Two systems that draw the same curve

The second rung of this anchor solved the two-mass lens exactly and found the caustic — the closed curve in the source plane across which two images appear out of nothing and the magnification formally diverges. It noted in passing that a binary lens has two caustics for most separations: a central one near the primary, and a planetary one further out — and that finite-source effects across a fold turn the source’s size into a ruler.

The two behave completely differently under a change of separation, and the difference is the subject of this rung. One of them reports the separation unambiguously. The other reports a combination of s and 1/s and cannot tell them apart.

What is degenerate and what is not #

A binary lens is described by two numbers: the mass ratio q and the separation s, in Einstein radii. Both appear in the light curve, and both are fitted.

The central caustic sits at the primary, where the magnification is already high, and it is a small four-cusped figure whose width goes as 4q/(s − 1/s)² for a wide binary. That expression depends on s only through s − 1/s, whose square is invariant under s → 1/s. The shape of the caustic, to the same order, is invariant too.

The planetary caustic is a different object. It sits at s − 1/s from the centre of mass for a wide binary — a location that moves with s and is nowhere near the origin — and for a close binary there are two planetary caustics, off the axis, in an entirely different place. Nothing about them is degenerate.

So the degeneracy is not a property of binary lenses. It is a property of central-caustic events: those in which the source track passes near the primary and never goes near the planetary caustic at all.

Why the invariance is s + 1/s rather than anything else is worth one line, because the algebra is short and the result is memorable. Expanding the binary lens equation about the primary, the leading perturbation from a companion of mass ratio q at separation s enters through the quantity s − 1/s — the shift of the companion’s own image — and the caustic’s size depends on its square. A reflection s → 1/s changes the sign of s − 1/s and leaves the square alone. The symmetry is a reflection across the Einstein ring, and the ring is the only length in the problem.

Why central-caustic events are the common ones #

The unfortunate part is that central-caustic events are not a minority.

The planetary caustic is small — its extent goes as √q for a wide binary — and it sits at a specific place, so a source track crosses it only by chance. The central caustic is smaller still, but it is at the primary, and the primary is where the source is going anyway: an event is only detected because the source passed close to the lens, so every event samples the central region and only some of them sample the planetary one.

Worse, the events most likely to be followed up intensively are the high-magnification ones, where the source passes very close to the primary — and those are exactly the events that sample the central caustic and nothing else. A deliberate strategy of monitoring high-magnification events, which is the most efficient way to find planets, is a strategy that maximises the fraction of detections subject to this degeneracy.

First order, and what the second order is worth #

The degeneracy is exact only in the limit q → 0, and the figures show what that means quantitatively rather than asserting it.

At q = 0.003 — a Neptune around an M dwarf — the two central caustics agree to about one per cent at wide separations and to thirteen per cent at s = 1.4. At q = 0.001 the disagreement at s = 1.4 falls to 4.8 per cent — a factor of 2.7 for a factor of three in the mass ratio, which is the first-order scaling behaving as it should. The pattern is what a first-order result predicts: the correction is of order q, so a smaller companion is more degenerate.

That is exactly the wrong direction for the science. The measurements that matter most — low-mass planets, where the census is thinnest — are the ones where the degeneracy is closest to exact, and the ones where it is breakable are the stellar-mass binaries nobody is trying to count.

There is a second reason the high-magnification strategy dominates, and it makes the situation worse rather than better. A source passing very close to the primary is magnified enormously, which means the photometry is precise in absolute terms, which means small anomalies are detectable — so the sensitivity to planets is highest exactly where the separation is least determined. The strategy that finds the most planets is the strategy that learns the least about where they are.

That is not an argument against it. A detection with an ambiguous separation is worth a great deal more than no detection, and the ambiguity is a factor of eight in one parameter rather than a factor of anything in the mass ratio, which is measured cleanly either way. But it is worth stating plainly, because the published planet counts and the published separation distributions rest on the same events and are not equally reliable.

What the degeneracy costs #

The two solutions are not neighbouring. They are s and 1/s, so a wide solution at 2.8 Einstein radii and a close one at 0.36 differ by a factor of nearly eight in projected separation.

