The one radius where the orbits stop mattering

A galaxy held up by the random motions of its stars can be weighed from those motions, but only up to a factor that depends on the shape of the orbits. Stars on mostly radial orbits, plunging in and out through the centre, look different in projection from stars on mostly circular ones, and the line-of-sight velocities that a spectrograph measures cannot tell the two apart from a change in the mass. That is the mass–anisotropy degeneracy, and it is the reason a dispersion is usually called a weaker measurement than a rotation curve.

For the Milky Way’s dwarf satellites the degeneracy ought to be crippling. These galaxies have no gas and no rotation; everything known about their mass comes from the line-of-sight velocities of a few hundred to a few thousand stars, and the stars’ orbits are unknown. Yet published masses for them are quoted to twenty or thirty per cent, and the argument that they are the most dark-matter-dominated systems known rests on those numbers.

The two statements are reconciled by a property of the degeneracy itself. Models that fit the same dispersion profile with different orbits disagree about the mass at the centre and disagree about it far out, in opposite directions. Somewhere between, they must agree. The radius where they do turns out to depend on almost nothing but where the stars are.

What the shape of the orbits does to a profile #

The anisotropy is conventionally written as $\beta = 1 - \sigma_t^2/\sigma_r^2$, comparing the dispersion of velocities tangential to the centre with the dispersion along the radius. Isotropic orbits have $\beta = 0$; purely radial ones $\beta = 1$; purely circular ones $\beta \to -\infty$. The spherical Jeans equation relates the radial dispersion to the mass:

$$\frac{GM(r)}{r} = -\sigma_r^2\left(\frac{d\ln\nu}{d\ln r} + \frac{d\ln\sigma_r^2}{d\ln r} + 2\beta\right),$$

where $\nu$ is the density of the stars. The mass appears on one side and the anisotropy on the other, and a measured dispersion alone cannot separate them.

Projection makes things worse, and the figure shows how. Near the centre of the galaxy as seen on the sky, the line of sight passes close to the centre, and a star’s radial direction points almost along it: what is measured there is mostly the radial dispersion. Far out, the line of sight passes the centre at a distance, and a star’s radial direction there points mostly across the sky: what is measured is mostly the tangential dispersion. Radial orbits therefore produce a high central line-of-sight dispersion that falls outward; tangential ones produce a low central dispersion that rises. One halo, with one mass profile, produces a falling, flat or rising profile depending on the orbits alone.

In the figure, a cusped halo with stars on radial orbits has a central dispersion of nearly nineteen kilometres a second; the same halo with tangential orbits has eight. Read without knowing the orbits, those two would imply central masses differing by a factor of five.

Twenty haloes, thirteen of them indistinguishable #

The real question is the inverse. Given a measured dispersion profile, how different can the haloes be that reproduce it?

The experiment drawn here takes a dwarf-like system: stars in a Plummer distribution with a half-light radius of three hundred parsecs, and a dispersion of ten kilometres a second, flat with radius, as measured in most of the Milky Way’s dwarf spheroidals. It then tries twenty dark haloes — cores and cusps of four inner slopes, each at five scale radii from a third of a kiloparsec to five — and for each finds the orbital anisotropy, allowed to differ between the centre and the outskirts, and the halo’s overall normalisation that best reproduce the flat profile. Each fit counts the stars at each radius as a real sample would, weighting the inner, more populous radii more heavily.

Thirteen of the twenty fit to within a few per cent everywhere. Those thirteen include cores and steep cusps, compact haloes and extended ones; their central densities differ by orders of magnitude. The seven that fail are the extremes — haloes so extended that the stars sit in a nearly uniform density, where a flat profile cannot be produced with any plausible orbits. Everything else is allowed. A dispersion profile measured to a few per cent says remarkably little about the shape of the halo that produced it, which is the same conclusion rotation curves reach about discs for a different reason.

Where they all agree #

Plot the enclosed mass of the thirteen surviving haloes against radius, and the degeneracy has a shape.

