Planet Occurrence: Doppler and Transit Surveys

Abstract

Prior to the 1990s, speculations about the occurrence of planets around other stars were based only on planet formation theory, observations of circumstellar disks, and the knowledge that at least one seemingly ordinary star had managed to make a variety of different planets. Since then, Doppler and transit surveys have revealed 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 have risen to new heights 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? (Feynman 1963). Here is one possibility: Most Sun-like stars have planets, which display a wider range of properties – size, mass, and orbital parameters – than the planets of the solar system.

To be quantitative, we could give the fraction of stars with planets, restricted to the types of planets we have managed to detect. But since this fraction is so close to unity, it might be more helpful to specify the average number n of planets per star: the total number of planets in the galaxy divided by the number of stars. Better still would be a mathematical function to show how occurrence depends on the planet’s properties. For example, we could supply a functional form for

$$\displaystyle \begin{aligned} \varGamma_{R,P} = \frac{\partial^2n}{\partial\log R~\partial\log P}, \end{aligned} $$

(1)

the average number of planets per star per log-intervals in radius and period. The number n is the occurrence rate, and the function Γ_R,P is an occurrence rate density.

Occurrence rates depend on other planetary parameters, such as orbital eccentricity, and on the characteristics of the star, such as mass and metallicity. Occurrence is also conditional on the properties of any other planets known to exist around the same star. No simple function could account for all these parameters and their correlations. Ideally, we would transmit a computer program that produces random realizations of planetary systems that are statistically consistent with everything we have learned from planet surveys. This would help our descendants design new instruments to detect planets and inspire their theories for planet formation.

What follows is an introductory review of the progress toward this goal that has come from Doppler and transit surveys. The basics of the Doppler and transit methods themselves are left for other reviews, such as those by Lovis and Fischer (2010), Winn (2010), and Wright (this volume). Here we will simply remind ourselves of the key properties of the Doppler and transit signals :

$$\displaystyle \begin{aligned} \begin{array}{rcl} K &\displaystyle = &\displaystyle \frac{0.64~{\text{m s}}^{-1}}{\sqrt{1-e^2}} \left( \frac{P}{1~{\mathrm{day}}} \right)^{1/3} \frac{(M/M_\oplus)\sin I}{(M_\star/M_\odot)^{2/3}}, \end{array} \end{aligned} $$

(2)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta &\displaystyle = &\displaystyle 8.4\times 10^{-5} \left( \frac{R/R_\star}{R_\oplus/R_\odot} \right)^2,~~p_{\mathrm{tra}} = \frac{0.0046}{1-e^2}~\frac{R_\star/a}{R_\odot/1~{\mathrm{AU}}} \end{array} \end{aligned} $$

(3)

where K is the radial velocity semiamplitude; δ is the fractional loss of light during transits; p_tra is the probability for a randomly oriented orbit to exhibit transits; a, P, e, and I are the orbital semimajor axis, period, eccentricity, and inclination; M and R are the mass and radius of the planet; and M_⋆ and R_⋆ are those of the star.

The next section describes methods for occurrence calculations. Because the surveys have shown major differences in occurrence between giant planets and small planets, with a dividing line just above 4 R_⊕ or 20 M_⊕, the results for giants and small planets are presented separately. After that comes a review of what is known about other types of stars, followed by a discussion of future prospects.

Methods

Life would be simple if planets came in only one type and we could detect them unerringly. We would search N stars, detect N_det planets, and conclude n ≈ N_det∕N. But detection is not assured: small signals can be lost in the noise. If the detection probability were p_det in all cases, then effectively we would only have searched p_detN stars, and the estimated occurrence rate would be N_det∕(p_detN).

In reality, p_det depends strongly on the characteristics of the star and planet (see Figs. 1 and 2). Detection is easier for brighter stars, shorter orbital periods, and larger planets relative to the star. For this reason, we need to group the planets according to orbital period and other salient characteristics for detection: the radius R, for transit surveys, and $m \equiv M\sin I$ for Doppler surveys. Then our estimate becomes

$$\displaystyle \begin{aligned} n_i \approx \frac{N_{{\mathrm{det}},i}} {\sum_{j=1}^N p_{{\mathrm{det}},ij}}, \end{aligned} $$

(4)

where the index i refers to a group of planets sharing the same characteristics and the index j specifies the star that was searched. Transit surveys have the additional problem that most planets produce no signal at all, because their orbits are not viewed at high enough inclination. Thus we must also divide by the probability p_tra for transits to occur.