Converted to physical distance, that is a planet at roughly four astronomical units against one at half. Those are different objects in every respect a formation theory cares about: one is beyond the snow line and is a cold giant of the kind microlensing exists to find; the other is inside it and is a different population.

The consequence for a census is a probability rather than an error bar. A published occurrence rate from a sample containing degenerate events has to assign each of them to one solution or the other, and the standard practice is to weight by a Galactic prior on the projected separation — which reintroduces the model dependence that a mass from θ_E and π_E was supposed to escape.

It is worth putting the numbers in astronomical units, since Einstein radii are not a unit anybody has an intuition for. For an M dwarf at four kiloparsecs lensing a bulge source, the Einstein radius projected to the lens is about two astronomical units. So a companion at s = 2 is at four AU and its degenerate twin at s = 0.5 is at one — the first is beyond the snow line and is the cold-giant population microlensing was built to census, and the second is inside it and belongs to a population the transit surveys already measure.

The two answers are therefore not two estimates of one thing. They are two different objects, and a census that assigns them by a prior is assigning membership of one population or the other on a model rather than on a measurement.

What breaks it #

Three things distinguish the two solutions, and each requires something beyond the basic light curve.

The planetary caustic. If the source crosses it, the separation is fixed unambiguously, because its position is s − 1/s for the wide case and a different place entirely for the close one. That is the clean answer and it depends on chance.

Higher-order terms in the light curve. The degeneracy is broken at order q, so a sufficiently precise light curve of a sufficiently large-q system distinguishes the solutions. In practice this works for mass ratios above about 10⁻², which is the brown-dwarf range.

Additional effects with their own geometry. The lens’s orbital motion during the event perturbs the caustic in a way that depends on the true separation rather than on the invariant combination, so a long event with detectable lens motion can be resolved. So can one where the finite source resolves the caustic structure at a level the first-order expansion does not describe.

None of the three is available on demand, and all three are more likely on long, bright, well-covered events — which reintroduces a selection: the events whose degeneracies are resolved are not a random sample of the events that have them.

A degeneracy in the equations, not in the data #

The distinction the title of this section draws is the one worth carrying out of the rung, and it is more general than microlensing.

Most degeneracies in astronomy are limitations of measurement. The mass a radial velocity gives is a lower bound because the inclination is unknown, and a transit or an astrometric orbit measures the inclination and removes it. The distance–reddening degeneracy in a photometric distance is broken by a second colour. In every such case the ambiguity comes from having taken one measurement where two were needed, and the remedy is another measurement.

This one is different. The two configurations produce the same central caustic because the lens equation possesses a symmetry, so improving the photometry does not help — a perfect light curve of a central-caustic event at q → 0 is exactly consistent with both. What is needed is not more precision but a different feature, and whether the event contains one is decided by where the source happened to pass.

Recognising which kind of degeneracy is at hand decides what to do about it. The first kind is an observing proposal. The second is a matter of luck, and of designing surveys that increase the chance of luck.

The other degeneracies, for contrast #

The close–wide one is the most famous and it is not the only one, and the family is worth listing because the members are of different kinds.

There is a jerk–parallax degeneracy, in which a light-curve distortion attributable to the observer’s acceleration can equally be attributed to the lens’s orbital motion. That one is a degeneracy between two physical effects rather than between two parameter values, and it is broken by observing long enough that the two diverge.

There is the u₀ sign degeneracy in space-based parallax, where two observatories cannot tell which side of the lens the source passed as seen from each. Four sign combinations, sometimes all four consistent, and it is discrete rather than continuous.

There is the ecliptic degeneracy, in which a field near the ecliptic makes the Earth’s projected motion nearly one-dimensional so the parallax vector’s component perpendicular to it is unconstrained. That one is a property of where the field is.

And there is the blending degeneracy, in which a fainter source more highly magnified and a brighter source less magnified fit the same curve. That one is a genuine measurement degeneracy and it is broken by resolving the source, which a space telescope does.

