Pairs of stars too loosely bound to survive heavy dark matter

The dark matter round the Milky Way is measured by its gravity, and its gravity says nothing about how it is divided. A tenth of a solar mass per cubic parsec near the Sun — the halo’s share is about a hundredth — could be an unimaginably fine mist of elementary particles, or a scattering of objects each as heavy as a star or a hundred stars, and the rotation curve would be the same. Telling those apart needs a measurement that responds to the graininess rather than to the total.

Microlensing was the first. A compact object passing in front of a background star brightens it for a time set by the object’s mass — a smooth rise and fall, or a light curve with a fold in it if the lens is double — and the surveys of the 1990s and 2000s, watching millions of stars in the Magellanic Clouds, found far too few events for objects between about a ten-millionth and a few tens of solar masses to make up the halo. Above that range the events last years and the surveys could not see them. In 2015 the first gravitational waves came from black holes of about thirty solar masses, merging into one that the recoil of its own merger can throw out of a small host, heavier than most expected, and the question of whether the dark matter could be black holes of tens to hundreds of solar masses, formed in the early universe, came back.

The constraint that closes that window from above uses no telescope time on lensing at all. It uses pairs of stars.

A binding a passing mass can undo #

Two stars of about a solar mass between them, a tenth of a parsec apart — twenty thousand astronomical units — orbit each other at about a hundred metres a second, once every few million years. Their binding energy per unit mass is about the square of that speed, a hundredth of a square kilometre per square second. A mass of a hundred suns passing a parsec away at three hundred kilometres a second gives each star a velocity kick of a few metres a second, and the two kicks differ by a fraction set by the ratio of the pair’s separation to the passing distance — the tidal part of the impulse, the only part that can change the pair, since a uniform kick moves the pair without heating it.

Each such passage changes the pair’s internal energy by a tiny random amount, and the changes accumulate. The net effect over many passages is a steady erosion of the binding energy, because a random walk in velocity adds energy on average. That is the diffusive regime, and in it the pair’s lifetime is inversely proportional to the mass of the passing objects at fixed total density: fewer, heavier objects deliver larger kicks, and the energy added grows as the square of the kick while the number of kicks falls only in proportion to the mass.

The lifetime falls with separation too, in proportion to its inverse, because a wider pair is both more weakly bound and a larger target for tides. A pair a tenth of a parsec wide in a halo of hundred-solar-mass objects lasts fifty billion years, comfortably longer than the halo has existed. A pair a parsec wide lasts five. If the halo were made of thousand-solar-mass objects, the tenth-of-a-parsec pair would last five billion years, and the halo’s oldest wide binaries would not exist.

For heavy enough objects the description changes. The kicks become rare and large, and the pair is dissolved by a single close passage rather than by accumulation. That is the catastrophic regime, and its timescale depends on the density of objects only through how often one comes close enough — it no longer depends on their mass at all, and the curves in the figure flatten onto a common floor at the largest masses and widest separations.

A pair loosened, then lost #

The erosion is not linear in what a survey sees. The binding energy falls at a steady rate, and the separation is inversely proportional to the binding energy, so the separation creeps at first and runs away at the end.

A pair among thirty-solar-mass objects barely notices them in the age of the halo. Among three-hundred-solar-mass objects the same pair has nearly tripled its separation and is approaching the point where the next few kicks unbind it entirely. The final stage is brief compared with the whole, so at any moment few pairs are caught in it: a survey does not see pairs coming apart, it sees the pairs that remain, and a shortage of pairs beyond some separation.

Where the census should end #

That shortage is the observable. Wide binaries are thought to form with a distribution of separations that is roughly flat per logarithmic interval, from a few astronomical units out to a parsec or so — the distribution called Öpik’s law, measured in young populations before much has happened to them. Ten billion years in a sea of compact objects would remove every pair wider than the separation at which the disruption time equals the age.

The real cut-off would be softer than the sharp step in the figure — pairs have different masses and different histories, and the diffusive erosion leaves a tail of survivors partly widened — but the step’s position is the quantity of interest. Heavier compact objects move the edge of the census inward. A survey of wide binaries that finds the distribution continuing undiminished out to some separation has measured a lower bound on that edge, and so an upper bound on the mass of the objects.

The ceiling the widest pair sets #

Turned round, the calculation gives the largest mass compact objects can have, given the widest pair observed to survive.

