A light curve with a fold in it

The rung below this one drew the single-lens curve: symmetric, achromatic, unrepeatable, and containing one number — the Einstein crossing time — that mixes the lens mass, its distance and its transverse velocity together and cannot be separated into them.

A single point mass is a well-behaved map. It takes each point of the source plane to two image positions, the total magnification is a smooth function of the source’s offset, and the only divergence is at the single point of perfect alignment.

A second mass changes the character of the map, not merely its parameters. The lens equation for two masses is not analytic — it involves the complex conjugate of the image position — and a non-analytic map can fold. Where it does, the Jacobian vanishes, the magnification diverges along a whole curve rather than at a point, and the number of images changes by two across it.

Where a fold comes from #

The lens equation maps an image position to a source position. For a single point mass it is a smooth two-to-one map and its Jacobian, which is the magnification’s reciprocal, vanishes on a circle — the Einstein ring — whose image in the source plane is the single point at the origin.

Add a second mass and that circle deforms. The set of image positions where the Jacobian vanishes is the critical curve, and its image in the source plane is the caustic. For a single lens the caustic degenerates to a point; for two masses it is a closed curve with cusps, and for most configurations there is more than one component.

Two facts about a fold matter observationally and neither depends on anything about the lens beyond its being a fold.

The image count changes by two. Approaching a caustic from outside, two images appear on the critical curve, at the same place and with equal and opposite parities. A binary lens has three images outside its caustics and five inside.

The magnification diverges as the inverse square root of the distance from the fold. That is a generic property of a fold catastrophe, not a property of gravity, and it is why a caustic crossing is a sharp rise with a characteristic shape rather than a symmetric peak.

What a crossing measures #

The planetary caustic’s position gives the projected separation $s$ in Einstein radii, from the relation that the caustic sits at $s - 1/s$ for a wide binary and at $-1/s + s$ folded inside for a close one. The caustic’s size gives the mass ratio: a planetary caustic’s width goes as $\sqrt{q}$ and a central caustic’s as $q$ divided by a function of $s$.

So an anomaly lasting a few hours in a month-long event gives two numbers, and neither of them is a mass. What comes out is $q$, a ratio, and $s$, an angle in units of $\theta_{\rm E}$ — and $\theta_{\rm E}$ is exactly the quantity a single-lens light curve cannot supply.

That is the position the method has always been in: microlensing measures mass ratios superbly and masses badly. A planet’s mass is $q$ times the host’s, and the host’s mass is buried in $t_{\rm E} = \theta_{\rm E}/\mu_{\rm rel}$ along with the relative distance and proper motion.

The degeneracy that is not a measurement error #

Two configurations produce nearly the same central caustic: a close binary at separation $s$ and a wide one at $1/s$.

This is not a coincidence of particular numbers. Expanded to the order at which a central-caustic anomaly is actually measured, the perturbation from a companion depends on $s$ only through the combination $s - 1/s$ — and that combination is invariant under $s \to 1/s$ up to a sign, which the geometry of the caustic does not resolve. The agreement is exact in a limit and approximate in practice, and it tightens as $|\log s|$ grows.

That last point is the sting. The degeneracy is worst exactly where the caustic is smallest, which is where the anomaly is hardest to measure and where a longer or better-sampled light curve does not help. Many published events have two solutions with separations reciprocal to each other, equally good fits, and physically different planets — one inside the snow line and one well outside it.

A degeneracy of this kind cannot be beaten with more of the same data. It has to be broken from outside: by the planetary caustic if the source happens to cross it too, since that one is not degenerate; by higher-order effects in the light curve; or by imaging the lens star directly years later, once it and the source have separated enough.

The nuisance that becomes a ruler #

A point source crossing a fold would go infinitely bright. A real source has an angular radius $\theta_*$, and what is observed is the magnification averaged over its disc — so the divergence is replaced by a rounded shoulder whose width in time is the time the source takes to cross the fold.

Everywhere else in this collection a star’s finite angular size is a complication to be removed: it smears a light curve’s ingress, it limits the resolution of an occultation, it blurs an interferometric visibility. Here it is the measurement, and the reason is that the fold is a straight edge of known sharpness sweeping across a disc of unknown size.

The chain is short. The crossing time $t_*$ and the Einstein time $t_{\rm E}$ give $\rho = \theta_*/\theta_{\rm E} = t_*/t_{\rm E}$. The source’s angular radius $\theta_*$ is obtained independently — from its dereddened colour and magnitude, through an empirical relation between colour and surface brightness that is calibrated on stars with interferometric diameters. Divide, and

$$\theta_{\rm E} = \frac{\theta_*}{\rho}.$$

That single quantity converts a mass ratio into a mass, because $\theta_{\rm E}$ and $t_{\rm E}$ together give the relative proper motion, and $\theta_{\rm E}$ with a lens distance gives the mass directly.

