Gravitational microlensing
Encyclopedia
Gravitational microlensing is an astronomical
phenomenon due to the gravitational lens
effect. It can be used to detect objects ranging from the mass of a planet to the mass of a star, regardless of the light they emit. Typically, astronomers can only detect bright objects that emit lots of light (star
s) or large objects that block background light (clouds of gas and dust). These objects make up only a tiny fraction of the mass of a galaxy. Microlensing allows the study of objects that emit little or no light.
When a distant star or quasar
gets sufficiently aligned with a massive compact foreground object, the bending of light due to its gravitational field, as discussed by Einstein in 1915, leads to two distorted unresolved images resulting in an observable magnification. The time-scale of the transient brightening depends on the mass of the foreground object as well as on the relative proper motion between the background 'source' and the foreground 'lens' object.
Since microlensing observations do not rely on radiation received from the lens object, this effect therefore allows astronomers to study massive objects no matter how faint. It is thus an ideal technique to study the galactic population of such faint or dark objects as brown dwarfs, red dwarfs, planets, white dwarfs, neutron stars, black holes, and
Massive Compact Halo Objects
. Moreover, the microlensing effect is wavelength-independent, allowing study of source objects that emit any kind of electromagnetic radiation.
Microlensing by an isolated object was first detected in 1993. Since then, microlensing has been used to constrain the nature of the dark matter
, detect extrasolar planets, study limb darkening
in distant stars, constrain the binary star
population, and constrain the structure of the Milky Way's disk. Microlensing has also been proposed as a means to find dark objects like brown dwarfs and black holes, study starspots
, measure stellar rotation, and probe quasars including their accretion disks.
effect. A massive object (the lens) will bend the light of a bright background object (the source). This can generate multiple distorted, magnified, and brightened images of the background source.
Microlensing is caused by the same physical effect as strong lensing and weak lensing, but it is studied using very different observational techniques. In strong and weak lensing, the mass of the lens is large enough (mass of a galaxy or a galaxy cluster) that the displacement of light by the lens can be resolved with a high resolution telescope such as the Hubble Space Telescope
. With microlensing, the lens mass is too low (mass of a planet or a star) for the displacement of light to be observed easily, but the apparent brightening of the source may still be detected. In such a situation, the lens will pass by the source in a reasonable amount of time, seconds to years instead of millions of years. As the alignment changes, the source's apparent brightness changes, and this can be monitored to detect and study the event. Thus, unlike with strong and weak gravitational lenses, a microlensing event is a transient phenomenon from a human timescale perspective.
Unlike with strong and weak lensing, no single observation can establish that microlensing is occurring. Instead the rise and fall of the source brightness must be monitored over time using photometry
. This function of brightness versus time is known as a light curve
. A typical microlensing light curve is shown below:
A typical microlensing event like this one has a very simple shape, and only one physical parameter can be extracted: the time scale, which is related to the lens mass, distance, and velocity. There are several effects, however, that contribute to the shape of more atypical lensing events:
Most focus is currently on the more unusual microlensing events, especially those that might lead to the discovery of extrasolar planets. Although it has not yet been observed, another way to get more information from microlensing events that may soon be feasible involves measuring the astrometric
shifts in the source position during the course of the event and even resolving the separate images with interferometry.
, although it shares some properties.) The optical depth is, roughly speaking, the average fraction of source stars undergoing microlensing at a given time, or equivalently the probability that a given source star is undergoing lensing at a given time. The MACHO project found the optical depth toward the LMC to be 1.2×10−7 or about 1 in 8,000,000, and the optical depth toward the bulge to be 2.43×10−6 or about 1 in 400,000.
Complicating the search is the fact that for every star undergoing microlensing, there are thousands of stars changing in brightness for other reasons (about 2% of the stars in a typical source field are naturally variable stars) and other transient events (such as nova
e and supernovae), and these must be weeded out to find true microlensing events. After a microlensing event in progress has been identified, the monitoring program that detects it often alerts the community to its discovery, so that other specialized programs may follow the event more intensively, hoping to find interesting deviations from the typical light curve. This is because these deviations – particularly ones due to exoplanets – require hourly monitoring to be identified, which the survey programs are unable to provide while still searching for new events. The question of how to prioritize events in progress for detailed followup with limited observing resources is very important for microlensing researchers today.
suggested that a light ray could be deflected by gravity. In 1801 Johann Georg von Soldner
calculated the amount of deflection of a light ray from a star under Newtonian gravity. In 1915 Einstein correctly predicted the amount of deflection under General Relativity
, which was twice the amount predicted by von Soldner. Einstein's prediction was validated by a 1919 expedition led by Arthur Eddington
, which was a great early success for General Relativity. In 1924 Orest Chwolson found that lensing could produce multiple images of the star. A correct prediction of the concomitant brightening of the source, the basis for microlensing, was published in 1936 by Einstein. Because of the unlikely alignment required, he concluded that "there is no great chance of observing this phenomenon". Gravitational lensing's modern theoretical framework was established with works by Yu Klimov (1963), Sidney Liebes (1964), and Sjur Refsdal
(1964).
