Finding Planets via Gravitational Microlensing

Abstract

Gravitational microlensing is a technique to probe compact objects toward the center of the galaxy, such as distant stars, planets, white and brown dwarfs, black holes, and neutron stars. Since the first microlensing planet discovered in 2003, more than 40 planets have been detected with this technique, as well as several black hole candidates, and a population of potential free-floating planets. This chapter first provides a presentation of the microlensing theory, including numerical aspects to solve binary and triple lens problems, and a discussion of the microlensing planetary detection efficiency, with a high potential regarding cold planets beyond the snow line. It also explains how the planetary characterization can be facilitated when the microlensing light curves exhibit distortions due to second-order effects, such as parallax, planetary orbital motion, and extended source, and how they can also introduce degeneracies in the models. The chapter then reviews the main discoveries to date and the recent statistical results from high-cadence ground-based surveys and space-based observations, especially on the planet mass function and the distance distribution of the microlensing planetary systems. Finally, future prospects are discussed, with the expected advances from dedicated space missions, extending the planet sensitivity range down to Mercury masses.

Introduction

Gravitational microlensing is not the easiest way to discover planets, and the development of this technique over the last 30 years required tremendous efforts and steady determination from the scientific community. Even Einstein, who was the first to really bring attention to this phenomenon Einstein (1936), i.e., the light of a distant source being bent by the gravitational field of a massive body, was pessimistic about the chances of observing such a phenomenon. Indeed, the microlensing optical depth toward the galactic center is of about v²∕c² ∼ 10⁻⁶, meaning millions of stars need to be monitored in order to catch a handful of magnified ones. Later on, several people considered experiments of microlensing observations (e.g., Refsdal 1964; Chang and Refsdal 1979), and finally Paczynski (1986) managed to convince the international community to develop the MACHO, EROS, and OGLE experiments to search for compact objects in the galactic halo (see acronyms at the end of the chapter). Hunting for microlensing events toward the galactic center with the intention to discover planets appeared a few years later, under the suggestion of Mao and Paczynski (1991) and Gould and Loeb (1992), who gave a detailed description of microlensing planetary perturbations.

Nowadays, thousands of microlensing events are observed per year, leading to the detection of up to ten exoplanets for each season, accounting for more than 40 planetary systems published to date and ten more under analysis. This amount is quite modest compared to the total amount of discovered exoplanets, but this technique offers a considerable advantage in probing a parameter space unaccessible to other methods.

Earth mass planets can be detected by microlensing (Bennett and Rhie 1996), within a range of separations to the host star that is complementary to other techniques. Microlensing is indeed mostly sensitive to planets orbiting just outside of the snow line, where planetary formation is predicted to be the most efficient according to the core accretion theory (Ida and Lin 2004). Moreover, as the only effect measured by microlensing is that of deflecting light by gravitational distortions of a given field, there is no need to measure the light of the lens system itself. This opens a domain of exploration down to very faint, even invisible objects, such as brown dwarfs, extremely far systems in the galactic bulge, as well as black holes, neutron stars, and free-floating planets.

More generally, the formation processes may be influenced by many environmental factors, such as star metallicity (Wang and Fischer 2015; Zhu et al. 2016), as well as the mass of the host star (Johnson et al. 2010), the star multiplicity (Kaib et al. 2013), and larger scale conditions like higher ambient radiations and star density in the galactic bulge (Thompson 2013). Thus, the distant population of planets unveiled by microlensing provides critical tests to the planetary formation theories.

Observations

The strategy to hunt for planets via microlensing has evolved over the last 10 years. The microlensing community has made a transition toward a network of wide-field imagers. They now conduct high-cadence surveys from different longitudes of the southern hemisphere. Since 2006, the MOA collaboration (Bond et al. 2001; Sumi et al. 2013) uses a 2.2-deg² FOV CCD camera (Sako et al. 2008) mounted on a 1.8 m MOA-II telescope in New Zealand. The OGLE collaboration (Udalski 2003) upgraded its camera in 2010 to a 1.4-deg² FOV on a 1.3-m telescope in Chile. These two groups monitor hundreds of fields in the bulge and alert 2000–3000 microlensing events each year. Additionally, (KMTNet, Kim et al. 2016) became operational in 2015, with three telescopes equipped with a 4 deg² camera, in Chile, South Africa, and Australia. The Wise observatory in Israel also conducts a survey (Gorbikov et al. 2010), despite its less favorable northern latitude.

In parallel to these surveys, several follow-up groups (e.g., PLANET, microFUN, RobotNet, MiNDSTEp) dedicate their telescopes to the most interesting events, with observatories at all longitudes to ensure a round-the-clock coverage of the planetary anomalies.

In 2014–2017, the microlensing campaigns from the ground were also combined with space-based observations using the telescopes Spitzer (Werner et al. 2004; Gould and Horne 2013) and Kepler (K2-C9) (Howell et al. 2014), as a pathway toward the future space missions WFIRST (Spergel et al. 2015) and Euclid (Penny et al. 2013) .

Microlensing Basics

The basic principle of a microlensing effect is illustrated in Fig. 1, showing a light ray coming from a source S being deflected by a point-like mass L, called “lens.” The light is deviated by an angle α, creating the illusion for the observer (O) that the source is at the position I (“image”) in the lens plane. Under the assumption of small angles, the three angles of this figure are linked by the lens equation, α(D_S − D_L) = θD_S − βD_S, where α = 4GM∕(c²D_Lθ), with M the lens mass, G the gravitational constant, and c the speed of light.

Fig. 1

Deviation of a light ray from a source S due to the gravitational field of a microlens L

The deviation of light does not affect only one ray but an ensemble of rays coming from the source, so when the source and the lens are perfectly aligned on the observer’s line of sight, β = 0 and the source image in the lens plan appears as a ring, called the “Einstein ring,” whose angular size θ_E is expressed as

$$\displaystyle \begin{aligned} \theta_{\mathrm{{E}}}=\sqrt{\kappa M\frac{D_S-D_L}{D_S D_L}},\quad \mathrm{{where}}\,\,\kappa=4G/c^2\simeq 8.14 \,\mathrm{kpc.M}^{-1}_\odot. \end{aligned} $$

(1)

The lens equation is thus commonly expressed as $\beta =\theta -\theta _{\mathrm {{E}}}^2/\theta$. The Einstein angular radius θ_E is used as a scale standard in microlensing units, and as long as its value remains unknown, most microlensing parameters will be expressed in units of θ_E.

