###### Abstract

Gravitational microlensing finds planets through their gravitational influence
on the light coming from a more distant background star. The presence of the
planet is then inferred from the tell-tale brightness variations of the
background star during the lensing event, even if no light is detectable from
the planet or the host foreground star. This review covers fundamental
theoretical concepts in microlensing, addresses how observations are performed
in practice, the challenges of obtaining accurate measurements, and explains
how planets reveal themselves in the data. It concludes with a presentation of
the most important findings to-date, a description of the method’s strengths
and weaknesses, and a discussion of the future prospects of microlensing.

###### keywords:

extrasolar planets; general relativity; microlensing; galactic bulge;
planetary systems; exoplanets

8 10 365 Received: 3 July 2018; Accepted: 26 September 2018; Published: 29
September 2018 Microlensing Searches for Exoplanets Yiannis Tsapras Yiannis
Tsapras Correspondence<EMAIL_ADDRESS>Tel.:
(+49)-(0-)6221-54-1819

##  1 Introduction

###  1.1 Discovering Exoplanets

Planets begin forming in the gaseous disks surrounding young stars, but the
exact physical processes that drive their formation and evolution are still
not fully understood Ford (2014). Theory predicts that these disks should last
for a few million years and that planetary embryos can migrate while they are
still embedded in them Kley and Nelson (2012). Some are eventually engulfed by
their host stars. Others survive and grow by accreting gas and dust from the
surrounding disk but end up far from where they first started forming. A
question may then be posed: How are these planets distributed around their
host stars and are there similarities with our own system or is the Solar
system in some ways unique?

The presence of liquid water on a planetary surface is no guarantee for the
existence of life, although it was a necessary ingredient for the development
of life on Earth. Liquid water can only exist at a certain range of distances
from the host star, the so-called habitable zone, but little is known about
the fraction of planets residing within the habitable zone of their host stars
and the stability of their orbits; nor is it yet possible to make definite
statements about the physical characteristics of even the closest exoplanets
de Wit et al. (2018); Anglada-Escudé et al. (2016). The situation becomes even
more challenging when looking for planets beyond the snow line, which is the
boundary distance from a star beyond which the ambient temperature drops below
$\sim$160 K and water turns to ice (Lecar et al., 2006; Min et al., 2011). The
exact location of the snow line in a protoplanetary disk depends on the mass
accretion rate but it roughly corresponds to distances of a few AU away from
the host. However, the sensitivity of the most successful planet-detection
methods to-date drops rapidly for orbits $\gtrsim$1 AU, making these planets
particularly hard to find.

Discovering planets and studying their distribution is the first goal of the
exoplanet community, from both the observational and theoretical perspectives,
in order to better understand their diversity and the physical processes that
drive their formation and evolution. To this effect, a number of different
techniques have been developed and employed over the past two decades. They
are as follows:

  * •

Radial velocity detects planets by measuring the periodic shifting of spectral
lines on the stellar spectrum that is induced by the orbital motion of the
planet around the host star. Discoveries from radial velocity surveys over the
past 20 years provide the first evidence that the incidence of giant planets
increases with increasing stellar mass, at least for planets with short
orbital periods Cumming et al. (2008). Although the radial velocity method has
been remarkably successful in identifying planets near their hosts,
discoveries of giant planets beyond 1 AU for low-mass stars have been
comparatively few Montet et al. (2014) because detecting planets at larger
orbital distances requires mission lifetimes that span several years to
decades.

  * •

Transit surveys, from the ground and from space, identify exoplanets with near
edge-on orbits that pass in front of their host stars causing them to appear
dimmer for the duration of the event. The most successful of these surveys to
date has been the Kepler space mission Batalha et al. (2013), having found
hundreds of large and small planets out to $\sim$1 AU from their hosts and
thousands of candidates. Although its primary mission was suspended in 2013
due to the failure of two reaction wheels, without which the telescope could
not point accurately, it provided strong evidence that terrestrial planets are
far more numerous than gas giants for periods less than 85 days Fressin et al.
(2013).

  * •

Astrometry involves very precise measurements of a star’s position in the sky
over a long period of time. If the star has planets orbiting it, then minute
periodic shifts in its measured position could be detectable. The difficulty
lies in extracting such highly accurate measurements, because the expected
shifts are minuscule. This is the reason that no astrometric planet candidates
have been confirmed to date. The Gaia space mission may change all that. It
launched on 19 December 2013, and it is estimated it will discover thousands
of planets in its expected 5-to-10-year lifetime Perryman et al. (2014). It is
worth noting that the strength of the astrometric signal is inversely
proportional to the distance of the planet and its host star from the Earth.

  * •

Pulsar timing provided the first successful exoplanet detections around
Pulsars in 1992 Wolszczan and Frail (1992). Pulsars are rapidly rotating
neutron stars; they are the superdense stellar remnants of supernova
explosions in the distant past. The beam of electromagnetic radiation they
emit as they rotate is detected on the Earth as it sweeps by, and is recorded
as a highly regular and ultra-precise pulsation signal. Planets orbiting the
pulsar reveal themselves through irregularities in the arrival time of the
pulses.

  * •

Direct imaging is the only method aiming to visually isolate an exoplanet from
its host star. Through high-contrast imaging using adaptive optic systems on
large telescopes, massive exoplanets orbiting at the outer reaches of their
systems have been discovered Macintosh et al. (2015). Only those planets that
lie very far from their host star can be found with this method, because a
star will outshine any nearby planets by a billion times in the optical part
of the spectrum.

  * •

Microlensing, in many ways the odd one out, is the topic of this review. It
discovers planets by measuring characteristic variations in the brightness of
a background star generated by the gravity of intervening objects along the
line of sight.

These are all complementary techniques that explore different regimes of the
planet mass versus orbit size distribution (see Figure 1). In order to better
appreciate the physical processes that form these exoplanets, theoretical work
is informed and validated through new discoveries over the full range of
planet mass and orbital separation. Transit missions and radial velocity
surveys have already established that the abundance of hot planets increases
dramatically towards lower mass objects and larger orbital separations for
FGK-host systems Mayor et al. (2011). There is also some evidence from
microlensing that the planet frequency continues to rise towards smaller
planet masses at even larger orbital radii Cassan et al. (2012):
$17^{+6}_{-9}$% of stars host a cool Jovian-type (0.3–10
$M_{\mathrm{Jupiter}}$) planet, $52^{+22}_{-29}$% have cool Neptunes (10–20
$M_{\mathrm{Earth}}$) and as many as $62^{+35}_{-37}$% have cool super-Earth
(5–10 $M_{\mathrm{Earth}}$) companions. This hints that there is a still
untapped large population of low-mass, cool planets, as predicted by planet
population synthesis models Ida et al. (2013), but the evidence remains
inconclusive since very few planets have been discovered beyond the snow line.
Furthermore, a recent study by Suzuki et al. (2016) found that the
distribution of mass ratios and separations of their microlensing planet
sample was better fit with a broken power-law model, implying that cold
Neptunes could well be the most common type of planets beyond the snow line.
The question is far from settled. These are the planets microlensing is
uniquely sensitive to, and finding them is critical in understanding the
physical processes that drive planet formation.

Figure 1: Planet-finding techniques are complementary. This “orbit size-vs-
planet mass” diagram shows all reported exoplanets to-date. The semi-major
axis has been scaled to the approximate location of the snow line of the
planet-hosting star (assuming $a_{snow}\approx 2.85M_{*}^{3/2}$AU). Transits
and radial velocity are exceptionally good at finding “hot” planets close to
their host stars, whereas microlensing and direct imaging are more efficient
in discovering “cold” planets. [Figure based on data from the NASA Exoplanet
Archive.]

###  1.2 What is Microlensing?

According to Einstein’s Theory of General Relativity, gravity may be thought
of as a curvature in space-time. Any massive stellar object will therefore
deflect the path of light-rays passing close to it, as famously demonstrated
by Eddington’s observations during the total Solar eclipse of 29 May 1919
Einstein (1911, 1915); Dyson et al. (1920). This phenomenon is called
gravitational lensing and it involves a foreground stellar mass object bending
light due to its gravity and a background stellar mass object which acts as
the source of light. Any sufficiently massive object can act as a
gravitational lens, from single stars to entire galaxies, and any object
emitting radiation can act as the source. For reasons of convenience, the
object doing the lensing is commonly simply referred to as the lens, while the
background object is the source. Gravitational lensing differs from
conventional optical lensing in that there is not a single point of focus; the
bending becomes more severe the closer the light-ray passes to the lens. As a
consequence, multiple distorted images of the source appear around the lens.
The number of images produced depends on the mass distribution doing the
lensing. Under the assumption of point lenses, a single lens will always
produce two images, whereas a binary lens will generate three or five images,
depending on the position of the source relative to the lens on the plane of
the sky. In the special case where lens and source are in perfect alignment as
viewed from the observer’s perspective, the multiple images all merge to form
a bright ring around the lens, the so-called ‘Einstein ring’. The radius of
this ring, as we shall discuss in Section 2, depends on the relative distances
between observer, lens and source, as well as the mass of the lens itself.

Gravitational microlensing, or simply microlensing, is a sub-category of
gravitational lensing in which both lens and source are stars and where the
angular distances between the images generated by the lensing effect are of
the order of milli-arcseconds. This means that no telescope in operation today
can individually resolve the images; only variations in the brightness of the
source are observed. Prompted by a conversation he had with Rudi Mandl, a
Czech engineer and amateur scientist, Einstein addressed this phenomenon in a
short article he published in 1936 Einstein (1936). In that article he worked
out the basic equations and concluded that “there is no great chance of
observing this phenomenon”. In fact, Renn et al. (1997) report that Einstein
had already derived the relevant mathematical relations in his research
notebooks dating back to 1912 (3 years before completing his general theory of
relativity), but had never sought to publish them thinking them of little
scientific value. This was not unreasonable given the technology available at
the time. Modern estimates of the optical depth to gravitational microlensing,
i.e., the probability that any given star is microlensed at any particular
time, are of order $10^{-6}$ in the direction of the Galactic bulge Kiraga and
Paczynski (1994); Udalski et al. (1994); Peale (1998); Sumi et al. (2013).
Finding microlensing events is like looking for the proverbial needle in the
haystack. Yet with today’s wide-field CCD cameras, astronomical surveys
monitor regularly about a billion stars and discover about 2000 microlensing
events annually Kim et al. (2016); Udalski et al. (2015). It is worth noting
that the number of reported events and the optical depth are not independent
quantities, as we shall see in Section 2.1.7; they are related by the event
duration. Fewer than 10 of these events every year will display the
unmistakable signs of planetary companions to the lens star, a situation
unlikely to change much until the launch of the WFIRST space mission in the
mid-2020s Poleski (2016); Ban et al. (2016).

