A black hole weighed by a wobble on the sky

For half a century every black hole of stellar mass known in the Milky Way was found the same way: by its X-rays. Gas pulled off a companion star spirals inward through an accretion disc, heats to millions of degrees and shines, and that light draws attention to a system that can then be studied in detail. The method is efficient and it has a blind spot the size of almost the entire population. A black hole emits X-rays only while it is being fed, and it is fed only if its companion is close enough to spill gas onto it — an orbit of hours to weeks. A black hole with a companion further away, or with none at all, is dark.

In 2023 the first of these dormant black holes was found by a completely different method. A Sun-like star 480 parsecs away, catalogued by the Gaia satellite, was seen moving round a loop on the sky every 185.6 days, a loop a few thousandths of an arcsecond across, with nothing visible at its centre. The loop’s size and shape, together with the star’s parallax, gave the mass of whatever it was orbiting directly: 9.6 solar masses, with no spectrum of the companion, no assumption about the inclination, and no X-rays. Gaia BH1, as it was named, is the nearest black hole known, and it was weighed by geometry alone.

The tilt the velocities cannot see #

A velocity curve measures the visible star’s motion along the line of sight, and so it measures the orbit only as it appears foreshortened by the tilt. The result is the mass function, a floor under the companion’s mass that becomes a measurement only once the inclination is known from somewhere else. Astrometry measures the motion in the two directions across the line of sight instead. The star traces out an ellipse on the sky, and that ellipse is the true orbit projected onto the plane of the sky: a circle tilted towards the observer appears as an ellipse whose axis ratio is the cosine of the inclination. The shape of the projected path, followed round a full orbit, gives the inclination and the orientation along with the angular semi-major axis.

The angular size of the orbit, $\alpha_1$, is not yet a length. It becomes one through the parallax $\varpi$, measured from the same data as the yearly loop the Earth’s motion imposes: the semi-major axis in astronomical units is $a_1 = \alpha_1/\varpi$, both angles in the same units. Kepler’s third law in solar units then reads

$$\frac{a_1^3}{P^2} = \frac{M_2^3}{(M_1+M_2)^2},$$

with the period $P$ in years. It is the same combination as the spectroscopic mass function, with one difference that changes everything: there is no $\sin^3 i$ on the right. The orbit’s tilt has been seen, not averaged over.

The figure shows what Gaia actually sees. The star drifts across the sky at a steady rate, wobbles once a year because the Earth is moving, and wobbles again, independently, twice a year because something heavy is pulling it round. The orbital loop is larger than the parallax loop. Separating the three motions takes a fit to a few dozen positions over several years, each measured to a small fraction of a milliarcsecond, and Gaia’s astrometric solutions do exactly that for millions of stars at once.

A position measured one direction at a time #

Gaia does not take pictures of the sky in the ordinary sense. It spins slowly, sweeping two fields of view around great circles, and each time a star crosses the focal plane its position is measured precisely in one direction only, along the scan, and only roughly across it. Over five years a typical star is crossed about seventy times, from directions that rotate as the spin axis precesses, and its two-dimensional motion is reconstructed from those one-dimensional measurements. For a star of the brightness of Gaia BH1’s companion each crossing fixes the position to about a tenth of a milliarcsecond along the scan.

A single star’s motion is described by five numbers: two coordinates, two components of proper motion and a parallax. An orbit adds seven more — period, eccentricity, time of periastron, and the size and orientation of the ellipse, which take four angles or lengths. Fitting twelve parameters to seventy one-dimensional measurements is routine for one star and a serious computation for millions, and the trick that makes it possible is a change of variables due to Thiele and Innes. Written in four combinations of the semi-major axis and the three orientation angles, the projected orbit is linear in its parameters once the period, eccentricity and time of periastron are fixed. The search becomes a scan over three non-linear parameters, with an exact least-squares solution for the other nine at each trial — fast enough to be run on every star whose motion does not fit the five-parameter model.

That is where the dormant black holes were found: among the stars whose single-star solution failed and whose orbital solution succeeded with a large photocentre orbit and a faint, low-mass visible star. A large orbit around a star of about a solar mass requires a heavy companion, and when no light from that companion appears in the spectrum, the companion is dark.

Orbits that are really the Earth’s #

The same scanning pattern makes some periods treacherous. An orbital period close to one year produces a motion on the sky with the same period as the parallax, and the fit cannot cleanly separate the two: some of the orbit can be absorbed into the parallax, and some of the parallax into the orbit. Periods near half a year, and near the precession period of Gaia’s scanning, are affected in subtler ways, because the directions from which the star is measured repeat on those cycles and some components of the motion are sampled poorly.

