Motions across the sky measure the orbits first

A spectrograph measures one component of a star’s velocity, the one along the line of sight, and in a dwarf galaxy a hundred kiloparsecs away that is all it measures. The dispersion of those velocities, as a function of distance from the dwarf’s centre, is then fitted with a dark halo and an assumption about the shape of the stars’ orbits — how much of their motion is radial, plunging through the centre, and how much tangential, circling it. Every choice of orbit shape picks a different halo, and a family of haloes from cored to steeply cusped fits the same flat profile, each with its own anisotropy. Only the mass inside about the half-light radius is common to all of them.

That degeneracy is not a failure of the data but of the projection. The line of sight is one direction, and the anisotropy is precisely a statement about how motion is shared between directions. Measuring the other two components — the proper motions across the face of the dwarf — removes the degeneracy in principle. The question is what it removes first, and how many stars that takes.

Three components instead of one #

Seen from a distance, a star in a spherical galaxy has three velocity components the observer can name: along the line of sight, along the direction on the sky pointing away from the galaxy’s centre, and across that direction. The orbits are described by two dispersions in the galaxy’s own frame — the radial one, toward and away from its centre in three dimensions, and the tangential one, around it — and their ratio is the anisotropy β, zero for isotropic orbits, positive for radial ones and negative for circular ones.

Each observed component mixes the two intrinsic dispersions differently, depending on where along the line of sight the star sits. The line-of-sight component, for a star near the projected centre, is mostly the star’s radial motion; for a star far out on the sky it is mostly its tangential motion. The component across the radius vector on the sky is always purely tangential to the galaxy. And the component along the radius vector on the sky mixes them the opposite way to the line of sight: mostly tangential near the centre, mostly radial far out.

The Jeans equation gives the radial dispersion at each radius for any halo and anisotropy, and projecting it with these three weightings gives the three observable profiles. The line of sight was all the halo family was fitted to. The other two profiles are predictions it was never asked to match, and the opening figure shows that the members disagree about them by up to thirty per cent at the half-light radius. That disagreement is the information the line of sight could not see.

A ratio that is the anisotropy #

The cleanest use of the two sky components does not involve the halo at all.

For isotropic orbits the two components on the sky are equal, whatever the mass. Radially biased orbits make the component along the sky’s radius vector larger, because far from the projected centre that direction is close to a star’s own radial direction; circularly biased orbits make the component across it larger. The ratio is close to a direct measurement of the anisotropy, and it is almost free of the halo: every halo in the family that needed a given β at a given radius predicts nearly the same ratio there.

That is why the first proper-motion measurements of dwarf galaxies were reported as anisotropies rather than as density slopes. The ratio of two dispersions measured from the same stars carries no distance, no mass and no knowledge of the halo; it carries the shape of the orbits. Once that shape is fixed, the line-of-sight profile no longer has a free function in it, and the halo has to account for the profile by itself.

Microarcseconds per year #

The motions are small. A velocity of 10 kilometres per second at a distance of a hundred kiloparsecs moves a star across the sky by 21 microarcseconds a year — about a two-thousandth of a pixel on a space telescope’s camera. A dispersion of that size has to be measured from the scatter of many such motions, each with its own error, and the errors add to the scatter before they can be subtracted from it.

The arithmetic is simple and unforgiving. A dispersion measured from N stars has a fractional error of one over the square root of twice N, multiplied by one plus the square of the ratio of the per-star error to the dispersion. When the per-star error equals the dispersion, that factor is two, and the number of stars needed is four times what perfect measurements would need; when the per-star error is three times the dispersion, it is a hundred times. Below some precision a measurement is not merely worse but impossible with the stars a dwarf has, since the faint dwarfs hold only a few hundred stars bright enough to measure at all.

The Milky Way’s dwarf satellites lie between twenty and two hundred and fifty kiloparsecs, with dispersions of a few to about ten kilometres a second. A single all-sky astrometric survey, measuring positions over a few years, reaches a few hundred microarcseconds a year for the faint giants of a dwarf galaxy — enough for the dwarf’s motion as a whole, not for its internal dispersion. The internal motions need repeated images from a space telescope separated by a decade or more, in which the positions of hundreds of stars are measured against distant background galaxies to a few hundredths of a pixel. That has now been done for the nearest and richest dwarfs, with a few hundred stars each, and it has measured their anisotropy.