Simple Lens

If we normalize the angles by θ_E and define the parameters u = β∕θ_E and z = θ∕θ_E, we can express the lens-source separation u as a function of the separation z between the image of the source and the lens, both being projected in the lens plane, u = z − 1∕z. If the lens and the source are not perfectly aligned (u ≠ 0), this equation has two solutions: $z_\pm =\pm (\sqrt {u^2+4}\pm u)/2$. The positive solution creates the major image, and the negative one the minor image, outside and inside of the Einstein ring, respectively.

The magnification of the source by the lens is then estimated as the area ratio of the images over the source. Since the images are tangentially stretched by a factor z_±∕u compared to the source and radially shrunk by a factor dz_±∕du, the magnification is then expressed as

$$\displaystyle \begin{aligned} A_\pm=\left|{\frac{z_\pm}{u}\frac{dz_\pm}{du}}\right|=\frac{1}{2}\left[\frac{u^2+2}{u\sqrt{u^2+4}}\pm 1\right],\quad \text{thus in total},\quad A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}. \end{aligned} $$

(2)

In its simplest form, the lens-source relative movement is rectilinear, and u can be expressed as a function of the impact parameter u₀ (in units of θ_E), the time of maximum magnification t₀, and the Einstein crossing time t_E:

$$\displaystyle \begin{aligned} u(t)=\left[u^2_0+\left(\frac{t-t_0}{t_{\mathrm{E}}}\right)^2\right]^{1/2}. \end{aligned} $$

(3)

The observed flux is a function of time, F(t) = A(t)F_s + F_b, where F_s is the source flux and F_b the blended light (any unresolved light) that is not magnified. It reaches its maximum at t₀, when the impact parameter is at its minimum u₀. The smaller u₀ is, the higher the maximum of magnification will be. For a point-source-point-lens (PSPL) event, the light curve can be fit by five parameters: t₀, u₀, t_E, F_s, and F_b. In practice, each observatory has its specific F_s and F_b, since different observatories may have different filter bandpasses and resolutions.

Multiple Lens

We now consider a lens composed of N_L objects, which total mass will be $M=\varSigma ^{N_L}_{i=1}m_i$. Let θ_i be the two-dimensional coordinates of these individual masses in the lens plane. The deflection angle of the source is then expressed by

$$\displaystyle \begin{aligned} \boldsymbol{\alpha}(\boldsymbol{\theta})=\frac{4G}{c^2}\left(\frac{1}{D_L}-\frac{1}{D_S}\right)\sum_i^{N_L}m_i\frac{\boldsymbol{\theta}-\boldsymbol{\theta}_i}{|\boldsymbol{\theta}-\boldsymbol{\theta}_i|{}^2}, \end{aligned} $$

(4)

where θ are the angular positions of the images. It is common to consider the lens equation in complex coordinates (Witt 1990; Rhie 1997). The source position u = (ξ, η) and the image position z = (x, y) can then be defined in the complex form, u = ξ + iη and z = x + iy, respectively. The lens equation thus becomes

$$\displaystyle \begin{aligned} u=z-\sum_i^{N_L}\frac{m_i/M}{\bar{z}-\bar{z}_i}, \end{aligned} $$

(5)

where $\bar {z}$ is the complex conjugate of z. Equation (5) can be solved numerically to find the image position z_j, where j is the index for the source images and i those of the individual lens masses. The magnification can be greater or less than 1, depending on the images area, and is equal to the inverse of the Jacobian matrix determinant,

$$\displaystyle \begin{aligned} A_j=\frac{1}{\mathrm{detJ}}\Big|{}_{z=z_j},\quad \mathrm{{detJ}}\equiv \frac{\partial(\xi,\eta)}{\partial(x,y)}=1-\frac{\partial u}{\partial z}\frac{\overline{\partial u}}{\partial z}. \end{aligned} $$

(6)

The total magnification is the sum of individual ones: A = Σ_j|A_j|. Singularities occur for positions of the source where detJ = 0, theoretically inducing an infinite magnification in the case of a point source. In practice, the source cannot be considered as a point in most cases, and the singularities appear as very high but finite magnifications.

From Eq. (5), $\,\partial u / \partial \bar {z}=\sum \left [\varepsilon _i/(\bar {z_i}-\bar {z})^2\right ]$, where ε_i = m_i∕M is the mass fraction of individual masses m_i. According to Eq. (6), the image positions for which detJ = 0 are given by

$$ \left|\sum\limits_i^{N_L}\frac{\varepsilon_i}{(\bar{z_i}-\bar{z})^2}\right|^2=1. $$

(7)

The solutions of Eq. (7) shape as closed contours in the lens plane, called “critical curves.” To these curves correspond an ensemble of source positions in the source plane, called “caustics.”

When the lens is composed of two masses, N_L = 2, the lens equation is expressed by

$$\displaystyle \begin{aligned} u=z+\frac{\varepsilon_1}{\bar{z_1}-\bar{z}}+\frac{\varepsilon_2}{\bar{z_2}-\bar{z}}. \end{aligned} $$

(8)

Equation (8) can be written as a 5-degree polynomial, which coefficients are given in Witt and Mao (1995). The solutions to this polynomial are not necessarily solutions to Eq. (8), because when the source is outside of the caustics, two of the five solutions yield to an imaginary magnification. Thus, the number of images in the lens plane varies from three to five when the source crosses a caustic.

Schneider and Weiss (1986) showed that there are three different topologies of caustics for a given value of mass ratio q between the two components of the lens. These three topologies present either one, two, or three caustics. As shown on Fig. 2, in the case of small separations between the two lenses, d ≪ 1 (in Einstein units), there are three caustics, one being the central caustic (with 4 cusps) and two secondary ones (with 3 cusps). The transition toward big separations goes through an intermediary phase that creates a single caustic with six cusps, of bigger size, called “resonant caustic.” This configuration happens when the separation is close to the Einstein radius. Finally, for large separations, this caustic splits into two caustics, with four cusps each. We then naturally call these three domains “close,” “resonant,” and “wide” separations.

Fig. 2

Topology of caustics for three regimes of separation, “close” (top), “resonant” (middle), and “wide” (bottom)

For triple lens modeling, Rhie (2002) gives the lens equation for point-source calculations. It is solved the same way as the binary lens equation, but the polynomial is of tenth order instead of fifth.

