A mass read from part of an orbit

Every mass in astronomy that is measured rather than inferred comes from the law linking an orbit’s size to its period. Kepler’s third law in Newton’s form says that the total mass of an orbiting pair is the cube of the semi-major axis divided by the square of the period, in suitable units, and nothing else is needed: no model of either body, no assumption about what it is made of. A binary star whose orbit is traced on the sky, at a known distance, weighs itself.

The law needs both quantities, and both come from following the orbit round. That is easy for a pair with a period of a few years and impossible, in a human lifetime, for one with a period of centuries. Between the two lies a large and increasingly important class: orbits that have been watched for part of their period — a third, a fifth, a tenth. Space astrometry has turned this from a nuisance into a central problem. Its catalogues contain hundreds of thousands of orbits fitted to a few years of data, many of them for companions whose periods are longer than the data, and the masses of the unseen companions among them — brown dwarfs, giant planets, black holes — depend on how much an arc can say.

This essay asks that question with the simplest honest tool: how precisely the mass is determined, given positions of a certain quality covering a certain fraction of an orbit. The answer has a definite shape. It depends on how much of the orbit was seen, on how eccentric the orbit is and, more than might be expected, on which part.

Three records of one orbit #

A record of positions along part of an orbit is an arc, and the question is how many different orbits could have produced it.

A whole orbit leaves no room: the ellipse is traced completely, its size is measured directly, and the period is the time to come back. Three-tenths of an orbit leaves some room; the arc fixes most of the ellipse’s shape but its far side has to be extrapolated. A tenth of an orbit is a short curve, and over a short stretch an orbit looks like a curve of nearly constant bending — the bending set by the gravitational acceleration at that point, which is the central mass divided by the square of the distance. Many ellipses pass through such a curve: a small, quick orbit seen near its far end; a large, slow one seen near its near end; and every combination in between that bends the same way at the same place with the same speed.

How the error grows #

The precision can be computed directly. Take forty positions of the orbiting body, evenly spaced through the record, each with an error of one per cent of the orbit’s semi-major axis in each direction — roughly the quality of a good space-astrometric orbit. Fit an ellipse with five free parameters: the period, the semi-major axis, the eccentricity, the orientation of the long axis and the phase. The curvature of the fit’s quality around the true values — the Fisher matrix — gives the uncertainty of each parameter and of any combination, and the mass is the combination three times the logarithm of the size minus twice the logarithm of the period.

With a whole orbit the mass comes out to half a per cent — better than the individual positions, because forty of them constrain the ellipse together. Covering more than one period hardly helps; once the orbit has closed, the extra data refine the period and the size equally and the mass with them, but slowly. Covering less than half a period hurts steeply. At half an orbit the error has risen by half; at a fifth it is five times worse; at a tenth, twenty times, a ten-per-cent mass. Below about half a period the error grows roughly as the inverse square of the fraction observed.

Eccentric orbits suffer less at short coverage, provided the arc includes the part where the orbit bends sharply. A more eccentric ellipse has more curvature to measure near pericentre, and the curvature’s change along the arc constrains the size of the orbit as a constant curvature cannot.

Which part of the orbit was seen #

The eccentric curves in the previous figure began their records near pericentre. Moving the same length of record round the orbit changes the answer.

A quarter of an orbit centred on pericentre, where the body sweeps through a large angle quickly and the path turns sharply, weighs an eccentric system three times better than the same quarter centred on apocentre, where the body creeps along a gentle curve. Kepler’s second law is at work: equal times sweep equal areas, so a fixed fraction of the period near pericentre covers a large fraction of the orbit’s angle, and near apocentre a small one. A survey of fixed length that happens to catch a pericentre passage has, in effect, seen more of the orbit than one that does not.

This is why the masses of long-period systems are often known from a single pericentre passage observed intensively — the star S2 at the Galactic centre, whose sixteen-year orbit was weighed most precisely in the months round its 2002 and 2018 pericentres, is the clearest example — and why astrometrists watching long-period binaries plan their campaigns round predicted periastra. The arc that contains the turn is worth more than the arc that does not.

