A floor under the unseen companion

In 1992 a team observing a faint orange star in Cygnus, in the aftermath of an X-ray outburst three years earlier, measured its velocity night after night and found it moving back and forth by more than four hundred kilometres a second, with a period of six and a half days. Nothing else in the spectrum moved with it. Whatever it was orbiting gave no light that the spectrograph could see. And from that velocity curve alone, without knowing the orbit’s tilt, the visible star’s mass or the distance, it was possible to say that the invisible companion weighed more than six times the Sun.

That is the power of a single number that comes out of any binary in which only one star is seen. The number is called the mass function, it is a precise combination of the period and the size of the velocity swing, and it has the unusual property of being a lower bound rather than an estimate. Every uncertainty in the problem — the inclination, the visible star’s mass — pushes the true answer upward from it. For the star in Cygnus, the floor stood far above the heaviest mass a neutron star can have. It was the cleanest proof yet that black holes exist.

Everything one velocity curve says #

Two bodies in orbit each move on an ellipse about their common centre of mass, with the sizes of the ellipses in inverse proportion to the masses. A spectrograph sees only the component of the visible star’s velocity along the line of sight, as the Doppler shift of its lines, and that component rises and falls once per orbit. Its period $P$ is the orbital period. Its half-range $K_1$ is the projected size of the visible star’s orbital speed: $K_1 = 2\pi a_1\sin i/\left(P\sqrt{1-e^2}\right)$, with $a_1$ the semi-major axis of the visible star’s orbit about the centre of mass and $i$ the inclination, the angle between the orbit’s axis and the line of sight.

Kepler’s third law connects the size of the relative orbit to the total mass, and the visible star’s share of the relative orbit is the companion’s share of the mass. Putting the three together eliminates everything that cannot be measured except one combination:

$$f(M) = \frac{P K_1^3 (1-e^2)^{3/2}}{2\pi G} = \frac{M_2^3\sin^3 i}{(M_1+M_2)^2}.$$

The left-hand side is made entirely of measured quantities. The right-hand side holds the unseen companion’s mass $M_2$, the visible star’s mass $M_1$, and the inclination. Nothing else about the companion — its size, its temperature, what it is — enters at all.

For V404 Cygni the numbers are a period of 6.4714 days and a semi-amplitude of 208.5 kilometres a second. Cubing the velocity and multiplying by the period gives 6.08 solar masses. The velocities are the only measurement, and they do not need the distance, the light of the companion or any model of either star.

A floor, not an estimate #

The structure of the right-hand side is what makes the mass function so useful. Since $\sin i \le 1$ and $M_1 \ge 0$, the right-hand side is never larger than $M_2$ itself: the expression $M_2^3/(M_1+M_2)^2$ equals $M_2$ only when the visible star weighs nothing, and is smaller otherwise. The mass function is therefore a strict lower bound on the companion’s mass. It is reached only in the unphysical limit of a massless visible star on an orbit seen exactly edge-on.

The figure shows how the bound behaves. A heavier visible star moves less for the same companion, so a given velocity swing implies a heavier companion; a more tilted orbit hides more of the motion, so it too implies a heavier companion. Both corrections run the same way. For V404 Cygni’s donor, a cool subgiant of about seven-tenths of a solar mass, the companion is at least 7.3 solar masses even if the orbit is edge-on.

The comparison that matters is with the largest mass a neutron star can have. A neutron star is held up by the pressure of matter at nuclear densities, and every equation of state for that matter has a maximum mass above which no static star exists — a limit analogous to the one a degenerate electron star meets, but set by physics nobody can reproduce in a laboratory. Weighing neutron stars has pinned the observed maximum at a little above two solar masses, and theory allows at most about three. A mass function of six solar masses clears that ceiling twice over. No assumption about the inclination or the visible star can bring the companion down to neutron-star territory. It is a black hole, by elimination of the only alternative.

The census that found the black holes #

The argument has a history, and the history is mostly of mass functions that were not quite large enough.

Cygnus X-1, identified in 1971 as a bright X-ray source with a blue supergiant companion, was the first serious black-hole candidate. Its mass function is a quarter of a solar mass, nowhere near the neutron-star limit, because the visible star weighs about forty solar masses and barely moves. The case for a black hole there had to be made the long way: estimate the supergiant’s mass from its spectrum and luminosity, estimate the inclination, and solve for the companion. Every step carried assumptions, and the case was argued for two decades — famously the subject of a bet that Stephen Hawking made against it and conceded in 1990.

