A stream is not the orbit it came from

The Milky Way’s halo is threaded with thin arcs of stars, some of them stretching a third of the way around the sky. Each is the remains of a globular cluster or a dwarf galaxy that the Galaxy has been pulling apart, and each is, on the face of it, the most direct measurement of the Galactic potential available: a set of test particles laid out along a trajectory, visible, and with proper motions.

The obvious analysis is to fit an orbit through them. It is also wrong, and the error is larger than the measurement precision.

The spread is under two per cent in apocentre. That is far too small to see in the drawing, and it is enough to invalidate the fit.

Two doors, and only two #

Start with why a cluster loses stars at all. The region within which a body can hold onto a satellite is set by the tide from the larger body it orbits, and a cluster in a galaxy is in exactly that position. Beyond its own tidal radius, the galaxy’s pull wins.

But “beyond the tidal radius” is not where a star leaves. It leaves through a saddle.

This is the same structure that appears in the corotating potential of any two-body system, and the two saddles are the inner and outer collinear Lagrange points. The cluster is not leaking in all directions. It has two exits, they are at opposite ends of a line pointing at the galactic centre, and which one a star uses decides everything about where it goes next.

An offset in energy, which is an offset in period #

A star leaving through the inner saddle is one tidal radius closer to the galactic centre than the cluster, and it is moving at the cluster’s angular rate. That combination has slightly less energy than the cluster’s own orbit. A star leaving outward has slightly more.

In a galaxy with a flat rotation curve — which is what rotation curves that refuse to fall mean in practice — a longer orbital period goes with a larger radius, so the two arms drift apart at a rate fixed by that offset. The inner star, on the shorter period, gets ahead. The outer star falls behind. After a few orbits the cluster has a lead and a trail, and after a few hundred it has a stream. The size of the offset is the tidal radius divided by the orbital radius — for a globular cluster at twenty kiloparsecs, about half a per cent. Multiply by four billion years of drift and it becomes a structure a third of the way around the sky.

Why fitting an orbit is wrong #

Here is the error, stated as precisely as the figures allow. A stream’s stars share the progenitor’s angular momentum to within half a per cent and its energy to within the same, but they do not share either exactly, and orbits with slightly different energies in a flattened potential do not have parallel tracks.

The consequence is that the stream’s track on the sky is misaligned with the orbit of any of its members, by an angle that depends on the potential and on the progenitor’s mass. Fitting an orbit to the track therefore returns a potential biased in a direction that can be computed but not ignored, and the bias grows with the mass of the progenitor because the mass sets the distance between the two doors.

The correct treatment is not to abandon the idea. It is to model the stream as what it is — a distribution of orbits with a known spread, generated by a known release process — and to fit the potential to the distribution rather than to a single trajectory. That is more work and it converges on the right answer.

What the stream measures that nothing else does #

The reason to do the work is that a stream is one of the very few probes of the shape of a dark matter halo rather than its mass.

Rotation curves measure the mass inside a radius and say nothing about whether the halo is round, oblate or triaxial, because a circular orbit in the plane samples only one direction. A stream on an inclined, eccentric orbit samples many radii and many directions at once, and the rate at which its plane precesses is a direct reading of the flattening. There is a second thing streams measure, and it is the reason a great deal of effort goes into finding more of them.

Holes left by things that emit nothing #

Cold dark matter predicts that a halo like the Galaxy’s contains many thousands of smaller subhaloes, most of them below the mass at which any stars form. If they exist, they are completely invisible: no light, no gas, no absorption. The only thing they do is gravitate.

A subhalo passing near a thin stream gives every star in it a small velocity kick — the impulse a distant encounter delivers, obtained by integrating along the unperturbed path — and a change of velocity in a stream is a change of orbital energy and therefore of the rate at which a star moves along the stream.

The logic of the measurement is worth spelling out because it is unusually clean. A gap’s depth measures the perturber’s mass over its impact parameter; a gap’s width measures the elapsed time; and the number of gaps along a stream of known age measures the number density of perturbers. None of those requires the perturber to be seen or even to still be nearby. The difficulty is not the theory. It is that a stream also has gaps for reasons that have nothing to do with dark matter: the progenitor sheds stars episodically rather than steadily, the Galactic bar and the spiral arms perturb the stream too, and a survey’s own coverage produces holes in any map. Distinguishing those is the open problem.