At a tenth of the half-light radius the enclosed masses span a factor of seventeen. At four times the half-light radius they span a factor of two and a half. Between, they pinch together. The tightest agreement in this family is at about one and a half times the stars’ three-dimensional half-light radius, where all thirteen agree to within fourteen per cent.

The pinch is not a coincidence of the haloes chosen; it is forced. Each model trades a guess about the orbits against a guess about the mass. A model with radial orbits near the centre attributes part of the high central line-of-sight dispersion to the orbits and so needs less central mass; to fit the same flat profile further out, where radial orbits depress the line-of-sight dispersion, it needs more mass there. A model with tangential orbits does the reverse. The mass curves of the two must therefore cross, and with many models fitting the same data they all cross near the same place.

Where that place is can be understood from the Jeans equation. If the line-of-sight dispersion is flat, the radial dispersion is roughly flat too, its logarithmic derivative is small, and the equation reduces to $GM/r \approx \sigma_r^2(-d\ln\nu/d\ln r - 2\beta)$. The anisotropy’s influence on the mass is then weakest where the stars’ density term dominates — and the projected, light-weighted average that a measurement actually constrains singles out the radius at which the stars’ density falls as $r^{-3}$, the steepness at which the average and the anisotropy decouple. At that radius, $r_3$, the mass is approximately $3\langle\sigma^2_{\rm los}\rangle r_3/G$, whatever the orbits.

For a Plummer distribution, $r_3$ is 0.94 of the three-dimensional half-light radius. The dot in the opening figure is $3\sigma^2 r_3/G$, and the fitted haloes lie between 0.65 and 1.03 times it there — a spread of under fifty per cent, against the factor of seventeen at the centre. The models in this calculation agree most tightly a little further out, at one and a half half-light radii; the published estimator, using the half-light radius itself because it is close to $r_3$ for most light profiles, quotes the mass there at an accuracy of twenty to thirty per cent. Both statements say the same thing. The degeneracy has a node, and the node is near the half-light radius.

Why the whole system has no anisotropy problem at all #

There is one quantity a dispersion measures with no anisotropy dependence whatsoever, and it is worth seeing why it is not enough.

The virial theorem says that for a system in equilibrium, twice the total kinetic energy of its stars equals minus their potential energy in the total gravitational field. The kinetic energy is the sum of the squared velocities in all three directions, and the line-of-sight component, averaged over every star in the system and every direction of view, is exactly a third of it whatever the orbits are. Radial orbits put more motion along some lines of sight and less along others; summed over the whole system they put exactly a third of it along any one. The theorem is an average over time and over the whole system, and in that average the shape of the orbits drops out.

So the light-weighted mean of the squared line-of-sight velocity, over every star in the galaxy, is fixed by the potential alone. The difficulty is that it is fixed by a weighted average of the potential over the stars’ distribution, not by the mass at any radius. A compact halo and an extended one can produce the same average with very different masses inside any given radius, and the virial theorem cannot say which. The global dispersion is anisotropy-free and mass-profile-blind.

The pivot is the bridge between the two. A dispersion profile adds radial information that the global average lacks, and the profile is where the anisotropy enters. The radius at which the profile’s anisotropy dependence and the global average’s profile dependence both become weak is the radius at which the stars’ density falls as the inverse cube — where, in the Jeans equation, the light-weighting and the geometry meet. The estimator $3\langle\sigma^2_{\rm los}\rangle r_3/G$ is the virial theorem’s anisotropy independence carried to a single radius by a profile flat enough not to undo it.

The node follows the light #

If the pinch is set by the stars’ density profile, it should move when the profile changes and not when the halo does.

The same experiment repeated with stars in a Hernquist profile — much more centrally concentrated, with extended wings — and in an exponential sphere gives the same structure: a deep minimum in the spread, at 1.04 and 1.26 half-light radii respectively, against 1.47 for the Plummer case. The masses there agree to between ten and twenty-five per cent; at a fifth of the half-light radius they disagree by factors of four to thirteen.