For mass ratios q ≪ 1 and a given separation d, the shape and size of the central caustic are invariant under the d↔d⁻¹ transformation. This duality is often called “close/wide degeneracy” (Griest and Safizadeh 1998; Dominik 1999; Albrow et al. 1999; An 2005). It comes from the fact that the Taylor development of the lens equation to the second degree are identical for d↔d⁻¹(1 + q)^1∕2, i.e., d↔d⁻¹ when q ≪ 1. Figure 3 gives an illustration of this degeneracy, where the source passes very close to the central caustic of a system with a q = 0.006 mass ratio. Two (d, 1∕d) models provide the same light curve. This figure also points out the importance of high-magnification events, for which u₀ ≪ 1. Indeed, when the source passes extremely close to the central caustic, it probes the effect of the planetary companion on the caustic shape. Significant efforts have been expended on the observation of high-magnification events (HME), because they probe the central caustic (Griest and Safizadeh 1998; Rhie et al. 2000; Rattenbury et al. 2002). Any planets in the system are likely to affect the central caustic, resulting in potentially high sensitivity to the presence of low-mass planets, whose signature can be detected at the peak of the microlensing event .

Fig. 3

Example of d↔d⁻¹ degeneracy

Finite Source Effects and Computation

As previously mentioned, in reality the source is a disk, and the point-like approximation is only valid when the source is far enough from the lens (projected in the lens plane). This means that when the impact parameter is very small, u₀ ≪ 1, finite source effects have to be taken into account at the peak of the microlensing event and require specific numerical treatments. It is also true when the source approaches or crosses a caustic in the case of multiple lens systems. In the first case, the corresponding effect on the light curve peak is very noticeable, with a rounded shape and a damped magnification. This effect was first analyzed by Gould (1994) and Nemiroff and Wickramasinghe (1994). They showed that the deviation from the standard PSPL light curve provides informations on the source proper motion, under assumptions on its brightness and color. Indeed, one can determine θ_E = θ_∗∕ρ, where ρ is the source radius in Einstein units and θ_∗ the physical source size in μas. The first parameter is derived from the light curve fit, and the latter from available informations on the source star color (Bensby et al. 2013) and brightness (Nataf et al. 2013), and empirical color/brightness relations (Kervella et al. 2004). Thus, the lens-source relative proper motion is derived by μ = θ_E∕t_E.

When the source crosses a caustic, the finite source effects start to be noticeable as the source projected distance from the caustic is of a few source radii (Pejcha and Heyrovsky 2009). Figure 4 gives an illustration of this effect.

Fig. 4

Zoom on microlensing light curves showing a caustic crossing (see the inset) with different source sizes: ρ = 0.0001, 0.0002, 0.0003, 0.0004, 0.0008, 0.002. The lens system here has a mass ratio q = 0.013 and a separation d = 0.914

Similarly, caustic crossings enable us to determine the source radius in Einstein units, as well as its luminosity profile, i.e., limb-darkening effects on its disk.

Formally, the source magnification can be calculated by integration over its entire surface, each point having its own magnification. Concretely, this approach is hard to implement due to divergences near the caustics. Several algorithms and techniques have been developed to solve the extended source case (Dominik 1995; Bennett and Rhie 1996; Wambsganss 1997; Gould and Gaucherel 1997; Gould 2008; Pejcha and Heyrovsky 2009; Bennett 2010). The most efficient approach seems to be the one that consists in cutting the light curve in segments as a function of the source distance to the caustics.

Two different methods are often applied, one when the source is relatively far from the caustics and the other for close approaches or crossings of caustics. The first method is based on an analytical approximation, such as the “hexadecapole” approximation (Gould 2008; Pejcha and Heyrovsky 2009) that evaluates the magnification of 13 points well distributed on the surface of the source.

The second one is a more complex numerical evaluation of the source magnification that integrates the images in the lens plane. One possibility is to compute an “inverse ray shooting,” i.e., to throw rays from the lens plane to the source plane, in order to calculate the images/source areas ratio. Indeed, if computing the source-to-lens path of light rays is difficult, the lens-to-source path is straightforward. One just needs to apply the equations of light deflection for a given lens model that is tested, and then count the amount of rays falling in the source disk. The ratio of images/source areas gives the total magnification. By Liouville’s theorem, the surface brightness is conserved; thus, the density of light rays in the image plane is uniform. This method requires a dense sampling of the image plane, a grid search on a large (d, q) parameter space, and an efficient algorithm to fit the other parameters (t₀, u₀, t_E, ρ, α), where α is the angle between the source trajectory and the lens system axis. Furthermore, triple lens modeling is often required and has been implemented in several codes, in particular those using a faster ray shooting technique, the image-centered ray shooting (Bennett and Rhie 1996; Bennett 2010).

Another method consists in using Green’s theorem to compute a 1-D mapping of the image contours, instead of 2-D (Dominik 1995, 1998; Gould and Gaucherel 1997). Each point requires to solve a fifth-order polynomial, which makes individual computations time-consuming, but this disadvantage is largely compensated by the faster 1-D integration. However, the task of connecting points in the lens plane in order to obtain closed contours is sometimes confusing, in particular when the source passes near a cusp. Such a code has been released to the public by Valerio Bozza in 2017, whose algorithm is described in Bozza (2010) (http://www.fisica.unisa.it/gravitationAstrophysics/VBBinaryLensing.htm) .

Microlens Parallax, π_E

The parallax effects count between the most remarkable and useful to solve microlensing problems. They appear as a distortion in the light curve, due to a nonuniform motion of the observer. For fairly long microlensing events (t_E ≥ 1year∕2π), the velocity of the Earth can no longer be considered as constant and rectilinear, and this induces asymmetries in the light curve called “orbital parallax” (e.g., Gould and Loeb 1992; Alcock et al. 1995; Mao 1999; Smith et al. 2002). Parallax effects can also be measured when the same event is observed from different observatories. Indeed, “terrestrial” parallax can result from simultaneous ground-based observations at different longitudes, for high-magnification events monitored at very high cadence. And “satellite” parallax can be measured when ground-based observations are combined with space-based observations.

