An event of a few hours and no host

Every planet detected by any other method is detected through its star. A transit is a dip in a star’s light; a reflex velocity is a star’s motion; a direct image is a planet beside a star that has been suppressed. Remove the star and every one of them fails completely rather than gradually.

Microlensing does not need the star, and the first rung of this anchor said so in passing: the measurement is of the lens’s mass through its gravity, and gravity does not care whether the lens is illuminated. This rung is what follows from taking that seriously.

Everything except the duration is the same #

The magnification of a point lens depends on one number, the separation in Einstein radii, and on nothing else. A source passing at u₀ = 0.1 is magnified by a factor of ten whether the lens is a red dwarf or a rock.

That is worth stating twice, because it is the whole content of the rung and it is surprising. Microlensing has no threshold in mass. It has a threshold in time: the event is over after a few times t_E, and t_E is θ_E divided by the relative proper motion, and θ_E goes as the square root of the mass. So the entire mass dependence of the observable is a duration.

The square root is worth dwelling on because it is unusually forgiving. A factor of a million in mass — from an M dwarf to a Moon — is only a factor of a thousand in duration, so a survey whose cadence spans three orders of magnitude in time covers six in mass. No other exoplanet method has that leverage: a transit depth goes as the square of the radius and a reflex velocity linearly in the mass, so both lose signal-to-noise proportionally and both have a hard floor where the signal disappears into the noise. Microlensing has no such floor, only a clock.

An M dwarf at half a solar mass gives an event lasting about three weeks. A Jupiter gives a day and a half. A Neptune gives about ten hours. An Earth gives ninety minutes.

Ninety minutes is a hard number for a survey. A telescope observing a field once a night sees nothing; one observing every twenty minutes sees four or five points on the curve and can fit it; one observing continuously sees the whole thing. The detection of terrestrial-mass free-floating objects is therefore a question of cadence, and it is the reason the surveys that find them observe a small number of fields extremely often rather than a large number occasionally.

The finite source, which is a defect and a gift #

There is a second thing that changes with the mass, and it is what saves the measurement from being a duration and nothing else.

The source star has an angular radius θ_, and what matters is ρ = θ_/θ_E — the source’s size in units of the Einstein radius. θ_E falls as the square root of the mass, so ρ rises as the mass falls, and below about a Jupiter mass ρ passes one: the source is larger than the ring magnifying it.

The flattening looks like a loss of signal and it is the opposite. ρ is a ratio of two angles, and the source’s own angular radius is known — its colour gives its surface brightness and its dereddened magnitude gives its flux, and the two together give θ_* to a few per cent by a relation calibrated on interferometrically measured stars. So a measured ρ delivers θ_E, which is the dimensioned quantity a photometric event otherwise cannot supply.

The events where ρ is measurable are exactly the low-mass ones. What ruins the photometry of a short event is what makes it a mass measurement.

It is worth putting a number on how well the source’s angular size is known, since the whole measurement rests on it. The surface-brightness relation gives θ_* from a dereddened colour and a magnitude, and it is calibrated on stars whose diameters have been measured directly by interferometry — a few hundred of them, to a few per cent. The scatter of the relation is about two per cent for giants and rather worse for main-sequence stars, and the dereddening is the dominant error toward the bulge, where the extinction is several magnitudes and variable on arcminute scales.

So a measured θ_E inherits an uncertainty of five to ten per cent from the source rather than from the light curve, and a mass derived from θ_E and π_E inherits it twice over. That is a good position to be in — it means the measurement is limited by a calibration that can be improved rather than by the event, which cannot be repeated.

What “free-floating” does and does not mean #

The events are real and the interpretation is contested, and the contest is worth laying out because it is a good example of what an unrepeatable measurement can and cannot settle.

A short event with no light from a lens is consistent with two things. The lens may be an isolated object, ejected from a system it formed in or formed in isolation. Or it may be a planet on a wide orbit around a star that is simply too far away, in Einstein radii, to leave any signature on the light curve — a bound planet at forty astronomical units produces a lens whose host is hundreds of Einstein radii away and contributes nothing.