The location depends on the light profile and moves by tens of per cent between profiles. It does not depend on the halo, because the haloes were the thing being varied. That is what makes the result usable: the light profile is measured directly, from star counts, and its half-light radius is one of the best-determined numbers about any galaxy. The dispersion and the half-light radius together give the mass inside the half-light radius, with an uncertainty set by the profile’s details rather than by the unknown orbits.

A mass inside a radius, not a total #

The pinch also says what not to quote. A dispersion of ten kilometres a second, measured out to three half-light radii, constrains the mass inside the half-light radius to tens of per cent, inside a tenth of it to a factor of ten or more, and the total mass of the halo not at all.

That last point is the one most often lost. A dwarf galaxy’s dark halo extends far beyond its stars, to tens of kiloparsecs, and its total mass is the quantity a cosmological simulation predicts. It is usually quoted by fitting a halo profile to the dispersion and extrapolating — which, given the family of thirteen haloes that all fit, is an extrapolation along whichever member the fit happened to prefer. At four half-light radii the members already differ by a factor of two and a half; at ten they differ by more. The measurement constrains one radius; the total is a model.

The central density is the other quantity the data cannot reach, and it is the quantity the longest-running argument about dark matter is about. Whether a dwarf’s dark matter has a cusp or a core is a question about the inner tenth of its half-light radius, exactly where the family of acceptable models spreads widest. A single population of stars with a single dispersion profile cannot settle it, however many stars are measured.

What the half-light masses say about the dwarfs #

Applied to the Milky Way’s satellites, the estimator gives the mass inside each galaxy’s half-light radius from two numbers each.

The mass inside the half-light radius, divided by half the luminosity, is the mass-to-light ratio there, and it is the cleanest measure of dark-matter domination the data allow. For an old stellar population, stars alone give two or three in solar units. Fornax and Leo I, the brightest of the classical dwarfs, give seven to nine: dominated by dark matter, but not overwhelmingly. Sculptor gives about twenty, Carina and Ursa Minor fifty to ninety, Draco over a hundred. Segue 1, one of the faintest galaxies known, with a few hundred solar luminosities in a half-light radius of thirty parsecs, gives about two thousand.

The ratio rises steadily as the luminosity falls, over four decades of luminosity, and the reason is that the masses inside the half-light radii vary far less than the light does. That observation, made with a related estimator in 2008, was stated as a common mass: the classical dwarfs all contain about ten million solar masses within three hundred parsecs of their centres, though their luminosities differ by a factor of ten thousand. It is one of the strongest pieces of evidence that the dwarfs live in dark haloes of similar inner density, with the visible galaxy a variable and sometimes tiny fraction of what the halo could have formed.

For the faintest dwarfs, two caveats have since grown. Binary stars inflate the measured dispersion of a system whose real dispersion is a few kilometres a second, and inflating it by a third doubles the mass. And the estimator assumes equilibrium, which a dwarf being tidally stripped by the Milky Way may not be in. Segue 1’s mass-to-light ratio has survived both checks; some of the other faintest systems have not.

The same node in a giant elliptical #

Nothing in the argument is specific to dwarfs. A giant elliptical galaxy is also held up by random motions, its dispersion profile is roughly flat over the radii a spectrograph reaches, and its mass inside its half-light radius is the quantity a dispersion measures cleanly.

For ellipticals the measurement is made differently — from the broadening of absorption lines in the integrated light of millions of unresolved stars, mapped across the galaxy with integral-field spectrographs — and the anisotropy is constrained better, because the shape of the line-of-sight velocity distribution, not just its width, carries information about the orbits. But the conclusion is the same: the mass inside the half-light radius is robust, the central density and the total mass are not.