Gravitational lensing was first observed in 1979, in the form of a quasar lensed by a foreground galaxy. That same year Kyongae Chang and Sjur Refsdal showed that individual stars in the lens galaxy could act as smaller lenses within the main lens, causing the source quasar's images to fluctuate on a timescale of months. Bohdan Paczyński
first used the term "microlensing" to describe this phenomenon. This type of microlensing is difficult to identify because of the intrinsic variability of quasars, but in 1989 Mike Irwin et al. published detection of microlensing in Huchra's Lens
.
In 1986, Paczyński proposed using microlensing to look for dark matter
in the form of massive compact halo objects (MACHOs) in the Galactic halo
, by observing background stars in a nearby galaxy. Two groups of particle physicists working on dark matter heard his talks and joined with astronomers to form the Anglo-Australian MACHO collaboration and the French EROS collaboration. In 1991 Paczyński suggested that microlensing might be used to find planets, and in 1992 he founded the OGLE microlensing experiment, which began searching for events in the direction of the Galactic bulge
.
The first two microlensing events in the direction of the Large Magellanic Cloud
that might be caused by dark matter were reported in back to back Nature
papers by MACHO and EROS in 1993, and in the following years, events continued to be detected. The MACHO collaboration ended in 1999. Their data refuted the hypothesis that 100% of the dark halo comprises MACHOs, but they found a significant unexplained excess of roughly 20% of the halo mass, which might be due to MACHOs or to lenses within the Large Magellanic Cloud itself.
EROS subsequently published even stronger upper limits on MACHOs, and it is currently uncertain as to whether there is any halo microlensing excess that could be due to dark matter at all. The SuperMACHO project currently underway seeks to locate the lenses responsible for MACHO's results.
Despite not solving the dark matter problem, microlensing has been shown to be a useful tool for many applications. Hundreds of microlensing events are detected per year toward the Galactic bulge, where the microlensing optical depth (due to stars in the Galactic disk) is about 20 times greater than through the Galactic halo. In 2007, the OGLE project identified 611 event candidates, and the MOA project (a Japan-New Zealand collaboration) identified 488 (although not all candidates turn out to be microlensing events, and there is a significant overlap between the two projects). In addition to these surveys, followup projects are underway to study in detail potentially interesting events in progress, primarily with the aim of detecting extrasolar planets. These include PLANET, RoboNet and MicroFUN.
, also called the Einstein angle, is the angular radius
of the Einstein ring
in the event of perfect alignment. It depends on the lens mass M, the distance of the lens dL, and the distance of the source dS:
(in radians)
For M equal to the mass of the Sun, dL = 4000 parsecs, and dS = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is 0.001 arcseconds (1 milliarcsecond). By comparison, ideal Earth-based observations have angular resolution
around 0.4 arcseconds, 400 times greater. Since
is so small, it is not generally observed for a typical microlensing event, but it can be observed in some extreme events as described below.
Although there is no clear beginning or end of a microlensing event, by convention the event is said to last while the angular separation between the source and lens is less than
. Thus the event duration is determined by the time it takes the apparent motion of the lens in the sky to cover an angular distance 
. The Einstein radius is also the same order of magnitude as the angular separation between the two lensed images, and the astrometric shift of the image positions throughout the course of the microlensing event.
During a microlensing event, the brightness of the source is amplified by an amplification factor A. This factor depends only on the closeness of the alignment between observer, lens, and source. The unitless number u is defined as the angular separation of the lens and the source, divided by
. The amplification factor is given in terms of this value:
This function has several important properties. A(u) is always greater than 1, so microlensing can only increase the brightness of the source star, not decrease it. A(u) always decreases as u increases, so the closer the alignment, the brighter the source becomes. As u approaches infinity, A(u) approaches 1, so that at wide separations, microlensing has no effect. Finally, as u approaches 0, A(u) approaches infinity as the images approach an Einstein ring. For perfect alignment (u = 0), A(u) is theoretically infinite. In practice, finite source size effects will set a limit to how large an amplification can occur for very close alignment, but some microlensing events can cause a brightening by a factor of hundreds.
Unlike gravitational macrolensing where the lens is a galaxy or cluster of galaxies, in microlensing u changes significantly in a short period of time. The relevant time scale is called the Einstein time
, and it's given by the time it takes the lens to traverse an angular distance 
relative to the source in the sky. For typical microlensing events, 
is on the order of a few days to a few months. The function u(t) is simply determined by the Pythagorean theorem:
The minimum value of u, called umin, determines the peak brightness of the event.
In a typical microlensing event, the light curve is well fit by assuming that the source is a point, the lens is a single point mass, and the lens is moving in a straight line: the point source-point lens approximation. In these events, the only physically significant parameter that can be measured is the Einstein timescale
. Since this observable is a degenerate function of the lens mass, distance, and velocity, we cannot determine these physical parameters from a single event.
However, in some extreme events,
may be measurable while other extreme events can probe an additional parameter: the size of the Einstein ring in the plane of the observer, known as the Projected Einstein radius: 
. This parameter describes how the event will appear to be different from two observers at different locations, such as a satellite observer. The projected Einstein radius is related to the physical parameters of the lens and source by
.
It is mathematically convenient to use the inverses of some of these quantities. These are the Einstein proper motion
and the Einstein parallax
.
