A cadence that finds a planet is not one that weighs it

A microlensing planet is found once and never again. The source star, the lens star and the observer drift out of alignment over a few weeks, and the brief deviation the planet adds to the light curve — an anomaly of hours or days on an event of a month — is all the information there will ever be. Everything a planet’s microlensing detection can say, it has to say from the points that happened to be taken while it lasted.

The earlier figures on this subject took the points for granted and asked what they contained: the fold in the lens mapping that makes a caustic, the Earth’s orbital motion that adds a second dimensioned number, the shift of the centre of light that adds a third, the brevity of an event with no host, and the symmetry under which two different planets draw the same curve. The question here is the one each of those raised and none of them answered: given a telescope and a finite number of exposures, where should they go?

The answer depends on what the survey is for, and the figures below show that it splits sharply. A survey can be designed to find planets — to put enough points on enough anomalies that each is recognised as one — or to weigh them, which takes points at particular moments within the anomaly. The two designs are not the same, and for the planets most microlensing has found, they are far apart.

What a mass ratio needs and what a mass needs #

A binary-lens fit to an anomaly returns the planet-to-star mass ratio $q$ and their projected separation $s$ in units of the Einstein radius, and it returns them from the shape of the anomaly whatever its sampling, provided the anomaly is recognised. Neither is a mass. The masses come from the Einstein radius $\theta_E$, an angle, which is

$$\theta_E = \sqrt{\kappa M \pi_\text{rel}},$$

with $M$ the lens mass and $\pi_\text{rel}$ the relative parallax of lens and source. The light curve measures time, not angle: the Einstein crossing time $t_E$ is $\theta_E$ divided by an unknown proper motion. In the photometry, an angle enters only through the size of the source star. When the source passes over a caustic, its finite disc rounds the infinitely sharp magnification of a point source, and the rounding depends on the ratio $\rho = \theta_\star/\theta_E$ of the source’s angular radius to the Einstein radius. Fit the rounding and $\rho$ is measured; read the source’s angular radius from its colour and brightness, by the surface-brightness relation an interferometer calibrates, and $\theta_E = \theta_\star/\rho$ follows. Combined with a parallax, it gives the mass.

So a detection gives $q$ and a measurement of $\rho$ turns it towards a mass. The difference matters because the rounding happens only at the caustic. A source crossing a fold takes as long to round it as it takes to cross its own diameter, $2\rho t_E$, and for a main-sequence source in the bulge, $\rho \approx 0.002$, on an event with $t_E = 25$ days, that is 2.4 hours. A survey that wants $\rho$ needs points inside those hours. A survey that wants only $q$ needs points anywhere in the anomaly.

One anomaly, four cadences #

The figure below is a single anomaly: an Earth-mass planet around a star of a third of a solar mass, $q = 10^{-5}$, at a projected separation of 1.25 Einstein radii, its planetary caustic crossed by a source of $\rho = 0.002$, computed as a finite-source light curve and divided by the single-lens curve of the star alone.

The source in the figure is comparable in size to the caustic it crosses, and that changes the shape. With a point source the light curve would jump at each fold — divergent in principle, limited only by the sampling — and dip between them. With a source of $\rho = 0.002$ the two crossings merge into one rounded hump, a little over five hours above five per cent, with shallow negative wings either side where the source passes the caustic’s outer influence. The hump’s width is set mostly by the source’s diameter, which is why its shape measures $\rho$.

The four sets of points are what four cadences would record at one arbitrary phase. Every fifteen minutes gives eighteen points more than five per cent off the single-lens curve and traces the shape. Every hour gives four, enough to recognise an anomaly and to estimate its width coarsely. Every four hours gives two, both of them in the shallow wings or on the edge, and daily sampling one. At those two cadences the anomaly is not merely poorly sampled; at no phase does it receive the three significant points that a detection asks for.

For an Earth, then, the cadence that finds the planet is already the cadence that weighs it, because the anomaly is hardly longer than the crossing. That is not true for larger planets.

How long anomalies last, and how long crossings last #

An anomaly lasts roughly as long as the source takes to cross the planet’s own Einstein ring, $2\sqrt{q}\,t_E$, and no less than the source takes to cross its own diameter.

The square root makes the range compressed: four decades in mass ratio are two decades in duration. A Jupiter around a third-solar-mass star, $q \approx 3 \times 10^{-3}$, makes an anomaly of about three days. An Earth around the same star makes one of about four hours, and below $q \approx 4 \times 10^{-6}$ the anomaly is no longer than the source crossing however small the planet. The fold crossing itself, the 2.4-hour dashed line, does not depend on $q$ at all. It depends on the source, the event and nothing about the planet.