Binaries, which a long baseline averages away #

Proper motions have one advantage over line-of-sight velocities that is easy to overlook. A star in a binary moves round its companion at a few kilometres a second, and a single spectrum catches it at one moment of that orbit. In a dwarf galaxy whose own dispersion is a few kilometres a second, those orbital motions inflate the measured dispersion by an amount that depends on the unknown binary population, and the faintest dwarfs’ masses have been overestimated that way before repeated spectra caught the binaries changing.

A proper motion is measured differently. It is the change in a star’s position between two images taken years apart, divided by the time between them, and the position is that of the light’s centre — the pair’s centre of mass, displaced by a wobble that is a tiny fraction of the pair’s separation. For a binary whose period is shorter than the baseline, the orbital motion averages away almost entirely: over ten years a pair with a period of one year has gone round ten times, and its centre of light ends close to where its centre of mass carried it. Only binaries with periods comparable to the baseline leave a trace, and at a hundred kiloparsecs their wobble is a few microarcseconds for any pair that a giant star can belong to.

So the component that arrives last in the history of these measurements is the one least affected by the nuisance that the first component suffers most. A dispersion across the sky is close to the dispersion of the stars’ centres of mass, which is the quantity the Jeans equation is written for. For the ultra-faint dwarfs, where binaries can double the apparent line-of-sight dispersion, that alone may eventually make proper motions the more reliable weight.

How many stars a slope needs #

The slope of the halo is a harder question, and the reason is that the anisotropy was not the only freedom. Each halo in the family was allowed an anisotropy that changes with radius, and two numbers’ worth of freedom in the orbits can absorb a good deal of a difference in the mass profile even when two more components are measured.

The two curves are straight lines on these axes, because the evidence against a wrong model grows in proportion to the number of stars. With line-of-sight velocities alone the evidence grows so slowly that no dwarf galaxy has enough stars to use it: a hundred thousand stars give a preference of two and a half standard deviations, and the richest dwarfs have a few thousand measurable giants. With proper motions the evidence grows fourteen times faster. Even so, telling a core from the best-fitting cusp at three standard deviations takes about eleven thousand stars with five-kilometre-per-second errors on every component.

At a hundred kiloparsecs, five kilometres a second is about ten microarcseconds a year, for each of eleven thousand stars. No dwarf has been measured to that standard, and only the most luminous have that many stars bright enough to try. The proper motions turn a question that line-of-sight data cannot answer for any dwarf into one that can be answered for a few, with an effort several times larger than anything yet attempted.

What the motions narrow #

The test can be turned round: rather than asking whether a particular cusp is excluded, ask which of the family survive, and what their mass profiles then have in common.

The light band is the degeneracy the line of sight leaves: a factor of nearly six in mass inside a quarter of the half-light radius, pinching to almost nothing near the half-light radius and opening again outside. That pinch is the node the line-of-sight analysis already found. The proper motions do not move it or sharpen it noticeably; the mass at the half-light radius was never in question. What they narrow is everything else — the inner mass, where a core and a cusp differ by a factor of several, and the outer mass, which sets how the dwarf responds to the Milky Way’s tides.

There is a pleasing symmetry in that. The half-light node is the radius at which the orbits’ shape stops mattering for the line of sight, which is why line-of-sight data measure the mass there and nowhere else. Proper motions measure the orbits’ shape, which is why they extend the measurement away from the node. The two kinds of data are complementary not because one is better than the other but because each measures what the other is blind to.

Why the inner slope is worth this much effort #

The inner slope of a dwarf’s halo has carried a disproportionate share of the argument about dark matter for thirty years. Simulations of cold dark matter alone produce haloes whose density rises towards the centre as a cusp — and a central black hole, where there is one, would steepen it further; rotation curves of gas-rich dwarf galaxies and many analyses of dwarf spheroidals have suggested flatter cores. Three classes of explanation compete. The dark matter could be cold and the cores made by the galaxy itself, as bursts of star formation drive gas in and out of the centre and drag the dark matter’s orbits outward — the reverse of the way a galaxy’s gas, settling inward, drags its halo in. Or the dark matter could be warm enough to stream out of the smallest fluctuations, which lowers the central densities of small haloes without making true cores. The dark matter could interact with itself, so that its particles exchange energy in dense centres and build cores without any help from the gas. Or the cores could be an artefact of the analysis — of the anisotropy in a dispersion-supported dwarf, or of non-circular motions in a rotating one.