##  2 Microlensing Basics

This section covers the theoretical background of the microlensing method,
without going into excessive detail. Readers interested in the fundamental
equations, the geometry and phenomenology of microlensing will find here
enough information to gain a good understanding of the particulars of this
technique. For those that want to delve deeper, the book Gravitational Lenses
by Schneider et al. (1992) provides detailed derivations and investigates the
underlying physics. Past reviews by Dominik (2010), Mao (2012), Gaudi (2012)
and the book Gravitational Lensing in Astronomy by Wambsganss (1998) are also
excellent sources of information. Students and researchers wanting to learn
more about microlensing can access educational resources and training
exercises on-line at http://microlensing-source.org/. Casual readers not
interested in the mathematics may skip this section without loss of
continuity.

Note that the thin-screen and small-angle approximations are used throughout
this section, though not explicitly mentioned. The thin-screen approximation
assumes that the deflection occurs instantaneously during the crossing of the
lens plane (at $D_{L}$). This is valid since the distances to the source and
the lens are much greater than the range within which the actual deflection
takes place. The small-angle approximation is also valid because all the
angles we are dealing with are very small for all cases of interest.

###  2.1 Single Lens

Gravitational microlensing will occur any time two stars are sufficiently well
aligned so that the light from the background ‘source’ star, at distance
$D_{S}$ from the observer, will bend under the influence of the gravitational
field of the intervening ‘lens’ star, at distance $D_{L}$, and make the source
appear temporarily brighter. As previously mentioned, microlensing events are
extremely rare occurrences, so most modern microlensing surveys point their
telescopes toward the very dense stellar fields of the Galactic bulge in order
to maximize their chances of finding them (The OGLE project, 2018; The MOA
project, 2018; The KMTNet project, 2018). Together, low-mass K and M-dwarf
stars make up about 90% of all stars in the Galaxy. Their relative abundance
is at least partly reflected in the type of lensing stars the surveys detect;
indeed, most lensing stars turn out to be faint K and M-dwarfs, rather than
the hotter G-type stars, like our Sun.

The basic concepts of microlensing are easier to introduce by considering a
simple scenario: a point lens in the foreground bending the light of a point
background source. This is commonly referred to as “point source-point lens”
(PSPL) lensing Einstein (1936); Paczynski (1986); Horne et al. (2009);
Hundertmark et al. (2018). It is a sufficiently good approximation for the
illustrative cases we will be considering here, but let us note now that for
certain geometric arrangements the finite extent of the source star becomes
important. Such considerations are covered in Section 2.2.

####  2.1.1 The Deflection Angle

The basic geometry of microlensing is illustrated in Figure 2: Were an
observer able to visually resolve arbitrarily small angular separations in the
sky, they would not be able to see the source star itself, but only two
distorted images generated by the lensing effect. In reality, the images are
so closely packed together that they appear as a single object on the
detector. According to general relativity, a light ray passing close to a lens
of mass $M_{L}$ at distance $\xi$, will be deflected by the “Einstein angle”
(Einstein, 1915; Schneider et al., 1992)

$\hat{\alpha}=\frac{4GM_{L}}{c^{2}\xi},$ (1)

where $c$ is the speed of light in vacuum and $G$ the gravitational constant.

Figure 2: The geometry of a single lens microlensing event. Light rays
originating from a source star are deflected near the lens. As a result, the
observer does not perceive the light of the source itself, but instead
measures the combined contribution of two distorted images of the source. The
distances from the observer to the lens and the source are $D_{L}$ and
$D_{S}$, respectively. The distance between lens and source is $D_{LS}$.
$\beta$ is the angular position of the unlensed source, $\alpha$ denotes the
deflection angle and $\theta$ is the angular position of one of the images.
The red and blue rectangles represent the parallel lens and source planes at
$D_{L}$ and $D_{S}$, respectively, which are perpendicular to the observer-
lens axis (dotted).

####  2.1.2 The Lens Equation

Let us consider Figure 2 again. A bright source at distance $D_{S}$ is being
observed from the Earth. Between the observer and the source lies a lens at
distance $D_{L}$ from the observer. The distance between lens and source is
$D_{LS}$. The angle between the true position of the source (in the absence of
lensing) is $\beta$, whereas the angle between the observer-lens optical axis
and the image considered is $\theta$. These quantities are related by

$\beta=\theta-\hat{\alpha}\frac{D_{LS}}{D_{S}}.$ (2)

This is commonly referred to as the lens equation. Given an image position
$\theta$, we can use the lens equation to evaluate the real position of the
source, $\beta$.

####  2.1.3 The Einstein Radius

Under our assumption of a point-mass lens, we can substitute in $\hat{\alpha}$
and write the lens equation as

$\beta=\theta-\frac{D_{LS}}{D_{S}D_{L}}\frac{4GM_{L}}{c^{2}\theta},$ (3)

where were have used the substitution $\xi=\theta D_{L}$.

Since the system is rotationally symmetric, a source located exactly on the
optical axis ($\beta=0$) will be imaged as a ring: the Einstein ring. By
setting $\beta=0$ in Equation (3) and solving for $\theta$, we obtain the
expression for the angular Einstein radius:

$\theta_{E}=\sqrt{\frac{D_{LS}}{D_{S}D_{L}}\frac{4GM_{L}}{c^{2}}}.$ (4)

The angular Einstein ring radius provides a natural scale for describing the
lensing geometry. The angular separation between the images of the source
generated by the lensing effect is of the order of 2$\theta_{E}$. The closer
the angular distance of the source to the observer-lens optical axis, the
stronger the magnification. In order to derive an average estimate for the
size of $\theta_{E}$, we can assume some typical values (Paczynski, 1986,
1991). Consider an M-dwarf lens star ($M_{L}=0.3M_{\odot}$) at a distance of
$D_{L}=6.5$ kpc from the Earth, and a source star at a distance
$D_{S}=8.5$kpc. The corresponding size of the angular Einstein ring radius is
then approximately

$\theta_{E}\approx
0.902\mathrm{mas}\left(\frac{M_{L}}{M_{\odot}}\right)^{1/2}\left(\frac{10\mathrm{kpc}}{D_{L}}\right)^{1/2}\left(1-\frac{D_{L}}{D_{S}}\right)^{1/2}\sim
300\mu\mathrm{as}.$ (5)

This corresponds to a physical distance on the lens plane of

$r_{E}=\theta_{E}D_{L}\sim
3\mathrm{AU}\left(\frac{M_{L}}{M_{\odot}}\right)^{1/2}\sim 1.6\mathrm{AU}.$
(6)

####  2.1.4 Image Positions

We can now simplify Equation (3) by substituting in $\theta_{E}$, and the lens
equation becomes $\beta=\theta-\theta_{E}^{2}/\theta$. Multiplying both sides
with $\theta$ yields a quadratic: $\theta^{2}-\beta\theta-\theta_{E}^{2}=0$
with solutions

$\theta_{\pm}=\frac{1}{2}\left(\beta\pm\sqrt{\beta^{2}+4\theta_{E}^{2}}\right).$
(7)

These solutions correspond to the angular positions of the images on the sky.
They appear close to the Einstein ring on opposite sides of the lens; the
minor image always inside the Einstein ring and the major image outside. The
angular separation between them is
$\Delta\theta=\theta_{+}-\theta_{-}=\sqrt{4\theta_{E}^{2}+\beta^{2}}\geq
2\theta_{E}$. As the apparent lens-source separation increases with time, the
minor image gets smaller and moves ever closer to the position of the lens,
until it disappears. On the other hand, the major image gradually approaches
the true position of the source until it coincides with it in terms of
position and brightness (see Figure 3).

Figure 3: (a) The face-on geometry of a single lens microlensing event. The
lens is the black dot at the center. The Einstein ring is indicated by the
edge of the larger gray circle. Source trajectories for five different impact
parameter values, $u_{0}$, are shown. For the trajectory passing closest to
the lens (blue line), 8 instantaneous positions for the source (unfilled blue
circles) and their corresponding images (red contours) are displayed. For the
source position highlighted (filled blue circle), the two images generated are
shown as filled red contours. The image centers always fall in a straight line
that passes through the position of the lens, as indicated by the dotted line.
The minor images move along inside the Einstein ring as the event progresses,
while the major images are always outside. Smaller lens-source separations on
the plane of the sky correspond to greater total magnifications. The $x$ and
$y$-axes mark distances on the lens plane in units of $\theta_{E}$; (b) Total
magnification as a function of time for the trajectories shown in panel a. The
filled blue circle marks the instantaneous magnification when the source is at
the corresponding position (filled blue circle) in panel a.

