A star magnified by a planet nobody will see again

Every other method in this field measures a planet by what it does to its own star: the light it removes, the wobble it induces, the heat it radiates. Microlensing measures a planet by what it does to the light of a star behind it, some thousands of parsecs further away, with which it has no connection whatever.

The measurement takes a few hours. Nothing about the geometry recurs. And the planet’s own star need not be detectable at all — in a good number of cases it never has been.

Light bends, and a point mass makes two images #

A mass deflects light passing at impact parameter $b$ by an angle $4GM/(bc^2)$ — twice the Newtonian value, which is the deflection Eddington’s expedition measured in 1919 and the reason general relativity is the theory it is.

When the lens is exactly on the line of sight the deflection is symmetric and the source is smeared into a ring, whose angular radius is

$$\theta_E = \sqrt{\frac{4GM}{c^2}\,\frac{D_S - D_L}{D_L D_S}}.$$

For a half-solar-mass star halfway to the Galactic bulge, that is about 0.3 milliarcseconds. Nothing resolves it, and nothing has to: the ring is a scale, not a picture.

Offset the source by $u$ Einstein radii and the ring breaks into two images, at $u_\pm = (u \pm \sqrt{u^2+4})/2$. Their combined magnification is

$$A(u) = \frac{u^2+2}{u\sqrt{u^2+4}},$$

which is 1.34 at $u = 1$, rises to 3.0 at $u = 0.35$, and diverges as $u \to 0$. The divergence is real for a point source and is cut off by the source’s finite size, which is one of the ways a real event carries more information than the idealised one.

Crucially, the magnification depends only on $u$. A star drifting past the line of sight at constant relative velocity has $u(t) = \sqrt{u_0^2 + (t/t_E)^2}$, so the light curve is symmetric, achromatic and completely specified by three numbers: the time of closest approach, the minimum separation $u_0$, and the Einstein crossing time $t_E$.

That achromaticity is the method’s signature and its principal defence against confusion with the enormous population of variable stars in the same fields. A Cepheid brightens and reddens; a lensed star brightens and does nothing else.

Where the magnification comes from #

A lens does not create photons, and it is worth being explicit about what it does instead, because “magnification” is a misleading word for it.

Gravitational deflection preserves surface brightness — a theorem of geometrical optics that holds for any lossless system — while changing the solid angle a source subtends. The two images together cover more sky than the source did, and since brightness per unit solid angle is unchanged, more light arrives. The photons that reach the observer are ones that would otherwise have gone elsewhere.

That has a consequence which is easy to miss and impossible to unsee: the light is taken from other lines of sight. Somewhere off to the side, an observer sees the same star slightly dimmed. Lensing is a redistribution, and its total over all directions is zero.

The deflection itself is the same inverse-square field a sphere makes, integrated along a light path rather than a trajectory. Nothing about the lens’s internal structure enters: a star, a black hole and a compact cloud of the same mass at the same distance produce identical light curves, which is exactly why microlensing was proposed in the 1980s as a way of detecting dark matter in the Galactic halo, and why the surveys that now find planets were built to look for something else entirely.

The planet is a lens inside a lens #

A planet orbiting the lens star sits somewhere in the same sky plane. If one of the two images happens to pass close to it, the planet lenses that image in its turn — and the total brightness jumps for as long as the coincidence lasts.

Two properties follow immediately from the mass ratio $q$, and both are worth more than the detection itself.

The planet’s own Einstein radius is $\sqrt{q}\,\theta_E$. For $q = 10^{-3}$ that is 3 per cent of the star’s; for an Earth-mass planet around an M dwarf, $q \approx 3\times10^{-5}$ and it is half a per cent. The deviation therefore lasts about $\sqrt{q}\,t_E$ — a day for a Jupiter, a couple of hours for an Earth, on an event lasting a month or two.

The amplitude of the deviation does not fall with $q$. This is the surprising half. A small planet produces a short deviation, not a shallow one: while the image is inside the planet’s own Einstein radius the perturbation is of order unity whatever the mass. So sensitivity to low-mass planets is limited by cadence, not by photometric precision — which is exactly the opposite of the transit and velocity methods, where the signal shrinks with the planet and the precision is everything.

The consequence for how the field is organised is direct. Microlensing surveys are cadence problems, and they are also crowding problems: an event requires two stars to line up to within a milliarcsecond, so the surveys point where the stars are densest, which is the Galactic bulge. The probability that any given bulge star is being lensed at a given moment is about one in a million — so OGLE and MOA monitor hundreds of millions of bulge stars nightly to find a thousand events a year, and a network of follow-up telescopes then observes the ongoing ones continuously, because the planet — if there is one — will announce itself for a few hours at an unpredictable moment.