That robust mass has been put to a use the dwarfs cannot match. Inside the half-light radius of an elliptical most of the mass is stars, so the dynamical mass there, divided by the light, is close to the stellar mass-to-light ratio — and that ratio depends on how many faint, low-mass stars the galaxy contains, which is set by the mass function its stars were born with. Large surveys found the dynamical ratio rising with the galaxy’s dispersion faster than a universal mass function allows: the most massive ellipticals appear to contain proportionally more dwarf stars than the Milky Way does. The inference is contested, because dark matter inside the half-light radius would produce the same trend, but it rests on the node: only at that radius is the dynamical mass well enough determined to compare with a stellar population’s.

What was actually measured #

The data behind each of these numbers are individual stellar velocities, from multi-object spectrographs on large telescopes that measure hundreds of stars per pointing to precisions of one or two kilometres a second.

Membership. A dwarf spheroidal is a faint overdensity of stars against the Milky Way’s own foreground, and deciding which stars belong to it uses position, velocity, and chemistry together. A foreground star at the wrong velocity inflates the dispersion; the selection is a probability, modelled jointly with the dispersion.

Binaries. Stars observed at several epochs reveal binaries by their changing velocities; stars observed once cannot be cleaned. For the classical dwarfs, with dispersions near ten kilometres a second and thousands of members, binaries matter little. For the faintest, with dispersions of a few and tens of members, they can dominate.

The profile. The dispersion is measured in radial bins and is usually flat within its uncertainties out to the outermost bin. The half-light radius comes from star counts fitted with a Plummer or exponential profile, and the luminosity from the counts integrated with a stellar population model.

Two populations. Several dwarfs contain two chemically distinct stellar populations, one more concentrated and colder, one more extended and hotter. Each gives its own half-light radius and its own dispersion, and therefore its own mass at its own pivot. Two masses at two radii give a slope of the mass profile between them — and that slope, in Fornax and Sculptor, came out shallower than a cusp allows, which is the closest the dispersion data have come to answering the core question.

Where the models stop #

Spherical, in equilibrium. Every model here is spherical and static. Dwarf spheroidals are flattened, some are tidally disturbed, and a non-equilibrium system’s dispersion is not a mass at all.

A restricted family. The twenty haloes are one family with two parameters, and the anisotropy is allowed one value at the centre and another far out. A wider family — haloes with more shape freedom, anisotropy free at every radius — would let more models fit and move the pinch slightly, and the tightest agreement found here, at 1.47 half-light radii for the Plummer profile, is a property of this family as much as of the physics.

A perfectly flat profile. The data are idealised as flat with no noise. Real profiles have uncertainties of a kilometre a second per bin and sometimes a gentle rise or fall, which widens every curve and leaves the pinch where it is.

A degeneracy with a node #

The general shape is that a degeneracy between two unknowns is often not uniform. Two parameters trade against each other along a curve, but the trade can be steep in one place and flat in another, and where it is flat one combination is measured well whatever the other is. The skill is in finding that combination and quoting it, rather than quoting each parameter with the full degeneracy as its error.

When every model that fits the data agrees about one quantity, that quantity is the measurement, whatever else the models disagree about. Here it is the mass inside the half-light radius. In cosmology it is the angle the acoustic scale subtends rather than any of the distances that make it; in planetary dynamics it is the combination of masses a timing curve fixes. The data rarely deliver the parameter that was asked for, and almost always deliver some parameter cleanly.

Still open: measuring the orbits directly #

The node exists because the line-of-sight velocities cannot see the tangential motions. The obvious way to break the degeneracy is to measure them: proper motions of individual stars within a dwarf galaxy, which at a distance of a hundred kiloparsecs and a dispersion of ten kilometres a second are about twenty microarcseconds a year. Space astrometry built for the stars of the Milky Way and repeated imaging with space telescopes over a decade have begun to measure them for the nearest dwarfs, and the first internal proper-motion dispersions have been published for Draco and Sculptor. With both components of the velocity measured, the anisotropy is an observable rather than a parameter, the family of acceptable haloes collapses, and the question the node was built to avoid — the density at the very centre — can finally be asked of the data.