####  2.1.5 Magnification

Gravitational light deflection does not involve emission or absorption, so the
surface brightness of either image is identical to that of the original
source: surface brightness is conserved. The total flux received from an image
is just the product of its surface brightness with the area it covers on the
sky. Since the former remains unchanged during microlensing, the brightness
change (magnification) of an image relative to the unlensed source, is simply
given by the ratio of their areas (see Figures 2 and 3):

$A_{\pm}=\left|\frac{\theta_{\pm}}{\beta}\frac{d\theta_{\pm}}{d\beta}\right|.$
(8)

Substituting $\theta_{\pm}$ from Equation (7), we obtain:

$A_{+}=\frac{\theta_{+}}{2\beta}\left(\frac{\beta}{\sqrt{\beta^{2}+4\theta_{E}^{2}}}+1\right),\quad
A_{-}=\frac{\theta_{-}}{2\beta}\left(\frac{\beta}{\sqrt{\beta^{2}+4\theta_{E}^{2}}}-1\right).$
(9)

By normalizing all angles by $\theta_{E}$ and defining $u=\beta/\theta_{E}$
for the (normalized) source position, $y_{\pm}=\theta_{\pm}/\theta_{E}$ for
the (normalized) image positions we can rewrite Equation (7) as

$y_{\pm}=\frac{u\pm\sqrt{u^{2}+4}}{2}.$ (10)

The magnification of each image is then given by

$A_{+}=\frac{u^{2}+2}{2u\sqrt{u^{2}+4}}+\frac{1}{2},\quad
A_{-}=\frac{u^{2}+2}{2u\sqrt{u^{2}+4}}-\frac{1}{2}.$ (11)

Note here that the first term on the right-hand side of the equation above is
always $>1/2$ and the minor ‘-’ image appears inverted, as shown in Figure 3.
The total magnification is given by the sum of the individual magnifications:

$A=A_{+}+A_{-}=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}.$ (12)

In the special case when the source lies exactly at the Einstein radius, we
have $\beta=\theta_{E}$, $u=1$, and the source is magnified by
$A=3/\sqrt{5}\approx 1.34$. In general, the total flux $F(t)$ recorded by the
detector is not proportional to the magnification because it contains
contributions from other sources of light. These are discussed in Section
2.2.1.

The magnification is a function of time since lens and source are moving
relative to each other with a relative proper motion
$\mu_{LS}=v_{\perp}/D_{L}$, where $v_{\perp}$ is the relative transverse
velocity of the lens with respect to the source. This gives rise to the
characteristic shape of the event light curve seen in Figure 3b, i.e., of how
the brightness of an event changes with time. This is sometimes referred to as
the Paczyński curve (Paczynski, 1986). In the simplest case, the lens-source
trajectory can be assumed to be rectilinear, and we may write

$u(t)=\sqrt{u_{0}^{2}+\left(\frac{t-t_{0}}{t_{E}}\right)^{2}},$ (13)

where $u_{0}$ is the minimum impact parameter of the event, when the apparent
separation between lens and source is shortest and the magnification is
greatest (denoted by $A_{0}$), $t_{0}$ is the time when $u=u_{0}$ and
corresponds to the time when the peak of the light curve is reached, and
$t_{E}$ is the characteristic event timescale, which is the time it takes for
the source to move with respect to the lens by one Einstein ring radius.

The impact parameter may be determined at any time from the magnification by
rewriting Equation (12) as

$u=\sqrt{\frac{2A}{\sqrt{A^{2}-1}}-2}.$ (14)

Modeling single-lens event light curves usually involves three parameters that
describe the shape of the curve, such as $u_{0},t_{0}$ and $t_{E}$. Often, the
baseline magnitude of the source is also included, which is obtained when the
source is far enough apart from the lens that lensing effects are not
relevant.

####  2.1.6 Microlensing Timescales

Since $u_{0}$ and $t_{0}$ only depend on the trajectory of the source relative
to the lens, the only parameter holding any physical information about the
lensing system is $t_{E}$, because it depends on the distances between
observer, lens and source, the mass of the lens, and the relative proper
motion between source and lens, $\mu_{LS}$. We can express the event timescale
as

$t_{E}=\frac{\theta_{E}}{\mu_{LS}}.$ (15)

If we use Equation (5) and assume a typical value for the proper motion
$\mu_{LS}\sim 15\mu{\mathrm{as/d}}$, then for lenses of about one Earth-mass
$t_{E}$ is of the order of hours, whereas in the case of Solar-mass objects
the lensing effect can last for a few months (Wood and Mao, 2005; Wyrzykowski
et al., 2015; Tsapras et al., 2016).

####  2.1.7 The Optical Depth

The optical depth to gravitational microlensing is the probability that a
given star at a specific instant in time has a magnification caused by
gravitational lensing that exceeds 1.34 (Paczynski, 1986), meaning that the
source lies inside the Einstein ring of a lens. To evaluate it for a given
source at distance $D_{S}$, we need to consider all potential lenses lying
along the line of sight to that source. The optical depth is then the integral
over the number density of lenses multiplied by the area enclosed by their
Einstein rings:

$\tau=\int_{0}^{D_{S}}\frac{4\pi
G\rho(D_{L})}{c^{2}}\frac{D_{L}D_{LS}}{D_{S}}dD_{L},$ (16)

where $\rho(D_{L})$ is the average mass density of lenses at $D_{L}$.

The mass distribution depends on the line of sight considered so an accurate
estimate of the optical depth requires the use of a Galactic model and the
evaluation of the integral over multiple directions. Note however that the
optical depth does not depend on the mass spectrum of lenses, but only on the
mass density along the line of sight.

To better understand the concept of optical depth, one can imagine an ensemble
of point lenses randomly distributed in the sky, each lens surrounded by a
small circle corresponding to its angular Einstein radius. Then, adding all
those areas up gives $\tau$, the fraction of the sky they cover up.

How is that related with the distribution of event durations and the event
rate? Continuing with the example above, each of these lenses is moving with
some three-dimensional velocity in space. Even though some fraction of them
moves relatively fast, they will not all generate microlensing events with
short timescales when they encounter a source star. This is because it is the
velocity vector projected on the plane of the sky that affects the measured
event duration, and if the magnitude of the projected vector is small, then
the event duration will be long. On the other hand, slow-moving lenses will
always generate long-timescale events. This means that the event duration
distribution will be broad and have tails of relatively long and relatively
short events.

The event rate is the expected number of microlensing events $N$ if $n$
sources are monitored over a time interval $\Delta t$. A simple expression may
be obtained under the assumption that all events have the same timescale
$t_{E}$. Then $N=(2n\tau\Delta t)/(\pi t_{E})$ (Paczynski, 1996). A detailed
analysis and the derivation of the full expression that takes into account the
distribution of event timescales may be found in Mao and Paczynski (1996).

The first observational evaluation of the optical depth toward the Galactic
bulge came from the OGLE survey in 1994 (Udalski et al., 1994), where it was
estimated to be in excess of $3.3\pm 1.2\times 10^{-6}$. By 2005, the MACHO
survey (Popowski et al., 2005; Alcock et al., 2000) had refined this value to
$\tau\sim 2.17^{+0.47}_{-0.38}\times 10^{-6}$. More recent estimates by Sumi
and Penny (2016); Sumi et al. (2013); Moniez (2010) and Ryu et al. (2008),
find similar but slightly lower values. This means that, when looking from the
Earth towards the Galactic bulge, only about one or two stars in every million
will be microlensed at any given time.

###  2.2 Higher-Order Effects

Most microlensing cases are adequately described by the standard point source-
point lens model we outlined in the previous sections, but that is not always
the case. Other, higher-order, effects may reveal themselves by perturbing the
light curve of a microlensing event. Some of these increase the uncertainties
in the measured parameters, whereas others may be used to extract extra
information about the physical properties of the lensing system.

####  2.2.1 Blending

Since microlensing events are such rare occurrences, surveys direct their
observing efforts towards those regions of the sky with the highest number of
stars. In practice, this means looking towards the very crowded stellar fields
in the direction of the Galactic bulge, where stars appear so numerous that it
often becomes difficult to disentangle them. However, it must be noted that
the detection efficiency drops rapidly when crowding becomes so extreme that
individual stars are no longer resolvable, such as when observing microlensing
events in other galaxies (Ingrosso et al., 2007; Tsapras et al., 2010; Calchi
Novati, 2010).

Regular blending refers to the case when two or more stars overlap on the same
image to such an extent that the software that performs the photometry cannot
identify them individually, but rather as a single blurred object. Blending is
a problem because it dilutes the true microlensing signal, which is now
‘buried’ in the light of all other stars contributing to the blend. As a
result, if one forgets to account for blending, the event appears less
magnified than it actually is, and the measured event duration is shorter
since blended events spend less time above a given threshold.

Taking blending into account involves adding a constant background to the
time-variable flux received from the microlensing event, so that the total
observed flux at any time is

$F(t)=F_{S}A(t)+F_{B},$ (17)

where $F_{S}$ is the source flux in the absence of lensing, $A(t)$ the
magnification at time $t$, and $F_{B}$ the total blending flux. For detailed
studies of how blending affects the determination of microlensing parameters,
see Han et al. (2010); Smith et al. (2007); Thomas and Griest (2006) and Han
(1999).

It is sometimes possible to constrain the contribution of the blend by
obtaining high-resolution images, either with adaptive optic systems on the
ground or from space with a telescope like the Hubble Space Telescope (HST).
Given that the typical lens-source proper motion is of the order of $\sim$5
mas/year, this can only be done about a decade after the event was originally
observed (Beaulieu et al., 2016; Koshimoto et al., 2017; Beaulieu et al.,
2018).

####  2.2.2 Parallax

So far we have assumed that the relative proper motion between a lens and a
source star can be approximated by a straight line, but the trajectory may
well be more complicated. The parallax effect can be measured from the Earth
or from space. It was first predicted for quasar microlensing in 1986 (Grieger
et al., 1986), and then evaluated for stellar microlensing in 1992 (Gould,
1992). The first observational confirmation came in 1995 (Alcock et al., 1995)
by the MACHO survey.

In ground-based microlensing observations, the parallax effect consists of a
subtle long-term distortion to the standard microlens light curve which arises
from the orbital motion of the Earth around the Sun. Parallax is a vector with
two components in the East and North directions, $\pi_{E,\mathrm{East}}$ and
$\pi_{E,\mathrm{North}}$, following the Geocentric formalism of Gould (2004).
The effect becomes particularly noticeable when the time scale of an event is
some non-negligible fraction of the Earth’s orbital period, i.e., when the
event lasts for a few months. It is sometimes referred to as ‘orbital
parallax’, and may be expressed as

$\pi_{E}=\frac{\pi_{L}-\pi_{S}}{\theta_{E}}=\frac{\theta_{E}}{\kappa
M_{L}};\quad\kappa\equiv\frac{4G}{c^{2}{\rm AU}}\simeq 8.1\frac{\rm
mas}{M_{\odot}},$ (18)

where $\pi_{L}$ and $\pi_{S}$ are respectively the lens and source parallaxes,
$M_{L}$ is the mass of the lens, $\theta_{E}$ the angular Einstein radius and
$\kappa$ a constant (Gould and Horne, 2013).

