A gap that counting can test and impostors can fill

The masses of black holes in X-ray binaries cluster between five and fifteen solar masses, and the masses of neutron stars between one and a little over two. Between the heaviest neutron stars and the lightest black holes, from about two to about five solar masses, the X-ray binaries left a gap: in a sample of a couple of dozen black-hole masses, none fell in it. If real, the gap says something about how massive stars die — that the collapse of a stellar core makes either a neutron star or a black hole of at least five solar masses, and almost never anything in between, which would favour supernova explosions that either succeed quickly or fail completely.

It might not be real. The black-hole masses in X-ray binaries are not measured directly; they come from the companion’s velocity curve, which gives a mass function and only a floor on the unseen mass, and from a model of the companion’s light curve, whose distortion by the black hole’s tide gives the orbit’s tilt. If those models systematically underestimate the tilt, the masses are systematically overestimated, and objects that belong in the gap are moved above it. Analyses that allowed for the effect found the gap partly filled.

Astrometric orbits avoid the problem entirely. The orbit traced on the sky by a star round a dark companion shows its tilt as the shape of the ellipse, and with the parallax it gives the companion’s mass outright, with no light curve and no model of the star’s distortion. The next astrometric catalogues are expected to contain dozens of such systems. The question is whether they can decide.

Where the X-ray masses could err #

The X-ray masses come from a chain worth spelling out, because each link is a place for the gap to have been made. The companion star, distorted by the black hole’s tide into a teardrop that fills its Roche lobe, shows a brightness that rises and falls twice an orbit as its long and short sides turn towards the observer. The amplitude of that ellipsoidal modulation depends on the inclination: an orbit seen edge-on shows the largest swing, one seen face-on none. Fitting the light curve gives the inclination, the inclination turns the mass function into a mass, and the mass function comes from the companion’s velocity curve, as in any single-lined binary.

The weak link is light that does not come from the companion. The accretion disc round the black hole adds light that does not vary with the ellipsoidal period, diluting the modulation and making the orbit look closer to face-on than it is. A face-on orbit needs a heavier companion to give the same velocities, so unaccounted disc light inflates the mass. In quiescence the disc is faint, but not always negligible, and it can flicker. Studies that measured the disc’s share directly from the depth of the companion’s absorption lines found that several masses had been overestimated, by enough to move objects near five solar masses down towards the gap. Whether by enough to fill it is the argument that astrometry is placed to end.

What precision the question asks for #

The gap is narrow on a logarithmic scale — a factor of two and a half — and its edges are sharp only if the masses are. Measured with errors of ten per cent, a sample from a world with an empty gap shows a clean trough: the neutron stars’ peak and the black holes’ peak stay apart, and fewer than one system in four hundred lands in the middle of the gap. Measured with thirty per cent errors, both peaks spread into it, and the trough fills to a third of what a genuinely filled gap would show.

So the precision that matters is about ten per cent in mass, per system. That is a modest target by the standards of eclipsing binaries, where masses are measured to a per cent, and a demanding one by the standards of X-ray binaries, where the inclination uncertainty alone is often larger. The question for astrometry is whether its masses reach it, and for which systems.

The error budget of one orbit #

The mass comes from three measured quantities and one assumed. The orbit’s angular size, divided by the parallax, gives the visible star’s orbit in astronomical units; the period comes from the timing; and the visible star’s own mass comes from its spectrum. Kepler’s third law then gives the companion’s mass, through a relation in which the angular size and the parallax enter cubed and the period squared.

Short orbits are small on the sky: a star in a one-month orbit round a three-solar-mass companion moves by a few tenths of a milliarcsecond at a kiloparsec, comparable to a single measurement’s error. Orbits much longer than the survey are followed for only part of a revolution, and an arc of an orbit does not separate the mass from the period. Between those limits, for periods from a few months to about the survey’s length, the error is set by distance. A companion at a kiloparsec is weighed to about four per cent; one at four kiloparsecs to no better than fifteen. The distance enters cubed, and the gap test is a local one.

The budget changes character with distance. Nearby, the limit is the visible star: its mass, inferred from its spectrum through stellar models, enters with a reduced weight but never vanishes. Beyond a kiloparsec or so, the parallax dominates, because its relative error grows in proportion to distance and it enters cubed. The parallax here is the system’s own, measured simultaneously with the orbit and correlated with it, and inverting it to a distance has its own biases once its error passes a few per cent. The ten-per-cent target is met comfortably within two kiloparsecs, which is the volume in which the gap test will be made.