The dwarf spheroidals are where these separate most cleanly. Their dark matter dominates their mass at every radius, so a cusp or core is the halo’s own; some of them formed their stars early and briefly, too briefly for bursty star formation to have made a core, so a core there would point to the dark matter itself. A measured inner slope in such a dwarf is one of the few observations that could distinguish a property of the dark matter from a property of the galaxies in it. And it is exactly the measurement the anisotropy degeneracy forbids to line-of-sight data.

The same question in the Milky Way’s own halo #

The degeneracy is not peculiar to dwarfs. The Milky Way’s stellar halo — the old stars on orbits that carry them far above the disc — is weighed the same way, by the dispersion of its stars’ velocities through a Jeans equation, and the anisotropy enters the same way. For decades only line-of-sight velocities were available for distant halo stars, and the Galaxy’s mass beyond a few tens of kiloparsecs was uncertain by a factor of two partly for that reason.

The all-sky astrometric survey changed this for the inner halo, where its proper motions are good enough for individual stars. The halo’s stars turned out to be strongly radially biased, with β around 0.7 in the inner tens of kiloparsecs, and much of that radial bias belongs to a single population on plunging orbits: the debris of one galaxy that merged with the Milky Way roughly ten billion years ago. A dispersion and an anisotropy measured together for that halo have narrowed the Galaxy’s mass inside fifty kiloparsecs to about twenty per cent, which a line-of-sight survey could not have done. It is the same argument as the dwarfs’, made where the stars are close enough for the motions to be easy, and it shows that the anisotropy, once measured, can itself be a fossil — of how the stars arrived, not only of where they are.

The dwarfs are the same problem at a distance where the motions are not easy, and in galaxies whose dark matter dominates everywhere. Stars that weigh and do not shine make a large galaxy’s stellar mass uncertain by a factor of two and its dark matter uncertain with it; in a dwarf spheroidal the stars of every mass together are a few per cent of the total, and the halo is all there is.

The first measurements #

The measurements now published use space-telescope images taken years apart, sometimes combined with an all-sky astrometric catalogue to fix the reference frame, for a few hundred stars in each of the nearest and richest dwarfs. They have measured the internal proper-motion dispersions, found them comparable to the line-of-sight ones, and constrained the anisotropy: broadly consistent with isotropic or mildly radial orbits, with uncertainties that still allow a range. Their verdicts on the inner slope depend on how much freedom the models give the anisotropy and the halo, which is what the figures here would predict for samples of that size.

The figures also say how those samples should grow. More stars help as the square root of their number; better per-star precision helps much faster while the per-star error is comparable to the dispersion. A decade’s further baseline on the same fields halves each star’s error, and for a dwarf where the per-star error and the dispersion are equal, that buys the equivalent of two and a half times as many stars. The richest nearby dwarfs, with a few thousand giants within reach and a twenty-year baseline, are where the core–cusp question will first be answered by motions alone.

What the models leave out #

The halo family is spherical, every halo has the same profile shape with two free numbers, and the anisotropy changes smoothly from one value at the centre to another far out. Real dwarfs are flattened, their haloes probably triaxial, and they orbit inside the Milky Way’s tidal field, which strips their outskirts and can inflate their outer dispersions — the same field that opens gaps in the streams of disrupted clusters when a dark subhalo passes. The stars are assumed to follow one population with one light profile, when several dwarfs show two populations of different metallicity with different spatial extents and dispersions — a structure that, used carefully, gives two tracers of one potential and is itself a way to constrain the slope. The per-star errors are Gaussian and equal; binary stars, which add their orbital motion to the line-of-sight dispersion, are ignored. And the chi-squared counts the best cusp in a fixed family, not every possible model; a family with more freedom would need more stars.

None of these changes the ordering. Motions across the sky measure the orbits’ shape with modest samples and the inner slope only with large ones, and the mass at the half-light radius was already known.

Still open: whether the slope differs between dwarfs that formed their stars early and late #

The prediction that separates the explanations is not a slope for one dwarf but a pattern across them. If cores are made by star formation, dwarfs that formed stars for billions of years should have them and dwarfs that formed stars only in the first billion years should keep their cusps. If cores are made by the dark matter itself, both kinds should have them, with sizes that depend on the halo’s density and the strength of the self-interaction. Proper-motion samples large enough to measure slopes in a handful of dwarfs with different star-formation histories would test that pattern directly — and a handful is about what eleven thousand stars per dwarf, at ten microarcseconds a year each, puts within reach of the next generation of space telescopes with wide fields and long baselines.