Nothing in the light curve distinguishes those. What can distinguish them is a search for the host: if the lens is bound, the host star is at the lens’s distance and will separate from the source over a decade or two, and adaptive optics or interferometry can look. Several candidates have been resolved that way and turned out to be bound; several have not.

The honest statement of the result is therefore about a population rather than about objects. Short-timescale events are commoner than the known distribution of bound planets and stars predicts, by an amount that requires a substantial population of objects unaccompanied on the scale a light curve can see — and whether “unaccompanied” means ejected or merely distant is a question about follow-up rather than about microlensing.

There is one further discriminant, and it is the reason the surveys observe as often as they do rather than merely as precisely. A bound planet at a wide separation produces a short event that sometimes shows a second, much longer bump — the host’s own event, offset in time by however long it takes the source to reach it. Catching that requires continuous coverage over weeks around a ninety-minute anomaly, which is a demanding observing pattern and is the only in-light-curve evidence a host exists.

Two such events have been reported with a plausible host bump. Neither is unambiguous, and both illustrate the general position: the negative statement — no host was seen — is much weaker than it sounds, because a host outside the region the source samples leaves nothing to see.

Why the abundance is a test of formation #

The reason anybody counts these events is that ejection is a prediction rather than a possibility.

Planet formation by core accretion in a disc produces several giant planets in mutually perturbing orbits, and dynamical instability among them is generic: the systems scatter, and scattering ejects. Simulations of the process typically eject one to a few objects per system, with a mass distribution weighted toward the smaller members because a light body is easier to throw out than a heavy one.

So the abundance of free-floating objects, as a function of mass, is a direct reading of how violent planet formation is — and it is a reading of a process that finished four billion years ago in any system that is now stable. The bound planets record what survived; the free ones record what did not.

That makes the measurement complementary to an occurrence rate from transits or velocities in a way no other pair of methods is. Two censuses of the same formation process, one of the products and one of the debris.

There is a second population the same measurement reaches and it is not planetary at all. Below about thirteen Jupiter masses an object cannot fuse deuterium and is called a planet; above it, up to about eighty, it fuses deuterium but not hydrogen and is a brown dwarf. Both are dark, both are found by microlensing, and the timescale distribution runs straight through the boundary without noticing it — a thirteen-Jupiter lens gives an event of about five days and a fourteen-Jupiter lens about the same.

That is worth stating because the boundary is a definition rather than a feature, and the microlensing census is one of the few that measures across it without a change of technique. Whether the mass function of unbound objects is continuous through the deuterium-burning limit is a question the method can answer and no other can, and the answer bears on whether the two populations form the same way.

What the picture cannot show #

The timescale relation is exact for the geometry it is given and the geometry is the whole problem.

t_E depends on the lens distance, the source distance and the relative proper motion as well as on the mass, and a survey measures only t_E. Converting a distribution of timescales into a distribution of masses therefore requires a Galactic model — the same model the parallax rung had to invoke — and the answer inherits its uncertainties.

For the short events the model matters more than usual, because a short event can be a light lens or a fast one, and the fast ones are not evenly distributed. A lens in the disc moving against a bulge source has a relative proper motion several times a typical bulge-bulge encounter’s, so disc lenses populate the short end of the timescale distribution without being low in mass at all. Separating the two requires the finite-source measurement, which is available on a minority of events.

There is a further difficulty specific to the shortest events, and it is instrumental rather than astrophysical. A ninety-minute event observed at a fifteen-minute cadence has six points, and a curve with three free parameters fitted to six points has very little redundancy against a spurious brightening from a variable star, a cosmic ray or a photometric artefact. Every published short-event sample rests on a vetting procedure, and the vetting procedure is where the systematics live.