A longer survey widens the window #

Every number above assumes a survey of five and a half years. An astrometric mission that runs for twice as long changes them in two ways at once. It takes twice as many measurements, so each orbit’s size and the parallax are known about thirty per cent better, and it follows orbits round for twice as long, so the periods at which the mass is well determined extend from about five years to about ten.

Both matter for the gap. A thirty-per-cent improvement in the astrometry moves the distance at which a three-solar-mass companion is weighed to ten per cent from about two and a half kiloparsecs to about three and a half, which nearly triples the volume searched. And longer periods are where the companions with wide orbits sit, which are the ones whose formation is least affected by the mass transfer and common envelopes that shape the short-period systems. A sample that spans periods from a few months to a decade samples the outcomes of stellar death with less interference from what happened to the binaries afterwards.

The cost of the longer baseline is time: the extended catalogue arrives years after the shorter one. The test is therefore likely to be made twice — first with the systems the shorter survey can weigh well, nearby and at periods of one to five years, and then with the larger sample. If the first already finds several systems squarely in the gap, the question will have moved from whether the gap is filled to what fills it, which only the second, with its wider range of orbits and the spectroscopy that follows, can answer.

How many systems decide it #

With the precision fixed, the number of systems needed follows from how different the two hypotheses’ predictions are.

The answer is asymmetric, and the asymmetry is instructive. If the gap is filled, a handful of systems with ten-per-cent masses will show it: each system that lands squarely in the gap is strong evidence against emptiness, and five are enough. If it is empty, confirming that takes six times as many, because the evidence is the absence of systems where a filled gap would have put a sixth of them, and an absence has to be accumulated. With errors of forty per cent — typical of some current estimates for distant or short-period systems — both numbers rise to a couple of hundred.

These numbers are small. A few dozen dark companions with good masses are within the reach of the astrometric catalogues of the coming years. If counting were all there was to it, the gap would be settled soon.

A survey that prefers heavy companions #

It is not all there is. The first complication is that the survey does not find every companion with equal probability.

The visible star’s orbit grows with the companion’s mass, so heavier companions produce larger wobbles and are detectable to greater distances, and the volume searched grows as the cube of that distance. A companion in the gap is found in a fifth of the volume a typical black hole is, and a neutron star in a seventeenth. A sample read naively would show fewer gap objects than exist, purely through selection.

The correction is known from geometry, which makes it harmless if it is applied — but it changes the arithmetic of the test. The sample of gap objects and neutron stars that the survey actually contains is smaller than the black-hole sample, and the systems that decide the question are the ones the survey is least efficient at finding.

What else weighs three solar masses in the dark #

The second complication is worse, and no amount of precision cures it.

An astrometric orbit measures the total mass of whatever the visible star orbits. If that is a close pair of compact objects, too close to be resolved and too faint to be seen, it appears as a single dark companion of their combined mass. Two white dwarfs reach nearly three solar masses; a neutron star and a white dwarf, more than three and a half; two neutron stars, more than four and a half. Every one of those combinations lands in the gap, and every one of them is a product of the same binary evolution that makes black holes with companions. A triple in which a star orbits a dark inner binary is a hierarchical system of a kind that is common among massive stars.

So a dark companion of three solar masses is not a gap object until a pair has been excluded, and excluding one is hard. A pair of white dwarfs gives no light at the wavelengths of the visible star; a pair of neutron stars gives none at all unless one is a pulsar beaming towards the observer. The arguments against a hidden pair are statistical — how often such triples form and survive, whether the inner pair’s orbit would be stable within the outer one — and they are exactly the kind of argument the astrometric method was supposed to replace.

There is a partial remedy. An inner binary of two compact objects has its own orbital motion, and if its period is not too short, the visible star’s orbit is perturbed by it in a measurable way: the outer orbit precesses, and its timing shows a small periodic residual at the inner period. A system weighed astrometrically and also followed spectroscopically at high precision can be checked for those perturbations, and a clean outer orbit with no residual limits the inner pair to periods short enough to escape detection — where, for white dwarfs and neutron stars, it would be shrinking by gravitational radiation and might eventually be found by that route instead.

