DC, Ford EB et al. (2013) On the Relative Sizes of Planets within KeplerMultiple-candidate Systems. ApJ763:41

Clanton C Gaudi BS (2016) Synthesizing Exoplanet Demographics: A Single Population of Long-period Planetary Companions to M Dwarfs Consistent with Microlensing, Radial Velocity, andDirect Imaging Surveys. ApJ819:125

Cumming A (2004) Detectability of extrasolar planets in radial velocity surveys.MNRAS354:1165–1176

Cumming A, Butler RP, Marcy GW et al. (2008) The Keck Planet Search: Detectability and theMinimum Mass and Orbital Period Distribution of Extrasolar Planets. PASP120:531

De Cat P, Fu JN, Ren AB et al. (2015) Lamost Observations in the Kepler Field. I. Database ofLow-resolution Spectra. ApJS220:19

Dong S Zhu Z (2013) Fast Rise of “Neptune-size” Planets (4-8 R ⊕) from P = 10 to 250 Days—Statistics of Kepler Planet Candidates up to 0.75 AU. ApJ778:53

Dong S, Xie JW, Zhou JL, Zheng Z Luo A (2017) LAMOST Reveals Neptune-size Cousins of hotJupiters, preferentially in “(metal-)rich” and “one-child” Kepler families. ArXiv e-prints

Dressing CD Charbonneau D (2015) The Occurrence of Potentially Habitable Planets Orbiting MDwarfs Estimated from the Full Kepler Dataset and an Empirical Measurement of the DetectionSensitivity. ApJ807:45

Fabrycky DC, Lissauer JJ, Ragozzine D et al. (2014) Architecture of Kepler’s Multi-transitingSystems. II. New Investigations with Twice as Many Candidates. ApJ790:146

Page 15: Planet Occurrence Rates from Doppler and Transit · PDF filePlanet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 1990s, speculations about

Planet Occurrence Rates 15

Fang J Margot JL (2013) Are Planetary Systems Filled to Capacity? A Study Based on KeplerResults. ApJ767:115

Figueira P, Marmier M, Boue G et al. (2012) Comparing HARPS and Kepler surveys. The align-ment of multiple-planet systems. A&A541:A139

Fischer DA Valenti J (2005) The Planet-Metallicity Correlation. ApJ622:1102–1117Foreman-Mackey D, Hogg DW Morton TD (2014) Exoplanet Population Inference and the Abun-

dance of Earth Analogs from Noisy, Incomplete Catalogs. ApJ795:64Foreman-Mackey D, Morton TD, Hogg DW, Agol E Scholkopf B (2016) The Population of Long-

period Transiting Exoplanets. AJ152:206Fressin F, Torres G, Charbonneau D et al. (2013) The False Positive Rate of Kepler and the Occur-

rence of Planets. ApJ766:81Fulton BJ, Tonry JL, Flewelling H et al. (2014) A Search for Planetary Eclipses of White Dwarfs

in the Pan-STARRS1 Medium-deep Fields. ApJ796:114Fulton BJ, Petigura EA, Howard AW et al. (2017) The California-Kepler Survey. III. A Gap in the

Radius Distribution of Small Planets. AJ154:109Gaidos E Mann AW (2014) M Dwarf Metallicities and Giant Planet Occurrence: Ironing Out Un-

certainties and Systematics. ApJ791:54Gilliland RL, Brown TM, Guhathakurta P et al. (2000) A Lack of Planets in 47 Tucanae from a

Hubble Space Telescope Search. ApJ545:L47–L51Gillon M, Triaud AHMJ, Demory BO et al. (2017) Seven temperate terrestrial planets around the

nearby ultracool dwarf star TRAPPIST-1. Nature542:456–460Grether D Lineweaver CH (2006) How Dry is the Brown Dwarf Desert? Quantifying the Relative

Number of Planets, Brown Dwarfs, and Stellar Companions around Nearby Sun-like Stars.ApJ640:1051–1062

Guo X, Johnson JA, Mann AW et al. (2017) The Metallicity Distribution and Hot Jupiter Rateof the Kepler Field: Hectochelle High-resolution Spectroscopy for 776 Kepler Target Stars.ApJ838:25

Hadden S Lithwick Y (2014) Densities and Eccentricities of 139 Kepler Planets from Transit TimeVariations. ApJ787:80