A size and a period that trade #

The degeneracy behind the growing error has a specific direction, and it is the worst possible direction for a mass.

From a short arc the fitted size and period are almost perfectly correlated: a larger orbit fits the data as well as a smaller one if its period is longer in the right proportion. The proportion is not arbitrary. What the arc measures well is the bending of the path — the acceleration, $GM/r^2$ — and the speed along it. Holding those fixed while changing the size of the orbit changes the mass the orbit implies, because the arc could be the near part of a large orbit around a larger mass or the far part of a small one around a smaller. The correlation runs along a line on which the size and period change together, and the mass, $a^3/P^2$, changes along it. If the two quantities were degenerate in the combination $a^3/P^2$ held fixed, a short arc would still give a good mass. They are degenerate in nearly the opposite sense.

When the orbit closes the correlation disappears, because the period is then measured directly as the time to return, independently of the size. The transition is abrupt rather than gradual: the correlation stays close to one until about half the orbit has been seen and collapses over the second half, as the far side of the ellipse comes into view and pins its extent.

What a survey of fixed length can weigh #

For a survey of a given duration the coverage fraction is the duration over the period, and the mass error becomes a function of the companion’s period.

Each survey weighs orbits shorter than itself to better than a per cent and loses precision steeply beyond. The three baselines drawn match the three main releases of the Gaia mission’s astrometry: about three years, five and a half, and ten and a half. The first release of orbits could weigh companions with periods up to about three years well and those of ten years poorly; the final one will weigh thirty-year orbits about as well as the first weighed ten-year ones. The shape of the curves explains why each release of a long astrometric survey is not just better than the last but qualitatively different: it moves the edge of the weighable population outward in period by a factor of two to four.

The edge matters for the black holes that space astrometry has found by their companions’ wobble. The heaviest stellar black hole yet found in the Galaxy by this method, Gaia BH3, has an orbital period of about eleven and a half years and was identified in data covering about half of it. Its mass of about thirty-three solar masses was well enough determined to be striking because the arc happened to include the companion’s periastron, and confirmed with ground-based radial velocities that pinned the period; the same arc centred on apastron would have left a much wider range.

When the arc is shorter still #

Below a tenth of an orbit the fit stops being a fit to an ellipse at all. What is left is a path that is almost straight, with a slight bend, and the bend is all that carries information about the companion. Astrometrists have learned to use exactly that. A star with an unseen companion on a long orbit moves across the sky with a proper motion that is slowly changing, and comparing the star’s mean motion measured by one mission with its mean motion measured by another, decades apart, gives the change — an acceleration. The comparison of the Hipparcos and Gaia catalogues, twenty-five years apart, has measured such accelerations for thousands of nearby stars.

An acceleration is the companion’s mass divided by the square of its distance from the star, projected on the sky. It is the bending of the path, and nothing more. It cannot give the mass alone, because a light companion close in and a heavy one far out bend the path equally. That is the degeneracy of the short arc in its purest form, and it is broken the same way: by adding a second measurement that depends on the mass and the separation in a different combination. A spectroscopic trend in the star’s radial velocity measures the line-of-sight acceleration; a direct image of the companion measures its separation; and two of the three, combined, give a mass. Several brown dwarfs and giant planets on orbits of decades have been weighed this way, from arcs too short to show any curvature in the companion’s own motion.

The arithmetic of the full fit says why this works. The fit’s error is not spread evenly over all combinations of size and period; it lies almost entirely along one line, the one on which the acceleration is constant. Any measurement that cuts across that line — a radial velocity, an image, a second epoch from another survey — collapses most of the uncertainty at once. A short arc is not a weak measurement of everything. It is a precise measurement of one thing and no measurement of another, and the right response is to find the other thing elsewhere.