The coordinates in which a stream is simple #

The misalignment described above is a statement in position space, and it is a statement that goes away in a different set of coordinates.

For a system whose motion is regular — bounded, non-chaotic, confined to a torus in phase space — there is a natural set of variables. The actions are three quantities conserved along every orbit; the angles are three that increase linearly with time at rates fixed by the actions. In those variables an orbit is a straight line, and a bundle of nearby orbits is a bundle of straight lines.

A stream is exactly such a bundle. The stars share the progenitor’s actions to within the small offset the escape process imposes, so in action space they occupy a compact clump; and because their angles advance at slightly different rates, they spread along a line in angle space whose direction is fixed by how the frequencies vary with the actions.

That is a much cleaner description than a track on the sky. The stream in angle space is a straight line whose direction is a derivative of the potential; the stream in position space is that line mapped through a transformation that depends on the potential, which is why it curves and why it is misaligned.

The practical consequence is a method. Choose a trial potential, transform the observed stars into its action–angle coordinates, and ask whether they form a compact clump in action and a straight line in angle. A wrong potential produces a smeared clump and a bent line, and the degree of smearing is a statistic to minimise.

The difficulty is that computing actions requires the potential to be integrable, and a realistic Galactic potential with a bar and a responding halo is not. What is done instead is to use an approximate action finder that is exact for a class of potentials and adequate nearby, which introduces its own error. The coordinates in which the problem is simple are the coordinates the problem’s answer is needed to construct, and every stream fit is a negotiation with that circularity.

What was actually measured #

Every claim above depends on measuring positions and velocities for individual halo stars, and the chain is worth stating.

A stream is found as an overdensity of stars in a narrow band of colour and magnitude, because its members formed together and therefore lie on a common sequence. That already needs photometry deep enough to place halo stars on a colour–magnitude diagram at twenty kiloparsecs, which puts them at nineteenth magnitude and fainter.

Distances come from that same sequence: matching the observed magnitude of the stream’s own turnoff or horizontal branch to its known absolute magnitude gives a distance modulus, which is a standard candle argument with all the standard candle’s assumptions in it.

Motions come from two places. Proper motions across the sky, which for a halo star at twenty kiloparsecs moving at two hundred kilometres a second amount to about two milliarcseconds a year, and radial velocities from spectra of individual members. The first requires an astrometric mission; the second requires a large telescope and a lot of nights.

Four streams, and what each has supplied #

The general account above is worth grounding in the particular objects, because the four best-studied streams have supplied four different kinds of result.

Palomar 5 is the archetype: a globular cluster still visible, with symmetric tails extending over twenty degrees. Because the progenitor is there, the release point is known and the modelling is as clean as it gets. Its tails contain density variations that have been read as subhalo encounters, as encounters with giant molecular clouds, and as the signature of the Galactic bar — three explanations for one set of features, which is a fair summary of where that argument stands.

GD-1 is the thin one: no surviving progenitor, a track eighty degrees long, and a width of under a hundred parsecs. Its thinness makes it the most sensitive probe of perturbations available, and it contains a gap and an adjacent spur of stars offset from the main track. Modelling reproduces both with a single passage of a compact object of a few million solar masses — which, if it is right, is a detection of a dark subhalo. The alternatives are a globular cluster that has since moved away, or the bar.

The Sagittarius stream is the largest: the debris of a dwarf galaxy, wrapping more than once around the sky, with a mass large enough that it perturbs the Galaxy rather than merely sampling it. Its two arms sit in planes that do not coincide, which is the clearest evidence available that the halo is not spherical — and the flattening inferred from it has disagreed between analyses by enough to be an open question.

The Magellanic Stream is the awkward one. It is gas rather than stars, it trails the Magellanic Clouds across a hundred and fifty degrees, and whether it is tidal, ram-pressure stripped, or both is unresolved. Its importance is that the Clouds are massive enough to have displaced the Galaxy’s own centre, so the stream is a record of a perturbation that every other stream in the halo is also responding to.

