A slope that is a whole curve of companions

A velocity survey that has watched a star for ten years can find a planet on a one-year orbit easily: ten cycles of a sinusoid, each one confirming the last. A companion on a hundred-year orbit has completed a tenth of a cycle, and the star’s velocity has changed along that tenth by a smooth, nearly straight curve. What the survey reports is a trend — a slope of a few metres a second per year, on top of whatever shorter signals the star carries.

Trends are common in long, precise velocity records, and each one says that something massive is out there. What it does not say is what. A trend is the star’s acceleration along the line of sight, and the acceleration a companion produces is its mass divided by the square of its distance. Double the distance and quadruple the mass, and the slope is the same. Over the range of distances a ten-year trend allows, that one slope is consistent with a Saturn-mass planet, a Jupiter, a brown dwarf or a small star.

The problem is therefore not detection but identification, and it has three solutions, each of which supplies one of the numbers the slope leaves out. Waiting supplies curvature. A picture supplies the separation. And a second measurement of the acceleration, in the plane of the sky rather than along the line of sight, supplies the direction — which, it turns out, is enough to fix the companion completely.

A tenth of an orbit is a straight line #

Any smooth curve looks straight over a short enough stretch, and a circular orbit traced for a tenth of its period is very nearly straight in velocity. The star’s line-of-sight velocity from a companion on a circular orbit is a sinusoid; near the point where that sinusoid crosses zero, it is a line whose slope is the full acceleration, and the departure from the line is of the order of the fraction of the orbit covered, cubed.

The figure shows a typical case. An inner planet with a forty-four-day period is recovered cleanly from the scatter; the outer companion, nine Jupiter masses on a 124-year orbit, is a straight line of 2.9 metres a second per year, fitted, against 2.6 true. Its own sinusoid, dashed, lies on the line to within the noise. The data contain two numbers about the companion — the line’s level, which is absorbed into the star’s systemic velocity, and its slope — and the companion has at least four: mass, distance, the orientation of its orbit, and where along the orbit it is.

That count is the whole difficulty. Two numbers cannot determine four. They can only define a family of companions consistent with them, and the family has a particular shape that the next figure draws.

A slope is measured better than anything else in the data #

The slope is also the best-measured thing a long velocity record contains, and that is why trends are found in such numbers. Fitting a straight line to $N$ velocities of noise $\sigma$ spread evenly over a span $T$ gives the slope with an uncertainty of $\sigma\sqrt{12/N}/T$. The span appears in the denominator alone: doubling the observing time halves the error on the slope, where doubling the number of velocities improves it only by the square root of two.

For the ninety velocities of two metres a second spread over ten years in the figure, the slope is measured to about seven-hundredths of a metre a second per year. A companion producing a trend of half a metre a second per year — Saturn, seen from outside — is detected at seven standard deviations. Thirty years observed at the same rate would detect a trend about a fifth as large, from a companion of Saturn’s mass more than twice as far away.

So the method is extremely sensitive to the existence of distant companions and nearly blind to their nature. The asymmetry is built in: the same span that measures the slope precisely is far too short to measure anything that would separate mass from distance. A record long enough to detect a companion by its trend is, almost by definition, too short to have seen it bend.

That is also why a trend changes how a planetary system is read. A massive companion at tens of astronomical units is not dynamically inert: its tidal pull on the inner planets drives their eccentricities and inclinations through slow secular cycles, and whether the inner system is stable, coplanar, or aligned with its star depends on a companion that the velocities know only as a slope. Interpreting the inner planets’ orbits therefore requires the very numbers the trend withholds.

The minimum mass grows as the square of the distance #

A companion of mass $M$ at distance $r$ accelerates its star by $GM/r^2$ towards itself. The velocity measures only the component of that acceleration along the line of sight, which is the full acceleration only when the companion lies in the plane of the sky and on an orbit seen edge-on, at the point where it is passing in front of or behind the star at its greatest projected distance — the configuration a velocity survey calls quadrature. Any other orientation or phase projects the acceleration and makes it smaller.

So the measured slope $\gamma$ gives a lower bound on the mass at each distance:

$$M \ge \frac{\gamma\, a^2}{G}.$$

For a trend of eight metres a second per year that lower bound is about one Jupiter mass at five astronomical units, eighteen at twenty and a hundred and sixty — a small star — at sixty. It is a lower bound in two senses at once: a lower bound on mass at each distance, like the $m\sin i$ of an ordinary velocity planet, and a family of such bounds, one for every distance, with nothing in the slope to choose between them.

The absence of curvature adds a constraint on distance. If ten years of data show no bending, the orbit is long compared with ten years — a reasonable rule is at least three times longer — and for a Sun-like star that places the companion beyond about ten astronomical units. Below that, the orbit would have curved visibly. The figure’s allowed region is everything above the line and to the right of the dashed limit, and it spans two decades in mass.