These vector quantities point in the direction of the relative motion of the lens with respect to the source. Some extreme microlensing events can only constrain one component of these vector quantities. Should these additional parameters be fully measured, the physical parameters of the lens can be solved yielding the lens mass, parallax, and proper motion as
. However, in some cases, events can be analyzed to yield the additional parameters of the Einstein angle and parallax: 
and 
. These include very high magnification events, binary lenses, parallax, and xallarap events, and events where the lens is visible.
If the lens passes directly in front of the source star, then the finite size of the source star becomes an important parameter. The source star must be treated as a disk on the sky, not a point, breaking the point-source approximation, and causing a deviation from the traditional microlensing curve that lasts as long as the time for the lens to cross the source, known as a finite source light curve. The length of this deviation can be used to determine the time needed for the lens to cross the disk of the source star
. If the angular size of the source 
is known, the Einstein angle can be determined as
.
These measurements are rare, since they require an extreme alignment between source and lens. They are more likely when
is (relatively) large, i.e., for nearby giant sources with slow-moving low-mass lenses close to the source.
In finite source events, different parts of the source star are magnified at different rates at different times during the event. These events can thus be used to study the limb-darkening
of the source star.
Caustic crossings in binary lenses can happen with a wider range of lens geometries than in a single lens. Like a single lens source caustic, it takes a finite time for the source to cross the caustic. If this caustic-crossing time
can be measured, and if the angular radius of the source is known, then again the Einstein angle can be determined.
As in the single lens case when the source magnification is formally infinite, caustic crossing binary lenses will magnify different portions of the source star at different times. They can thus probe the structure of the source and its limb darkening.
An animation of a binary lens event can be found at this YouTube video.
perpendicular to the motion of the lens while the difference in the time of peak amplification yields the component parallel to the motion of the lens. This direct measurement was recently reported using the Spitzer Space Telescope
. In extreme cases, the differences may even be measurable from small differences seen from telescopes at different locations on the earth.
More typically, the Einstein parallax is measured from the non-linear motion of the observer caused by the rotation of the earth about the sun. It was first reported in 1995 and has been reported in a handful of events since. Parallax in point-lens events can best be measured in long-timescale events with a large
-- from slow-moving, low mass lenses which are close to the observer.
If the source star is a binary star
, then it too will have a non-linear motion which can also cause slight, but detectable changes in the light curve. This effect is known as Xallarap (parallax spelled backwards).
The first success of this technique was made in 2003 by both OGLE and MOA of the microlensing event OGLE 2003–BLG–235 (or MOA 2003–BLG–53). Combining their data, they found the most likely planet mass to be 1.5 times the mass of Jupiter. As of January 2011, eleven exoplanets have been detected by this method, including OGLE-2005-BLG-071Lb
, OGLE-2005-BLG-390Lb
, OGLE-2005-BLG-169Lb
, two exoplanets around OGLE-2006-BLG-109L
, and MOA-2007-BLG-192Lb
. Notably, at the time of its announcement in January 2006, the planet OGLE-2005-BLG-390Lb probably had the lowest mass of any known exoplanet orbiting a regular star, with a median at 5.5 times the mass of the Earth and roughly a factor two uncertainty. This record was contested in 2007 by Gliese 581 c
with a minimal mass of 5 Earth masses, and since 2009 Gliese 581 e
is the lightest known "regular" exoplanet, with minimum 1.9 Earth masses.
Comparing this method of detecting extrasolar planets with other techniques such as the transit
method, one advantage is that the intensity of the planetary deviation does not depend on the planet mass as strongly as effects in other techniques do. This makes microlensing well suited to finding low-mass planets. One disadvantage is that followup of the lens system is very difficult after the event has ended, because it takes a long time for the lens and the source to be sufficiently separated to resolve them separately.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
phenomenon due to the gravitational lens
Gravitational lens
A gravitational lens refers to a distribution of matter between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer...
effect. It can be used to detect objects ranging from the mass of a planet to the mass of a star, regardless of the light they emit. Typically, astronomers can only detect bright objects that emit lots of light (star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s) or large objects that block background light (clouds of gas and dust). These objects make up only a tiny fraction of the mass of a galaxy. Microlensing allows the study of objects that emit little or no light.
When a distant star or quasar
Quasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...
gets sufficiently aligned with a massive compact foreground object, the bending of light due to its gravitational field, as discussed by Einstein in 1915, leads to two distorted unresolved images resulting in an observable magnification. The time-scale of the transient brightening depends on the mass of the foreground object as well as on the relative proper motion between the background 'source' and the foreground 'lens' object.
Since microlensing observations do not rely on radiation received from the lens object, this effect therefore allows astronomers to study massive objects no matter how faint. It is thus an ideal technique to study the galactic population of such faint or dark objects as brown dwarfs, red dwarfs, planets, white dwarfs, neutron stars, black holes, and
Massive Compact Halo Objects
Massive compact halo object
Massive astrophysical compact halo object, or MACHO, is a general name for any kind of astronomical body that might explain the apparent presence of dark matter in galaxy halos. A MACHO is a body composed of normal baryonic matter, which emits little or no radiation and drifts through interstellar...