Parallax is significant because for those microlensing events where both
$\pi_{E}$ and $\theta_{E}$ can be measured, it is possible to determine the
mass of the lens and the its distance from the Earth with great accuracy. The
distance to the lens is given by

$D_{L}=\frac{{\rm AU}}{\pi_{E}\theta_{E}+\pi_{S}},$ (19)

where $\pi_{S}={\rm AU}/D_{S}$ is the parallax of the source star, which is
usually known.

The problem is that most microlensing events last only for a few weeks, so the
effect of ‘orbital parallax’ is in most cases negligible, unless the
magnification gradient is large with the source passing near the caustics (see
Section 2.5). It is also possible that simultaneous observations from
different observatories on the Earth may record small displacements to the
event light curve due to the difference in perspectives. This is called
‘terrestrial parallax’.

‘Satellite parallax’, on the other hand, relies on having two observers, one
on the Earth and a satellite in Solar orbit, observe the same microlensing
event simultaneously. Since their points of view are slightly shifted with one
another, the time when the event reaches its peak brightness and the perceived
magnification will be different for each observer. In microlensing parlance,
their $t_{0}$ and $A_{0}$ values will not be the same. Calchi Novati et al.
(2015) and Yee et al. (2015) first demonstrated this using simultaneous ground
and space-based observations with the Spitzer satellite.

A source with a binary companion will also experience accelerated motion which
can affect a light curve in a way analogous to ground-based parallax. This
effect is called ‘xallarap’ (parallax spelled backwards) and if the orbit of
the source matches the reflex motion of the Earth around the Sun, then the two
effects manifest identically (Bennett, 1998).

Finally, ‘diurnal parallax’ is a one-day modulation resulting from the
rotation of the Earth around its axis An et al. (2002).

####  2.2.3 Finite Source

Stars have a finite size but up to now we treated the source as point-like in
the calculations. For the most part this approximation is sufficient for
single-lens events, but it breaks down when the impact parameter, i.e., the
minimum distance between the lens and the source star, is comparable to or
smaller than the size of the source (as a rule of thumb when $u_{0}\lesssim
3\rho_{S}$, where $\rho_{S}$ is the angular size of the source in units of
$\theta_{E}$). To obtain the magnification for a source of finite size, one
needs to integrate over the area of the source (Lee et al., 2010, 2009;
Dominik, 2007), or integrate over the source boundary using Green’s theorem
(Dominik, 1998). It is worth mentioning that the source can have a brightness
profile and might not be circular in projection.

When finite source effects are detected in the light curve on an event, they
can be used to calculate the lens-source relative proper motion and evaluate
the Einstein radius of the lens, since $\theta_{E}=\theta_{S}/\rho_{S}$.
$\theta_{S}$ and $\rho_{S}$ are the angular radius of the source star and
normalized source radius respectively. The former can be obtained from
determining the spectral type of the source star, while the latter is
evaluated during modeling of the event light curve. This is what is required
for an accurate determination of the mass of the lens, as previously described
in Section 2.2.2.

####  2.2.4 Binary Source

The case of a binary source is simple, as it generates a linear sum of two
single-lens light curves. Naturally, since the two stellar components may have
different luminosities and colors, the resulting light curve may well be
chromatic. If the distance between the two source stars is large and the lens
trajectory is along the line joining the two, we may have the perception of
two independent microlensing events separated by a few months or years, which
are both produced by the same lens. Of course, if the binary source is a
gravitationally bound system, it is possible that orbital motion effects may
also be detectable.

The effect was first mentioned in a paper by Griest and Hu (1992) in 1992.
Some years later Gaudi (1998) pointed out that there exist a subset of binary
source events than can mimic the main features of planetary microlensing and
thus masquerade as planetary events, contaminating the statistical sample.
Fortunately, this degeneracy can often be broken by dense observations during
the time of the perturbation and by using color information. Han (2002)
pointed out that if a lensing event is observed astrometrically, it is
possible to unambiguously break the degeneracy.

Binary source events are not as commonly detected as originally predicted
(Griest and Hu (1992) estimated that $\sim$10% of all events should show
binary source features). This apparent paucity was explained by Dominik (1998)
and Han and Jeong (1998); they are often mistaken for single source events.

All aforementioned higher-order effects have been detected and characterized
in hundreds of microlensing light curves. The ways in which they affect or
dilute the microlensing signal are generally well understood. Sometimes their
presence can even be used to break degeneracies between the estimated
parameters of the system, and especially in the cases of finite-source and
parallax, to determine the mass and distance to the lens.

###  2.3 Binary Lens

We now turn to the case when the lens is not a single body, but two: a binary
star or a star with a planet. This is called ‘binary lensing’, and the light
curves it produces can show extraordinarily diverse morphologies (Liebig et
al., 2015).

Let us define our lens as two point masses $M_{1}$ and $M_{2}$ at a distance
$D_{L}$ from the observer (see Figure 4). On the lens plane, we choose a
coordinate system ($x,y$), where the $x$-axis passes through both masses and
the origin is defined as the midpoint of the line joining them. $D_{S}$ and
$D_{LS}$ are the observer-source and lens-source distances respectively. We
define another coordinate system ($u,v$) on the source plane, which is
parallel to ($x,y$), and whose origin lies on the point of intersection with
the optical axis.

Figure 4: The geometry of gravitational microlensing by a binary lens. The two
components of the lens are at positions $\xi_{1}$ and $\xi_{2}$ on the lens
plane, and the origin is chosen to be in the middle of the line joining the
two masses. The distances from the observer to the lens and the source are
$D_{L}$ and $D_{S}$, respectively. The distance between lens and source is
$D_{LS}$. A ray of light originating from the source plane at point
$\eta(u,v)$ at an angular position $\beta$, is gravitationally deflected by an
angle $\alpha$ on its way to the observer. The position it hits the lens plane
is denoted by $\xi(x,y)$, at an angular position $\theta$.

For any light ray, the deflection angle due to a single lensing mass is given
by Equation (1). Assuming geometrically-thin lenses, when more than one
lensing masses are involved, the total deflection angle is the vector sum of
all individual deflections (Schneider et al., 1992; Bourassa et al., 1973).
When the contributions of each deflecting mass are added up, we can write the
deflection angle for such a composite lens as

$\vec{\hat{\alpha}}(\vec{\xi})=\frac{4G}{c^{2}}\sum_{i}M_{i}\frac{\vec{\xi}-\vec{\xi}_{i}}{\left|\vec{\xi}-\vec{\xi}_{i}\right|^{2}},$
(20)

where $\vec{\xi}$ is the position of the light ray on the lens plane, and
$\vec{\xi}_{i}$ that of the mass $M_{i}$.

For the binary-lens case, this reduces to

$\vec{\hat{\alpha}}(\vec{\xi})=\frac{4G}{c^{2}}\left[M_{1}\frac{\vec{\xi}-\vec{\xi}_{1}}{\left|\vec{\xi}-\vec{\xi}_{1}\right|^{2}}+M_{2}\frac{\vec{\xi}-\vec{\xi}_{2}}{\left|\vec{\xi}-\vec{\xi}_{2}\right|^{2}}\right],$
(21)

where $\vec{\xi}_{1},\vec{\xi}_{2}$ are lens-plane vectors pointing to
$M_{1},M_{2}$ respectively.

The binary-lens equation can be written as

$\vec{\eta}=\vec{\xi}\frac{D_{S}}{D_{L}}-D_{LS}\vec{\hat{\alpha}}(\vec{\xi}),$
(22)

where the light ray originating from a source at $\eta(u,v)$ hits the lens
plane at image position $\xi(x,y)$. As before, $D_{S},D_{L}$ and $D_{LS}$
denote the observer-source, observer-lens and lens-source distances.

Following Schneider and Weiss (1986), we can use the definitions
$r_{E}=\theta_{E}D_{L}$, $\vec{z}=\vec{\xi}/r_{E}$,
$\vec{w}=(D_{L}/D_{S})\cdot\vec{\eta}/r_{E}$, and express the masses as
fractions of the total lens mass, $\epsilon_{1}=M_{1}/M_{L}$,
$\epsilon_{2}=M_{2}/M_{L}$, so that Equation (22) can be written as

$\vec{w}=\vec{z}-\epsilon_{1}\frac{\vec{z}-\vec{z_{1}}}{|\vec{z}-\vec{z_{1}}|^{2}}-\epsilon_{2}\frac{\vec{z}-\vec{z_{2}}}{|\vec{z}-\vec{z_{2}}|^{2}},$
(23)

with $\vec{w}(u,v)$ on the source plane mapping to $\vec{z}(x,y)$ on the lens
plane. In other words, through Equation (23), we can find the source point
$\vec{w}(u,v)$ at which a light ray which hits the lens plane at
$\vec{z}(x,y)$ originates.

Witt (1990) rewrote Equation (23) using complex notation, which makes it more
convenient to use. If $w=u+iv$ and $z=x+iy$ are now complex numbers and
$\bar{w},\bar{z}$ their complex conjugates, we can obtain $w$ as a function of
$z$ and $\bar{z}$:

$w=z-\frac{\epsilon_{1}}{\bar{z}-\bar{z_{1}}}-\frac{\epsilon_{2}}{\bar{z}-\bar{z_{2}}}.$
(24)

Here, we have adopted the notation of Rhie (1997), where the general case for
$N$ lensing bodies is described.

We are usually more interested in knowing where a source, located at $w(u,v)$,
can be seen on the lens plane; thus we need to invert Equation (24). This is
non-trivial as it is of fifth degree in $z$ and the inversion cannot be done
analytically for arbitrary $w$. However, the roots may be found using standard
numerical recipes, such as the ZROOTS routine from Press et al. (2002), or the
optimized algorithm of Skowron and Gould (2012).