That last point is the method’s real floor, and it is worth being clear that it is a floor of a peculiar kind. Microlensing has no sensitivity limit in mass in the ordinary sense: a lighter lens does not give a weaker signal, it gives a shorter one, until the Einstein radius drops below the source’s own angular size and the magnification is diluted. Where that happens depends on the source, so the floor is different for every event, and it is around a lunar mass for a bulge giant and rather lower for a main-sequence source.

What the surveys actually do #

The observing strategy that finds these events is worth describing, because it is the strategy the arithmetic above forces and it looks nothing like the rest of exoplanet astronomy.

A ninety-minute event requires a cadence of minutes, and a cadence of minutes over a field large enough to contain millions of monitored stars requires a wide-field camera on a dedicated telescope. So the surveys — OGLE, MOA, KMTNet — observe the same handful of bulge fields, every clear night, for years, at intervals of ten to twenty minutes. KMTNet uses three telescopes at three longitudes so the bulge is never below the horizon at all of them, which removes the daily gap that would otherwise chop every short event in half.

That is a survey optimised for time rather than for depth or area, and it is the opposite of a transit survey’s problem, where the signal is small, repeats, and can be recovered by folding years of data. A microlensing event does not repeat. Everything about it has to be caught while it happens, and a gap in the coverage is a gap in the result.

The consequence for the free-floating census is that its completeness is a function of the weather. A published occurrence rate is corrected for it, by injecting simulated events into the real photometric time series and counting how many are recovered — which is the same injection-and-recovery machinery an occurrence rate from transits uses, applied to a much less forgiving signal.

The habit #

The structure worth extracting is that a method with no threshold in the quantity of interest still has a threshold, and it is in a different quantity entirely.

Microlensing’s sensitivity to mass is not a sensitivity at all: the signal’s amplitude does not depend on the mass. What depends on the mass is the duration, so the method’s limit is set by how often a telescope can point at the same star — an entirely instrumental number that has nothing to do with photons or with gravity. A survey twice as fast reaches four times lower in mass, and a survey twice as precise reaches no lower at all.

That pattern is rarer than it sounds and it is worth recognising when it appears. A pulsar timing measurement has the same character: the precision is in a time rather than in an intensity, so the limit improves with the length of the baseline rather than with the size of the telescope. In both cases the useful investment is patience and scheduling, not aperture.

What the count currently says #

It would be dishonest to end without the numbers, and equally dishonest to present them as settled.

The surveys report an excess of short events — timescales below about two days — over what the known populations of stars and bound planets predict. Estimates of the implied abundance have ranged from about two objects of Jupiter mass per main-sequence star, in an early analysis, down to fractions of one per star in later ones with larger samples and better completeness corrections. The direction of that revision is the usual one: the first measurement of a rare population is high, because the systematics that would remove events are found afterwards.

What has survived every revision is an excess at the lowest masses, around and below the Earth, where the events last hours and where the finite-source effect makes θ_E measurable. Those are the best-characterised free-floating candidates and there are of order ten of them.

Ten objects is not a mass function. It is the beginning of one, and the reason the subject is worth watching is that the sample size is governed by observing cadence and field coverage rather than by anything fundamental — which means the next order of magnitude is a matter of building the right survey rather than of waiting for the objects to become brighter.

Where this ladder goes next #

Five rungs have taken this anchor through a planetary spike, a caustic crossing, a parallax, a centroid and an event with no host. What is left is the one degeneracy that none of them resolves.

The next rung is the close–wide degeneracy: a binary lens with separation s and one with separation 1/s have central caustics that agree to first order in the mass ratio, so the same event is a planet outside the Einstein ring or one inside it — four astronomical units or less than one, from a curve that does not distinguish them. It is a symmetry of the lens equation rather than a limitation of the data, and no amount of photometric precision removes it.

Beyond it: the survey design question, which is what cadence and which fields maximise the number of anomalies actually caught; and what a space survey of the bulge changes, which is the answer to nearly every limitation this ladder has recorded.