The first release of orbital solutions included many candidate dark companions that did not survive follow-up, and a large share of the spurious ones clustered near those periods. A false orbit of a few milliarcseconds around a star of a solar mass, read through the third law, produces a companion of several solar masses with no light, which is exactly what a black-hole search looks for. The confirmed systems were vetted by spectroscopy: the visible star’s velocity, followed over its orbit, has to reproduce the semi-amplitude the astrometric orbit predicts. Gaia BH1 passed that test to within its errors. Its 185.6-day period, close to half a year, made the vetting essential rather than a formality. The mass a parallax and an orbit give together is exact only if the orbit is real, and the one instrument that can say so independently is the spectrograph the method was meant to do without.

One point on the family the velocities allowed #

Placing the astrometric result beside what a velocity curve alone would give shows the difference in kind.

The velocity curve of Gaia BH1’s visible star, measured afterwards, gives a mass function of about four solar masses, already above the neutron-star limit. The astrometric orbit picks a single point on the family of solutions that mass function allows, at a tilt of 53 degrees, and that point is nearly ten solar masses. The two measurements agree, and the agreement is a test: the astrometric semi-major axis and inclination predict a velocity semi-amplitude, and the spectrograph measured it.

The construction is related to the orbital parallax of a double-lined binary, which uses the same orbit seen as an angle and as a length to get a distance. Here the distance is taken from the parallax instead, and what the orbit returns is a mass. The two methods trade one unknown for the other, and each fixes the quantity the other needs as input.

The distance enters cubed #

The step from angle to length is where the precision of the parallax matters, and it matters more than at first appears.

The semi-major axis is the angle divided by the parallax, and the third law cubes it, so the total mass goes as the parallax to the minus third power. A five per cent error in the distance becomes a thirteen per cent error in the black hole’s mass. For Gaia BH1, whose parallax of 2.09 milliarcseconds is measured to a fraction of a per cent, that is not the limiting factor. For a star several kiloparsecs away, with a parallax of a few tenths of a milliarcsecond measured to ten per cent, it would be — and inverting a noisy parallax is not as simple as taking one over it. The method weighs nearby black holes precisely and distant ones loosely, which is the opposite of the X-ray method, whose reach depends on luminosity rather than distance.

The visible star’s own mass enters too, through the denominator, but weakly when the companion is much heavier. Gaia BH1’s visible star is a slightly metal-poor G dwarf of about 0.93 solar masses, its mass known from its spectrum to a few per cent, and a ten per cent error in it moves the black hole’s mass by about two per cent.

A bright companion makes the mass smaller, not larger #

Gaia does not measure the position of the visible star. It measures the position of the centre of light — the photocentre — of whatever is in its point-spread function, and if the companion gives light too, the two are not the same.

The photocentre lies on the line between the two stars, at a position weighted by their light, while the barycentre lies on the same line weighted by their mass. The photocentre’s orbit about the barycentre therefore has a size proportional to the difference between the companion’s share of the mass and its share of the light. A dark companion contributes no light, and the photocentre follows the visible star’s whole orbit. A luminous companion pulls the photocentre back towards the barycentre and shrinks the loop. Two identical stars produce no photocentre motion at all, however large their orbit.

This has a reassuring consequence for the search. Mistaking a luminous companion for a dark one makes the inferred mass come out too small, so it cannot manufacture a black hole. What can imitate a heavy dark companion is a companion that is heavy but faint for its mass: a close pair of dimmer stars, or a pair of white dwarfs or neutron stars. A spectrum excludes a luminous main-sequence companion of the required mass, because a star of ten solar masses would outshine the visible one many times over. Excluding a hidden inner binary requires further work — looking for its light in the infrared and in the ultraviolet, and arguing about the stability of the triple — and it is the residual doubt in every system of this kind.

Where the dormant ones live #

The two methods see black holes in different parts of the same space.

The X-ray binaries sit at the left of the figure, at periods short enough for the companion to fill its Roche lobe and feed the black hole. For a Sun-like donor that means less than about half a day; giant donors fill their lobes at periods of up to a month or so. Astrometry reaches from about a day, where the signature becomes measurable, to the length of the survey, ten years, where an orbit can no longer be followed round. Its window covers four orders of magnitude in period at which a black hole and a normal star can sit indefinitely without interacting, and in which there is no other way to find them.