Fig. 1

Idealized Doppler survey of 10⁴ identical Sun-like stars. Each star has one planet on a randomly oriented circular orbit, with a mass and period drawn from log-uniform distributions between the plotted limits. Each star is observed 50 times over 1 year with 1 m s⁻¹ precision. The small gray dots are all the planets; the blue circles enclose those detected with 10σ confidence. For periods shorter than the survey duration, the threshold mass is proportional to P^1∕3, corresponding to a constant Doppler amplitude. For longer periods, the threshold mass increases more rapidly, with an exponent depending on the desired false-alarm probability (Cumming 2004)

Fig. 2

Idealized transit survey of 10⁴ identical Sun-like stars. Each star has one planet on a randomly oriented circular orbit, with a radius and period drawn from log-uniform distributions between the plotted limits. Each star is observed continuously for 1 year with a photon-limited photometric precision corresponding to 3 × 10⁻⁵ over 6 h. The small gray dots are all the planets; the blue circles enclose those detected with 10σ confidence based on at least two transit detections. Compared to the Doppler survey, the transit survey finds fewer planets and is more strongly biased toward short periods, because of the geometric transit probability is low and proportional to P^−2∕3. For orbital periods shorter than survey duration, the threshold radius varies as P^1∕6 (Pepper et al. 2003). For longer periods, it is impossible to observe more than one transit

This conceptually simple method has been the basis of many investigations. The results of Doppler surveys are presented as a matrix of occurrence rates for rectangular regions in the space of $\log m$ and $\log P$; for transit surveys, the regions are in the space of $\log R$ and $\log P$. Ideally, each region is large enough to contain many detected planets and yet small enough that the detection probability does not vary too much from one side to the other.

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

$$\displaystyle \begin{aligned} \varGamma_{m,P} = \frac{\partial^2n}{\partial\log m~\partial\log P} = C\,m^\alpha P^\beta, \end{aligned} $$

(5)

and use it to construct a likelihood function for the outcome of a survey. This function must take into account the detection probability, the properties of the detected systems, and the properties of the stars for which no planets were detected. Then the values of the adjustable parameters are determined by maximizing the likelihood. Details are provided by Tabachnik and Tremaine (2002), Cumming et al. (2008), and Youdin (2011). Foreman-Mackey et al. (2014) cast the problem in the form of Bayesian hierarchical inference, emphasizing the importance of accounting for observational uncertainties in the planet and stellar properties.

Most studies report the occurrence rate density as a function of planet properties, regardless of any other planets in the system. It is more difficult to quantify the multiplicity of planetary systems, the number of planets that orbit together around the same star. For Doppler surveys, one problem is that the star is pulled by all the planets simultaneously. As a result, the detectability of a given planet depends on the properties of any other planets – especially their periods – and on the timespan and spacing between the data points. This makes it difficult to calculate the detection probabilities. For transit surveys, the overlap between different planetary signals is minimal; instead the problem is a degeneracy between multiplicity and inclination dispersion. A star with only one detected planet could lack additional planets, or it could have several planets only one of which happens to transit. In principle, this degeneracy can be broken by combining the results of Doppler and transit surveys (Tremaine and Dong 2012).

Doppler surveys have uncovered a total of about 500 planets. The most informative surveys for planet occurrence were based on observations with the High Resolution Echelle Spectrometer (HIRES) on the Keck I 10-m telescope (Cumming et al. 2008; Howard et al. 2010) and the High Accuracy Radial velocity Planet Searcher (HARPS) on the La Silla 3.6-m telescope (Mayor et al. 2011). Both instruments were used to monitor ∼10³ stars for about a decade, with a precision of a few meters per second. Additional information comes from a few lower-precision and longer-duration surveys (see, e.g., Lovis and Fischer 2010).