Hansen BMS Murray N (2012) Migration Then Assembly: Formation of Neptune-mass Planetsinside 1 AU. ApJ751:158

He MY, Triaud AHMJ Gillon M (2017) First limits on the occurrence rate of short-period planetsorbiting brown dwarfs. MNRAS464:2687–2697

Hirano T, Dai F, Gandolfi D et al. (2017) Planetary Systems around Low-mass Stars Unveiled byK2. ArXiv e-prints

Howard AW, Marcy GW, Johnson JA et al. (2010) The Occurrence and Mass Distribution of Close-in Super-Earths, Neptunes, and Jupiters. Science 330:653

Howard AW, Marcy GW, Bryson ST et al. (2012) Planet Occurrence within 0.25 AU of Solar-typeStars from Kepler. ApJS201:15

Huang C, Wu Y Triaud AHMJ (2016) Warm Jupiters Are Less Lonely than Hot Jupiters: CloseNeighbors. ApJ825:98

Ida S Lin DNC (2008) Toward a Deterministic Model of Planetary Formation. V. AccumulationNear the Ice Line and Super-Earths. ApJ685:584-595

Johnson JA, Howard AW, Bowler BP et al. (2010) Retired A Stars and Their Companions. IV.Seven Jovian Exoplanets from Keck Observatory. PASP122:701

Kerr M, Johnston S, Hobbs G Shannon RM (2015) Limits on Planet Formation Around YoungPulsars and Implications for Supernova Fallback Disks. ApJ809:L11

Lopez ED Fortney JJ (2013) The Role of Core Mass in Controlling Evaporation: The Kepler RadiusDistribution and the Kepler-36 Density Dichotomy. ApJ776:2

Lovis C Fischer D (2010) Radial Velocity Techniques for Exoplanets, pp 27–53Mann AW, Gaidos E, Vanderburg A et al. (2017) Zodiacal Exoplanets in Time (ZEIT). IV. Seven

Transiting Planets in the Praesepe Cluster. AJ153:64Masuda K Winn JN (2017) Reassessment of the Null Result of the HST Search for Planets in 47

Tucanae. AJ153:187

Page 16: Planet Occurrence Rates from Doppler and Transit · PDF filePlanet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 1990s, speculations about

16 Joshua N. Winn

Mayor M, Marmier M, Lovis C et al. (2011) The HARPS search for southern extra-solar planetsXXXIV. Occurrence, mass distribution and orbital properties of super-Earths and Neptune-mass planets. ArXiv e-prints

Mazeh T, Holczer T Faigler S (2016) Dearth of short-period Neptunian exoplanets: A desert inperiod-mass and period-radius planes. A&A589:A75

Mulders GD, Pascucci I Apai D (2015) An Increase in the Mass of Planetary Systems aroundLower-mass Stars. ApJ814:130

Mulders GD, Pascucci I, Apai D, Frasca A Molenda-Zakowicz J (2016) A Super-solar Metallicityfor Stars with Hot Rocky Exoplanets. AJ152:187

Owen JE Wu Y (2013) Kepler Planets: A Tale of Evaporation. ApJ775:105Owen JE Wu Y (2017) The Evaporation Valley in the Kepler Planets. ApJ847:29Pepper J, Gould A Depoy DL (2003) Using All-Sky Surveys to Find Planetary Transits. Acta

Astron53:213–228Petigura EA, Howard AW Marcy GW (2013) Prevalence of Earth-size planets orbiting Sun-like

stars. Proceedings of the National Academy of Science 110:19,273–19,278Petigura EA, Howard AW, Marcy GW et al. (2017a) The California-Kepler Survey. I. High-

resolution Spectroscopy of 1305 Stars Hosting Kepler Transiting Planets. AJ154:107Petigura EA, Marcy GW, Winn JN et al. (2017b) The California-Kepler Survey. IV. Metal-Rich

Stars Host a Greater Diversity of Planets. ApJTBDReffert S, Bergmann C, Quirrenbach A, Trifonov T Kunstler A (2015) Precise radial velocities of

giant stars. VII. Occurrence rate of giant extrasolar planets as a function of mass and metallicity.A&A574:A116