Gaia’s third data release contained the first two such systems. Gaia BH1 has a 185-day orbit. Gaia BH2, found soon afterwards, is a red giant in a 1,277-day orbit about a black hole of about nine solar masses at 1.2 kiloparsecs. Gaia BH3, announced in 2024 from preliminary data for the fourth release, is the most striking: a metal-poor giant 590 parsecs away, in an orbit of 11.6 years about a black hole of 33 solar masses — by far the heaviest stellar black hole known in the Galaxy, and three times heavier than any found through X-rays.

What the silent ones say #

Each of the three has said something the X-ray sample could not.

Gaia BH1’s orbit is a puzzle. Its black hole formed from a star of perhaps twenty or more solar masses, which during its life swelled into a supergiant several astronomical units across, larger than the present orbit of its companion. The standard route to a black hole with a close companion runs through a common envelope, in which the companion spirals inside the supergiant’s envelope and ends on a much tighter orbit. That route cannot easily leave a Sun-like star at 1.4 astronomical units on an eccentric orbit. The alternatives — the companion was captured later in a dense star cluster, or the black hole formed without ever swelling into a supergiant — each have their difficulties. A single system on an orbit no X-ray binary occupies has reopened the question of how black holes acquire their companions.

Gaia BH3’s mass connects to a different population entirely. Black holes of thirty solar masses and more were first found by gravitational-wave detectors, merging in distant galaxies, and their masses were a surprise, since the black holes in X-ray binaries cluster below twenty. The explanation offered was metallicity: stars poor in heavy elements lose less mass to winds, keep more of it to the end, and leave heavier remnants. Gaia BH3’s companion is a metal-poor star belonging to an old stellar stream in the Galactic halo, exactly the environment that explanation requires, and its black hole has exactly the mass it predicts. It is the first such black hole close enough to examine at leisure.

And together the three say something about numbers. Population models predict that the Milky Way holds of order a hundred million stellar black holes, of which a few dozen are known through X-rays. If even a small fraction have stellar companions in the astrometric window, Gaia should find dozens to hundreds in its full data. That it found only a handful in its first three years of data, all with relatively long periods, is already informative: black holes with companions at a few astronomical units appear rarer than some models predicted, which constrains the kicks black holes receive at birth and how often their binaries survive them.

What the method cannot see #

Astrometry finds a dark companion only if there is a luminous star orbiting it. Isolated black holes, which are presumably most of them, remain invisible to it; the only one so far identified was found by the way it bent and shifted the light of a background star during a microlensing event, a method that measures a mass once and never again. Black holes with companions too faint for precise astrometry — distant ones, or ones whose companions are low-mass dwarfs — are also missed, and the census is therefore a census of black holes with bright, nearby partners, not of black holes.

Within that census, the method assumes a two-body orbit. A hierarchical triple, with an outer star orbiting a close inner pair, can produce an astrometric orbit whose mass belongs to the inner pair, and if that pair is faint the result looks exactly like a dark companion. Most candidate systems are argued to be clean on the grounds that the inner pair would have to be implausibly faint for its mass, but the argument is a statement about likelihood rather than a measurement.

Still open: how many, and with what masses #

Gaia’s fourth data release, covering five and a half years of measurements, will extend the window to longer periods and fainter stars, and it is expected to contain many more dormant black holes. The questions they will answer are the ones the X-ray sample could not. Whether the dormant black holes show the same gap between two and five solar masses that the X-ray binaries do — a gap that might come from how the X-ray masses were modelled — will be tested with masses that need no light-curve modelling at all. Their eccentricities and orbital separations record the kicks the black holes received at birth, which the X-ray binaries, circularised by tides and transformed by mass transfer, have long since forgotten. And their numbers, set against the predictions of population models, will say how often a massive star’s binary survives its core’s collapse.

The census of black holes without companions will come from the same kind of measurement turned around. When a dark object passes almost in front of a background star, its gravity bends the star’s light, brightening it for weeks and shifting its apparent position by a fraction of a milliarcsecond. The brightening alone gives a combination of mass, distance and velocity; the positional shift, measured astrometrically during the event, breaks that combination and gives the mass. The single isolated black hole identified so far was weighed this way, with the Hubble Space Telescope following the shift over six years. Surveys of the Galactic bulge planned for the coming decade will measure the shifts for hundreds of events. Between them, astrometric orbits of the companions and astrometric microlensing of the lone ones will put masses on the dark population that the X-ray binaries sampled only at its brightest corner, and the two censuses can be compared mass for mass.

The first three systems have each overturned an expectation. Whether they are the vanguard of a population that resembles the X-ray binaries in its masses and differs only in its orbits, or of one that differs in both, is not yet known, and the answer is a few years of position measurements away.