What the measurement actually delivers #

A well-sampled planetary event gives $q$ and $s$, the projected separation in Einstein radii, both to a few per cent. Neither is a mass or a distance. In particular there is no orbital period, so the harmonic law — which supplies the missing quantity in almost every other method in this field — has nothing to work on: a single event sees a fixed geometry, not a motion.

To get a planet mass in kilograms, the lens mass and the Einstein radius in physical units are both needed, and the light curve gives neither directly. Two effects supply them when they are measurable: the finite size of the source, which resolves the sharp features and gives $\theta_E$ in absolute terms; and microlensing parallax — the Earth’s own orbital motion distorting the curve over a long event, which is the same parallax argument used on nearby stars applied to a lens rather than to a source. Where both are measured, the mass and distance follow exactly. Where neither is, the answer is a probability distribution over the Galactic model, and the published mass is a median with a factor-of-two spread. The model in question is a description of where stars are and how fast they move — built from the distance ladder and from surveys of the bulge — so the planet’s mass inherits the uncertainties of Galactic structure, which is an unusual sentence to have to write about a planet.

This is a genuinely different epistemic position from the rest of the field. A transiting planet’s radius is uncertain because a stellar radius is uncertain; a microlensing planet’s mass is uncertain because the lens itself is often unobservable, and what is being used instead is a statistical model of where stars are in the Galaxy.

The region of sensitivity, and why it is the interesting one #

The method is most sensitive when an image passes near the planet, and the images sit near the Einstein ring. So the sensitivity peaks for planets at projected separations near $\theta_E$ — which for a typical bulge lens corresponds to two or three astronomical units.

That is not a convenient accident. It is just beyond the snow line, the distance at which water condenses in a protoplanetary disc, where the solid surface density jumps and the theory of core accretion says giant planets should preferentially form. Every other method has its own inconvenient blind spot there: a radial-velocity survey needs decades to reach that orbital period, a transit survey needs the alignment and a mission lasting years, and direct imaging cannot resolve it at those distances.

Microlensing also reaches something no other method can: planets with no star at all. An isolated lensing event of very short duration — a few hours to a couple of days, with no long-timescale stellar signal around it — is a free-floating object of planetary mass. Several dozen candidates are known, and a survey-scale statement about how many such objects the Galaxy contains is one of the things the Roman Space Telescope was designed to settle.

What was actually measured #

OGLE-2003-BLG-235/MOA-2003-BLG-53, 2004. The first microlensing planet: a deviation of about seven days on a longer event, giving $q = 0.004$ and a planet of roughly 2.6 Jupiter masses at 4.3 AU. Two surveys observing the same field independently caught the same anomaly, which is what made it credible.

OGLE-2005-BLG-390Lb, 2006. The one that demonstrated the method’s reach: $q = 7.6\times10^{-5}$, a planet of about 5.5 Earth masses at 2.6 AU from a 0.22-solar-mass star, at a distance of 6.6 kpc. The deviation lasted about half a day, and it was caught because three telescopes on three continents were watching. At the time it was by a large margin the least massive planet known around a normal star, and it was found around a star nobody has ever seen.

The statistical results. Because each detection is an independent draw with a computable sensitivity, microlensing gives occurrence rates for cold planets that nothing else can. The surveys imply that low-mass planets beyond the snow line are common — of order one per star, within a factor of a few — and that Jupiters there are much rarer, at roughly one in six. The most striking statistical claim, that free-floating Jupiters outnumber stars, has been substantially revised downward as the samples grew; the current estimates put free-floating objects at lower masses and lower abundances than the first analyses suggested, which is what happens when a rate is estimated from a handful of events.

The mass of a lone white dwarf, 2017. The astrometric version of the same effect: the lens shifts the source’s apparent position as well as brightening it, by of order $\theta_E$, and Hubble measured a two-milliarcsecond deflection of a background star by the white dwarf Stein 2051 B. That gives the mass directly — 0.675 solar masses — with no orbit and no companion required. It was the first mass measured for a star by the bending of light from another, which is the experiment Eddington performed on the Sun with the same equation.

A test of general relativity, incidentally. The deflection of starlight is the same physics as the 1919 eclipse expedition and the same as galaxy-scale lensing. Nothing in a microlensing analysis is calibrated against a laboratory; the geometry is taken from the theory and the theory is taken as settled.

The rate is a measurement too #

The individual events are what the field publishes, and the number of them is a separate measurement that gets far less attention.