That is the asymmetry the whole design problem rests on. A Jupiter’s anomaly is thirty times longer than the crossings within it. A daily survey puts several points on it and recognises it easily; the same survey puts a point inside a 2.4-hour crossing about one time in ten. An Earth’s anomaly is barely longer than its crossings, so whatever cadence samples the anomaly samples them too.

The cadence that finds and the cadence that weighs #

The question can be put to a population of crossings rather than to one. In the next figure, seven source tracks cross the planetary caustic of each planet at offsets spanning it, and each is sampled at 160 random phases for each cadence. An anomaly is counted as detected if three points deviate from the single-lens curve by more than five times a one per cent photometric error, and as weighed if at least two of those points fall where the source’s finite size changes the magnification by more than five per cent — that is, during a fold crossing.

For the Jupiter the two curves separate by more than an order of magnitude in cadence. Detection is complete from five minutes to four hours and falls only beyond: half the crossings are still found at one point a day. The measurement of the source size is at its maximum of 43 per cent from five minutes to an hour — the maximum is set by the three tracks in seven that cross a fold at all, the others skimming past the caustic without entering it — and falls to 15 per cent at four hours and to nothing at a day. A survey sampling every four hours finds every Jupiter crossing in the figure and weighs fewer than one in six.

For the Earth the two curves fall together, and the limit on weighing is the track rather than the cadence. Any crossing whose track enters the caustic is weighed whenever it is detected, at every cadence where it is detected at all; the tracks that miss the caustic are found at fast cadences and never weighed. Detection falls steeply between thirty minutes and four hours because the whole anomaly is only five hours long.

The two planets therefore ask different things of a survey. A cadence of a few hours is enough to count Jupiters and too slow to weigh them. For Earths, the cadence needed to count them is the cadence needed to weigh them, and anything slower than about an hour does neither.

A crossing in is a promise of a crossing out #

The weighing problem for giant planets has one feature that makes it less hopeless than the figure suggests, and it comes from the geometry of the caustic rather than from the survey. A caustic is a closed curve. A source that has crossed into one must cross out again, and between the two crossings it is inside, where the lens makes five images instead of three and the magnification stays high. The first crossing announces the second.

For a Jupiter’s planetary caustic the interval between them is a substantial fraction of the anomaly, a day or more, and the time of the exit can be predicted from the entry and from the light curve before it, which fixes the source’s track and the caustic’s size to within hours. A survey that sees a point jump above the single-lens curve can therefore raise its own cadence on that field, or alert others, before the exit happens. That is how many of the best-measured giant planets were weighed: found by a survey at a counting cadence, then sampled densely at the second crossing because the first had made it inevitable. Some of the first binary-lens events with a measured source size, in 1998, were caught this way, by networks of telescopes that scheduled their observations around a predicted exit.

The figure above uses a fixed cadence and so does not credit this. A survey that adapts its cadence to what it has seen weighs more than any fixed cadence of the same cost would, and the gain is largest exactly where the fixed-cadence weighing is worst, for long anomalies with well-separated crossings. For an Earth there is nothing to adapt to: the entry and exit are a few hours apart, the whole anomaly is over before an alert could be acted on, and only the cadence already running counts.

What the source size buys #

The trouble taken to measure $\rho$ is worth stating in what it delivers. Without it, a planetary event gives $q$, $s$ and $t_E$, and the host’s mass and distance are a probability distribution drawn from a model of the Galaxy — where stars are, how fast they move, how their masses are distributed — and typically uncertain by a factor of two. With $\rho$, the Einstein radius $\theta_E$ is measured, and the mass and distance are confined to a single curve, $M = \theta_E^2/(\kappa\,\pi_\text{rel})$; a Galactic model then places the lens along that curve far more tightly. With a parallax as well, the curve becomes a point.

And $\theta_E$ gives one more number that matters later: the relative proper motion of lens and source, $\mu = \theta_E/t_E$. That is the rate at which the two stars will separate on the sky, so it says when an image taken years afterwards will be able to resolve the lens from the source and weigh the host directly by its light. A planet detected without $\rho$ has an unknown proper motion, and the later imaging has to be scheduled blind. A cadence that weighs a planet at the time therefore also tells the future observer when to look.