For transits, the ground-based surveys have discovered about 200 planets but are not well suited to occurrence calculations because the sample of searched stars and the detection probabilities are poorly characterized. Instead our most important source is the NASA Kepler mission, which used a 1-m space telescope to measure the brightness of 150,000 stars every 30 min for 4 years (Borucki 2016). The typical photometric precision over a 6-h time interval was of order 10⁻⁴. 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 found around ≈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 M_Jup and P from 2–2000 days. They fitted a power law of the form given by Eq. 5, finding α = −0.31 ± 0.20 and β = 0.26 ± 0.10, normalized such that 10.5% of Sun-like stars have such a planet. (Technically, Cumming et al. (2008) calculated the fraction of stars with planets, rather than the average number of planets per star. The value of C is 1.04 × 10⁻³ when mass is measured in M_Jup and period is measured in days.) They found the data to be equally well described by a distribution uniform in $\log P$ from 2–300 days (i.e., β = 0), followed by a sharp increase by a factor of 4–5 for longer periods. The Kepler data are also consistent with the latter description (Santerne et al. 2016). This uptick in planet occurrence at long periods might be related to the location of the “snow line” in protoplanetary disks, which plays a role in the theory of giant planet formation via core accretion; beyond this line, there is enough snow (frozen volatiles) to pack onto a growing protoplanet and help it to achieve the critical mass for runaway accretion of hydrogen and helium gas (Pollack et al. 1996; Lecar et al. 2006).

Fig. 3

Occurrence of giant planets as a function of orbital period, from the Kepler transit survey and two independent Doppler surveys. Data from Cumming et al. (2008) refer to planets with minimum mass m > 0.3 M_Jup; data from Mayor et al. (2011) are for planets with m > 0.16 M_Jup; data from Santerne et al. (2016) are for planets with a radius in the approximate range 0.5–2 R_Jup. Rates are reported in planets per star per $\varDelta \log P = 0.23$ (the range indicated by the horizontal error bars). Downward-pointing arrows indicate upper limits

Metallicity

The earliest Doppler surveys revealed the occurrence of giant planets with periods shorter than a few years to be a steeply rising function of the host star’s metallicity (Santos et al. 2003; Fischer and Valenti 2005). This too is widely interpreted as support for core accretion theory. The logic is that the rapid assembly of a massive solid core – an essential step in the theory – is easier to arrange in a metal-rich protoplanetary disk. Fischer and Valenti (2005) found their sample of Doppler-detected giant planets to be compatible with n ∝ z², where z is the iron-to-hydrogen abundance relative to the solar value. Most recently, Petigura et al. (2018) used Kepler data to determine the best-fitting parameters of

$$\displaystyle \begin{aligned} \varGamma_{P,z} = \frac{\partial^2N}{\partial\log P~\partial\log z} = C\,P^\alpha z^\beta. \end{aligned} $$

(6)

For hot Jupiters, they found β = 3.4 ± 0.9, a remarkably strong dependence. However, for companions more massive than 4 M_Jup, Santos et al. (2017) found the association with high metallicity to be much weaker or absent, suggesting that such objects do not form through core accretion. Schlaufman (2018) reached the same conclusion with a sample spanning a larger range of companion masses and went so far as to say that companions more massive than 10 M_Jup should not be considered planets. The metallicity effect is also weaker for planets smaller than Neptune (Buchhave et al. 2012), although for orbital periods shorter than about 10 days, even small planets are associated with elevated metallicity (Mulders et al. 2016; Wilson et al. 2018; Petigura et al. 2018).

Hot Jupiters

Easy to detect, but intrinsically uncommon, hot Jupiters have an occurrence rate of 0.5–1% for periods between 1 and 10 days. They are even rarer for periods shorter than 1 day (Howard et al. 2012; Sanchis-Ojeda et al. 2014). There is a ≈2σ discrepancy between the rate of 0.8–1.2% measured in Doppler surveys (Wright et al. 2012; Mayor et al. 2011) and 0.6% measured using Kepler data (Howard et al. 2012; Petigura et al. 2018). This is despite the similar metallicity distributions of the stars that were searched (Guo et al. 2017). While we should never lose too much sleep over 2σ discrepancies, it might be caused by misclassified stars and unresolved binaries in the Kepler sample (Wang et al. 2015).