The limit scales inversely with the separation of the widest surviving pair and with the fraction of the halo in compact objects. A pair half a parsec wide caps the objects at about a hundred solar masses if they make up the whole halo; if they make up only a fifth, the cap is five times higher. Each curve stops where the catastrophic regime takes over — where a single close passage would have dissolved the pair whatever the objects’ mass — and beyond that point the survival of the pair limits the fraction of compact objects rather than their mass.

The structure of the argument has a vulnerability that is easy to state. The whole bound rests on the widest pairs, a handful of systems at the edge of the distribution, and it is only as strong as the evidence that those pairs are genuinely bound. Two unrelated stars that happen to lie close on the sky and share a similar motion through the Galaxy look exactly like a wide binary in a catalogue of positions and proper motions, and at separations of a parsec the chance of such coincidences is not small.

The pair that moved the limit #

That vulnerability is not hypothetical. The first analysis, in 2004, used a catalogue of wide binaries on halo orbits identified from their common proper motions and found the distribution continuing to separations of about a parsec. It concluded that compact objects heavier than about forty solar masses could not make up the halo. In 2009 radial velocities were measured for the widest pairs in the sample, which proper motions alone cannot supply, and the widest turned out to have components moving at different speeds along the line of sight: a chance alignment, not a binary. With it removed, the next widest pairs set the limit, and the bound relaxed to about five hundred solar masses.

The episode is the method’s character in miniature. A bound from the widest member of a distribution is a bound from the member most likely to be spurious. Later analyses with larger samples, better distances and more careful treatment of chance alignments have tightened the limit again, to a few tens to a hundred solar masses depending on how the initial distribution of separations is assumed, and the Gaia catalogue, with its precise parallaxes and motions for a billion stars, has made it possible to test each candidate pair’s distances directly. But the lesson of 2009 stands: the constraint is a statement about a few objects, and each must be shown to be bound by more than its position on the sky.

Why the pairs are taken from the halo #

There is a reason the test uses binaries on halo orbits — old, metal-poor stars on eccentric, inclined orbits that carry them far from the Galactic disc — rather than the much more numerous binaries in the disc near the Sun.

A binary in the disc is bombarded by everything in the disc, and the giant molecular clouds — a hundred thousand solar masses each, slow-moving — are by far the most destructive. They would dissolve a tenth-of-a-parsec pair within a billion years whatever the dark matter is made of, so the wide binaries in the disc tell more about the clouds than about the halo. A binary on a halo orbit crosses the disc quickly and spends most of its life far from it, so for it the clouds and the disc stars are minor, and the halo’s own population of compact objects, if it existed, would dominate. The test needs the pairs that are least exposed to everything except the thing being tested — the same separation of perturbers that every impulse argument has to make, from the stars a passing cloud can strip without adding energy to the heating of a cluster’s halo by the disc it crosses.

That choice has a cost: halo binaries are rare, a few per cent of nearby stars, and their wide pairs are fewer still. A sample of a hundred wide halo pairs is large. The statistics of the widest few, which carry the bound, are correspondingly thin.

A smaller fraction, a larger ceiling #

The constraint is usually quoted for compact objects making up the whole halo. The question raised by the gravitational-wave black holes is different: whether some fraction of the dark matter, not all of it, might be black holes of tens of solar masses.

Since the disruption time is inversely proportional to the density of perturbers, a population that is a fifth of the halo needs objects five times heavier to do the same damage. The census then constrains the product of mass and fraction rather than either alone, and the wide binaries’ reach falls off for small fractions: they can rule out black holes of thirty solar masses as the whole of the dark matter, but not as a tenth of it. Other constraints — from the microwave background, which accreting black holes would have heated, from the survival of star clusters in dwarf galaxies, and from the rate of black-hole mergers itself — cover different parts of the same plane, and the wide binaries are one edge of the region they leave open.

Soft pairs and the rule they obey #

The wide binaries in this argument are what dynamicists call soft: their binding energy is far smaller than the kinetic energy of the objects passing them, so every encounter, on average, loosens them. The opposite kind, hard binaries — pairs bound more tightly than the typical passing energy — are hardened by encounters instead: a passing star tends to carry away energy and leave the pair more tightly bound than before. The dividing line between the two is a statement about energies, not separations, and it is the rule that governs a pair that heats what is trying to cool in the core of a star cluster, where hard binaries are the cluster’s energy source.