###  2.4 The Magnification and Image Positions for the Binary Lens

As in the single lens case, the magnification of an image $i$ is obtained by
taking the ratio of the flux density of the lensed image and the flux density
of the unlensed source. This is determined by evaluating the Jacobian
determinant, $J$, of the mapping by the binary-lens equation:

$A_{i}=|J|^{-1};\quad J=1-\left|\frac{\partial
w}{\partial\bar{z}}\right|^{2};\quad\frac{\partial
w}{\partial\bar{z}}=\frac{\epsilon_{1}}{(\bar{z}-\bar{z_{1}})^{2}}+\frac{\epsilon_{2}}{(\bar{z}-\bar{z_{2}})^{2}}.$
(25)

The need to evaluate the Jacobian arises because we want to know how an
infinitesimal area element in the lens plane is distorted during mapping
through the lens equation.

The total magnification is simply given by the sum of the magnifications of
all images:

$A=\sum_{i=1}^{n}A_{i}=\sum_{i=1}^{n}|J|^{-1}.$ (26)

###  2.5 Critical Curves and Caustics

Certain values of $\bar{z}$ in Equation (25) will make the Jacobian
determinant vanish. These positions trace out closed curves on the lens plane
called critical curves. Mapping those through the lens equation onto the
source plane, one obtains another set of closed curves called caustics.
Critical curves and caustics are of fundamental importance in understanding
microlensing by two or more lensing bodies. That is, at these positions the
magnification diverges and reaches infinity for a point source (see Figure 5).
In reality however, because real sources are extended and the magnification is
a weighted mean over the source, the magnification is always finite.

Figure 5: (a) Critical curves (black) and caustics (red) of a binary lens with
$s$ = 1, $q$ = 0.1, assuming a point source. The origin is the center of mass
of the system. When the lens is composed of two massive objects, the images of
the source generated by the lensing effect are no longer two, but either three
of five, depending on the location of the source relative to the caustics. For
each of the four source trajectories considered, two new images get created or
destroyed whenever the source crosses the caustic, producing the sudden
‘jumps’ in magnification seen in panel b. The positions of the two lensing
masses are marked by the black dots and the $x$ and $y$-axes mark distances on
the lens plane in units of $\theta_{E}$; (b) Corresponding light curves for
each of the trajectories shown in panel a. Note the absence of sudden ‘jumps’
in magnification for the trajectories shown in orange and gray, where the
source remains far from the caustics at all times.

The significance of the caustics becomes easier to appreciate when one
considers that whenever the source crosses a caustic, the number of images
changes by two, leading to abrupt ‘jumps’ in magnification. When the source is
outside the area enclosed by the caustic curve, the number of images is always
three. As the source crosses the caustic, two more images are produced at
diametrically opposite locations (Paczynski, 1996), and for as long as the
source remains enclosed within the caustic structure, the total number of
images is always five. Generalizing, the total number of images generated by a
lens consisting of $N$ lensing masses correspond to the solutions of the
complex polynomial obtained from inverting the lens equation and therefore
cannot exceed $N^{2}+1$ (Rhie, 1997). In fact, not all solutions of the
complex polynomial correspond to real images; some are spurious. Rhie (2003)
demonstrated that at most $5(N-1)$ real images will be generated. As in the
case of a single lensing mass, the image separations are too small to resolve
them individually, but binary lens light curves exhibit much more complicated
structures.

The shape of the caustics is very sensitive to the mass ratio between the two
components of the lens, $q$, and different mass ratios generate different
caustic structures. The shape of the caustics is also sensitive to the angular
separation, $s$ (in units of $\theta_{E}$), between the two components of the
lens, with larger separations ‘stretching’ the caustics along the binary axis,
while beyond a certain point the caustics split up. Different values of $q$
and $s$ will produce caustics that belong to one of three possible topologies:
close, intermediate/resonant and wide (Schneider and Weiss, 1986; Erdl and
Schneider, 1993; Han, 2006; Cassan, 2008; Gaudi, 2010; Bozza et al., 2016).
These are illustrated in Figure 6. Close topologies are characterized by three
separate caustic curves: a central caustic and two smaller symmetrical ones on
either side of the binary axis. These are usually called planetary caustics
since they are associated with the smaller of the two masses. In wide
topologies, there is a central caustic and an isolated secondary caustic that
appears along the binary axis. Finally, an intermediate/resonant topology
produces a single large central caustic. The shape of the observed light curve
will then depend on exactly how close the trajectory of the source brings it
to these curves.

Figure 6: Three possible caustic topologies: close, intermediate/resonant and
wide, with black lines separating the three regimes. A close binary (in the
example plotted here $s=0.75$, $q=10^{-2}$) will produce a central four-cusped
caustic and two smaller three-cusped symmetrical ones. An intermediate
resonant binary ($s=1$, $q=10^{-2}$) has a single large six-cusped central
caustic, while a wide binary ($s=1.5$, $q=10^{-2}$) has a central and an
isolated secondary caustic, both featuring four cusps.

As can be seen in Figure 5, when the source trajectory is far from the
caustics, the resulting light curve is similar to the single lens case. Only
those trajectories that pass close to or cross the caustics will produce
significant deviations. Observations of microlensing events only provide a
light curve, from which the size and structure of the caustics must be
inferred. Provided the photometry is good, the exact morphology of the light
curve deviations can be used to tell whether a microlensing event is caused by
a star-star binary lens or by a star and its planet. The duration of a
deviation scales roughly as $\Delta t_{p}\sim\sqrt{q}t_{E}$, so for a typical
event timescale $t_{E}=30$ days, a Jupiter-Sun equivalent (mass ratio
$q=10^{-3}$) would produce a deviation lasting for about a day, whereas the
deviation caused by an Earth-Sun equivalent ($q=3\times 10^{-6}$) would only
last for about two hours.

An alternative way to think about light curve deviations is to consider what
happens to the images while the lensing event is ongoing. Let us for example
consider a binary lens composed of a star hosting a planet. The planet will
only be revealed when one of the images generated by the lensing event happens
to sweep past its location on the lens plane. The gravity of the planet will
then act as a mini-lens, further perturbing the image and producing a
‘planetary anomaly’ in the light curve.

###  2.6 Binary Light Curve Degeneracies

As we saw in Section 2.1.5, three parameters are enough to describe the shape
of a point-source, point-lens light curve: the minimum impact parameter,
$u_{0}$, the time of maximum magnification, $t_{0}$, and the event timescale,
$t_{E}$. Binary lenses require three additional parameters: the angle $\alpha$
at which the source trajectory crosses the binary axis, the mass ratio between
the two components, $q$, and the separation between them, $s$, projected on
the lens plane. If the source at any point approaches the caustics, its
angular size, $\rho_{S}=\theta_{S}/\theta_{E}$, must also be included as an
extra parameter, bringing the total number of parameters required to describe
the shape of the binary light curve to seven.

Determining these parameters through light curve modeling is sometimes
complicated due to a number of well-studied degeneracies (Gaudi and Gould,
1997; Dominik, 1999, 2009). A binary lens where one of the components is a
planet, is subject to two discrete degeneracies. The first relates to which
image the planet is perturbing; Is it the major image outside the Einstein
ring or the minor image inside? Since these result in different light curve
perturbations, this degeneracy can easily be broken for the majority of cases.
The second degeneracy has to do with whether the planet lies closer to or
further from the star than the position of the image it is perturbing. This is
harder to break but does not affect the determination of the mass ratio. In
addition to these discrete degeneracies, there is a continuous degeneracy
arising from finite-source effects being misinterpreted as a larger mass
ratio, because both of these are related to the duration of the planetary
perturbation. If, for example, the size of the source is larger than the
Einstein ring of the planet, then the duration of the perturbation will be
determined by the crossing time of the source, not the Einstein ring of the
planet. Another type of degeneracy, whose origin lies in the lens equation
itself, exists between close and wide binary lenses (Dominik, 1999). Gaps in
the observations, especially when the source is traversing the caustics, may
also permit multiple degenerate model solutions since the shape of the light
curve during the anomaly is only loosely constrained (Skowron et al., 2018).

In general, dense and precise sampling of the light curve with high-cadence
observations will place tight constraints on possible degenerate solutions. As
we shall see next, when higher-order effects are detected in the light curve,
it is often possible to break degeneracies and arrive at unique solutions for
the physical parameters of the lensing system.

###  2.7 Higher-Order Effects for Binary Lenses: Orbital Motion

All higher-order effects that apply to single lenses (see Section 2.2), can
also affect binary lenses. In fact, the finite size of the source $\rho_{S}$
is even more important in binary lenses since its effects need to be taken
into account whenever the source approaches a caustic, whereas for single lens
events it is rarely required. Similarly, parallax effects are much more
frequently detected in binary than in single lens events. In addition to
these, it is sometimes possible in binary lens events to detect evidence of
orbital motion between the lens components.

Typically, microlensing events last only for a few weeks, so the positions of
the two masses can be considered fixed on the lens plane. This means that our
observations will only get a ‘snapshot’ of the system. However, when the
duration of the microlensing event is some non-negligible fraction of the
orbital period of the binary and the source trajectory passes close to the
caustics, the orbital motion of the two bodies comprising the lens will leave
its traces in the event light curve (Dominik, 1998; Albrow et al., 2000).

Orbital motion manifests itself by changing the shape of the caustics with
time and rotating them, but since only a small part of the orbit is covered
during the microlensing event, it is usually only a small effect. However, the
effects can be particularly dramatic on the planetary caustics for close
binaries. To first order approximation, orbital motion can be accounted for by
introducing two more parameters to the binary lens model: the rate of change
of separation between the lens components $ds/dt$ (projected on the lens
plane), and the angular rotation rate $d\alpha/dt$ (relative to the source
trajectory). It is worth noting that parallax and orbital motion effects are
partly degenerate but can often be disentangled with careful observations
(Bachelet et al., 2012; Park et al., 2013). Remarkably, for a few systems it
has been possible to derive complete Keplerian solutions for the orbits (Shin
et al., 2011, 2012).