Jupiter analogs

A perennial question is whether the solar system is typical or unusual in some sense. It is difficult to answer because the current Doppler and transit surveys are only barely sensitive to the types of planets found in the solar system: the inner planets are too small and the outer planets have periods that are too long. The most easily detected planet in an extraterrestrial Doppler survey would probably be Jupiter; hence, a few groups have tried to quantify the occurrence of solar-like systems by searching for Jupiter-like exoplanets. Wittenmyer et al. (2016) presented the latest effort, finding the occurrence rate to be $6.2_{-1.6}^{+2.8}$% for planets of mass 0.3–13 M_Jup with orbital distances from 3–7 AU and eccentricities smaller than 0.3. Of course the rate depends on the definition of “Jupiter analog,” a term without a precise meaning. The same problem arises when trying to measure the occurrence of “Earth-like” planets.

Long-period giants

Regarding the more general topic of wide-orbiting giant planets, Foreman-Mackey et al. (2016) measured the occurrence of “cold Jupiters” with periods ranging from 2–25 years, by searching the Kepler data for stars showing only one or two transits over 4 years. For planets with R = 0.4–1.0 R_Jup, they found

$$\displaystyle \begin{aligned} \varGamma_{R,P} = \frac{\partial^2n}{\partial\log R~\partial\log P} = 0.18\pm 0.07. \end{aligned} $$

(7)

Integrating over the specified ranges of radius and period gives a total occurrence rate of 0.42 ± 0.16 planets per star.

Bryan et al. (2016) studied the occurrence of long-period giants conditioned on the detection of a shorter-period giant. Using high-resolution imaging and long-term Doppler monitoring, they searched for wide-orbiting companions to 123 giant planets with orbital distances ranging from 0.01 to 5 AU. They found the occurrence of outer companions to be higher than would be predicted by extrapolating the power law of Cumming et al. (2008) to longer periods. They also found $dn/d\log P$ to decline with period, unlike the more uniform distribution observed for closer-orbiting giant planets. The occurrence rate was (53 ± 5)% for outer companions of mass 1–20 M_Jup and orbital distance 5–20 AU.

Other properties

The giant planet population is distinguished by other features. Their orbits show a broad range of eccentricities (see, e.g., Udry and Santos 2007). Their occurrence seems to fall precipitously for masses above ≈10 M_Jup, at least for orbital distances shorter than a few AU. Because of this low occurrence , the mass range from 10–80 M_Jup is often called the “brown dwarf desert” (Grether and Lineweaver 2006; Sahlmann et al. 2011; Triaud et al. 2017). As mentioned earlier, the inhabitants of this desert are not strongly associated with high-metallicity stars, unlike Jovian-mass planets (Santos et al. 2017; Schlaufman 2018). Occasionally we find two giant planets in a mean-motion resonance (Wright et al. 2011). The rotation of the star can be grossly misaligned with the orbit of the planet, especially if the star is more massive than about 1.2 M_⊙ (Triaud, this volume). These and other topics were reviewed recently by Winn and Fabrycky (2015) and Santerne (this volume).

Smaller Planets

Overall occurrence

About half of Sun-like stars have at least one planet with an orbital period shorter than 100 days and a size in between those of Earth and Neptune. Planet formation theories generally did not predict this profusion of close-orbiting planets. Indeed some of the most detailed theories predicted that close-in “super-Earth” or “sub-Neptune” planets would be especially rare (Ida and Lin 2008). Their surprisingly high abundance led to new theories in which small planets can form in short-period orbits, rather than forming farther away from the star and then migrating inward (see, e.g., Hansen and Murray 2012; Chiang and Laughlin 2013).

Doppler surveys provided our first glimpse at this population of planets and then Kepler revealed it in vivid detail. For planets with periods shorter than 50 days and minimum masses between 3 and 30 M_⊕, two independent Doppler surveys found the occurrence rate 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 and 4 R_⊕, analysis of Kepler data gave an occurrence rate of (13.0 ± 0.8)% (Howard et al. 2012). The results of these surveys are compatible, given reasonable guesses for the relation between planetary mass and radius (Howard et al. 2012; Figueira et al. 2012; Wolfgang and Laughlin 2012).