Detection of such effects yields the Einstein radius determination, projected in the observer plane, $\tilde {r_{\mathrm {{E}}}}\equiv (2R_{\mathrm {{Sch}}}D_{\mathrm {{rel}}})^{1/2}$, where R_Sch ≡ 2GMc⁻² is the lens Schwarzschild radius and $D_{\mathrm {{rel}}}^{-1}\equiv D_L^{-1}-D_S^{-1}$. If in addition the Einstein angular radius θ_E = (2R_Sch∕D_rel)^1∕2 is measured from finite source effects, then the lens mass can be deduced (Gould and Loeb 1992)

$$\displaystyle \begin{aligned} M=\frac{c^2}{4G}\tilde{r_{\mathrm{{E}}}}\theta_{\mathrm{{E}}}=0.1227M_\odot\left(\frac{\tilde{r_{\mathrm{{E}}}}}{1\,\mathrm{{AU}}}\right)\left(\frac{\theta_{\mathrm{{E}}}}{1\,\mathrm{{mas}}}\right). \end{aligned} $$

(9)

Parallax effects are generally defined by the vector π_E, called microlens parallax, which magnitude gives the Einstein radius projected in the observer plane,

$$\displaystyle \begin{aligned} \pi_{\mathrm{{E}}}\equiv \frac{\pi_L-\pi_S}{\theta_{\mathrm{{E}}}}:\frac{1\,\mathrm{{kpc}}/D_{\mathrm{{rel}}}}{\theta_{\mathrm{{E}}}/1\,\mathrm{{mas}}}=\frac{1\,\mathrm{{AU}}}{\tilde{r_{\mathrm{{E}}}}}, \end{aligned} $$

(10)

where π_L∕1 mas = 1 kpc∕D_L and π_S∕1 mas = 1 kpc∕D_S. The orientation of π_E is that of the lens-source relative proper motion. Knowing θ_E and π_E gives a relation between the lens and source distances, D_L = (π_Eθ_E + 1∕D_S)⁻¹.

Measuring the microlensing parallax π_E and the Einstein radius θ_E represents the optimal way to derive physical parameters of the lens system. As orbital and terrestrial parallaxes are only detectable for rare high-magnification or long microlensing events, the idea of measuring a satellite parallax has recently increased importance within the microlensing community, as was first suggested by Refsdal (1966) and Gould (1999). The first space-based microlensing observations were done with Spitzer for a Small Magellanic Cloud event (OGLE-2005-SMC-0001 Dong et al. 2007). Such observations were also conducted for a microlensing event toward the bulge (MOA-2009-BLG-266 Muraki et al. 2013) using the Deep Impact spacecraft.

Spitzer appeared to be an excellent microlensing parallax satellite being on a solar orbit at ∼1 AU from the Earth, and the microlensing community, under a pilot program led by A. Gould (Gould and Horne 2013; Gould et al. 2014a, 2015), carried out Spitzer observations in 2014, 2015, and 2016. These campaigns resulted in a fair amount of highly precise Earth-Spitzer microlens parallaxes, e.g., the case of an isolated star (Yee et al. 2015), a binary star (Han et al. 2016a), and several exoplanets (e.g., Udalski et al. 2015; Street et al. 2016). However, a recurrent obstacle encountered with Spitzer microlensing light curves has been the faintness of the observed targets leading to a low S/N and the fact that only fragments are being observed, with limited baseline data.

Figure 5 shows an example of a microlensing event, OGLE-2014-BLG-0124Lb, observed simultaneously with the OGLE telescope and Spitzer. The planetary perturbation is due to a ∼ 0.5 M_jup planet orbiting a 0.7 M_⊙ star. The two distinct light curves provided a parallax measurement, to a precision ≤7%. Unfortunately, the error on the mass determination turned out to be much higher (30%) due to poor constraints from finite source effects (Udalski et al. 2015).

Fig. 5

OGLE-2014-BLG-0124Lb: a ∼ 0.5 M_jup planet orbiting ∼ 0.7 M_⊙ star, at a_⊥∼ 3.1 AU. The planetary anomaly and the peak of the event have been observed by Spitzer simultaneously to ground-based observations by OGLE (Figure from Udalski et al. 2015)

Space-based microlensing observations have also been conducted in 2016 with the Kepler telescope in its K2-C9 campaign (Howell et al. 2014), with the aim of detecting free-floating planets. In the longer term, this mass measurement method will play an even bigger role in the context of microlensing dedicated space missions like WFIRST (Spergel et al. 2015).

However, to fully exploit the potential of this effect, further investigations have been carried out to understand an underlying fourfold discrete degeneracy associated with the Earth-space microlensing parallax. This degeneracy was already discussed in the original paper by Refsdal (1966). Since the same microlensing event is seen from two different observatories, the two observers obtain distinct light curves with different t₀ and u₀, and the parallax is given by the difference in these quantities (Gould 1994):

$$\displaystyle \begin{aligned} {\boldsymbol{\pi}_{\mathrm{{E}}}}=\frac{AU}{{\mathbf{D_\perp}}}\left(\frac{\Delta t_0}{t_{\mathrm{{E}}}},\Delta u_0\right), \end{aligned} $$

(11)

where Δt₀ = t_0,sat − t_0,⊕, Δu₀ = u_0,sat − u_0,⊕, and D_⊥ is the physical vector between the two observers projected in the lens plane. The direction of D_⊥ gives the one of π_E. Since u₀ is a signed quantity which magnitude can be derived from the single lens light curve, but not the sign, there are actually four solutions Δu_0,±,± = ±|u_0,sat|±|u_0,⊕|. Hence, one degeneracy creates four different orientations of the parallax vector, and a second, two different magnitudes of this vector. Several attempts have been made to break these degeneracies under special circumstances (e.g., Calchi Novati et al. 2015; Yee et al. 2015), but it might actually require a revision of the parallax formalism itself. Calchi Novati and Scarpetta (2016) revisited the analysis of the microlensing parallax from a heliocentric reference frame instead of the commonly adopted geocentric frame and extended the Gould (1994) expression of the parallax to observers in motion, as opposed to observers at rest. They showed that this new approach provides a clearer understanding of the fourfold degeneracy and interesting leverages to break it.

In a similar way to parallax effects, if the source is animated by a significant acceleration and nonrectilinear motion during the microlensing event, in other words if the source is a binary, the light curve can be affected by additional distortions. These effects are called “xallarap” by symmetry with the parallax phenomenon. Unless the source motion coincidentally mimics the Earth’s motion, these two effects are generally well distinguishable (Poindexter et al. 2005). Such effects have been detected in the light curve of OGLE-2007-BLG-368Lb, a cold Neptune around a K-star (Sumi et al. 2010) .