###  2.8 Finding Planets

Planetary microlensing events are just binary lenses where the mass ratio
between the two components of the lens is very small. Strictly speaking, they
can be (and indeed have been) found in more complex systems involving three or
more lensing objects, but these are much rarer occurrences. Because the mass
of the planet is so small compared to the host star, their light curves for
the most part resemble those of single lenses. The planet will reveal itself
when one of the images of the source generated by the lensing effect happens
to ‘sweep’ past its location. This will produce brief binary-lens type
perturbations on the light curve, usually referred to as ‘planetary
anomalies’, which after careful observations and analysis can be used to
derive the characteristics of the system. Since the images appear close to the
Einstein ring of the lens, that is where the probability of detecting a
planet, if it is there, is highest. For typical lens and source distances,
this corresponds to a physical distance from the host star of about 1–10 AU
(Gaudi et al., 2002; Tsapras et al., 2003; Gould et al., 2010). Although the
amplitude of the planetary signal can be large irrespective of the mass of the
planet, the detection probability scales approximately as $q^{1/2}$ in the
vicinity of the planetary caustics or as $q$ close to the central caustic
(Griest and Safizadeh, 1998; Peale, 2001).

The analysis of microlensing event light curves featuring perturbations due to
the presence of multiple planets can be a daunting task requiring optimization
over a large parameter space. Multi-planet systems form several disconnected
sets of caustics; there is a ’central caustic’ located very close to the
primary lens and multiple ’planetary caustics’ further away. Perturbations
related to the source interacting with the central caustic appear near the
peak of the light curve, whereas those related to interactions with the
planetary caustic manifest at the wings on either side. Provided the
deviations induced by the individual planets do not occur at similar places,
most cases of planetary caustic perturbations can be adequately addressed
using the binary-superposition approximation, whereby the perturbations are
treated as the sum of the individual perturbations caused by each planet.
These can be investigated separately using standard binary lens analysis
methods and repeated for each planet (Han et al., 2001; Han, 2005). Although
the approximation still holds for many cases of central caustic perturbations,
the situation becomes more complex since these always occur in the same region
regardless of the number of planets (Rattenbury et al., 2002). This means that
it can be difficult to isolate the individual contribution of each planet in
the pattern of anomalies observed in the light curve, especially when the
separation between the planets is small. In general, the binary-superposition
approximation is valid for $q\ll 1$ and $|s-1|\gg q$ and will provide a rough
first estimation of the parameters of the event which can then be refined by
performing the full analysis in a greatly narrowed-down parameter space.

This concludes the theory section. What ultimately determines the degree of
success in detecting and characterizing planetary signals is the timeliness,
quantity and quality of observations, and it is to this we turn to next.

##  3 Microlensing Observations in Practice

What we actually observe during a microlensing event is an increase in the
brightness of the source star as the lens approaches it on the plane of the
sky, followed by a gradual dimming back to its normal brightness as the lens
appears to move away (see Figure 3). The relative proper motions between the
stars in the Galaxy produce microlensing events that typically last a few
weeks to a few months. If the lensing star happens to host a planet, there is
a chance that the planet itself may also perturb the light coming from the
source star, producing brief but intense variations, or ‘anomalies’, in the
event light curve, i.e., in the measurements of how the source brightness
varies with time. These anomalies typically last for a few days in the case of
Jupiter-mass planets and only for a few hours for Earth-mass planets, but the
amplitude of the anomaly can be substantial in both cases. Planetary
deviations are detected in $<1\%$ of all microlensing events discovered.

As mentioned in Section 1.2, microlensing is a rare phenomenon; only about one
star in every million undergoes microlensing at any given time in the Galaxy.
Therefore, in order to maximize their chances of finding these elusive events,
microlensing surveys have been targeting those regions of the sky with the
highest density of stars per square degree; and this means looking in the
direction of the Galactic bulge.

###  3.1 Surveys and Follow-Up

The first microlensing surveys were actually set up to investigate whether a
significant fraction of the dark matter in the Milky Way halo was made up of
massive compact objects (MACHOs) that emit little or no radiation, like black
holes, neutron stars and brown dwarfs (Udalski et al., 1992; Alard et al.,
1995; Alcock et al., 1996; Renault et al., 1997), but little evidence was
found to support this hypothesis. It wasn’t until Mao and Paczynski (1991) and
Gould and Loeb (1992) pointed out that planets could be discovered in this
manner that efforts started to move in that direction.

In the early to mid ’90s, microlensing survey telescopes only covered what by
today’s standards is but a small area of the sky, of the order of $\sim$1
degree, finding few microlensing events. For example, the OGLE-I survey that
lasted from 1992 until 1995 discovered 19 microlensing events, which at the
time was a resounding success. The pivotal point came with the introduction of
the OGLE early-warning system in 1994, which allowed early detection of on-
going microlensing events in their data stream (Udalski et al., 1994). They
decided to release this information publicly and promptly to the astronomical
community, enabling follow-up observations from other telescopes, so that as
much information about these one-off events could be obtained. This was
necessary given that surveys at that time lacked the observing cadence
required to constrain the parameters of the most interesting among the events
they were discovering.

This decision, soon emulated by other surveys, enabled small dedicated teams
around the world to strongly contribute to the science (Pratt et al., 1995;
Albrow et al., 1998; Yoo et al., 2004; Burgdorf et al., 2007; Tsapras et al.,
2009; Dominik et al., 2010). Under this tacit agreement, surveys provided the
alerts and daily monitoring necessary to discover the microlensing events,
whereas follow-up teams concentrated their efforts on obtaining observations
every few hours for a small subset of events that were either highly-
magnified, which implied that the probability of discovering potential
planetary companions was high, or that were already exhibiting anomalous
features (Han, 2007; Abe et al., 2013). As pointed out by Dominik et al.
(2002), the OGLE-III survey made the crucial difference for the efficiency of
follow-up campaigns by providing a much larger number of useful events on
brighter targets (a factor 5–8 as compared to OGLE-II). Some follow-up teams
removed the human-decision element altogether and introduced automated
processes that could detect and assess ongoing anomalous features in real
time, relegating key observing decisions to software agents (Tsapras et al.,
2002; Dominik et al., 2007, 2008).

This arrangement benefited all parties for a number of reasons: First, surveys
possessed cameras with a wide field of view, allowing them to discover and
issue alerts about ongoing microlensing events. This was something follow-up
teams could not do due to limitations imposed by the designs of the
telescopes, which generally had much smaller fields of view. Secondly, follow-
up teams independently operated different telescopes around the world,
allowing for near-continuous observations of a microlensing event, when
surveys could only obtain observations when it was a clear night at the
location of their telescope. This made it much more likely that all
interesting features of an event could be observed. Thirdly, unlike surveys,
follow-up teams had the option of tailoring the exposure times of their
observations to the current brightness of the individual event they were
monitoring, thereby maximizing the signal-to-noise ratio while avoiding
saturation.

One of the difficulties of this approach was combining disparate data sets,
given that the telescopes and instrumentation used by each team had different
characteristics. In order to achieve the required accuracy to detect planetary
signals ($\leq$1% photometry), all data sets had to be carefully aligned and
the noise properties of each evaluated (Bachelet et al., 2015). What’s more,
observations were reduced in real time and light curves made promptly
available to the community so that putative signals could be independently
assessed. Based on these ‘live’ data, a quick decision could be made on
whether individual observing schedules required adjustments. For many of the
anomalous microlensing events observed during this period (roughly between
2005 to 2015), up to 20 telescopes around the world contributed observations,
providing 24-h coverage of the light curve and independent confirmation of the
anomalous features.

###  3.2 Real-Time Modeling

Precise modeling of individual microlensing events is a process that can last
several months and is usually performed on computing clusters with hundreds of
CPUs because a large parameter space needs to be thoroughly explored. The
nature of anomalous features in the light curve of an event while the anomaly
is still ongoing is very hard to interpret. Since the main feature containing
information about the mass of the secondary component of the lens is the
duration of the anomaly, it is often impossible to tell whether it is a
planet, a brown dwarf, or a companion star causing it. As the event progresses
and more observations are obtained, the likelihood of the different
interpretations changes, but estimating exactly by how much requires the
ability to model events in real-time as they are happening, and to constantly
re-evaluate how well these models represent the data. This was computationally
prohibitively expensive until 2010 when, based on previous work in the field,
Bozza (2010) came up with a conceptually simple method that could evaluate
competing models in-real time and help follow-up teams decide whether to
continue intensive observations or to reallocate their resources to a
different target (Ryu et al., 2010; Bozza et al., 2012). Furthermore, the
software product was made publicly available and could be run on an average
laptop. This made a complicated process accessible to the entire community and
this code is now at the core of most publicly available microlensing modeling
software (Bachelet et al., 2017; Poleski and Yee, 2018; Ranc and Cassan,
2018).

###  3.3 Second-Generation Surveys

The OGLE and MOA microlensing surveys have gradually upgraded their
instrumentation, installing new wide-field cameras on their telescopes, and
can currently cover several square degrees on the sky with a single pointing
(Sako et al., 2008; The OGLE project, 2010; The MOA project, 2018; The OGLE
project, 2018). Hundreds of millions of stars are now monitored as often as
every 10–30 min in the direction of the Galactic bulge, and this substantially
improved survey sensitivity to planetary signals in the light curve. Follow-up
observations from different longitudes are still useful for independently
confirming the signals, as well as providing light curve coverage when the
Galactic bulge is not visible from the survey sites, either due to inclement
weather or during daytime (Muraki et al., 2011; Miyake et al., 2011; Kains et
al., 2013; Tsapras et al., 2014). Survey efforts were greatly augmented in
2016, when the Korea Microlensing Telescope Network (KMTNet) commenced full
operations with its three 1.6 m survey telescopes in Chile, Australia and
South Africa, which were deployed and commissioned during 2015 (Kim et al.,
2011; Henderson et al., 2014; The KMTNet project, 2018). The cameras on each
of the KMTNet telescopes have a field of view of $\sim$4 square degrees and
their $\sim$10 min cadence observing zone covers about 16 square degrees on
the sky.