The probability that a given background star is lensed at any moment — the optical depth — is the integral along the line of sight of the lensing cross-section times the density of lenses. It depends on how much mass is in compact objects between here and the source, and on how that mass is distributed in distance.

Measuring it means counting events and dividing by the number of stars monitored and the time monitored, which is a straightforward exercise made difficult by the fact that a survey’s detection efficiency depends on the event’s duration, its peak magnification and the source’s brightness. The corrections are large.

The answer for the Galactic bulge came out somewhat higher than the pre-survey models predicted, and the excess was one of the earlier pieces of evidence that the inner Galaxy contains a bar rather than a smooth axisymmetric bulge — since a bar viewed near end-on puts more material along the line of sight than a round distribution of the same mass.

The same statistic answered the question the surveys were built for. Monitoring the Magellanic Clouds rather than the bulge measures the optical depth of the Galactic halo, and if the halo’s dark matter were in compact objects of stellar mass, the event rate would have been large. It was small — enough to exclude compact objects as the dominant halo component over a wide mass range.

A null result from counting events is a statement about what the halo is not made of, and it is one of the strongest such statements in the subject.

Where the picture stops #

The event never repeats. The lens and source are unrelated stars whose paths crossed once. There is no possibility of a confirming observation of the same planet, no follow-up spectrum, no second season. Every microlensing planet rests permanently on the photometry taken during those hours, and where the sampling had a gap, the parameters have a corresponding hole in them.

Degeneracies are common and sometimes exact. The best known is the close–wide degeneracy: a planet at separation $s$ and one at $1/s$ produce nearly identical central caustics, so a well-measured event often yields two solutions with separations differing by a factor of several and no way to choose. It is not a modelling failure but a symmetry of the lens equation. Where a third body is involved — a second planet, or a binary lens star — the equation is a genuine many-body problem with no closed solution for the image positions, and the family of light curves it can produce is correspondingly wild: the same situation, in a different guise, as what “unsolvable” means for three gravitating bodies.

Nothing else can be learned about the planet. There is no spectrum, no radius, no density, no atmosphere — the two-measurement combination that gives a composition has no analogue here. A microlensing planet is a mass ratio, a projected separation and, in the good cases, a mass. It is the least informative detection per planet and the most informative per unit of survey effort, which is a trade the field makes deliberately.

A single event constrains only one point of a system. Whatever the planet’s orbit is, the measurement is a snapshot of one projected separation at one instant. The orbital period is unknown, the eccentricity is unknown, and whether there are other planets is unknown unless they too happened to be in the way.

The lens is usually invisible. In many events the lens star is too faint or too blended to detect even years later, when the two stars have separated enough to try. Where it can be detected — with adaptive optics after eight or ten years of relative proper motion — the lens mass is measured directly and the statistical estimate is checked. Those checks have largely held, which is the reason to trust the ones that cannot be checked.

The event’s two parameters are the impact parameter and the mass ratio, and each of them changes the light curve in a way the other cannot imitate.

The generalisation #

Lensing is a single phenomenon observed across about fifteen orders of magnitude in mass.

Clusters of galaxies produce strong lensing: multiple images, arcs and Einstein rings large enough to photograph, from which the cluster’s total mass follows — and the discrepancy between that mass and the visible one is a substantial part of the case for dark matter. Individual galaxies lens quasars into four-image configurations whose relative time delays measure the Hubble constant. Weak lensing measures the mass distribution of the universe from the statistical distortion of the shapes of millions of background galaxies.

Microlensing is the same equation with the images unresolvable, so the only observable left is the total brightness. That is a reduction of the information by an enormous factor, and what makes it work is that the geometry sweeps through: time does the job that angular resolution cannot, and a light curve is a one-dimensional scan of a two-dimensional magnification map.

And the image geometry at a large impact parameter, which is the configuration in which nothing is detectable at all.

Where this goes next #

The obvious extension is to stop treating the planet as a small perturbation and solve the two-mass lens exactly — which produces caustics, closed curves in the source plane across which the magnification is formally infinite, and a taxonomy of light-curve shapes that is far richer than the single spike drawn here.

Later rungs on this anchor: the binary lens equation and its caustics. Finite-source effects, and the angular Einstein radius. Microlensing parallax from the Earth’s orbit and from a second observatory in solar orbit. The close–wide degeneracy. Free-floating planets and their mass function. Astrometric microlensing, which is how a lone white dwarf was weighed. Lensing of one star by another with a known orbit. Cold-planet occurrence rates. And what a space survey of the bulge changes, which is the answer to nearly every limitation listed above.