Orbital Motion

In the case of multiple lenses, the orbital motion of the lens components can create deviations in the light curve (Dominik 1998) due to variations in the shape and orientation of the caustics. These effects are generally negligible as microlensing events mostly involve companions with long periods. Although a Keplerian orbit would require five parameters to be characterized, microlensing light curve perturbations can only constrain two additional parameters. For a binary system, one reflects the variation of the separation d between the two bodies, called s here for clarity, and the second being the variation ω of the angle α between the source trajectory and the binary lens axis

$$\displaystyle \begin{aligned} s=s_0+\mathrm{{d}}s/\mathrm{{d}}t(t-t_0), \quad \alpha=\alpha_0+\omega(t-t_0). \end{aligned} $$

(12)

We consider r_⊥ = sD_Lθ_E the projected separation of the lens system and evaluate the instantaneous projected velocity of the lens components as the quadratic sum of r_⊥γ_⊥ = r_⊥ω and r_⊥γ_|| = r_⊥(ds∕dt)∕s, perpendicular and parallel to the projected binary axis, respectively. Unfortunately, there is an important degeneracy in the microlensing models between π_E,⊥ and γ_⊥ (Batista et al. 2011; Skowron et al. 2011), where π_E,⊥ is the component of π_E that is perpendicular to the instantaneous direction of the Earth’s acceleration.

For bright lenses, radial velocity (RV) measurements can be a good test of microlensing orbital models, as it was done by Yee et al. (2016) for OGLE-2009-BLG-020, confirming the model published by Skowron et al. (2011).

Mass Estimates from Galactic Models

The lens mass and distance have been directly determined in some microlensing events, thanks to microlens parallax measurements combined with Einstein angular radius estimates (e.g., Bennett et al. 2008, 2016; Gaudi et al. 2008; Batista et al. 2009; Muraki et al. 2013; Kains et al. 2013; Furusawa et al. 2013). However, this is not the case for many microlensing events, and in most cases, in addition to informations provided by the light curve, mass and distance estimates require galactic models that generate distributions of disk and bulge stars according to some empirical constraints (e.g., density of stars in the galaxy, proper motion, mass function, binary stars abundances) (Han and Gould 1995; Duquennoy and Mayor 1991; Raghavan et al. 2002; Duchene and Kraus 2013).

In the absence of parallax and/or θ_E measurements, additional observations with high angular resolution imaging are strongly advocated, in order to derive constraints on the lens brightness and thus on its mass with mass-luminosity relations. It has been done for many microlensing systems, reducing drastically the uncertainties on their physical properties, using the VLT, Keck, and Subaru telescopes equipped with adaptive optics (AO) or the Hubble Space Telescope (HST) (Bennett et al. 2006, 2014, 2015, 2016; Dong et al. 2009; Janczak et al. 2010; Sumi et al. 2010; Batista et al. 2011, 2014, 2015; Kubas et al. 2012; Beaulieu et al. 2016; Fukui et al. 2015; Koshimoto et al. 2017).

Detections Overview

More than 40 exoplanets have been discovered via microlensing since the first discovery 13 years ago (Bond et al. 2004). Although this number is relatively modest compared to transit and RV methods, the microlensing technique explores a population of planets that is hardly accessible, in the medium term, to other techniques. It is sensitive to planets down to Earth mass beyond the snow line, where the temperature is cool enough for ices to form. According to the core accretion theory (e.g., Ida and Lin 2004), planetary formation is predicted to be the most efficient in this region of the protoplanetary disk. The planet sample provided by microlensing offers a range of various planetary systems and shows the potential of the technique to bring important constraints to the planetary formation theories. It also revealed a population of free-floating planet candidates, down to Jupiter-masses (Sumi et al. 2011).

Cold Super-Earths and Neptunes

Although the microlensing method is, like any other technique, more sensitive to massive planets, with a sensitivity decreasing with mass as $\sim \sqrt {m_p}$, the third and fourth microlensing detections were super-Earth and Neptune-like planets, suggesting that these planets are more common than gas giants. OGLE-2005-BLG-390Lb was the first super-Earth detected via microlensing (Beaulieu et al. 2006), with light curve shown in Fig. 6. A Bayesian analysis using a galactic model, combined with a θ_E measurement indicated that the planet is likely to be a ∼ 5.5 M_⊕ super-Earth orbiting a M ∼ 0.22 M_⊙ star, at a projected separation of ∼2.6 AU. The equilibrium temperature of the planet is ∼50 K, making it the first cold super-Earth discovered at that time.

Fig. 6

OGLE-2005-BLG-390Lb light curve (Beaulieu et al. 2006)

This discovery was immediately followed by the detection of a Neptune-/Uranus-like planet, OGLE-2005-BLG-169Lb (Gould et al. 2006), confirming that such low-mass planets should be common. The mass and distance estimates of this system were confirmed and refined down to a 10% precision (m_p = 13.2 ± 1.3 M_⊕) with high angular resolution observations, taken 6 and 8 years after the event using HST (Bennett et al. 2015) and the Keck telescope in Hawaii (Batista et al. 2015), respectively. It is the first case of a planetary microlensing event for which the source and the lens are resolved, being separated by ∼62 mas on the Keck images. It allowed the detection of the lens brightness and the lens-source relative proper motion. The latter provides an additional mass-distance relation of the lens system.

Several other cold super-Earths and Neptunes have been detected (Muraki et al. 2013; Sumi et al. 2010, 2016; Furusawa et al. 2013), as well as two very low-mass planets of m_p ∼ 3.2 M_⊕ (Bennett 2008b; Kubas et al. 2012) and m_p ∼ 1.66 M_⊕ (Gould et al. 2014b) .

Triple Lens Systems

The first triple lens system discovered by microlensing is the two-planet one-star system, OGLE-2006-BLG-109Lb,c, observed in 2006 (Gaudi et al. 2008; Bennett et al. 2010). This system presents a remarkable similarity to a scaled version of Jupiter and Saturn. It is also one of the most complex microlensing light curves observed to date, exhibiting five distinct features from the two planets (Fig. 7). Finite source and parallax effects were detected for this event, as well as orbital motion of the Saturn-like planet. This led to a complete solution to the lens system and provided the physical characteristics of the two planets.

Fig. 7

OGLE-2006-BLG- 109Lb,c: a Jupiter/Saturn analog (Figure from Gaudi et al. 2008)

The next published multiple planet event was OGLE-2012-BLG-0026Lb,c (Han et al. 2013), composed of a Jupiter-like planet and a sub-Saturn orbiting a solar-mass star. This result was confirmed and refined by Beaulieu et al. (2016) who detected the flux coming from the lens using the Keck telescope.