. Moreover, the microlensing effect is wavelength-independent, allowing study of source objects that emit any kind of electromagnetic radiation.
Microlensing by an isolated object was first detected in 1993. Since then, microlensing has been used to constrain the nature of the dark matter
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
, detect extrasolar planets, study limb darkening
Limb darkening
Limb darkening refers to the diminishing of intensity in the image of a star as one moves from the center of the image to the edge or "limb" of the image...
in distant stars, constrain the binary star
Binary star
A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
population, and constrain the structure of the Milky Way's disk. Microlensing has also been proposed as a means to find dark objects like brown dwarfs and black holes, study starspots
Sunspot
Sunspots are temporary phenomena on the photosphere of the Sun that appear visibly as dark spots compared to surrounding regions. They are caused by intense magnetic activity, which inhibits convection by an effect comparable to the eddy current brake, forming areas of reduced surface temperature....
, measure stellar rotation, and probe quasars including their accretion disks.
How it works
Microlensing is based on the gravitational lensGravitational lens
A gravitational lens refers to a distribution of matter between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer...
effect. A massive object (the lens) will bend the light of a bright background object (the source). This can generate multiple distorted, magnified, and brightened images of the background source.
Microlensing is caused by the same physical effect as strong lensing and weak lensing, but it is studied using very different observational techniques. In strong and weak lensing, the mass of the lens is large enough (mass of a galaxy or a galaxy cluster) that the displacement of light by the lens can be resolved with a high resolution telescope such as the Hubble Space Telescope
Hubble Space Telescope
The Hubble Space Telescope is a space telescope that was carried into orbit by a Space Shuttle in 1990 and remains in operation. A 2.4 meter aperture telescope in low Earth orbit, Hubble's four main instruments observe in the near ultraviolet, visible, and near infrared...
. With microlensing, the lens mass is too low (mass of a planet or a star) for the displacement of light to be observed easily, but the apparent brightening of the source may still be detected. In such a situation, the lens will pass by the source in a reasonable amount of time, seconds to years instead of millions of years. As the alignment changes, the source's apparent brightness changes, and this can be monitored to detect and study the event. Thus, unlike with strong and weak gravitational lenses, a microlensing event is a transient phenomenon from a human timescale perspective.
Unlike with strong and weak lensing, no single observation can establish that microlensing is occurring. Instead the rise and fall of the source brightness must be monitored over time using photometry
Photometry (astronomy)
Photometry is a technique of astronomy concerned with measuring the flux, or intensity of an astronomical object's electromagnetic radiation...
. This function of brightness versus time is known as a light curve
Light curve
In astronomy, a light curve is a graph of light intensity of a celestial object or region, as a function of time. The light is usually in a particular frequency interval or band...
. A typical microlensing light curve is shown below:
- Lens mass distribution. If the lens mass is not concentrated in a single point, the light curve can be dramatically different, particularly with causticCaustic (optics)In optics, a caustic or caustic network is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an...
-crossing events, which may exhibit strong spikes in the light curve. In microlensing, this can be seen when the lens is a binary starBinary starA binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
or a planetary systemExtrasolar planetAn extrasolar planet, or exoplanet, is a planet outside the Solar System. A total of such planets have been identified as of . It is now known that a substantial fraction of stars have planets, including perhaps half of all Sun-like stars...
. - Finite source size. In extremely bright or quickly-changing microlensing events, like caustic-crossing events, the source star cannot be treated as an infinitesimally small point of light: the size of the star's disk and even limb darkeningLimb darkeningLimb darkening refers to the diminishing of intensity in the image of a star as one moves from the center of the image to the edge or "limb" of the image...
can mollify extreme features. - ParallaxParallaxParallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...
. For events lasting for months, the motion of the Earth around the Sun can cause the alignment to change slightly, affecting the light curve.
Most focus is currently on the more unusual microlensing events, especially those that might lead to the discovery of extrasolar planets. Although it has not yet been observed, another way to get more information from microlensing events that may soon be feasible involves measuring the astrometric
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...
shifts in the source position during the course of the event and even resolving the separate images with interferometry.
Observing microlensing
In practice, because the alignment needed is so precise and difficult to predict, microlensing is very rare. Events, therefore, are generally found with surveys, which photometrically monitor tens of millions of potential source stars, every few days for several years. Dense background fields suitable for such surveys are nearby galaxies, such as the Magellanic Clouds and the Andromeda galaxy, and the Milky Way bulge. In each case, the lens population studied comprises the objects between Earth and the source field: for the bulge, the lens population is the Milky Way disk stars, and for external galaxies, the lens population is the Milky Way halo, as well as objects in the other galaxy itself. The density, mass, and location of the objects in these lens populations determines the frequency of microlensing along that line of sight, which is characterized by a value known as the optical depth due to microlensing. (This is not to be confused with the more common meaning of optical depthOptical depth
Optical depth, or optical thickness, is a measure of transparency. Optical depth is defined by the negative logarithm of the fraction of radiation that is not scattered or absorbed on a path...