Four kinds — a symmetry of the equations, a coincidence of two effects, a discrete sign ambiguity and an ordinary parameter correlation — and each requires a different remedy. Calling all of them “degeneracies” hides that.

What the picture cannot show #

The comparison drawn here is of caustic widths, and a caustic is a shape rather than a length.

Two caustics of the same width and different shapes would produce different light curves, and the figures do not test that. The full statement of the degeneracy — Dominik’s result — is that the whole central caustic, shape included, is invariant to first order, and checking that properly means comparing the curves point by point rather than comparing their extents. The width comparison is a necessary condition and is what the figures verify.

The second omission is that a light curve is not a caustic. What is observed is the magnification along one track, so two lenses with identical caustics certainly give identical curves, but two lenses with slightly different caustics may still give indistinguishable curves if the track samples only the part where they agree. The observational degeneracy is therefore broader than the mathematical one, and how much broader depends on the track — which is another way of saying the answer differs event by event.

And the whole treatment is of a static binary. A real lens is two bodies in orbit, and over a long event the separation changes, so “the separation” is a value at a reference epoch and the light curve constrains its rate of change as well. That extra parameter is what breaks the degeneracy in the best-observed events, and it is also one more thing that can absorb a mismodelled feature.

How often it actually bites #

The honest question is what fraction of published microlensing planets carry an unresolved close–wide ambiguity, and the answer is a large minority.

Roughly a third to a half of detections come from central-caustic anomalies in high-magnification events, and of those the majority do not have a resolving feature. Published catalogues report both solutions with their relative likelihoods from the fit, and the likelihoods are often close enough that neither is excluded — a Δχ² of a few over hundreds of data points is not a discrimination.

The remainder come from planetary-caustic crossings, where the separation is unambiguous, and those are the events that anchor the separation distribution. So the census has a well-determined subset and a larger ambiguous one, and the standard treatment is to fold both in with weights. Whether that is a fair procedure depends on whether the ambiguous events are drawn from the same underlying distribution as the clean ones, which is not obviously true: a planetary-caustic crossing requires the caustic to be large, which favours a particular range of s.

That is the shape of a selection effect rather than a degeneracy, and the two compound. The events whose separations are known are selected for having a certain separation.

What a space survey changes about this one #

It is worth being specific about which of the difficulties above a space-based bulge survey removes, because the answer is: not this one.

Continuous coverage removes the daily gaps, so more anomalies are caught and more of them are fully sampled. Diffraction-limited imaging resolves the source from its neighbours, which removes the blending degeneracy outright. Photometric precision of a millimagnitude makes the finite-source measurement routine rather than fortunate, so θ_E is available on most events instead of a few. A parallax comes free from the spacecraft’s own orbit.

None of that touches the close–wide degeneracy, because it is not a limitation of the data. A perfect light curve of a central-caustic event at small q is exactly consistent with both solutions, and the only escape is a track that samples a planetary caustic — which is a matter of geometry and remains a matter of luck.

What a space survey does change is the number. With ten or twenty times as many detections, the subset with resolving features is ten or twenty times larger, and a separation distribution measured on that subset alone becomes statistically useful without needing the ambiguous events at all. The remedy for an irreducible ambiguity is not to resolve it but to have enough events that the unambiguous ones suffice.

Where this ladder goes next #

Six rungs have taken this anchor from a single magnification through a caustic, a parallax, a centroid, a host-less event and a symmetry of the equations. What is left is not a new physical effect but a question about how to observe.

The next rung is survey design: given that anomalies are rare, brief and unrepeatable, and that the degeneracies above are broken by features a track either does or does not sample, what cadence and which fields maximise the number of usable detections rather than the number of detections? The answer is not obvious and it has changed twice — from wide-and-shallow to a survey-plus-follow-up model to a return to high-cadence surveys with no follow-up at all.

Beyond it: what a space survey of the bulge changes, which is the answer to nearly every limitation this ladder has recorded — no weather, no daytime, no seeing, source stars resolved rather than blended, and a photometric precision that turns the finite-source measurement from a fortunate case into a routine one.