The three other triple lens microlensing systems are one-planet two-star systems. OGLE-2007-BLG-349(AB)c is the first circumbinary planet found by microlensing (Bennett et al. 2016). The main features of the light curve are two cusp approaches that reveal the presence of a companion with a Saturn-to-Sun mass ratio, but only a third body could explain the slope between the two bumps. At that point of the analysis, it is not possible to disentangle the scenario of an additional planet, or an additional star. HST high-resolution observations ruled out the two-planet scenario because the lens flux could not explain the brightness of the host star in such a configuration. The system is then likely to be a Saturn-like planet orbiting two tight M dwarfs, separated by only ∼0.08 AU, at ∼2.6 AU. Figure 8 compares this system to the known circumbinary systems, making it the most compact regarding the binary stars, and the largest regarding the planetary orbit.

Fig. 8

OGLE-2007-BLG-349(AB)c: the first circumbinary planet found via microlensing compared to known circumbinary planet systems. The black dots show the orbital separation of the stars, and the crosses represent the planet separation from the stars center of mass (except for the microlensing one that shows a black dot with error bars) (Figure from Bennett et al. 2016)

OGLE-2013-BLG-0341Lb,c is a triple lens system with a very small planetary signal, and a large signature from the crossing of a resonant caustic due to a binary star (Gould et al. 2014b). A low-amplitude bump in a very early phase of the event enabled the authors to identify the binary as being wide, separated by ∼15 AU. The planet orbits one of them at ∼1 AU. It is also the smallest microlensing planet discovered to date (∼ 1.66 M_⊕). A similar system has been published by Poleski et al. (2014), OGLE-2008-BLG-092, i.e., a wide stellar binary in which the primary star is orbited by a Uranus-like planet (∼ 4 M_Uranus).

All these triple lens events (except the last one) exhibited finite source and parallax effects, allowing to derive precise estimates of their physical properties from the best models .

An Exomoon Around a Free-Floating Planet?

Bennett et al. (2014) has shown that it is already possible to detect exomoons of terrestrial mass, orbiting either free-floating planets or planets distant from their star. MOA-2011-BLG-262 is a system composed of two bodies with a mass ratio of 3.8 × 10⁻⁴ and a very short duration. The analysis led to two scenarios, a nearby 0.5 M_⊕ exomoon orbiting a gaseous rogue planet of ∼ 5 M_J or a high-velocity system composed of a Neptune orbiting an M dwarf or a brown dwarf in the galactic bulge. Adaptive optics observations did not allow to discriminate these two models. However, if exomoons are abundant, with more systematic space-based observations in the future, it will be possible to break the mass/distance degeneracy and identify such objects.

Massive Planets Around Very Low-Mass Stars/BDs

Determining the frequency of planets orbiting low-mass stars is of interest because these systems provide important tests to planet formation theories. In particular, the core accretion theory predicts that gas giants should be less common around low-mass stars (Laughlin et al. 2004; Ida and Lin 2004). Indeed, in smaller protoplanetary disks, hydrogen and helium might dissipate before the ∼ 10 M_⊕ planet cores grow enough to trigger rapid accretion. This lack of efficiency in the accretion process would be responsible for a larger population of failed gas giants around low-mass stars. Numerous microlensing discoveries seem to confirm these expectations, being cold 10–40 M_⊕ planets around M dwarfs (e.g., Gould et al. 2006; Muraki et al. 2013; Furusawa et al. 2013; Sumi et al. 2016). However, microlensing discoveries also revealed a population of giant planets >1 M_J around very low-mass stars (< 0.2 M_⊙) (e.g., Batista et al. 2011; Jung et al. 2015; Han et al. 2016b) and even around brown dwarfs (BDs) (Han et al. 2013; Choi et al. 2013). For these particular systems, it is still not clear if they have been formed through gravitational instabilities (Boss 2006) or core accretion. Moreover, the microlensing BD systems present an unusual morphology compared to most of those detected with other methods, since they are very tight binary systems (< 1 AU versus ∼10–45 AU). Here again, it is not clear if they result from the same formation mechanisms as the wide systems, that are believed to come from the same process as gravitational fragmentation for stellar binaries (Lodato et al. 2005).

Stellar Remnants

Aside from searching for exoplanets, microlensing is also one of the rare ways to explore the population of black holes and neutron stars. Three black hole candidates had been found in the past, MACHO-98-6, MACHO-96-5, MACHO-99-22, but they could not be confirmed because there was no measurement of θ_E (Bennett et al. 2002; Poindexter et al. 2005; Mao et al. 2002).

From the OGLE-III database of 150 million objects observed from 2001 to 2009, Wyrzykowski et al. (2016) published a list of 15 stellar remnants candidates, combining parallax and brightness measurements from microlensing light curves. The most massive black hole of their sample is estimated to be 8.3 M_⊙ at a distance of 2.4 kpc. Recent high-resolution images of these candidates should provide constraints on their lens brightness (or lack of light).

Shvartzvald et al. (2015) analyzed a binary microlensing event, OGLE-2015-BLG-1215La,b, that was observed simultaneously from space and Earth, during the 2015 Spitzer campaign. They found that the lens system is likely to lie in the galactic bulge, with the more massive component being > 1.35 M_⊙, either a black hole or a neutron star. The authors advocate future high-resolution observations of this system, in order to measure the proper motion of the bright companion, likely to be a low-mass main-sequence star or a white dwarf.

In addition to photometric searches, recent projects have also sought to detect stellar remnants, in particular single black holes and neutron stars, using astrometric microlensing. Indeed, in addition to the well-known photometric effect, a microlensing event also leads to a small apparent change in the position of the source (Hog et al. 1995; Dominik and Sahu 2000). This effect is small but, if detected, leads to direct constraint on θ_E, facilitating direct mass measurements via microlensing. With recent improvements in technology and astrometric precision capabilities, several projects are now underway to detect this effect, mostly using HST. These projects aim to detect stellar-mass black holes, neutron stars, and even probe for the existence of intermediate-mass black holes in globular clusters (Kains et al. 2016). Future missions like WFIRST will offer enhanced astrometric precision, allowing us to constrain the demographics of stellar remnants in the Milky Way .

Exoplanet Mass-Ratio Function

More than 3500 planets have been discovered to date, in about 2700 planetary systems. The vast majority of them were detected by the transit and RV surveys, i.e., two detection methods that are most sensitive to close-in planets. From these surveys, Winn and Fabrycky (2015) estimated that half of the sunlike stars should host a small planet (Earth to Neptune) within 1 AU and 10% of them a giant planet within a few AU. Although only about 40 microlensing planets have been discovered, this technique is complementary to others because it probes a different parameter space, being sensitive to cold low-mass planets beyond the snow line, at ∼1 to 10 AU from their host star (Bennett 2008b; Gaudi 2012).