For comparison, Jupiter itself accelerates the Sun by about six and a half metres a second per year at its quadratures, and Saturn by about six-tenths. An observer elsewhere watching the Sun for a decade would see nearly all of Jupiter’s orbit, and Saturn’s as a trend: a slope of six-tenths of a metre a second per year, consistent with Saturn and equally with a planet of more than a Jupiter mass at twice Saturn’s distance.

When the line begins to bend #

Waiting is the first remedy, and it works slowly. As the observing span lengthens, the companion’s orbit covers more of its cycle and the velocity curve departs from a line: first a curvature, the change of the acceleration, then a turning point, then eventually a full period.

Each new derivative is a new number, and each scales differently with distance. The acceleration goes as $M/a^2$. Its rate of change, for a circular orbit, goes as $M/a^2$ times the orbital frequency, which itself goes as $a^{-3/2}$, so the curvature scales as $M/a^{7/2}$. Two derivatives with different powers of distance determine both the mass and the distance, given an assumed orientation.

The opening figure is a direct calculation of that process. For each companion on a grid of separations and masses — on edge-on circular orbits, with phase and the star’s systemic velocity fitted — the agreement with simulated data is computed, and every companion within a fixed threshold of the best fit is shaded. With six years of data the allowed companions form a band three decades long in mass along the line $M \propto a^2$. With twelve, the band shortens as curvature enters. With twenty-four, a fifth of the true 103-year orbit, it has contracted to an island one decade across, centred near the true companion.

This is why the longest velocity records are among the most valuable data in exoplanet science. Surveys begun in the late 1980s and 1990s now span thirty years and more, and companions on orbits of fifty to a hundred years are emerging from trends into curves — a population of cold giants at distances where the velocity method and direct imaging meet.

A picture from above #

The second remedy is to look. A companion that is massive enough to produce a large trend and far enough away to show no curvature is, in many cases, both bright enough and well enough separated from its star to be imaged directly. The two measurements bound it from opposite sides.

A trend says the companion is at least so massive at each distance; an image that fails to find it says it is at most so massive — so faint — at each distance outside the region the star’s glare hides. Above the trend line and below the imaging limit is a wedge, and because the trend’s lower bound rises as the square of the distance while the imaging limit is roughly flat, the wedge closes. Beyond a certain separation, the trend demands a companion bright enough to have been seen.

The imaging limit depends on age, which is the weak point of the argument. A young planet still glows with the heat of its formation and is easy to see; an old one has cooled and is faint by orders of magnitude more. Converting a brightness limit to a mass limit requires an age and a cooling model, and the older the star, the higher the mass limit and the wider the wedge.

When the image does find the companion, the trend and the picture together give a mass. The picture fixes the projected separation; the trend, at that separation, gives a minimum mass; and if the companion’s brightness is known, a comparison with cooling models for the star’s age tests whether that minimum is consistent with what the companion’s light says it should weigh. Surveys built to image the companions behind velocity trends have found a series of brown dwarfs and low-mass stars this way, several of them the faintest companions ever imaged with a dynamical mass.

The acceleration is a vector, and it points at the companion #

The third remedy is the most complete. The velocity measures one component of the star’s acceleration. The other two lie in the plane of the sky, and they appear as a change in the star’s proper motion.

An astrometric satellite measures a star’s motion across the sky over its mission, a few years, and gives a proper motion that is effectively an instantaneous velocity in the plane of the sky. Two such missions twenty-five years apart give two such velocities, and the difference between them, divided by the interval, is the star’s acceleration on the sky. A third velocity — the long-term average, from the difference of the two missions’ positions divided by the time between them — makes the measurement robust. The difference between the short-term and long-term proper motions, called the proper-motion anomaly, is now catalogued for over a hundred thousand nearby stars, from the same astrometric solutions that give their parallaxes.

With all three components, the acceleration is a vector, and a gravitational acceleration points exactly at the body producing it. If the companion has also been imaged, its position on the sky is known, and the only unknown is how far in front of or behind the star it lies. The ratio of the line-of-sight acceleration to the sky-plane acceleration fixes that depth: it is the ratio of the companion’s line-of-sight offset to its projected separation. With the depth known, the three-dimensional distance is known, and Newton’s law turns the size of the acceleration into a mass.

No orbit is fitted. No inclination is assumed, no phase, and the orbit need not be circular. The figure shows the crossing directly: the velocity trend implies one mass for every assumed depth and so does the proper-motion change, but they imply different masses except at one depth, and there the companion is.