Candidates that turned out to be stars #

The impostor problem is not hypothetical, and the recent record is sobering. Several of the most celebrated dark companions found by radial velocities in the last few years were later shown to be something else: not black holes at all, but stars that had been stripped of most of their envelopes by a companion and were faint, hot and unusual enough to be missed in the spectrum. A system claimed to hold a black hole of seventy solar masses, another claimed as the nearest black hole to the Sun, and a pair of red giants whose companions were proposed as black holes of about three solar masses — exactly in the gap — were each reinterpreted as a binary containing a stripped star, once the spectra were disentangled and the faint component found.

Those were velocity-curve systems, where the mass is a floor, and the misidentification inflated a floor into a black hole. Astrometric masses are not floors, and a stripped star of the required mass would usually show up in the spectrum once looked for. But the lesson carries over: the first claimed objects in any new class are disproportionately the ones that were misread, because the misreadings are what make an ordinary system look extraordinary. The velocity curve’s floor was the first trap; the hidden companion — luminous but missed, or dark but double — is the next.

The gap seen in other ways #

Astrometry is not the only new route. Gravitational-wave detectors measure the masses of merging compact objects directly from the waveform, and they have found a few objects in the gap: a companion of about two and a half solar masses in one merger, and a component of between two and a half and four and a half in another. Those detections show that objects in the gap exist, at least in merging binaries, without saying how common they are among compact objects generally; and they too face a question of identity, since an object of two and a half solar masses could be the heaviest neutron star or the lightest black hole, and the maximum mass a neutron star can have is itself uncertain at about that value.

Microlensing gives masses for isolated dark objects, and a few events have been interpreted as dark remnants of a few solar masses. Each such mass rests on a combination of the event’s duration, the lens’s distance and its velocity, and is uncertain by tens of per cent. The astrometric companions are, in principle, the cleanest of all the routes, which is why the impostor problem matters most there: it is the method’s one systematic error, and it lands exactly on the question.

What a filled or empty gap would mean #

The theoretical stakes are about how a massive star’s core collapses. In models in which the explosion is driven by neutrinos depositing energy behind a stalled shock, success or failure is decided within the first second, and the outcome is sharply bimodal: a successful explosion leaves a neutron star, a failed one lets the whole envelope fall back and leaves a black hole of most of the star’s mass. Such models produce a gap naturally. Models in which explosions are weaker or slower, so that part of the envelope falls back onto a newly formed neutron star after a partial explosion, fill the gap with remnants of intermediate mass. A gap that is empty within the errors of a few dozen good masses would favour the first kind of mechanism; a filled one, the second.

The answer also feeds forward. The heaviest stellar black holes and the lightest ones bracket the range of outcomes of stellar death, and the distribution between them is what population models of merging black holes and of X-ray binaries take as input. A mass distribution measured without light-curve modelling, and corrected for its own selection, would be the first that does not inherit the systematic errors of the methods that discovered the gap.

What the figures assume #

The error budget uses scalings rather than a survey’s error model: errors per measurement that do not depend on the star’s brightness, a coverage penalty for orbits longer than the survey that stands in for the real degeneracy of a partial orbit, and a visible star whose mass is known to five per cent from its spectrum. The hypotheses for the mass distribution are deliberately simple — two log-normal populations and a flat filling — and a real analysis would fit the shapes as well as the gap. The selection factor assumes a fixed period and precision; in practice both vary, and the correction is computed system by system. And the impostor ranges use the standard maximum masses of white dwarfs and neutron stars, the second of which is uncertain by a tenth of a solar mass or so.

Still open: how often a star orbits a dark pair #

The test’s arithmetic is easy and its sample is near. What it needs from elsewhere is the rate at which visible stars orbit close pairs of compact objects, because that rate, folded through the survey’s selection, is the background against which any gap objects must be counted. The rate is not known. It depends on how often triple systems form among massive stars, how often their inner binaries survive two supernovae or two common envelopes, and how often the outer star survives the same events. Those numbers are exactly what population models of massive-star evolution are least certain about. A gap test that does not know its background can find that the gap is filled without knowing what fills it, and the first few dark companions of three solar masses will be argued over not for their masses, which will be good, but for what they are.