Size, mass, and period

The surveys also agree that within this range of periods and planet sizes, the occurrence rate is higher for the smallest planets, roughly according to power laws (Howard et al. 2010, 2012):

$$\displaystyle \begin{aligned} \frac{dn}{d\log m} \propto m^{-0.5},~~\frac{dn}{d\log R} \propto R^{-2}. \end{aligned} $$

(8)

For even smaller or longer period planets, Kepler provides almost all the available information. Figure 4 shows some of the latest results (see also Fressin et al. 2013; Burke et al. 2015). The period distribution $dn/d\log P$ rises as ≈P² between 1 and 10 days, before leveling off to a nearly constant value between 10 and 300 days.

Fig. 4

Planet occurrence around FGK dwarfs (top) and M dwarfs (bottom) based on Kepler data. The blue dots represent a random sample of planets around 10³ stars, drawn from the occurrence rate densities of Petigura et al. (2018) and Dressing and Charbonneau (2015). Compared to FGK stars, the M stars have a higher occurrence of small planets and a lower occurrence of giant planets. For the M dwarfs, occurrence rates for planets larger than 4 R_⊕ were not reported because only four planet candidates in that range were detected

Multiple-planet systems

Small planets occur frequently in closely spaced systems (Mayor et al. 2011; Lissauer et al. 2011), with as many as eight planets with periods shorter than a year (Shallue and Vanderburg 2018). The period ratios tend to be in the neighborhood of 1.5–5 (Fabrycky et al. 2014). In units of the mutual Hill radius,

$$\displaystyle \begin{aligned} a_{\mathrm{H}} \equiv \left( \frac{M_{\mathrm{in}} + M_{\mathrm{out}}}{3M_\star} \right)^{1/3} \left( \frac{a_{\mathrm{in}} + a_{\mathrm{out}}}{2} \right), \end{aligned} $$

(9)

more relevant to dynamical stability, the typical spacing is 10–30 (Fang and Margot 2013). At the lower end of this distribution, the systems flirt with instability (Deck et al. 2012; Pu and Wu 2015). A few percent of the Kepler systems are in (or near) mean-motion resonances, suggesting that the orbits have been sculpted by planet-disk gravitational interactions. These systems offer the gift of transit-timing variations (Agol & Fabrycky, this volume), the observable manifestations of planet-planet gravitational interactions that sometimes allow for measurements of planetary masses as well as orbital eccentricities and inclinations. Such studies and some other lines of evidence show that the compact multiple-planet systems tend to have orbits that are nearly circular (Hadden and Lithwick 2014; Xie et al. 2016; Van Eylen and Albrecht 2015) and coplanar (Fabrycky et al. 2014).

Radius gap

The radius distribution of planets with periods shorter than 100 days shows a dip in occurrence between 1.5 and 2 R_⊕ (Fulton et al. 2017; Van Eylen et al. 2017, see Fig. 5). Such a feature had been anticipated based on theoretical calculations of the photo-evaporation of the atmospheres of low-mass planets by the intense radiation from the host star (Owen and Wu 2013; Lopez and Fortney 2013). Thus, the radius gap or “evaporation valley” 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).

Fig. 5

From Fulton et al. (2017). Occurrence as a function of radius based on Kepler data, for orbital periods shorter than 100 days. The dip in the occurrence rate density between 1.5 and 2 R_⊕ has been attributed to the erosion of planetary atmospheres by high-energy radiation from the star

Hot Neptunes

As mentioned above, $dn/d\log P$ changes from a rising function for $P \lesssim 10$ days to a nearly constant value for P = 10–100 days. The critical period separating these regimes is longer for larger planets. The effect is to create a diagonal boundary in the space of $\log R$ and $\log P$, above which the occurrence is very low (see Fig. 4). The same phenomenon is seen in Doppler data (Mazeh et al. 2016). This “hot Neptune desert” may be another consequence of atmospheric erosion. Interestingly those few hot Neptunes that do exist are strongly associated with metal-rich stars (Dong et al. 2017; Petigura et al. 2018), making them similar to giant planets and unlike smaller planets. The hot Neptunes are also similar to hot Jupiters in that they tend not to have planetary companions in closely spaced coplanar orbits (Dong et al. 2017). All this suggests that the hot Neptunes and close-in giant planets originate in similar circumstances, possibly from some type of dynamical instability.