, although it shares some properties.) The optical depth is, roughly speaking, the average fraction of source stars undergoing microlensing at a given time, or equivalently the probability that a given source star is undergoing lensing at a given time. The MACHO project found the optical depth toward the LMC to be 1.2×10−7 or about 1 in 8,000,000, and the optical depth toward the bulge to be 2.43×10−6 or about 1 in 400,000.
Complicating the search is the fact that for every star undergoing microlensing, there are thousands of stars changing in brightness for other reasons (about 2% of the stars in a typical source field are naturally variable stars) and other transient events (such as nova
Nova
A nova is a cataclysmic nuclear explosion in a star caused by the accretion of hydrogen on to the surface of a white dwarf star, which ignites and starts nuclear fusion in a runaway manner...
e and supernovae), and these must be weeded out to find true microlensing events. After a microlensing event in progress has been identified, the monitoring program that detects it often alerts the community to its discovery, so that other specialized programs may follow the event more intensively, hoping to find interesting deviations from the typical light curve. This is because these deviations – particularly ones due to exoplanets – require hourly monitoring to be identified, which the survey programs are unable to provide while still searching for new events. The question of how to prioritize events in progress for detailed followup with limited observing resources is very important for microlensing researchers today.
History
In 1704 Isaac NewtonIsaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
suggested that a light ray could be deflected by gravity. In 1801 Johann Georg von Soldner
Johann Georg von Soldner
Johann Georg von Soldner was a German physicist, mathematician and astronomer, first in Berlin and later in 1808 in Munich.-Life:...
calculated the amount of deflection of a light ray from a star under Newtonian gravity. In 1915 Einstein correctly predicted the amount of deflection under General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, which was twice the amount predicted by von Soldner. Einstein's prediction was validated by a 1919 expedition led by Arthur Eddington
Arthur Stanley Eddington
Sir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...
, which was a great early success for General Relativity. In 1924 Orest Chwolson found that lensing could produce multiple images of the star. A correct prediction of the concomitant brightening of the source, the basis for microlensing, was published in 1936 by Einstein. Because of the unlikely alignment required, he concluded that "there is no great chance of observing this phenomenon". Gravitational lensing's modern theoretical framework was established with works by Yu Klimov (1963), Sidney Liebes (1964), and Sjur Refsdal
Sjur Refsdal
Prof Sjur Refsdal was a Norwegian astrophysicist, born in Oslo. He is best known for his pioneer work on gravitational lensing, including the Chang-Refsdal lens....
(1964).
Gravitational lensing was first observed in 1979, in the form of a quasar lensed by a foreground galaxy. That same year Kyongae Chang and Sjur Refsdal showed that individual stars in the lens galaxy could act as smaller lenses within the main lens, causing the source quasar's images to fluctuate on a timescale of months. Bohdan Paczyński
Bohdan Paczynski
Bohdan Paczyński or Bohdan Paczynski was a Polish astronomer, a leading scientist in theory of the evolution of stars, accretion discs and gamma ray bursts....
first used the term "microlensing" to describe this phenomenon. This type of microlensing is difficult to identify because of the intrinsic variability of quasars, but in 1989 Mike Irwin et al. published detection of microlensing in Huchra's Lens
Huchra's Lens
Huchra's lens is the lensing galaxy of the Einstein Cross ; it is also called ZW 2237+030 or QSO 2237+0305 G. It exhibits the phenomenon of gravitational lensing that was postulated by Albert Einstein when he realized that gravity would be able to bend light and thus could have lens-like effects....
.
In 1986, Paczyński proposed using microlensing to look for dark matter
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
in the form of massive compact halo objects (MACHOs) in the Galactic halo
Dark matter halo
A dark matter halo is a hypothetical component of a galaxy, which extends beyond the edge of the visible galaxy and dominates the total mass. Since they consist of dark matter, halos cannot be observed directly, but their existence is inferred through their effects on the motions of stars and gas...
, by observing background stars in a nearby galaxy. Two groups of particle physicists working on dark matter heard his talks and joined with astronomers to form the Anglo-Australian MACHO collaboration and the French EROS collaboration. In 1991 Paczyński suggested that microlensing might be used to find planets, and in 1992 he founded the OGLE microlensing experiment, which began searching for events in the direction of the Galactic bulge
Galactic Center
The Galactic Center is the rotational center of the Milky Way galaxy. It is located at a distance of 8.33±0.35 kpc from the Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest...
.
The first two microlensing events in the direction of the Large Magellanic Cloud
Large Magellanic Cloud
The Large Magellanic Cloud is a nearby irregular galaxy, and is a satellite of the Milky Way. At a distance of slightly less than 50 kiloparsecs , the LMC is the third closest galaxy to the Milky Way, with the Sagittarius Dwarf Spheroidal and Canis Major Dwarf Galaxy lying closer to the center...
that might be caused by dark matter were reported in back to back Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
papers by MACHO and EROS in 1993, and in the following years, events continued to be detected. The MACHO collaboration ended in 1999. Their data refuted the hypothesis that 100% of the dark halo comprises MACHOs, but they found a significant unexplained excess of roughly 20% of the halo mass, which might be due to MACHOs or to lenses within the Large Magellanic Cloud itself.