What was actually measured #

A microlensing planet detection is a light curve, sampled by a survey that monitors hundreds of millions of bulge stars every fifteen to sixty minutes, with follow-up photometry triggered when an event brightens.

The anomaly is short. For a mass ratio of $10^{-3}$ — roughly Jupiter around an M dwarf — the planetary caustic crossing lasts $t_{\rm E}\sqrt{q}$, about a day out of a month. For an Earth-mass planet, $q \approx 3\times10^{-6}$ and the anomaly is a couple of hours. Missing it is the normal outcome, and the entire architecture of the field — continuous longitude coverage from several continents, automated alerts, and now a survey cadence fast enough not to need follow-up at all — exists to avoid missing it.

What the fitted parameters are, for a typical published event: $t_{\rm E}$ to a few per cent, $u_0$ and the peak time precisely, $q$ to ten or twenty per cent, $s$ to a few per cent with a twofold ambiguity, and $\rho$ only when a caustic is crossed — which happens in perhaps a third of planetary events.

The masses that result carry error bars of tens of per cent and are quoted with a distance that is often a probability distribution rather than a number. The census microlensing has produced is a census of mass ratios and separations, and it is turned into one of masses through a model of where the lenses are in the Galaxy.

Cusps, and the anomalies that have no fold in them #

A caustic is not made only of folds. Where two folds meet, the curve has a cusp — a point rather than an arc, at which three images merge instead of two — and cusps behave differently enough to be worth separating.

Passing near a cusp without crossing the caustic still produces an anomaly, because the magnification is enhanced in a whole region around a cusp rather than only inside the caustic. That matters practically: a substantial fraction of planetary detections are cusp approaches rather than caustic crossings, and they produce a smooth bump rather than the sharp double-shouldered feature a fold pair gives.

The distinction has consequences for what can be measured. A fold crossing has a sharp edge whose rounding measures the source size, which is what supplies the angular Einstein radius; a cusp approach has no such edge, so the source size is unconstrained and the mass stays unmeasured. The events that are easiest to detect are not the events that yield the most, and a survey’s catalogue is therefore split into a minority with masses and a majority with only mass ratios.

There is a second structure worth naming because it is the source of a persistent ambiguity. Near a cusp, the three merging images have magnifications that satisfy a relation with no free parameters: the sum of the two of one parity equals the third. For a resolved lens that relation is a test — a system violating it has substructure between the images — and for a microlensing event it is one of the few internal consistency checks available on a fit whose likelihood surface has multiple minima.

Cusps also sharpen the close–wide problem rather than resolving it. The central caustic of a close binary and that of a wide one agree in size and shape to the order that matters, and they differ in the arrangement of their cusps — but the difference appears in the light curve only if the source track happens to pass near the cusps that differ, which is a matter of geometry the observer does not control. Some events break the degeneracy and most do not, and which is which is decided by where the source happened to go.

Where the model stops #

A binary lens has more parameters than a light curve constrains. Beyond $q$ and $s$ there is the angle of the source track relative to the binary axis, the source size, limb darkening, and — for long events — the orbital motion of the lens itself and the parallax from the Earth’s motion during the event. Fits routinely explore a likelihood surface with a dozen dimensions and several isolated islands.

The lens is often not two masses. Triple lenses exist, and a caustic structure from a star with two planets, or from a binary star with one, can imitate a single planet’s anomaly closely enough to be published as one.

And the source is often blended. A bulge field at one arcsecond resolution has several stars per resolution element, so the baseline flux attributed to the source includes light that is not being lensed. Blending is degenerate with the magnification, and it is the reason a colour–magnitude estimate of $\theta_*$ is done on the source’s colour recovered from the event rather than on the star’s apparent colour.

The picture cannot show any of the images. Everything drawn in the source plane above is a construction: the images are milliarcseconds apart, the caustic subtends microarcseconds, and the only thing ever measured is a total flux against time. The whole geometry is inferred, and the confidence in it comes from the fact that a fold is a generic mathematical object whose signature could not plausibly be produced by anything else.

A detection that cannot be followed up #

Every other planet-detection method produces a target. A transiting planet can be re-observed at another wavelength, its atmosphere probed during a later transit, its mass measured by a spectrograph, its orbit refined over decades. A radial-velocity planet can be watched for a second orbit.