The X-ray transients changed that. These are binaries in which a low-mass star feeds a compact companion through its Roche lobe, the flow erupting in an X-ray outburst every few decades. Between outbursts the accretion disc is faint and the donor star can be seen and tracked. In 1986 A0620−00 gave a mass function of about three solar masses — above the neutron-star limit, if only just, and with a donor light enough not to matter. In 1992 V404 Cygni gave 6.08, which ended the argument. Since then a few dozen transients have had their mass functions measured, and most of them lie above the line in the figure. Their companions are black holes not on the strength of a model, but on the strength of an inequality.

How close the floor usually lies #

A lower bound is only as useful as it is close to the truth, and here a simple geometric fact helps. The inclinations of binary orbits are random, in the sense that their axes point in random directions in space. For directions uniform on the sphere, it is the cosine of the inclination that is uniformly distributed, not the angle itself, and that makes nearly edge-on orbits more common than nearly face-on ones.

Half of all randomly oriented binaries have a true mass within a factor of about one and a half of the edge-on value. The same geometry governs the minimum masses of planets found by the wobble they induce: a radial-velocity planet’s quoted $m\sin i$ is usually close to its true mass, and the planets that are much heavier than their minimum masses are the rare, nearly face-on systems. For the black-hole transients the selection runs the other way in one respect: systems seen close to edge-on are more likely to show eclipses and strong orbital modulation, and are more likely to be well studied, so the observed sample is tilted towards inclinations that put the floor close to the truth.

The same floor at every scale #

Nothing in the mass function knows it is being applied to a black hole. The same expression, with the same inequality, describes every binary in which one member is tracked and the other is not, and running it down the scale of masses is the best way to see what it measures.

The Sun is itself the visible member of such a binary. Jupiter pulls it round their common centre of mass, just outside the Sun’s surface, every 11.86 years, at 12.5 metres a second. An observer on a distant star who measured only that velocity curve would compute a mass function of $8.7\times10^{-10}$ solar masses. When the companion is much lighter than the visible star the denominator is just the visible star’s mass squared, so the mass function’s cube root, in solar masses, is the companion’s minimum mass. The cube root here is $9.55\times10^{-4}$, which is Jupiter’s mass to three figures, with the inclination folded in. This is exactly the calculation behind every planet found by radial velocity, and the familiar $m\sin i$ of the exoplanet catalogues is the mass function in disguise: the floor, read off a star that barely moves, under a companion it cannot see.

Between the planets and the black holes sit the binary pulsars, where the visible member is a neutron star and its motion is measured not from a spectrum but from the arrival times of its pulses, which run early and late as it swings towards and away from the observer. Pulse timing gives the period and the projected semi-major axis to many significant figures, and so the mass function with almost no error. For most millisecond pulsars it is a few thousandths of a solar mass, which, with the pulsar’s mass taken as about 1.4, gives companions of a fifth of a solar mass or so — the helium white dwarfs left behind by the mass transfer that spun the pulsar up. For the first binary pulsar, PSR B1913+16, the mass function is 0.13 solar masses. That was not enough to say what the companion was. It took the relativistic effects in the orbit to give the two masses separately, and they turned out nearly equal: another neutron star.

The reach of one formula, from a Jupiter-mass planet to a nine-solar-mass black hole, comes from the cube on the velocity. The mass function goes as $K_1^3$, so a companion a thousand times heavier needs only ten times the velocity amplitude at the same period, and the velocities a spectrograph can measure span a factor of a hundred thousand, from metres to hundreds of kilometres a second. The cube also sets the error budget. A three per cent uncertainty in the semi-amplitude is a nine per cent uncertainty in the floor, while the period, measured over many cycles, contributes almost nothing. In every application the velocity amplitude is the number that has to be got right.

Turning the floor into a mass #

To get a mass rather than a bound, two more numbers are needed: the visible star’s mass, or equivalently the mass ratio, and the inclination.

The mass ratio comes from the donor’s spin. A star that fills its Roche lobe in a close binary is tidally locked, rotating once per orbit, and the width its rotation gives its spectral lines, $v\sin i$, is then fixed by its size, which is fixed by the Roche geometry, which depends only on the mass ratio. The ratio of that rotational broadening to the orbital semi-amplitude is a known function of $q = M_1/M_2$ alone. For V404 Cygni it gives a mass ratio of about 0.06: the donor is a sixteenth of the companion’s mass.

The inclination comes from the donor’s shape. A star filling its Roche lobe is drawn into a teardrop pointing at its companion, and as it orbits it presents its long side to the observer twice per orbit and its narrow end twice. Its brightness therefore varies with two maxima and two minima per cycle — the ellipsoidal variation — with an amplitude that goes roughly as $\sin^2 i$. A model of the distorted star, fitted to the light curve, gives the inclination.

With the inclination near 67 degrees, V404 Cygni’s companion comes out at about nine solar masses, the value now quoted. The figure also shows the weakness of the method. The mass goes as $1/\sin^3 i$, and below about fifty degrees it climbs steeply while the ellipsoidal amplitude falls, so the inclination is hardest to measure exactly where it matters most. An edge-on system is weighed precisely; a face-on one is barely weighed at all.

