A reference built from other stars

A star subtracted using the star ends with a defect it cannot fix. The reference used to subtract a star’s scattered-light halo is built from the target’s own exposures, so any real companion is in every frame of the reference, and close to the star — where the field rotation has not carried the planet far — the planet subtracts most of itself.

The cure is to build the reference out of something the planet is not in.

Why the throughput is exactly one #

That self-subtraction is not a defect of any algorithm. It is a consequence of the reference containing the signal: a linear combination of frames chosen to minimise a residual will use whatever freedom it has to reduce the planet’s own flux, because the planet is part of the residual.

Build the reference from exposures of a different star and that freedom disappears. There is no planet in any of the library frames, so no combination of them contains one, and the subtraction removes exactly the halo and none of the companion.

The throughput is one, at every separation, by construction. That is a much stronger statement than a large number: it is not a throughput that has been measured to be high but a throughput that cannot be anything else, and it removes an entire step — the injection-recovery campaign — from the estimation of a detection limit.

What replaces it is a different problem with a different shape.

The match, and why it is worse #

A speckle pattern is a snapshot of the instrument’s wavefront error. It depends on the telescope’s flexure with elevation, the temperature of the optics, the adaptive-optics system’s state, and the atmosphere above the aperture at that moment.

Two stars observed an hour apart at different elevations have different speckle patterns, and the difference is larger than a planet. That is exactly the objection made against reference stars, and it is correct — for a single reference star.

The technique works because it does not use a single reference. It uses a library of thousands of archival exposures, accumulated over years of everyone’s observations with the same instrument, and constructs the reference as a combination of whichever ones happen to match. With enough frames, some of them were taken in a state close to the target’s, and the closer the library’s best match the smaller the residual.

The improvement is not unlimited. Beyond a certain library size the limiting residual is not the quality of the match but the part of the wavefront error that is static — the difference between what the wavefront sensor measures and what the science camera sees, which is a property of the instrument rather than of the epoch and is therefore in every frame of the library too. That floor is where the curves flatten.

What the trade is actually about #

Setting the two side by side makes the choice legible, and it is a choice about where a planet is expected rather than about which technique is better.

Far from the star, the field rotation has carried a companion many resolution elements, self-subtraction is negligible, and the target’s own frames are the better reference because they match. Angular differential imaging wins, and it is what the surveys for wide young giants use.

Close to the star, the arc a companion travels is a fraction of a resolution element, the throughput collapses, and no library is needed to beat it. Reference-star differential imaging wins.

And the boundary between them is not a property of the telescope. It depends on how much field rotation the observation happened to get — which depends on when in the night the target was observed and on where it is on the sky, and on how much of the atmosphere’s own blurring the adaptive optics removed — and on how many archival frames exist. Two identical observations of the same star on two nights can want different techniques.

What the detections say about the trade #

Fourteen years of high-contrast surveys have produced a short list of imaged companions, and their separations say which technique found them.

Almost every one sits beyond ten resolution elements, which for a young nearby star is tens of astronomical units — and every one of them was found against a halo whose brightness the inverse-square law fixes. At those separations self-subtraction is negligible and the target’s own frames are the better reference, so the catalogue is a catalogue of angular differential imaging’s successes — and it is also a catalogue of what the technique’s inner working angle allowed anybody to look for.

That is a selection effect with a specific shape, and it matters for what the surveys conclude. A survey reports an occurrence rate as a function of separation, obtained by dividing detections by what the survey could have seen — and at small separations the completeness is near zero, so the occurrence rate there is unconstrained rather than measured as low.

The reference technique’s contribution is therefore not primarily the objects it has found. It is the region of parameter space it opens: separations of a few resolution elements, which for a nearby star is a few astronomical units, which is where the giant planets of the solar system are. The imaged planets known so far are all at tens of astronomical units, which is a population with no counterpart here, and closing the gap between the two is what the inner working angle is for.

Where it changed the field #

The technique’s importance grew for a reason that has nothing to do with algorithms: field rotation requires an atmosphere to be observed through.