For the halo test the rule fixes which pairs are informative. A binary of two suns a few astronomical units apart is hard against anything the halo could contain and is unaffected by compact objects of any mass. Only pairs so wide that their orbital speeds are a few hundred metres a second, against passing speeds of hundreds of kilometres a second, are soft enough to be eroded at all — which is why the argument is made with pairs thousands of astronomical units apart and why its whole weight falls on the widest few. The impulse approximation is excellent for them: a passing object at three hundred kilometres a second crosses a tenth of a parsec in three centuries, while the pair takes millions of years to complete an orbit, so every kick is delivered to a pair that is effectively frozen.

It is also why the microlensing surveys and the binaries meet in the middle. Microlensing is most sensitive to lenses light enough that their events last days to months, which a few years of monitoring can catch; it loses objects of more than a few tens of solar masses because their events last years and are rare. The binaries are blind to light objects, whose kicks are too small to matter, and most sensitive to heavy ones. Together they bracket the compact-object hypothesis from both sides, and the gap between them is where the gravitational-wave black holes sit. The short events that microlensing does find — some lasting only a few hours with no host star in sight — are planets and dwarfs, far too light for either test to count as dark matter.

An edge the Galaxy draws anyway #

Even with no compact objects at all, the census of wide pairs cannot run on for ever. The Galaxy’s own tidal field — the difference between its pull on one star of a pair and on the other — limits the separation at which a pair can stay bound, in the same way that the Sun’s tide limits the region a planet may keep a moon in. At the Sun’s distance from the Galactic centre that limit, the pair’s Jacobi radius, is about a parsec and a half for a pair of one solar mass. Pairs wider than that are not pairs; they are two stars drifting apart at the speed of their separation’s growth, and catalogues of comoving stars at several parsecs contain many such dissolving pairs and loose groups.

So the observed census has a natural edge near a parsec whatever the dark matter is, and the compact-object test is a question of whether the edge falls short of that — whether pairs disappear well inside the separation at which the Galaxy’s tide would remove them anyway. For objects of a few hundred solar masses the predicted edge is a tenth of a parsec, an order of magnitude inside the tidal limit, and the test is clean. For objects of thirty solar masses the predicted edge has moved out to near the tidal limit itself, and the two effects can no longer be separated by the position of the edge alone; the shape of the distribution just inside it, softened by partial erosion, becomes what distinguishes them. That is why the limit is firm at hundreds of solar masses and argued about at tens.

The same pairs have been used for a quite different test — whether the orbital speeds of the widest binaries, where the acceleration of each star towards the other falls below a nanometre per second squared, follow Newton’s law or the modified law proposed to replace dark matter in galaxies. That test asks how fast the stars move; this one asks whether they are still together. Both depend on the same catalogues and on the same discipline about chance alignments.

Formulae for the average pair, applied to the widest few #

The disruption times use the standard approximate formulae for the diffusive and catastrophic regimes and take the larger, where the real transition between them is gradual. All the pairs have one solar mass and start on circular orbits; real pairs have a range of masses and eccentricities, and an eccentric pair spends much of its time at a separation larger than its semi-major axis, which makes it more fragile. The census assumes a flat initial distribution of separations out to a parsec, an assumption that cannot be checked in the halo’s own population because the halo has had ten billion years to modify it. The Galaxy’s own tide, which limits how wide any pair can be at the Sun’s distance from the centre to about a parsec, and the pairs’ passages through the disc, are left out of the halo calculation. Each of these moves the numbers by a factor of order unity, and the published limits differ by about that much for exactly these reasons.

Still open: whether the widest pairs are all real #

The bound now rests on samples of wide halo binaries tens of times larger than the one that first set it, drawn from the Gaia catalogue with distances good enough to reject most chance alignments outright. The remaining question is the one that moved the limit in 2009, in statistical form: at separations of a parsec, what fraction of apparent pairs are unrelated stars with similar motions? Models of the halo’s population of single stars predict that fraction, and it can be tested directly by searching for apparent pairs with mismatched radial velocities. If the widest pairs survive that test in numbers, the census continues to a parsec, and compact objects of more than a few tens of solar masses are excluded as the bulk of the dark matter by the stars themselves. If too many of them turn out to be coincidences, the edge of the census moves inward, the ceiling rises — and the window in which the mass that is not the light could be black holes of a hundred suns stays open a little longer.