Earth-like planets

A goal with broad appeal is measuring the occurrence rate of Earth-sized planets orbiting Sun-like stars within the “habitable zone,” the range of distances within which a rocky planet could plausibly have oceans of liquid water. The Kepler mission provided the best-ever data for this purpose. However, even Kepler was barely sensitive to such planets. The number of detections was of order 10, depending on the definitions of “Earth-sized,” “Sun-like,” and “habitable zone.” The desired quantity can be understood as an integral

$$\displaystyle \begin{aligned} \eta_\oplus \equiv \int_{R_{\mathrm{min}}}^{R_{\mathrm{max}}} \int_{S_{\mathrm{min}}}^{S_{\mathrm{max}}} \frac{\partial^2n}{\partial\log S~\partial\log R}~d\log S~d\log R, \end{aligned} $$

(10)

where S is the bolometric flux the planet receives from the star. The integration limits are chosen to select planets likely to have a solid surface with a temperature and pressure allowing for liquid water. These limits depend on assumptions about the structure and atmosphere of the planet and the spectrum of the star (see, e.g., Kasting et al. 2014).

Even if we set aside the problem of setting the integration limits, the measurement of

$$\displaystyle \begin{aligned} \varGamma_\oplus \equiv \left.\frac{\partial^2n}{\partial\log P~\partial\log R}\right|{}_{P=1~{\mathrm{yr}},~R=R_\oplus} \end{aligned} $$

(11)

has proven difficult and may require extrapolation from measurements of larger planets at shorter periods. The Kepler team has published a series of papers reporting steady advancement in the efficiency of detection, elimination of false positives, and understanding of instrumental artifacts. The most recent effort to determine Γ_⊕ found the data to be compatible with values ranging from 0.04 to 11.5 (see Fig. 6). Since then the Kepler team and other groups have clarified the properties of the stars that were searched (Petigura et al. 2017; 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 for planet detection and validation. These developments have brought us right to the threshold of an accurate occurrence rate for Earth-like planets.

Fig. 6

From Burke et al. (2015). Estimates for Γ_⊕ based on Kepler data. The orange histogram is the posterior probability distribution considering only the uncertainties from counting statistics and extrapolation. The other curves illustrate the effects of some systematic errors: uncertainty in the detection efficiency, orbital eccentricities, stellar parameters, and reliability of weak planet candidates. These systematic effects led to a range in Γ_⊕ spanning an order of magnitude. The vertical lines show other estimates of Γ_⊕ by Foreman-Mackey et al. (2014), Petigura et al. (2013), Dong and Zhu (2013), and Youdin (2011)

Other Types of Stars

Almost all the preceding results pertain to main-sequence stars with masses between 0.5 and 1.2 M_⊙, i.e., spectral types from K to late F. Stars with masses between 0.1 and 0.5 M_⊙, the M dwarfs, are not as thoroughly explored, especially near the low end of the mass range. However these stars are very attractive for planet surveys because their small masses and sizes lead to larger Doppler and transit signals and because planets in the habitable zone have conveniently short orbital periods.

Giant planets are relatively rare around M dwarfs, at least for periods shorter than a few years. Cumming et al. (2008) showed that if planet occurrence is modeled by the functional form of Eq. 5, then planets with masses exceeding 0.4 M_Jup and periods shorter than 5.5 years are 3–10 times less common around M dwarfs than around FGK dwarfs. Similar results were obtained by Bonfils et al. (2013).

On the other hand, for smaller planets over the same range of periods, M dwarf occurrence rates exceed those of FGK dwarfs by a factor of 2–3 (Howard et al. 2012; Mulders et al. 2015). This result is based on Kepler data, which remains our best source of information on this topic despite the fact that only a few percent of the Kepler target stars were M dwarfs. Comprehensive analyses have been performed by Dressing and Charbonneau (2015) and Gaidos et al. (2016). Their results differ in detail but agree that, on average, M dwarfs have about two planets per star with a radius in between those of Earth and Neptune and an orbital period shorter than 100 days (see Fig. 4). One implication of this high occurrence rate is that the nearest habitable-zone planets are almost surely around M dwarfs. Indeed, Doppler surveys have turned up two candidates for “temperate” Earth-mass planets around M dwarfs within just a few parsecs: Proxima Cen (1.3 pc, Anglada-Escudé et al. 2016) and Ross 128 (also known as Proxima Vir; 3.4 pc, Bonfils et al. 2017).