Several statistical microlensing studies have been published, before and after the survey groups had updated their observational strategy and moved to high-cadence monitoring with wide-field imagers. The first studies were done even before the first planet was discovered, computing detection efficiency on microlensing events, to derive upper limits on the frequency of giant cold planets (Gaudi et al. 2002; Snodgrass et al. 2004). Then, from the ten first planetary detections, Sumi et al. (2010) analyzed the distribution of planet mass ratios beyond the snow line and found that it could follow the power law $dN/d\log q=q^{-0.68\pm 0.20}$, implying that Neptunes would be much more common than Jupiters.

Gould et al. (2010) used the argument of a non-biased systematic monitoring of high-magnification events (HME), to derive statistics from a sample of the 13 observed HME at that time, that involved 6 planets. They found a ∼36% absolute frequency of planets, for a mass-ratio centered on q = 5 × 10⁻⁴ and a projected separation beyond the snow line (∼ 3a_snow).

Cassan et al. (2012) published an analysis based on the PLANET follow-up data during 6 years and combined their estimate with the previous results. They found a planet frequency that can be described by the power law $d^2N/d\log M\,d\log a\sim 0.80(M/M_\oplus)^{-0.65\pm 0.34}$, being consistent with every star hosting a super-Earth on average.

Clanton and Gaudi (2016) searched for a common power law to connect the results from five independent surveys, microlensing (Gould et al. 2010; Sumi et al. 2010), RV (Montet et al. 2014) and direct imaging (Lafrenière et al. 2004; Bowler et al. 2015). They derived a planet distribution of the form $d^2N_{\mathrm {{pl}}}/(d\log m_p d \log a)=\mathscr {A}(m_p/M_{\mathrm {{Sat}}})^\alpha (a/2.5\,AU)^\beta$, where α ∼−0.57, β ∼ 0.49, $\mathscr {A}\sim 0.26\,\mathrm {{dex}}^{-2}$, and with an outer cutoff at a_aout ∼ 9.7.

All the statistical results presented above were based on a relatively small number of microlensing events and ≤10 planets. Within the last decade, the number of microlensing events has substantially increased due to the new generation of high-cadence wide-field surveys (OGLE, MOA, Wise, KMNet), and these planet abundances have been updated recently. Shvartzvald et al. (2016) estimated the planet mass ratio function from a 4-year data survey of a 8 deg² galactic field by OGLE, MOA, and Wise. Their results show that 55% of the stars host at least one planet beyond the snow line, with a mass-ratio $-4.9<\log (q)<-1.4$, with Neptunes being ∼10 times more common than Jupiters.

Finally, Suzuki et al. (2016) conducted a statistical analysis over 6 years of the MOA-II microlensing survey, being the study that covers the biggest amount of microlensing events (1474) and planetary detections (23) to date. They find that the power law describing the planetary mass-ratio function is no longer a single power law as previously published, but a broken power law of the form $d^2N/(d\log q\,d\log s)=\mathscr {A}(q/1.7\times 10^{-4})^{n}\,s^{m}\,\mathrm {{dex}}^{-2}$, where $\mathscr {A}=0.61^{+0.21}_{-0.16}$ and $m=0.49^{+0.47}_{-0.49}$. For the low mass-ratio regime, q < 1.7 × 10⁻⁴, $n=0.6^{+0.5}_{-0.4}$, and for the bigger mass ratios, q > 1.7 × 10⁻⁴, n = −0.93 ± 0.13. Their distribution thus reaches a peak around the Neptune mass ratio, as shown in Fig. 9, with a slightly lower abundance of Earthlike planets than Neptunes, beyond the snow line. Suzuki et al. (2016) compared their microlensing mass ratio function to previous RV results. For low mass ratios, q ∼ 3 × 10⁻⁵, it is consistent with Mayor et al. (2009) and Howard et al. (2010), the latter being for period orbits < 50 days. For bigger mass ratios, ≳ 10⁻³, their values are higher than Cumming et al. (2008), and their function rises more steeply toward Neptune mass ratios. That might indicate that there are more Neptunes beyond the snow line than at ∼1 AU and that cold Neptunes would not be highly subject to migration .

Fig. 9

Planetary mass ratio distribution from the 2007–2012 MOA-II microlensing survey, compared to previous microlensing and RV results (Mayor et al. 2009; Howard et al. 2010; Cumming et al. 2008; Johnson et al. 2010; Bonfils et al. 2013; Montet et al. 2014) (Figure from Suzuki et al. 2016)

Galactic Distribution of Planets

Planets found via microlensing are distributed over a wide range of distances, from the Earth to the galactic center. According to the lens distance estimates of the actual ∼40 microlensing discoveries, about 10% are likely to be closer than 1 kpc, and ∼10% in the bulge, >7 kpc. Disk and bulge populations of stars might have distinct properties (e.g., age, metallicity, density of nearby stars, intensity of radiations) that could affect the planet formation process. Therefore, comparing the planet frequencies and system morphologies of these two populations may be of great interest.

The actual microlensing planet sample is not ideal for building objective distance distributions because it has involved heterogeneous hunting and modeling strategies over the last decade. Some planetary anomalies were covered by follow-up groups, others only by survey groups. Some of the events benefitted from a dense observation coverage, parallax, and finite source constraints, while others required Bayesian analyses using galactic models to be characterized.

Penny, Henderson et al. (2016) compared the 31 planets discovered before 2016 to a simulated lens distance sensitivity distribution. They used simulations made by Henderson et al. (2014) for the KMNet survey that follow the Han and Gould (2003) galactic model. Their result reveals a discrepancy between the distance estimates from data and their model, expecting a larger amount of systems at further distances (6–7 kpc) and less nearby systems ($\lesssim$2 kpc). They examined different factors that could be responsible for this discordance and found several possible explanations, aside from the quick conclusion that the galactic bulge would be devoid of planets. Indeed, their “model vs data” shows a better agreement when they take into account additional dependences of planet detection efficiency on some microlensing parameters, such as the event timescale.