EROS subsequently published even stronger upper limits on MACHOs, and it is currently uncertain as to whether there is any halo microlensing excess that could be due to dark matter at all. The SuperMACHO project currently underway seeks to locate the lenses responsible for MACHO's results.
Despite not solving the dark matter problem, microlensing has been shown to be a useful tool for many applications. Hundreds of microlensing events are detected per year toward the Galactic bulge, where the microlensing optical depth (due to stars in the Galactic disk) is about 20 times greater than through the Galactic halo. In 2007, the OGLE project identified 611 event candidates, and the MOA project (a Japan-New Zealand collaboration) identified 488 (although not all candidates turn out to be microlensing events, and there is a significant overlap between the two projects). In addition to these surveys, followup projects are underway to study in detail potentially interesting events in progress, primarily with the aim of detecting extrasolar planets. These include PLANET, RoboNet and MicroFUN.
Mathematics
The mathematics of microlensing, along with modern notation, are described by Gould and we use his notation in this section, though other authors have used other notation. The Einstein radiusEinstein radius
The Einstein radius is the radius of an Einstein ring, and is a characteristic angle for gravitational lensing in general, as typical distances between images in gravitational lensing are of the order of the Einstein radius.- Derivation :...
, also called the Einstein angle, is the angular radius
Angular diameter
The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its...
of the Einstein ring
Einstein ring
In observational astronomy an Einstein ring is the deformation of the light from a source into a ring through gravitational lensing of the source's light by an object with an extremely large mass . This occurs when the source, lens and observer are all aligned...
in the event of perfect alignment. It depends on the lens mass M, the distance of the lens dL, and the distance of the source dS:
(in radians)For M equal to the mass of the Sun, dL = 4000 parsecs, and dS = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is 0.001 arcseconds (1 milliarcsecond). By comparison, ideal Earth-based observations have angular resolution
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...
around 0.4 arcseconds, 400 times greater. Since
is so small, it is not generally observed for a typical microlensing event, but it can be observed in some extreme events as described below.Although there is no clear beginning or end of a microlensing event, by convention the event is said to last while the angular separation between the source and lens is less than
. Thus the event duration is determined by the time it takes the apparent motion of the lens in the sky to cover an angular distance . The Einstein radius is also the same order of magnitude as the angular separation between the two lensed images, and the astrometric shift of the image positions throughout the course of the microlensing event.During a microlensing event, the brightness of the source is amplified by an amplification factor A. This factor depends only on the closeness of the alignment between observer, lens, and source. The unitless number u is defined as the angular separation of the lens and the source, divided by
. The amplification factor is given in terms of this value:This function has several important properties. A(u) is always greater than 1, so microlensing can only increase the brightness of the source star, not decrease it. A(u) always decreases as u increases, so the closer the alignment, the brighter the source becomes. As u approaches infinity, A(u) approaches 1, so that at wide separations, microlensing has no effect. Finally, as u approaches 0, A(u) approaches infinity as the images approach an Einstein ring. For perfect alignment (u = 0), A(u) is theoretically infinite. In practice, finite source size effects will set a limit to how large an amplification can occur for very close alignment, but some microlensing events can cause a brightening by a factor of hundreds.
Unlike gravitational macrolensing where the lens is a galaxy or cluster of galaxies, in microlensing u changes significantly in a short period of time. The relevant time scale is called the Einstein time
, and it's given by the time it takes the lens to traverse an angular distance relative to the source in the sky. For typical microlensing events, is on the order of a few days to a few months. The function u(t) is simply determined by the Pythagorean theorem:The minimum value of u, called umin, determines the peak brightness of the event.
In a typical microlensing event, the light curve is well fit by assuming that the source is a point, the lens is a single point mass, and the lens is moving in a straight line: the point source-point lens approximation. In these events, the only physically significant parameter that can be measured is the Einstein timescale
. Since this observable is a degenerate function of the lens mass, distance, and velocity, we cannot determine these physical parameters from a single event.However, in some extreme events,
may be measurable while other extreme events can probe an additional parameter: the size of the Einstein ring in the plane of the observer, known as the Projected Einstein radius: . This parameter describes how the event will appear to be different from two observers at different locations, such as a satellite observer. The projected Einstein radius is related to the physical parameters of the lens and source by.It is mathematically convenient to use the inverses of some of these quantities. These are the Einstein proper motion
Proper motion
The proper motion of a star is its angular change in position over time as seen from the center of mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600 arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of change in...
and the Einstein parallax
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...