In practice the measurements are combined with whatever part of the orbit the velocities and the imaging positions have traced, in a joint fit that returns a full orbit with uncertainties. But the core of the result is the vector, and it is why dynamical masses for wide companions, unobtainable for decades, are now routine.

What was actually measured #

Gliese 229 B is the case that shows what a dynamical mass can do. It was the first brown dwarf ever imaged, in 1995, a methane-rich object orbiting an M dwarf nineteen light-years away, and for a quarter of a century its mass was estimated from its brightness and an assumed age. When velocity trends, imaging positions and the proper-motion anomaly were combined, its dynamical mass came out near seventy Jupiter masses — far too heavy for an object so faint. Either the cooling models were badly wrong or something about the companion was not as it appeared. In 2024 interferometric imaging and high-resolution spectroscopy resolved it into two brown dwarfs, of about thirty-eight and thirty-four Jupiter masses, orbiting each other every twelve days. The total mass was right all along; it was the assumption that it belonged to one object that had failed.

Companions found by following trends. Imaging surveys targeted at stars with long-term velocity trends have discovered dozens of brown dwarfs and low-mass stellar companions at separations of ten to a hundred astronomical units, each with a dynamical mass from the combination. Several have masses that test cooling models at ages of billions of years, where the models had no other calibration — doing for substellar objects what eclipsing binaries do for stars. The imaging itself depends on removing the star’s own light to a part in a million, using the star’s own image as the template.

Planets weighed in three dimensions. The same combination has given true masses for giant planets found by velocity alone, removing the $\sin i$ ambiguity for planets on orbits of decades, and has found that some of them are inclined enough to be brown dwarfs.

A drift with nothing behind it #

Not every trend is a companion, and the impostors are worth knowing because they are of the same size.

A magnetic cycle. A star’s activity rises and falls over years, like the Sun’s eleven-year cycle, and the suppression of convective blueshift rises and falls with it. Over a fraction of a cycle the result is a slow velocity drift of a few metres a second, indistinguishable from a trend in velocity alone. It is distinguished by the activity indicators measured from the same spectra, which track the cycle and not a companion.

The star’s own motion. A nearby star moving fast across the sky has a radial velocity that changes with time even if nothing pulls on it, because the direction to it changes. As it approaches its closest point, its velocity turns from approaching to receding. The rate of change is the square of its transverse velocity divided by its distance, and for the nearest fast-moving stars it is metres a second per year. Barnard’s star, the second-nearest stellar system, has a perspective acceleration of about four and a half metres a second per year — comparable to the trend Jupiter imposes on the Sun, and produced entirely by geometry. It is computed from the star’s parallax and proper motion and subtracted before any companion is sought.

The observer. Every velocity is first corrected to the barycentre of the solar system, and an error in that correction — in the Earth’s ephemeris or in the time — that grows linearly would appear as a trend in every star. The corrections are now good to centimetres a second, and the check is that the stars with no companions show no common drift.

Where the model stops #

Circular, edge-on orbits. The minimum-mass curves and the island of allowed companions are computed for circular orbits seen edge-on. Inclined orbits need more mass for the same trend and extend every region upwards; eccentric ones change the relation between slope and separation by factors depending on phase.

A representative imaging limit. The contrast limit drawn is a deep search of a young, nearby star, flattened outward; real limits depend on the star’s brightness, age, the instrument and the wavelength.

Astrometric accelerations without error. The vector figure assumes both components are measured perfectly. In reality the sky-plane acceleration from two catalogues decades apart has uncertainties comparable to its size for all but the largest companions, and the crossing becomes a region.

An acceleration is a ratio #

The general shape is that a derivative of a motion constrains a ratio, not its parts. An acceleration is a mass over a distance squared; the curvature is a mass over a distance to the seven-halves power; a full orbit gives the mass and distance separately because it supplies all derivatives at once. A partial record gives a partial ratio, and the honest report of a trend is a curve rather than a companion.

Every component of a vector that is measured is worth a separate constraint, and the direction is worth as much as the size. The line-of-sight acceleration alone defines a curve; the sky-plane acceleration alone defines another; together they define the direction, and for a central force the direction is the position of the source. The same principle finds the Sun’s acceleration about the Galaxy from the drift of distant quasars — a vector, measured by its effect on light, pointing at the mass that causes it.

Still open: the orbit, measured position by position #

The proper-motion anomaly compares two snapshots taken a quarter-century apart. The astrometric satellite that supplied the second snapshot measured each star’s position many dozens of times over a decade, and when those individual measurements are released, the star’s motion will be traced along its curve rather than inferred from its endpoints. For companions with periods of a few to a few tens of years, that is the orbit of the barycentre drawn on the sky directly, giving inclination, period and mass with no velocity at all — and for the tens of thousands of stars with both astrometric orbits and velocity trends, the two together will turn curves of companions into companions.