###  3.4 From Digital Images to Light Curves

Telescopes carrying out microlensing observations are equipped with CCD or
EMCCD cameras that produce high-quality digital images. On a typical night
during the microlensing season, lasting approximately from April to September
each year, when the Galactic bulge is visible from the Southern hemisphere for
six or more hours, dozens to hundreds of images will be obtained. The
achievable photometric precision depends on the performance of the camera at
the chosen optical wavelength, the observing conditions, the brightness of the
target, the degree to which its light is blended with that of other nearby
stars on the image, and the method used to extract the photometric
measurements.

All images undergo standard preliminary processing to remove noise due to the
camera electronics and to correct for irregularities in the optical path, such
as specks of dust on the optics (Martinez et al., 1997; Howell, 2006). Because
telescope pointing is not perfect, all images are aligned to a common
reference so that stars occupy the same pixel areas on every image. Photometry
is performed by measuring the intensity of each star, i.e., its flux, at the
given position. This is typically done in one of three ways: aperture
photometry, point-spread function (PSF) fitting, or difference imaging.

Aperture photometry is conceptually the simplest and involves summing up the
flux contribution of each pixel within an aperture of fixed radius centered on
the position of the star to be measured (Mighell, 1999). Comparison stars of
constant brightness on the same image are also measured and used to remove
artificial trends in the data. While this method works well when there are few
isolated stars on the image, it is not designed to perform well in crowded
field conditions, which are characteristic of microlensing observations. In
crowded fields, such as those in the direction of the Galactic bulge, stellar
profiles are too blended on the images and it is hard to disentangle the
individual contributions to the total flux measured within the aperture.

PSF fitting photometry does better when there is a moderate degree of blending
and is still used extensively for many astronomical projects. About 20–30
isolated stars are selected on an image and a scalable mathematical stellar
profile is constructed by fitting a model to the image data. The model can be
a sum of 2D Gaussian or Moffat functions. This PSF profile is then scaled to
fit all stars identified on the image (Heasley, 1999), and the volume under
the profile corresponds to the total flux of the star. PSF-fitting was
routinely used for the analysis of microlensing observations until Tomaney and
Crotts (1996) and Alard and Lupton (1998) proposed the more efficient
difference imaging method for measuring stellar flux variability in crowded
fields.

The technique of difference imaging involves the subtraction of all constant
features from the images, using them to self-calibrate the photometry, and
then using the difference images to measure the brightness fluctuations of a
variable star, such as a microlensing target. Any number of observations taken
under excellent observing conditions can be combined to produce a single
reference image with sharp features. This reference is then adjusted to match
the observing conditions of every other image of the target, taken at
different times. The adjustment involves a convolution that shifts the
coordinates of the stars to match small shifts in telescope pointing between
the images and a ‘blurring’ and scaling of stellar profiles in order to match
different atmospheric seeing conditions and transparency. The relevant
transformations are applied to the reference, and it is then subtracted from
each other image, removing all constant features and leaving behind only the
signals of stars that have varied between different exposures. The brightness
variations of microlensing events show up prominently on the residual images
and can by measured by PSF fitting or aperture photometry (see Figure 7).
Difference imaging is, on average, a factor of $\sim$2 more accurate than PSF
fitting, and can even perform $\sim$7 times better for very faint targets
(Alard, 1999). The technique has been improved upon and extended in recent
years (Bramich, 2008; Albrow et al., 2009; Bramich et al., 2013) and is now
the preferred choice for analyzing microlensing observations.

Figure 7: [Top] 30$\times$30 pixel thumbnails of an image sequence with the
microlensing target at the center. [Bottom] Residual difference images
corresponding to each image above after subtraction of the reference image.
The change in brightness of the microlensed star is hard to spot in the top
row but is clearly visible in the bottom.

##  4 Results from Microlensing

Microlensing has detected more than 60 exoplanets to date (see Figure 1),
including two systems with multiple planets (Gaudi et al., 2008; Han et al.,
2013), a planet orbiting a double star system (Bennett et al., 2016), and the
first possible detections of exo-moons (Bennett et al., 2014; Hwang et al.,
2018). The majority of these discoveries have been the result of close
collaboration between survey and follow-up teams and, in most cases, the
planetary signals have been confirmed in multiple data sets. The published
parameters for all of these systems can be found in the NASA exoplanet archive
(NASA Exoplanet Archive, 2018) or the Extrasolar Planet Encyclopaedia (The
Extrasolar Planets Encyclopaedia, 2018), which also lists companions in the
brown dwarf mass range. Both archives are regularly updated.

These microlensing planets orbit their stars (predominantly K and M-dwarfs) at
separations between 0.5 and 18 AU, a regime that remains largely unexplored by
radial velocity and transit surveys, and their masses range from $\sim$1.4
$M_{\mathrm{Earth}}$ to $\sim$13 $M_{\mathrm{Jupiter}}$. Beyond
$13M_{\mathrm{Jupiter}}$, which is the approximate minimum mass required to
start deuterium fusion, the distinction between planet and brown dwarf becomes
difficult (Spiegel et al., 2011). Furthermore, it must be noted that all
microlensing planets have been discovered at distances of several kilo-parsec
away from the Solar system, along the line-of-sight to the Galactic center.
They are much further away than the planets found by any other method (see
Figure 8). This is particularly interesting because the Galaxy has a
metallicity gradient (Daflon and Cunha, 2004), and a link has been established
between stellar metallicity and the frequency of giant planets (Gonzalez,
1997; Valenti and Fischer, 2008). This could mean that the statistical
properties of the systems microlensing is discovering do not follow the same
distributions as those found by other search methods, which probe only nearby
stars. Thus, microlensing is currently the only way to explore and understand
the true Galactic population of planets. At the current rate, only a few
microlensing planets are discovered every year, so building a large enough
statistical sample in order to address this question might take decades.

Figure 8: Almost all known exoplanets lie within a few hundred parsecs from
the Sun. The exception are planets found with microlensing, since most lens
stars are at distances of several kilo-parsec. Microlensing is therefore the
only technique capable of exploring the true Galactic population of planets.
Stars closer to the Galactic center are generally more metal-rich than those
in the halo, and this can have implications on the type of planetary systems
they host.

###  4.1 Highlights

It is instructive to consider two microlensing events of note and examine what
has been learned from studying the morphology of their light curves.

####  4.1.1 A Cold Super-Earth Orbiting an M-Dwarf Star

The discovery of a planet in microlensing event OGLE-2005-BLG-390 in 2005 was
noteworthy for three reasons (Beaulieu et al., 2006). First, at the time of
publication, it was one of the lowest-mass planets known. Second, with a mass
of $\sim$5.5 $M_{\mathrm{Earth}}$ and a distance from its M-dwarf host of
$\sim$2.6 AU, which implied a surface temperature of $\sim$50 K, it was the
only planet known with a potentially solid surface composed of rock and ice.
Third, the discovery gave credence to the idea that terrestrial-mass planets
orbiting low-mass stars between 1 and 10 AU are far more common than Jupiter-
mass planets, as predicted by the core-accretion model of planet formation
(Laughlin et al., 2004).

The planet produced an anomaly in the event light curve that lasted about a
day, but the signal was picked up and confirmed by four different telescopes
independently (see Figure 9a). The signal itself was generated by the source
passing over a wide planetary caustic, which caused a brief boost in the
observed magnification about 10 days after the main peak of the event.

Figure 9: (a) The light curve of planetary microlensing event
OGLE-2005-BLG-390. Six telescopes around the world participated in this
discovery, with the individual contributions represented by the data points in
different colors. The inset on the top right shows the planetary anomaly with
the best-fit planetary model indicated by the solid black line. Single lens
and binary source models, respectively represented by the orange and gray
dashed lines, cannot account for the observed deviation. The planet has a mass
of $\sim$5.5 $M_{\mathrm{Earth}}$ and lies at a distance of $\sim$2.6 AU from
its host star. [Figure adapted with permission from Beaulieu et al. (2006),
Figure 1]; (b) The light curve of planetary microlensing event
OGLE-2006-BLG-109. The event was observed from 11 different telescopes,
represented by the data points in different colors. Five distinct anomalous
features are apparent, produced by the source crossing the caustic of the two-
planet system (shown in the inset on the left). The gray caustics show how the
structure of the caustics appears at different times due to the orbital motion
of the outer planet. The source trajectory is slightly curved due to the
effect of annual parallax. [Figure adapted with permission from Gaudi et al.
(2008), Figure 1].

####  4.1.2 A Jupiter-Saturn Analog

In 2008, Gaudi et al. (2008) published the first microlensing discovery of a
two-planet system (see Figure 9b). The host is about half as massive as the
Sun and the two gas giant planets are $\sim$0.71 and $\sim$0.27 times the mass
of Jupiter with orbital separations of $\sim$2.3 and $\sim$4.6 AU
respectively, both beyond the star’s snow line. The system therefore appears
like a scaled-down version of our Solar system with regard to two of the outer
giant planets. Even though only two planets were detected, the presence of
more planets with very short (<0.4 AU) or very long (>10 AU) orbits cannot be
excluded.

The estimated mass of the host was confirmed in 2010 by Bennett et al. (2010),
who used the adaptive optics system of the Keck 10 m telescope to measure the
flux of the lens in the $H$-band.

####  4.1.3 Limits on the Frequency of Planets

Each data point in a given light curve corresponds to a particular position of
the source. As the event progresses, the images of the source generated by the
lensing effect scan the periphery of the lens, their size growing larger the
shorter the apparent lens-source separation (see Figure 3). Since the observed
magnification is given by dividing the surface area of the images by the
surface area of the unlensed source, any perturbations to one of the images
caused by the presence of a planet at the corresponding location will affect
the measured magnification and produce a deviant data point. To state it
differently, each data point that does not deviate from the general shape of
the light curve of a single lens, excludes the possibility that there is a
planet at the location of the images at the time of observation. This means
that for events with no observed planetary anomalies, it is possible to place
limits on the presence of planets in the system.

The limits obtained from analyzing events with no observed deviations can be
compared with the actual detections to place limits on the number and
frequency of planets in the Galaxy (Albrow et al., 2000; Sumi et al., 2010;
Dominik, 2011; Udalski et al., 2018). Early attempts to do this, even before
any microlensing planets were reported, concluded that less than 1/3 of all
stars host Jupiter-mass planets between $\sim$1–4 AU (Gaudi et al., 2002;
Snodgrass et al., 2004). The light curves available for those studies had a
mean sampling frequency of $\sim$1 day, so it was not possible to derive
meaningful limits for lower-mass planets.