.These vector quantities point in the direction of the relative motion of the lens with respect to the source. Some extreme microlensing events can only constrain one component of these vector quantities. Should these additional parameters be fully measured, the physical parameters of the lens can be solved yielding the lens mass, parallax, and proper motion as
Extreme microlensing events
In a typical microlensing event, the light curve is well fit by assuming that the source is a point, the lens is a single point mass, and the lens is moving in a straight line: the point source-point lens approximation. In these events, the only physically significant parameter that can be measured is the Einstein timescale. However, in some cases, events can be analyzed to yield the additional parameters of the Einstein angle and parallax: and . These include very high magnification events, binary lenses, parallax, and xallarap events, and events where the lens is visible.Events yielding the Einstein angle
Although the Einstein angle is too small to be directly visible from a ground-based telescope, several techniques have been proposed to observe it.If the lens passes directly in front of the source star, then the finite size of the source star becomes an important parameter. The source star must be treated as a disk on the sky, not a point, breaking the point-source approximation, and causing a deviation from the traditional microlensing curve that lasts as long as the time for the lens to cross the source, known as a finite source light curve. The length of this deviation can be used to determine the time needed for the lens to cross the disk of the source star
. If the angular size of the source is known, the Einstein angle can be determined as .These measurements are rare, since they require an extreme alignment between source and lens. They are more likely when
is (relatively) large, i.e., for nearby giant sources with slow-moving low-mass lenses close to the source.In finite source events, different parts of the source star are magnified at different rates at different times during the event. These events can thus be used to study the limb-darkening
Limb darkening
Limb darkening refers to the diminishing of intensity in the image of a star as one moves from the center of the image to the edge or "limb" of the image...
of the source star.
Binary lenses
If the lens is a binary star with separation of roughly the Einstein radius, the magnification pattern is more complex than in the single star lenses. In this case, there are typically three images when the lens is distant from the source, but there is a range of alignments where two additional images are created. These alignments are known as caustics. At these alignments, the magnification of the source is formally infinite under the point-source approximation.Caustic crossings in binary lenses can happen with a wider range of lens geometries than in a single lens. Like a single lens source caustic, it takes a finite time for the source to cross the caustic. If this caustic-crossing time
can be measured, and if the angular radius of the source is known, then again the Einstein angle can be determined.As in the single lens case when the source magnification is formally infinite, caustic crossing binary lenses will magnify different portions of the source star at different times. They can thus probe the structure of the source and its limb darkening.
An animation of a binary lens event can be found at this YouTube video.
Events yielding the Einstein parallax
In principle, the Einstein parallax can be measured by having two observers simultaneously observe the event from different locations, e.g., from the earth and from a distant spacecraft. The difference in amplification observed by the two observers yields the component of perpendicular to the motion of the lens while the difference in the time of peak amplification yields the component parallel to the motion of the lens. This direct measurement was recently reported using the Spitzer Space TelescopeSpitzer Space Telescope
The Spitzer Space Telescope , formerly the Space Infrared Telescope Facility is an infrared space observatory launched in 2003...
. In extreme cases, the differences may even be measurable from small differences seen from telescopes at different locations on the earth.
More typically, the Einstein parallax is measured from the non-linear motion of the observer caused by the rotation of the earth about the sun. It was first reported in 1995 and has been reported in a handful of events since. Parallax in point-lens events can best be measured in long-timescale events with a large
-- from slow-moving, low mass lenses which are close to the observer.If the source star is a binary star
Binary star
A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
, then it too will have a non-linear motion which can also cause slight, but detectable changes in the light curve. This effect is known as Xallarap (parallax spelled backwards).
Detection of extrasolar planets
If the lensing object is a star with a planet orbiting it, this is an extreme example of a binary lens event. If the source crosses a caustic, the deviations from a standard event can be large even for low mass planets. These deviations allow us to infer the existence and determine the mass and separation of the planet around the lens. Deviations typically last a few hours or a few days. Because the signal is strongest when the event itself is strongest, high-magnification events are the most promising candidates for detailed study. Typically, a survey team notifies the community when they discover a high-magnification event in progress. Followup groups then intensively monitor the ongoing event, hoping to get good coverage of the deviation if it occurs. When the event is over, the light curve is compared to theoretical models to find the physical parameters of the system. The parameters that can be determined directly from this comparison are the mass ratio of the planet to the star, and the ratio of the star-planet angular separation to the Einstein angle. From these ratios, along with assumptions about the lens star, the mass of the planet and its orbital distance can be estimated.OGLE-2005-BLG-071Lb
OGLE-2005-BLG-071Lb is an planet discovered by the Optical Gravitational Lensing Experiment and others in 2005, using gravitational microlensing. According to the best fit model, it has about 3.5 times the mass of Jupiter and a projected separation of 3.6 astronomical units from the star...
, OGLE-2005-BLG-390Lb
OGLE-2005-BLG-390Lb
OGLE-2005-BLG-390Lb is a 'super-Earth' extrasolar planet orbiting the star OGLE-2005-BLG-390L, which is situated 21,500 ± 3,300 light years away from Earth, near the center of the Milky Way galaxy...
, OGLE-2005-BLG-169Lb
OGLE-2005-BLG-169Lb
OGLE-2005-BLG-169Lb is an extrasolar planet located approximately 2700 parsecs away in the constellation of Sagittarius, orbiting the star OGLE-2005-BLG-169L. This planet was discovered by the OGLE project using the gravitational microlensing method. Based on a most likely mass for the host star of...
, two exoplanets around OGLE-2006-BLG-109L
OGLE-2006-BLG-109L
OGLE-2006-BLG-109L is a dim magnitude 17 unclassified galactic bulge star approximately 4,920 light-years away in the constellation of Sagittarius.- Planetary system :...