A microlensing planet cannot be any of those things. The alignment that produced the signal will not recur — the lens and the source are unrelated objects passing at a relative proper motion of a few milliarcseconds a year — and once the event is over the system is a faint star in a crowded field with nothing to distinguish it. The planet has been measured once and will never be measured again.

That shapes the field in ways worth stating, because it explains choices that otherwise look strange. It is why the modelling is done so exhaustively on each event: there is no prospect of a second data set to settle an ambiguity, so every degeneracy has to be explored and reported rather than resolved later. It is why the published parameters are quoted as multi-modal distributions rather than as values with error bars. And it is why the field’s results are framed as occurrence rates rather than as catalogues of objects — the individual detections are not follow-up targets, so their value is entirely statistical.

There is one exception and it is the reason large telescopes are pointed at old event fields. The lens and the source separate at a few milliarcseconds a year, so after a decade or two they can be resolved from one another by adaptive optics or from space. Measuring the lens star’s own brightness then gives its mass and distance directly, which converts the event’s mass ratio into a planet mass with none of the modelling above.

That is a strange kind of observation: the decisive measurement of a planet is made ten or twenty years after the only opportunity to observe the planet has passed, on a star that was invisible at the time and is now merely faint. A few dozen events have been resolved that way, and the masses they return have in several cases selected between the degenerate solutions that the light curve alone could not separate. The method’s one route to certainty runs through waiting.

The caustic’s size and the images that produce it are the two halves of the geometry, and each is worth drawing at a second configuration.

The generalisation #

The mathematics here is not specific to gravity. A fold is the simplest of the catastrophes: the generic way a smooth map from one plane to another can fail to be locally invertible, with a square-root divergence in the density of images and a change of two in their number.

The same structure appears wherever a wave or a ray family is focused by a smooth but non-uniform medium. The bright lines on the bottom of a swimming pool are caustics of the water surface; a rainbow is the caustic of refraction through a sphere, and its brightness diverges in exactly the same square-root way at the geometric-optics level; a mirage is the caustic of a temperature gradient. In each case the pattern of the singularity is universal and only its scale depends on the physics.

That universality is what makes the microlensing interpretation robust. The specific caustic shape depends on the two masses and their separation, and getting it wrong gets those wrong. But that there is a curve, that crossing it adds two images, and that the magnification goes as an inverse square root, are consequences of the map being smooth and non-analytic — and no astrophysical alternative reproduces them.

The shortest events, and what they are not #

One class of detection deserves separating out, because it is the only one for which the finite-source measurement is not an improvement but a requirement.

An event with no host at all — a single lens whose light curve rises and falls within hours rather than weeks — implies a lens of planetary mass, because the Einstein crossing time scales as the square root of the mass and a day-long event corresponds to something below a Jupiter. Several dozen such events have been reported, and they are read as free-floating planets: objects ejected from the systems that made them, or formed in isolation, wandering the Galaxy unattached.

The reading is only as good as the ruling-out of a host, and a host is ruled out by not seeing an anomaly. That is an argument from absence, and it is weak in a specific way: a wide-separation companion produces a caustic far from the source track and no signal at all, so an event with no anomaly is consistent with a bound planet at a large separation as readily as with a free one. What the non-detections constrain is the separation, not the boundness — a survey can say that no companion lies within a few tens of astronomical units and can say nothing beyond that.

The finite-source effect is what makes such an event worth anything quantitatively. For an event lasting a day the source’s angular size is a substantial fraction of the Einstein radius, so the light curve is visibly rounded rather than pointed, and the rounding gives the ratio of the two. That converts an unremarkable brief brightening into a measurement of an angular Einstein radius, and therefore into a mass–distance relation rather than a single degenerate timescale.

So the shortest events are the ones where the source’s size stops being a correction and becomes the whole measurement, and it is the same fold physics as the rest of this essay operating on a lens with no fold in it at all.

And a longer event at a larger impact parameter, which is the shape most of the survey’s detections actually have.

Where this ladder goes next #

Later rungs on this anchor: microlens parallax, in which the Earth’s own motion during a long event distorts the light curve and gives a second mass–distance constraint; space-based parallax, where a satellite an astronomical unit away sees a different light curve and the difference is the geometry; astrometric microlensing, where the image centroid shifts by hundreds of microarcseconds and is now measurable, giving $\theta_{\rm E}$ without needing a caustic crossing; free-floating planets, which produce very short events with no host and whose abundance is a direct test of planet formation; and the survey design problem, which is the question of what cadence and which fields maximise the number of anomalies that are actually caught.