In a 2010 article, Gould et al. (2010) derived a planet frequency beyond the
snow line from microlensing that was a factor 7 larger than the one derived
from radial velocity studies for planets with much shorter periods, from 2 to
2000 days. This difference was consistent with the gradient derived from
radial velocity results when these were extrapolated well beyond the
separations they measured. In the same study, Gould also concluded that only
about 1/6 of all planet-hosting stars host planets analogous to our Solar
system.

In an independent analysis published in 2012, Cassan et al. (2012) arrived at
an estimate for the fraction of bound planets at distances of 0.5–10 AU from
their host stars. He found that only $17^{+6}_{-9}$% of stars host Jupiter-
mass planets (in the mass range 0.3–10 $M_{\rm Jupiter}$), but also that
Neptunes (10–30 $M_{\rm Earth}$) and super-Earths (5–10 $M_{\rm Earth}$) are
much more common beyond the snow line; their respective abundances being
$52^{+22}_{-29}$% and $62^{+35}_{-37}$%. The remarkable conclusion was that,
on average, every star in the Galaxy hosts a planet, and that planets are the
rule rather than the exception. Although the analysis has received some
criticism for relying on a small sample of events and for its use of Bayesian
methods to convert from mass ratios to masses (Suzuki et al., 2016), the main
results have been confirmed by more recent studies (Shvartzvald et al., 2016;
Tsapras et al., 2016). Interestingly, a 2016 analysis by Suzuki et al. (2016)
which relied exclusively on MOA-II microlensing survey data obtained between
2007 and 2012, reported consistent results but found a broken power-law mass
function with a change of slope at $q\sim 10^{-4}$. This “turnover” in the
mass function was recently confirmed by Udalski et al. (2018). Concrete
numbers for the required sample sizes for constraining planetary mass
functions are given in Dominik (2011).

##  5 Strengths and Weaknesses of the Microlensing Method

Having reviewed the particulars of the method, we can summarize its strengths
and address its weaknesses.

###  5.1 Weaknesses

  * •

The probability of observing planet-lensing events is low, $\sim$$10^{-8}$ per
star. In practice, this does not constitute a problem because surveys monitor
about a billion stars regularly and there are $\sim$3–5 new microlensing
planets discovered every year.

  * •

Planetary anomalies last only a few hours for Earths and a few days for
Jupiters. Dozens of telescopes around the world coordinate their efforts to
provide high-cadence observations of even the briefest anomalies.

  * •

Microlensing planets are too far away and are not good targets for searching
for life with next-generation space missions. Although these planets cannot be
the studied further, they provide the missing piece of the puzzle when it
comes to understanding how planetary systems form and evolve: what happens
beyond the snow line.

  * •

Microlensing events are one-off occurrences, with no possibility of observing
them again. Yet their light curves are observed from dozens of telescopes
around the world, providing independent confirmation of the features detected.
They are typically very well sampled, so anomalies can be well constrained.

  * •

It cannot find planets very close ($\lesssim$0.4 AU) or too far ($\gtrsim$100
AU) from their host stars. Those regimes are better explored by transits,
radial velocity and direct imaging. Microlensing is uniquely capable of
exploring intermediate distances.

###  5.2 Strengths

  * •

It does not have a bias for nearby stars, thus it explores the true Galactic
population of planets.

  * •

It does not have a bias for the type of host star and can equally well find
planets around brown or red-dwarf stars, main-sequence stars, stellar
remnants, or event detect planets that have been ejected from their systems
and are no longer gravitationally bound to their stars.

  * •

It does not require many years of observations to discover a planet. The
typical microlensing event lasts for less than a month and provides a
‘snapshot’ of the system, with a good chance of revealing any planets close to
the Einstein ring of the lens.

  * •

It is very cheap in terms of the resources it requires, as it doesn’t rely on
having access to massive telescopes. 1m-class telescopes are routinely used
for the observations, and for the brighter events even smaller amateur
telescopes have contributed useful data.

  * •

It is sensitive to Earth-mass planets from ground-based observations.

  * •

It is uniquely capable of finding planets at and beyond the snow line of their
host stars.

##  6 Future Prospects

The population of close-in planets with orbits of less than 1 AU has been
extensively explored through ground and space-based transit surveys, such as
the Kepler mission, and through the radial velocity method. This regime will
continue to be investigated, with improved sensitivity to planets with lower
masses, by the newly launched (18 April 2018) Transiting Exoplanet Survey
Satellite (TESS) and the PLAnetary Transits and Oscillations of stars (PLATO)
mission, set to launch in 2026. Meanwhile, ground-based direct imaging efforts
using adaptive optics on large telescopes will keep on finding large planets
at orbits of hundreds of AU. It then depends on microlensing observing
campaigns to fill the gap at intermediate distances. Having a complete picture
is crucial for testing models of planet formation and developing our
understanding about how planetary systems originate and evolve.

While there are plans to continue microlensing observations using ground-based
facilities in the near future, finding planets of Earth-mass and below, as
well as increasing the overall rate of discoveries, can be better achieved by
going to space.

The NASA WFIRST space mission, set to launch in the mid-2020s, includes a
microlensing planet-finding program. Going to space for a microlensing survey
comes with several advantages (besides a stable PSF and uninterrupted
coverage). First, WFIRST will rival Kepler in number of planets detected.
Simulations suggest it will detect thousands of microlensing exoplanets,
thereby significantly shortening the time required to understand the
distribution of planets between $\sim$1–100 AU, even for low-mass planets
(Bennett and Rhie, 2002). Second, provided the events are also observed from
the ground, it is possible to routinely measure the parallax effect, placing
tight constraints on the physical parameters of the event (Green et al.,
2012). Third, it will be sensitive to, and is expected to discover, planets
that have been ejected from their systems, commonly referred to as ‘rogue’ or
‘solivagant’ planets (Barclay et al., 2017).

Throughout this review we considered the photometric signature of microlensing
events, but it is worth mentioning that they can also be detected through
their astrometric signature, i.e., through small shifts in the light centroid
of the observed star as the event progresses (Paczyński, 1998; Safizadeh et
al., 1999; Dominik and Sahu, 2000; Honma, 2001). At lens-source separations
greater than $\sim\theta_{E}$, the centroid shift caused by astrometric
microlensing is proportional to $1/u$ while the photometric magnification is
proportional to $1/u^{4}$. Because the astrometric signature of a microlensing
event precedes and lasts longer than the photometric one, it is possible to
anticipate a photometric signal when an expected high magnification event has
been detected astrometrically. Combining photometric and astrometric
microlensing can be useful in resolving some types of binary-lens
degeneracies, although this still remains to be demonstrated (Han, 1999; Han
et al., 1999).

Paczynski (1995) first pointed out in 1995 that it is possible to predict
astrometric microlensing events from precise proper motions and positions of
lens and source stars. In light of this, a number of independent studies have
used existing catalogs to select faint high-proper motion stars and, by
comparing their trajectories with background stars, predict astrometric
microlensing events (e.g., Salim and Gould (2000); Proft et al. (2011)). By
far the most accurate measurements of positions and proper motions to-date are
those contained in the recent second data release of the Gaia space
observatory (Gaia DR2). Klüter et al. (2018a), Klüter et al. Klüter et al.
(2018b) and Bramich and Nielsen (2018) used them to predict astrometric events
in the coming decades, until the end of the century. A small fraction of these
are likely to produce detectable photometric deviations and will inevitably
attract the interest of observers.

The prospects of resolving individual images of stellar microlensing events
using long-baseline interferometric observations have been improving with
every new generation of instruments. Such observations could be used to
estimate the angular Einstein radius $\theta_{E}$ directly from the images.
Typical microlensing values ($M_{L}\sim$ 0.5–1$M_{\oplus}$,
$\pi_{\mathrm{rel}}\sim$ 0.03–0.5 mas) result in Einstein radii in the range
$\theta_{E}\sim$ 0.3–2.0 mas; This is very close to the limits of current VLTI
instrumentation, such as PIONIER and GRAVITY, which can already achieve
resolutions between 2.5 and 3 mas at maximum baseline (Delplancke et al.,
2001; Cassan and Ranc, 2016).

New ideas are also being considered to improve the accuracy of ground-based
photometric microlensing observations using Lucky Imaging techniques that
compensate for atmospheric turbulence and deliver significantly sharper images
when compared to the performance of standard CCD cameras, but so far they have
only been tested with a relatively narrow field of view (Mackay, 2013;
Sajadian et al., 2016). Mackay et al. (2017) recently argued that by combining
several such cameras, wide-field high-resolution imaging is possible, which
would provide enough sensitivity to detect bodies down to Lunar-mass with
ground-based microlensing observations. While the construction of such an
instrument comes with many technical challenges, it would be able to detect
microlensing events in the most crowded regions of the Galactic bulge and
compete with and complement WFIRST observations for the faintest targets.

The author acknowledges the support of DFG priority program SPP 1992
“Exploring the Diversity of Extrasolar Planets” (WA 1047/11-1).

###### Acknowledgements.

I would like to thank Etienne Bachelet for preparing Figure 6, Markus
Hundertmark for providing feedback on the text and the three anonymous
referees for making helpful comments, corrections and suggestions that have
greatly improved this work. The author declares no conflict of interest. The
following abbreviations are used in this manuscript:
CCD | Charge-coupled device
---|---
AU | Astronomical unit. The average distance between the Sun and the Earth
WFIRST | Wide Field Infrared Survey Telescope
PSPL | Point source - Point lens
MACHO | Massive compact halo object
MOA | Microlensing observations in astrophysics
OGLE | Optical gravitational lensing experiment
KMTNet | Korea Microlensing Telescope Network
CCD | Charge-Coupled Device
EMCCD | Electron Multiplying Charge Coupled Device
VLTI | Very Large Telescope Interferometer
PIONIER | Precision Integrated-Optics Near-infrared Imaging ExpeRiment
GRAVITY | a VLTI instrument for precision astrometry and interferometric imaging
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