Recently, Zhu et al. (2017) analyzed all 41 microlensing events from the 2015 Spitzer campaign that were simultaneously observed by OGLE and KMNet. They computed a planetary detection efficiency on this free-planet sample and combined these calculations with their determination of the microlensing event rate. It amounts to adding planet sensitivity to the expected distance distribution of observed lenses. From this, they were able to build a distribution of planet sensitivity (i.e., no longer host star sensitivity) as a function of the lens distance. Their total distribution divides into a disk and a bulge distribution, plus the contribution of high-magnification events (A_max > 8), computed separately, as such events generally provide much stronger constraints. They predict that if the planet formation is equally efficient in the disk and in the bulge, ∼1/3 of the planet population unveiled by an automatic high-cadence survey with parallax measurements should be in the bulge. They also find a lens mass distribution peaking at ∼ 0.5 M_⊙, as anticipated.

The methodology developed by Zhu et al. (2017) could be used on large samples of planets that will be found by dedicated space missions (Euclid and WFIRST), supported by ground-based observations. They should provide much better constraints on the planet distribution as a function of their host’s location within the galaxy .

Free-Floating Planets

From the 2006–2007 MOA-II survey, Sumi et al. (2010) (S11 hereafter) identified an overabundance of short-timescale microlensing events (t_E < 2 days) that could not be explained by known stellar and brown dwarf populations. They interpreted these light curves as signatures of either free-floating planets (FFP) or wide-orbit Jupiters (separation ≳ 10 AU from their host star). Bennett et al. (2012) estimated that if all of this population was due to planets bound in wide orbits, their semimajor axis would likely be > 30 AU.

Clanton and Gaudi (2017) generated a population of distant bound planets with distributions of mass and distance consistent with microlensing, RV, and direct imaging surveys (Clanton and Gaudi 2016) and estimated which fraction of them would not show any signature of the host star in a microlensing light curve. They tried to fit the microlensing event timescale distribution measured by S11 with a lens mass function composed of brown dwarfs, main-sequence stars, and remnants, augmented by the distribution of these bound planets with no star signature. They found that this fraction does not entirely explain the excess of short-timescale events from S11 and concluded that ∼60% of it should be due to FFP. Figure 10 shows their timescale distribution, consistent with S11 data and composed of three contributors including their predicted population of FFP. This would imply that FFP are ∼1.3 times more abundant than main-sequence stars in the galaxy.

Fig. 10

Best fit model from Clanton and Gaudi (2017) to the observed timescale distribution of S11 (black histogram), showing the individual contributors: brown dwarfs, main-sequence stars, remnants (thin pink and blue lines centered on long timescales), bound planets at wide separations (thin pink and blue lines centered on short timescales), and FFP (dashed black lines). The total is shown in thick lines. Blue and pink colors correspond to two formation processes, disk instability (hot start) and core accretion (cold start), respectively (Figure from Clanton and Gaudi 2017)

K2-C9 was designed by the microlensing community as a program with the main purpose of detecting FFP with Kepler-Earth parallax measurements (Henderson et al. 2016). Penny et al. (2016) expected a handful of FFP detections with Kepler according to the predictions of S11. With the revised numbers of Clanton and Gaudi (2017), this amount should be reduced by a factor of ∼ 0.6, and it might be unlikely to derive reliable statistics from this short survey. WFIRST combined with ground-based surveys will surely provide stronger constraints on the abundance and characteristics of free-floating planets.

Conclusion and Future Prospects

Microlensing is currently capable of detecting low-mass planets beyond the snow line, as well as Jupiter-mass free-floating planets, with a network of wide-field telescopes strategically located around the world. This technique is almost uniformly sensitive to planets orbiting all types of stars and remnants, while other methods are most sensitive to FGK dwarfs and are now extending to M dwarfs. It is therefore an independent and complementary detection method for aiding a comprehensive understanding of the planet formation process.

Ultimately, comprehensive demographics of planets down to below an Earth mass at separations ranging from the habitable zone to infinity will be achieved from a space-based microlensing survey, such as the WFIRST mission (Spergel et al. 2015) and hopefully the Euclid mission (Penny et al. 2013). The concept was initiated with a dedicated mission (Bennett et al. 2002) called the Microlensing Planet Finder (MPF), which has been proposed to NASA’s Discovery program but not selected. The objective is to be able to monitor turnoff stars in the galactic bulge as microlensing sources. Bennett and Rhie (1996) had shown that the detectability of exoplanets via microlensing depends strongly on the size of the source star, lower-mass planets being detectable only with small source stars. The ideal microlensing planet hunter is a 1–2 m class telescope, with a wide-field near-infrared imager (about ∼ 0.5 deg²) and high angular resolution (0.1–0.3 arcsec/pixel), to observe the galactic bulge with a sampling rate better than 20 min.

Euclid is an ESA medium-size mission, scheduled to launch in 2020, that will measure parameters of dark energy using weak gravitational lensing and baryonic acoustic oscillation and test the general relativity and the cold dark matter paradigm for structure formation. Since its original submission in 2007 to ESA, a microlensing planet hunting program has been listed as part of the Legacy science (Beaulieu et al. 2008). The vision adopted by the Europeans of a joint mission with Dark Energy probes and microlensing has been promoted and adopted by the Astro 2010 Decadal Survey when it created and ranked as top priority the WFIRST mission.

A baseline for the design of the Euclid microlensing survey is described by Penny et al. (2013) with a detailed simulation demonstrating the capabilities and the expected scientific outcomes. They conducted the same simulations for the WFIRST survey (Spergel et al. 2015) and predicted that such a mission will detect several thousand bound planets, in addition to several thousand free-floating planets. Moreover, for a planet separation of ∼2–5 AU, i.e., the range of highest microlensing sensitivity, it will be possible to detect masses lower than Mercury, and even down to the mass of Ganymede (see Fig. 11). Combined with the Kepler survey, it will determine how common Earthlike planets are over a wide range of orbital parameters.

Fig. 11

Sensitivity range of the WFIRST and Euclid missions in a planet-mass versus semi-major axis diagram, complementary to the parameter space probed by the Kepler mission (Figure courtesy of M. Penny (Ohio State University) & J. D. Myers (JPL))

Acronyms

MACHO::

MAssive Compact Halo Objects

EROS::

Expé rience de Recherche d’Objets Sombres

OGLE::

Optical Gravitational Lens Experiment

MOA::

Microlensing Observations in Astrophysics

PLANET::

Probing Lensing Anomalies NETwork

μFUN::

μlensing Follow-up Network

MiNDSTEp::

Mincrolensing Network for the Detection of Small Terrestrial Exoplanets

KMTNet::

Korean Microlensing Telescope Network

WFIRST::

Wide-Field InfraRed Survey Telescope