A history of orbits weighed before they closed #

The problem is old. The first unseen companion found by its gravity was Sirius B: Bessel noticed in 1844 that Sirius’s proper motion was not straight but wavy, with a period of about fifty years, and inferred a companion before it was seen. The wobble’s period was fixed from less than a full cycle, and the companion’s mass waited until Alvan Clark saw it in 1862 and its orbit had been followed long enough to close. For a century afterwards, the masses of visual binaries were the slow output of patient observers, published orbit by orbit as each closed, with the long-period ones carried for generations as preliminary.

What has changed is not the arithmetic but the scale. A space survey measures every star in the sky at dozens of epochs, so its catalogue of partial orbits is large enough that the shape of the errors matters as much as their size. A single preliminary orbit could be refined later; a population of a hundred thousand preliminary orbits, each with a banana-shaped uncertainty, has to be handled statistically as it stands.

The eccentricity is lost first #

Among the five orbital elements, the eccentricity is the one a short arc determines worst.

From a whole orbit the eccentricity is known to a few thousandths; from a tenth it is uncertain by a fifth of its value, and it gets worse quickly below that. A short arc of an orbit seen near one end cannot tell a circle from a long ellipse, because near any one point the two can have the same curvature and speed; it is the change of curvature along the path, invisible over a short stretch, that distinguishes them. Since the period and size are tied to the eccentricity through the orbit’s geometry, the eccentricity’s uncertainty is a large part of what the mass inherits.

The same problem appears in its sharpest form for an orbit whose eccentricity is close to zero: noise always pushes a fitted eccentricity upward, and a short arc makes the bias worse. Population studies that draw eccentricity distributions from partial orbits have to correct for both effects together.

What the calculation leaves out #

The orbit is face-on, so that the arc on the sky is the true orbit; a real orbit is inclined, and the projection keeps everything but the focus. Fitting an inclined orbit adds two more parameters, the inclination and the node, and a short arc constrains them poorly too, so the true errors for short arcs are larger than drawn. The distance is taken as known; in practice the parallax is fitted at the same time, and its error adds three times its fractional value to the mass error. The positions are evenly spaced with uniform errors, where real surveys have gaps and scanning patterns that sample some parts of an orbit better than others.

For an unresolved binary, the positions are of the light centre rather than either star, and the orbit measured is a scaled-down version of the relative orbit, scaled by a factor that depends on the masses and brightnesses of both components. Radial velocities from spectroscopy break that degeneracy and, measuring the line-of-sight motion directly, constrain the orbit in a different direction; combining the two is much more powerful than either for short arcs.

Still open: masses for a population seen in arcs #

The individual orbit is not the main use of these catalogues. The masses of hundreds of thousands of companions, most of them seen over partial orbits, make a population — the distribution of companion masses for stars of each kind, the brown-dwarf desert, the black holes in binaries — and the population is what most of the science wants. A population drawn from fits whose errors grow steeply below half a period, and depend on whether each arc happened to catch a periastron, is a population with a selection function that depends on the companions’ periods and phases in a complicated way.

The population also has a bias built into it by the phase effect. Among companions with periods longer than a survey, those whose periastron fell inside the survey’s span are weighed well and those whose span fell near apastron are weighed badly or not recognised as orbits at all. Since eccentric orbits spend most of their time near apastron, most long-period eccentric companions are caught on the slow, uninformative part of their orbits. A naive census of well-determined masses would over-represent the companions that happened to be passing periastron — a small, random subset whose eccentricities, inferred from the same arcs, are themselves biased.

How to infer the population without fitting each orbit to a single best mass — by carrying the full, banana-shaped uncertainty of each partial orbit into a statistical model of the whole sample — is an active problem. The orbits determined from three observations and the error ellipses that are nearly all in one direction are its relatives in the solar system, where the same lesson was learned for asteroids: an orbit known only over an arc is best described not by one set of elements but by the whole family that fits, and the family is the measurement.