A ground-based telescope tracking a target in pupil-stabilised mode rotates the sky across the detector because the Earth turns and the telescope’s own pupil is held fixed. A space telescope has no such constraint and is generally operated with the field fixed, so there is no rotation to exploit and angular differential imaging is simply unavailable.

So for space-based high-contrast imaging the reference-star technique is not an alternative — it is the only option, and the whole observing strategy is built around it. A target is observed, then the telescope is rolled by a few degrees and it is observed again, which gives a very small amount of rotation; and a separate reference star is observed immediately afterward, giving a reference taken minutes rather than months away.

That last point is the one that makes space observations work. The residual in the reference match is dominated by how much the instrument changed between the target and the reference, and in space the instrument barely changes at all — there is no atmosphere, no flexure from a moving telescope tube, no elevation-dependent gravity load. A reference taken twenty minutes later is a very good reference, and the technique reaches contrasts on a two-metre telescope that a ground-based eight-metre cannot.

The reference that is the target #

There is a third construction that sits between the two and is worth naming, because it shows what the trade is really about.

A companion at a given separation is present in the target’s own frames at a position angle that rotates. Rather than building a reference from frames in which the companion is elsewhere, one can build it from the same frames with the companion’s expected position masked out — so the reference is the target’s own halo with a hole in it, and the hole means the planet cannot subtract itself.

That is the idea behind the modern hybrid reductions, and it works because the halo is smooth on the scale of the mask while the companion is not. What it costs is the information in the masked region, which is exactly the region the reference most needs to be accurate in — so the masked reference is a slightly worse match near the planet, which is where it matters.

The three constructions therefore form a sequence in what they trade. The target’s own frames give the best match and lose the planet; the archival library loses no planet and gives the worst match; the masked target’s frames lose no planet and give a match that is degraded only locally.

The best available answer is usually the third kind, and it exists because somebody noticed that the two obvious options are two ends of a continuum rather than two alternatives. The generalisation is worth stating as a habit: when two methods fail for opposite reasons, the construction that fails for neither is often obtainable by restricting one of them rather than by combining the two.

What was actually measured #

A sequence of exposures of the target, a library of archival exposures of other stars taken with the same instrument in the same configuration, and a reduction that chooses a combination of the library to subtract.

Three practical difficulties decide whether it works, and none is in the figures.

The library has to be homogeneous. Frames taken with a different coronagraph, a different filter, a different exposure time or after a realignment are not usable, so a decade of archive may yield a few hundred usable frames for a given configuration rather than thousands. Instruments designed after the technique was established keep their configurations fixed for exactly this reason.

The selection of the match matters more than the algorithm. Choosing which library frames to build the reference from — by correlation with the target’s own halo outside the region of interest, which is a region the planet is not in — does most of the work, in the same way a comparison star is chosen for a differential light curve, and the principal-component decomposition applied afterwards does the rest.

And the brightness has to be matched. Two stars of different magnitudes give haloes of different amplitude, so the reference has to be scaled, and the scaling is fitted on an annulus. An error in the scaling is a residual proportional to the halo, which is largest exactly where the technique is being used.

What is published is a contrast curve, and unlike the one the target’s own frames produce, it does not need an injection-recovery campaign to establish the throughput. It still needs one to establish the noise, because the residual after subtracting an imperfect reference is not Gaussian and its statistics have to be measured — and at a few resolution elements there are only a dozen independent samples to measure them from.

Where the model stops #

The curves here are a model with stated parameters. The throughput of the reference technique is exactly one and that is exact; everything about the residual — how it falls with library size, where it floors — is a two-parameter description of a behaviour that is measured empirically for each instrument. What the figures establish is the shape of the trade and the direction the crossing moves, not a number anybody should quote.

The two techniques are not exclusive. The usual modern reduction combines them: the library is augmented with the target’s own frames, and the algorithm chooses. That inherits a self-subtraction proportional to how much of the target’s own data it used, which has to be measured, and it outperforms either alone.

Spectral differential imaging is a third axis and is omitted entirely. A speckle’s position scales with wavelength while a planet’s does not, so observing at several wavelengths and rescaling gives another reference — with its own self-subtraction, worst for planets with featureless spectra.