Ricker GR, Winn JN, Vanderspek R et al. (2015) Transiting Exoplanet Survey Satellite (TESS).Journal of Astronomical Telescopes, Instruments, and Systems 1(1):014003

Sahlmann J, Segransan D, Queloz D et al. (2011) Search for brown-dwarf companions of stars.A&A525:A95

Sanchis-Ojeda R, Rappaport S, Winn JN et al. (2014) A Study of the Shortest-period Planets Foundwith Kepler. ApJ787:47

Santerne A, Moutou C, Tsantaki M et al. (2016) SOPHIE velocimetry of Kepler transit candidates.XVII. The physical properties of giant exoplanets within 400 days of period. A&A587:A64

Santos NC, Israelian G, Mayor M, Rebolo R Udry S (2003) Statistical properties of exoplanets. II.Metallicity, orbital parameters, and space velocities. A&A398:363–376

Schlaufman KC Winn JN (2016) The Occurrence of Additional Giant Planets Inside the Water-IceLine in Systems with Hot Jupiters: Evidence Against High-Eccentricity Migration. ApJ825:62

Tabachnik S Tremaine S (2002) Maximum-likelihood method for estimating the mass and perioddistributions of extrasolar planets. MNRAS335:151–158

Thompson SE, Coughlin JL, Hoffman K et al. (2017) Planetary Candidates Observed by Kepler.VIII. A Fully Automated Catalog With Measured Completeness and Reliability Based on DataRelease 25. ArXiv e-prints

Tremaine S Dong S (2012) The Statistics of Multi-planet Systems. AJ143:94Triaud AHMJ, Martin DV, Segransan D et al. (2017) The EBLM Project IV. Spectroscopic orbits

of over 100 eclipsing M dwarfs masquerading as transiting hot-Jupiters. ArXiv e-printsTwicken JD, Jenkins JM, Seader SE et al. (2016) Detection of Potential Transit Signals in 17

Quarters of Kepler Data: Results of the Final Kepler Mission Transiting Planet Search (DR25).AJ152:158

Udry S Santos NC (2007) Statistical Properties of Exoplanets. ARA&A45:397–439Van Eylen V Albrecht S (2015) Eccentricity from Transit Photometry: Small Planets in Kepler

Multi-planet Systems Have Low Eccentricities. ApJ808:126Van Eylen V, Agentoft C, Lundkvist MS et al. (2017) An asteroseismic view of the radius valley:

stripped cores, not born rocky. ArXiv e-printsWang J, Fischer DA, Horch EP Huang X (2015) On the Occurrence Rate of Hot Jupiters in Differ-

ent Stellar Environments. ApJ799:229

Page 17: Planet Occurrence Rates from Doppler and Transit · PDF filePlanet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 1990s, speculations about

Planet Occurrence Rates 17

Weiss LM, Marcy GW, Petigura EA et al. (2017) The California-Kepler Survey V. Peas in a Pod:Planets in a Kepler Multi-planet System are Similar in Size and Regularly Spaced. ArXiv e-prints

Winn JN (2010) Exoplanet Transits and Occultations, University of Arizona Press, pp 55–77Winn JN Fabrycky DC (2015) The Occurrence and Architecture of Exoplanetary Systems.

ARA&A53:409–447Wittenmyer RA, Butler RP, Tinney CG et al. (2016) The Anglo-Australian Planet Search XXIV:

The Frequency of Jupiter Analogs. ApJ819:28Wolszczan A (2012) Discovery of pulsar planets. New A Rev56:2–8Wright JT, Upadhyay S, Marcy GW et al. (2009) Ten New and Updated Multiplanet Systems and

a Survey of Exoplanetary Systems. ApJ693:1084–1099Wright JT, Veras D, Ford EB et al. (2011) The California Planet Survey. III. A Possible 2:1 Reso-

nance in the Exoplanetary Triple System HD 37124. ApJ730:93Wright JT, Marcy GW, Howard AW et al. (2012) The Frequency of Hot Jupiters Orbiting nearby

Solar-type Stars. ApJ753:160Xie JW, Dong S, Zhu Z et al. (2016) Exoplanet orbital eccentricities derived from LAMOST-Kepler

analysis. Proceedings of the National Academy of Science 113:11,431–11,435Youdin AN (2011) The Exoplanet Census: A General Method Applied to Kepler. ApJ742:38