Many fields seldom, or few fields often #

A survey’s cadence is not chosen in isolation, because the exposures are a fixed budget. A telescope that can take one exposure of one field every fifteen minutes can watch one field every fifteen minutes or two fields every thirty, or ninety-six fields once a day; the number of events it sees grows in proportion to the number of fields, and the cadence on each falls in the same proportion.

The product of the number of fields and the per-crossing efficiency gives the yield, in units of what one field watched continuously would produce. For Jupiters the detection yield rises almost in proportion to the number of fields as long as the per-field efficiency stays near one, and peaks at 96 fields — one visit a day to each — before the cadence becomes too slow to put three points on even a three-day anomaly. The yield of weighed Jupiters peaks at 12 fields, a cadence of three hours, and its peak is about a sixteenth of the detections’.

Neither design is wrong. A survey that wants the statistics of planets — how common giant planets are beyond the snow line, the peak in their abundance that neither radial velocities nor imaging can see on its own — wants the mass ratio and the separation for as many planets as possible, and mass ratios are what a detection gives. A survey that wants planets with masses, to compare with the planets of other methods, wants $\rho$, and should cover eight times less sky to get it. The history of the field went back and forth between them. The early surveys covered wide areas about once a night and alerted other telescopes to follow promising events densely, a division of labour that worked for the brightest events and was at the mercy of whoever was awake and clear. The surveys since the mid-2010s cover their central fields every fifteen minutes and their outer fields every few hours, largely without follow-up — a choice of a few fields at the weighing cadence and many at the counting one, made with the trade-off in this figure in mind.

Who is watching when the source crosses #

A cadence is a property of a camera when the camera is observing. On the ground it is observing for part of each night, and only on clear nights.

The figure averages over the time of day at which a crossing happens. One site with eight and a half hours of usable darkness in the bulge season, clear six nights in ten, sees 22 per cent of fold crossings — a fifteen-minute cadence is a fifteen-minute cadence for about a fifth of the time. Three sites spread in longitude, as the southern network that has operated in Chile, South Africa and Australia since 2015 is, cover 61 per cent; they hand the bulge from one to the next as the Earth turns and suffer only when weather and daylight coincide. A telescope in space observes continuously within the season the bulge is away from the Sun and is limited by nothing but its pointing constraints.

That is the case for the space survey that the last essay on this subject raised: no weather, no daytime and no seeing, so the cadence in the earlier figures is the cadence the survey actually achieves, and the source star is resolved rather than blended, so the photometric precision assumed above is closer to what it gets. It turns the measurement of $\rho$ from the fortunate case it is on the ground into a routine one, for exactly the planets — cold, low-mass, a few astronomical units from their stars — that no other method reaches.

What the model leaves out #

The figures are calculations with stated simplifications, not a survey simulation. Every track crosses a planetary caustic of a wide planet, perpendicular to the star–planet axis; real anomalies also come from central caustics in high-magnification events, where the source passes close to the host star and the sampling problem is different and often easier, because the timing is predictable from the event’s rise. The source size is measured here only from fold crossings, with a threshold on the finite-source signature, and a real fit also extracts $\rho$ from cusp approaches and from smoothly passing near a caustic, so the weighing fractions are lower limits. Noise is a fixed one per cent and the detection criterion is three points at five times it, where real pipelines fit the whole event and use the statistical improvement of the fit; blending, crowding and the faintness of the source at the start of the event are absent. The figure of observatory longitudes uses idealised nights and a single clear-sky probability.

What is robust is the comparison: a fold crossing lasts a time set by the source, an anomaly a time set by the planet, and for any planet much more massive than the Earth the second is many times the first.

Still open: whether a census can be weighed afterwards #

The surveys now running have found planets mostly by their mass ratios, and the planets’ masses mostly remain unknown, because the Einstein radius and the parallax that turn a mass ratio into a mass are measured for a minority. There is a second route to the mass that does not depend on the cadence at the time: wait. Lens and source separate at a few milliarcseconds a year, and a decade after the event a space telescope or an adaptive-optics camera can resolve the lens star from the source, measure its brightness and its direction of motion, and weigh the host directly; the mass ratio then gives the planet. That route has been taken for dozens of events and is being planned for the thousands a space survey will find. Whether it can be applied to every detection, including the faint lenses that are M dwarfs or white dwarfs, decides whether the census that the fastest cadences cannot weigh at the time can be weighed afterwards — and whether the trade between finding and weighing drawn here is a permanent cost or a delay of ten years.