And none of this is the coronagraph. Everything here is about what happens to the light that gets past the starlight suppression, and the suppression itself — which is where most of the contrast is won — is an optical problem rather than an algorithmic one.

The library as a shared instrument #

There is an institutional consequence and it is not a digression, because it changed how the instruments are run.

An archival library is worth more to everybody than to anybody, and it accumulates only if raw frames are kept, calibrations are recorded, and the configuration is not changed. Each of those is a cost borne by whoever is operating the instrument now, in exchange for a benefit accruing to whoever uses it in five years.

The high-contrast community made those choices largely because the reference technique made the benefit legible: a contrast curve improved by a factor of two, at the separations the instrument was built to reach, from frames that had already been taken. That is an unusually direct argument for an archive, and it is more persuasive than the general case for preserving data because the gain is measurable on a plot.

Two consequences followed. Instruments now hold their coronagraph and filter configurations fixed for years at a time, at some cost in flexibility. And the observations taken to build the library — deliberate exposures of stars known to have no companions, taken purely as reference material — are now a scheduled part of an instrument’s programme rather than a by-product of somebody else’s.

An observation taken for no scientific purpose can be the limiting component of somebody else’s measurement, and the practice has spread: the same argument is made for reference spectra in high-precision radial velocity work, for dark frames in space photometry, and for the calibrator catalogues that make an interferometric position possible at all.

The number that decides an instrument’s reach #

It is worth extracting the one quantity that both techniques are competing to minimise, because it is not the one either is usually described by.

A detection limit is a noise divided by a throughput. The reference techniques differ in both terms, and the product is what matters — so a technique with a throughput of a tenth and a residual half as large is exactly as good as one with a throughput of one and a residual five times larger. Quoting either number alone is meaningless, and the older literature quoted the residual.

That is the reason contrast curves published before the field adopted throughput correction are not comparable with later ones. The correction is a factor of two to five at small separations, always in the direction of the earlier curves being optimistic, and it is the single largest revision the field has made to its own published limits.

The same arithmetic decides what an instrument should be built for. Reducing the residual is an optical problem — better wavefront control, a more stable bench, a colder telescope. Raising the throughput is an observational and algorithmic one. An instrument that has minimised the wrong one of the two has spent its money on the part of the product that was not limiting, and which is limiting depends on the separation, which depends on what the instrument is for.

The generalisation #

The shape to carry is that two ways of estimating the same nuisance can fail in disjoint places, and that the right response is to locate the crossing rather than to pick a winner.

Both techniques estimate the same thing — the halo that would have been there without a planet — and both are wrong. One is wrong because its estimate contains the signal; the other because its estimate comes from a different time. Those errors are not two sizes of one error; they have different dependences on separation, so neither dominates everywhere.

When two estimators of one nuisance have different arguments, the useful question is where they cross, and the answer is usually a function of something operational — here, the field rotation and the archive’s size — rather than of anything physical. That makes the choice a decision about the observation rather than about the object, and a survey that fixes its reduction in advance is choosing a range of separations to be good at.

The second reading is about what an archive is. A library of thousands of exposures of other people’s targets is not data in the ordinary sense — nobody took them for this purpose, and each was taken to answer a different question. It is nonetheless the component that sets the achievable contrast at small separations, and its value grows with every night anybody uses the instrument. An instrument’s calibration can be an emergent property of its own history, and designing for that — keeping configurations fixed, archiving raw frames, publishing the pipeline — is a design decision that pays off years later and is easy not to make.

Still open: not subtracting the halo at all #

What comes next stops estimating the halo and starts removing it optically. An interferometric nuller combines the light from two apertures so that the star’s own light interferes destructively while a companion’s, arriving at a slightly different angle, does not — so the starlight never reaches the detector and there is nothing to subtract.

That changes the inner working angle by an order of magnitude, because the null’s width is set by the interferometric baseline rather than by a single aperture’s diffraction. It also brings an entirely different set of systematics, dominated by how stably the null can be held, and it is the route the next generation of instruments is being designed around.