The floor holds and the mass may not #

The asymmetry between the two halves of this procedure is the most important thing about it. The mass function depends only on the velocities, and velocity measurements of a star’s absorption lines are hard to bias by much. The step from floor to mass depends on modelling the donor’s light curve, and that is where the errors have been.

The trouble is that the accretion disc contributes light too, even in quiescence. The disc’s light does not vary with the donor’s orbital phase in the same way, so it dilutes the ellipsoidal modulation: the double wave looks shallower than it is, and a shallow double wave reads as a low inclination. A low inclination, through the $1/\sin^3 i$, reads as a large mass. A0620−00, the first transient above the neutron-star line, was assigned masses as high as eleven solar masses from early light curves. Later work that measured the disc’s contribution spectroscopically and modelled it together with the donor brought it down to about six and a half. The mass function, three solar masses throughout, never moved.

The same asymmetry has decided several more recent claims, in the opposite direction. In 2019 a bright star called LB-1 was reported to orbit a black hole of about seventy solar masses, an object too heavy for any accepted account of how black holes form. Its mass function was small, about a solar mass. The large mass came entirely from the step beyond the floor: an assumed mass for the visible star of about eight solar masses, taken from its spectral type, and a mass ratio read from the motion of an emission line attributed to the companion. Both failed. The visible star turned out to be a stripped star, the remnant of a star that had lost its outer layers to a companion, with a mass nearer one solar mass than eight, and the emission line’s motion turned out to be an artefact of the absorption lines on top of it. With the visible star’s mass corrected, the floor no longer required anything exotic, and the companion is now thought to be an ordinary rapidly rotating star. Similar claims for several other systems have gone the same way. In each case the velocity curve was right and the conclusion was wrong, because the conclusion rested on a mass for the visible star that its spectrum appeared to give and did not.

That is the sense in which the floor is the trustworthy number. A claim that a companion is a black hole because its mass function exceeds the neutron-star limit rests on velocities and an inequality. A claim that the black hole weighs nine solar masses rests on a light curve and a model of the light that is not from the donor. Both are useful. Only the first is as secure as the orbit itself, and the stars whose masses are known without any such model are the ones in which both stars are seen.

What the inequality cannot say #

The mass function is silent about everything except mass. It cannot distinguish a black hole from any other dark object of the same mass, and the argument for a black hole is entirely an argument by elimination: nothing else that dark and that heavy is known to be stable. It is possible in principle that the companion is a close pair of fainter objects, two neutron stars or a neutron star and a white dwarf, whose combined mass is large and whose combined light is small; the velocities would look identical. Such configurations are argued against by their rarity and by the X-ray behaviour of the transients, which is characteristic of accretion onto a single compact object with no surface, rather than by the orbit.

The inequality also needs the visible star to be the only source of the lines being measured, and to be moving with its centre of mass. A donor heated on the side facing an X-ray source has its absorption lines weakened there, shifting the centre of light away from the centre of mass and biasing the measured semi-amplitude. Corrections for this are routine and small for the quiescent transients, but they are model-dependent, and in the brightest systems they are not small.

And it needs a visible star close enough and bright enough to follow. The X-ray transients are found because they erupt, which selects black holes with close companions that feed them. A black hole with a companion on a wider orbit, too far away to be fed, makes no X-rays and is found only if someone happens to measure the companion’s velocity — or its position. That population was invisible until astrometry could weigh a dark companion by its partner’s wobble on the sky, and it turns out to be the larger one.

Still open: the gap between the heaviest neutron stars and the lightest black holes #

When the masses of the black-hole transients are plotted together, there is a conspicuous absence: almost none lies between about two and five solar masses. The heaviest neutron stars sit just above two; the lightest black holes in the sample sit near five. Whether that gap is real matters for how stars die, because it would mean that the collapse of a massive star’s core either stops at a neutron star or overshoots by a factor of two, with nothing in between — a constraint on the explosion mechanism itself.

The mass function cannot answer the question, because the gap is in masses rather than floors, and the masses inherit the inclination’s systematic errors. The same disc dilution that once inflated A0620−00 operates in every system, and if it has not been fully corrected it pushes every mass upward — emptying exactly the region between two and five. Some reanalyses that treat the disc light more carefully move several black holes towards the gap without filling it. Gravitational-wave detections of merging compact objects have since found a few with masses inside the gap, which the X-ray sample may have missed for reasons of selection rather than physics. Whether the electromagnetic gap survives better light-curve modelling, or was always a product of it, is not settled, and the floor under each companion, which is secure, does not reach high enough to say.