, and MOA-2007-BLG-192Lb
MOA-2007-BLG-192Lb
MOA-2007-BLG-192Lb, occasionally shortened to MOA-192 b, is an extrasolar planet approximately 3,000 light-years away in the constellation of Sagittarius. The planet was discovered orbiting the brown dwarf or low-mass star MOA-2007-BLG-192L. At a mass of approximately 3.3 times Earth, it is one of...
. Notably, at the time of its announcement in January 2006, the planet OGLE-2005-BLG-390Lb probably had the lowest mass of any known exoplanet orbiting a regular star, with a median at 5.5 times the mass of the Earth and roughly a factor two uncertainty. This record was contested in 2007 by Gliese 581 c
Gliese 581 c
Gliese 581 c or Gl 581 c is a planet orbiting the red dwarf star Gliese 581. It is the second planet discovered in the system and the third in order from the star. With a mass at least 5.6 times that of the Earth, it is classified as a super-Earth...
with a minimal mass of 5 Earth masses, and since 2009 Gliese 581 e
Gliese 581 e
Gliese 581 e or Gl 581 e is an extrasolar planet found around Gliese 581, an M3V red dwarf star approximately 20.5 light-years away from Earth in the constellation of Libra...
is the lightest known "regular" exoplanet, with minimum 1.9 Earth masses.
Comparing this method of detecting extrasolar planets with other techniques such as the transit
Astronomical transit
The term transit or astronomical transit has three meanings in astronomy:* A transit is the astronomical event that occurs when one celestial body appears to move across the face of another celestial body, hiding a small part of it, as seen by an observer at some particular vantage point...
method, one advantage is that the intensity of the planetary deviation does not depend on the planet mass as strongly as effects in other techniques do. This makes microlensing well suited to finding low-mass planets. One disadvantage is that followup of the lens system is very difficult after the event has ended, because it takes a long time for the lens and the source to be sufficiently separated to resolve them separately.
Microlensing experiments
There are two basic types of microlensing experiments. "Search" groups use large-field images to find new microlensing events. "Follow-up" groups often coordinate telescopes around the world to provide intensive coverage of select events. The initial experiments all had somewhat risqué names until the formation of the PLANET group. There are current proposals to build new specialized microlensing satellites, or to use other satellites to study microlensing.Search collaborations
Photographic plate search of bulge. Remarkable for largely being the work of a single graduate student, Christophe Alard, for his Ph.D. Thesis.- Experience de Recherche des Objets Sombres (EROS) (1993–2002) Largely French collaboration. EROS1: Photographic plate search of LMC: EROS2: CCD search of LMC, SMC, Bulge & spiral arms.
- MACHO (1993–1999) Australia & US collaboration. CCD search of bulge and LMC.
- Optical Gravitational Lensing Experiment (OGLE)Optical Gravitational Lensing ExperimentThe Optical Gravitational Lensing Experiment or OGLE is a Polish astronomical project based at the University of Warsaw that is chiefly concerned with discovering dark matter using the microlensing technique. Since the project began in 1992, it has discovered several extrasolar planets as a side...
( 1992 – ), Polish collaboration established by Paczynski and Udalski. Dedicated 1.3m telescope in Chile run by the University of Warsaw. Targets on bulge and Magellanic Clouds. - Microlensing Observations in Astrophysics (MOA)Microlensing Observations in AstrophysicsMicrolensing Observations in Astrophysics is a collaborative project between researchers in New Zealand and Japan, led by Professor Yasushi Muraki of Nagoya University. They use microlensing to observe dark matter, extra-solar planets, and stellar atmospheres from the Southern Hemisphere...
(1998 – ), Japanese-New Zealand collaboration. Dedicated 1.8m telescope in New Zealand. Targets on bulge and Magellanic Clouds. - SuperMACHO (2001 – ), successor to the MACHO collaboration used 4 m CTIO telescope to study faint LMC microlenses.
Follow-up collaborations
- Probing Lensing Anomalies Network (PLANET)Probing Lensing Anomalies NetworkThe Probing Lensing Anomalies NETwork collaboration coordinates a network of telescopes to rapidly sample photometric measurements of the magnification of stars in the galactic bulge undergoing gravitational microlensing by intervening foreground stars...
Multinational collaboration. - Microlensing Follow Up Network, μFUN
- Microlensing Planet Search (MPS)
- Microlensing Network for the Detection of Small Terrestrial Exoplanets, MiNDSTEp
- RoboNet-II. Searching for planets using a global network of robotic telescopes
Andromeda galaxy pixel lensing collaborations
- MEGA
- AGAPE (in French)
- WeCAPP
- The Angstrom Project
- PLAN
Proposed satellite experiments
- Galactic Exoplanet Survey Telescope (GEST)
- SIM Microlensing Key Project would have used the extremely high precision astrometryAstrometryAstrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...
of the Space Interferometry MissionSpace Interferometry MissionThe Space Interferometry Mission, or SIM, also known as SIM Lite , was a planned space telescope developed by the U.S. National Aeronautics and Space Administration , in conjunction with contractor Northrop Grumman...
satellite to break the microlensing degeneracyDegeneracyDegeneracy may refer to:* DegenerationIn science and mathematics:* Degeneracy , a property of quantum states sharing the same energy levels...
and measure the mass, distance, and velocity of lenses. This satellite was postponed several times and finally cancelled in 2010.