Some other comparisons with FGK dwarfs have been made. Among the similarities are that M dwarfs often have compact systems of multiple planets (Muirhead et al. 2015; Gaidos et al. 2016; Ballard and Johnson 2016) and that high metallicity is associated with giant planet occurrence (Johnson et al. 2010; Neves et al. 2013). It remains unclear whether or not the occurrence of smaller planets is associated with high metallicity for M dwarfs (see, e.g., Gaidos and Mann 2014). There is also evidence that the planet population around M dwarfs exhibits both the “evaporation valley” between 1.5 and 2 R_⊕ and the “hot Neptune desert” (Hirano et al. 2017).

Beyond the scope of this review, but nevertheless fascinating, are the occurrence rates that have been measured in Doppler and transit surveys of other types of stars: evolved stars (Johnson et al. 2010; Reffert et al. 2015), stars in open clusters (Mann et al. 2017) and globular clusters (Gilliland et al. 2000; Masuda and Winn 2017), binary stars (Armstrong et al. 2014), brown dwarfs (He et al. 2017), and white dwarfs (Fulton et al. 2014; van Sluijs and Van Eylen 2018).

Future Prospects

The data from recent surveys offer opportunities for progress for at least another few years. The ongoing struggle to measure the occurrence rate of Earth-like planets has already been described. Another undeveloped area is the determination of joint and conditional probabilities; for example, given a planet with radius R₁ and period P₁, what is the chance of finding another planet around the same star with radius R₂ and period P₂? Conditional rates, or the relative occurrence of different types of systems, may be more useful than overall occurrence rates for testing planet formation theories. Only a few cases have been studied, such as the mutual radius distribution of neighboring planets (Ciardi et al. 2013; Weiss et al. 2018) and the probability for giant planets to have wider-orbiting companions (Huang et al. 2016; Bryan et al. 2016; Schlaufman and Winn 2016).

New data are also forthcoming. The stellar parallaxes soon to be available from the ESA Gaia mission (Gaia Collaboration et al. 2016) will clarify the properties of all the Kepler stars as well as the targets of future surveys. Among these future surveys is the Transiting Exoplanet Survey Satellite (TESS), which was launched in April 2018 (Ricker et al. 2015). This mission was not designed to measure planet occurrence rates but rather to pluck low-hanging fruit: short-period transiting planets around bright stars. Less well appreciated is that TESS may be superior to Kepler for measuring the occurrence of planets larger than Neptune with periods shorter than 10 days. When TESS was conceived, it was expected that limitations in data storage and transmission would restrict the search to ≈10⁵ preselected stars, as was the case with Kepler. Later it became clear that entire TESS images could be stored and transmitted with 30-min time sampling. As a result, although ≈10⁵ Sun-like stars will still be selected for finer time sampling, it should be possible to search millions of stars for large and short-period planets. This includes hot and massive stars, for which comparatively little is known. TESS should excel at finding 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 advances in the study of low-mass stars, using new Doppler spectrographs operating at far-red and infrared wavelengths and ground-based transit surveys focusing exclusively on low-mass stars. Particularly encouraging was the discovery of TRAPPIST-1, a system of seven Earth-sized planets orbiting an “ultracool dwarf” that barely qualifies as a star (Gillon et al. 2017). This system was found after searching ≈50 similar objects with a detection efficiency of around 60% (Burdanov et al., this volume; M. Gillon, private communication), and the transit probability for the innermost planet is 5%. This suggests that 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 outcome of planet formation around ultracool dwarfs.

In the decades to come, the domains of all the planet detection techniques – including direct imaging, gravitational microlensing, and astrometry – will begin overlapping. Some efforts have already been made to determine occurrence rate densities based on data from very different techniques (see, e.g., Montet et al. 2014; Clanton and Gaudi 2016). We can look forward to a more holistic view of the occurrence of planets around other stars, barring any civilization-ending cataclysm.