Each is the best available instrument for a different question, and none of them answers any of the others.

There is a fifth entry that belongs in the list by absence. Simulations of a galaxy like the Milky Way predict many more thin streams than have been found — the halo should be criss-crossed with the debris of every cluster it has destroyed, and most of them should still be coherent. Surveys find dozens where the models suggest hundreds.

Whether that is a shortfall in the surveys or in the models is unresolved, and it matters for the subhalo counting: a stream that has been heated past recognition is a stream that was perturbed, so the missing population is either an observational selection effect or a measurement of how much perturbing has gone on. The two readings have opposite implications, and telling them apart requires knowing what a survey would have detected — which is the completeness calculation every population argument in this collection ends up needing.

The same construction on a different orbit and a different progenitor separates what the stream records from what the host does.

The width, and what sets it #

A stream’s width is the one property that is easy to measure and hard to interpret, and it is worth separating the three things that contribute to it.

The first is the progenitor’s own size. Stars leave with the velocity dispersion of the cluster they left, a kilometre or two a second for a globular, and that dispersion spreads them perpendicular to the orbit at a rate that does not grow with time — the spread is an epicyclic oscillation about the mean track rather than a drift. So a cluster’s internal motions set a floor on the width and nothing more.

The second is the release geometry. The two doors are separated by twice the tidal radius, which is a real physical width at the moment of release and which the drift then stretches along the stream rather than across it. That is why the arms are thin: the offset that separates them acts almost entirely in the direction of motion.

The third is everything that has happened since. Encounters with subhaloes heat the stream perpendicular as well as along it; passages through the disc do the same; and a stream that has been around for ten billion years is wider than one that has been around for two, whatever it started as. Reading a stream’s width as a heating history therefore needs the intrinsic width subtracted, and the intrinsic width needs the progenitor’s mass — which is often the thing being inferred. That circularity is real and it is handled by fitting the whole stream at once rather than by measuring one number at a time.

The progenitor, and whether it is still there #

A stream can outlive its parent. Some have an identifiable cluster still sitting in them, which fixes the release point and makes the modelling much easier; others do not, and for those the release history has to be inferred from the stream’s own density profile.

That connection is the one worth holding onto. The escape of stars through the two saddles is the same escape that drives a cluster’s evaporation; the difference is only whether the escaping star is followed or counted. A stream is a cluster’s evaporation, seen from outside and laid out in time.

One more thing the hero figure quietly assumes is worth flagging, because it is the assumption most likely to be wrong for the real Galaxy. The integration treats the host as static and spherical, and the Milky Way is neither. Its disc has a bar whose pattern rotates, its halo is flattened by an amount nobody has measured well, and the Large Magellanic Cloud is massive enough that the inner Galaxy is accelerating with respect to the outer one. Each of those effects displaces a stream from where a static model puts it, and each does so by an amount comparable to the offset this essay is about. A stream fitted in a static potential therefore returns a potential that is wrong in a way the fit cannot report, because the model has no parameter for the thing that is missing. The response has been to fit several streams at once and demand that they agree, which is a weak test and the only one available.

Two more readings, of the escape geometry and of the perturbation the whole method exists to detect.

The inference runs backwards from a shape to a mass and it is not unique: a wider gap can be a heavier subhalo passing further away or a lighter one passing closer, and the two are separated only by the gap’s detailed profile. What makes the measurement worth attempting anyway is that the alternative is no measurement at all — there is no other observable consequence of a dark subhalo below the mass at which it holds stars.

Where the ladder goes #

Two things this rung has assumed are the next rung’s subject.

The first is that the host potential is static. It is not: the Galaxy’s disc responds to the Magellanic Clouds, the halo’s centre of mass moves, and a stream on a five-hundred-million-year orbit records all of it. Streams have already been used to detect the Clouds’ effect on the halo, which is a measurement of the halo’s response rather than of its shape.

The second is that the release is smooth. Real clusters shed stars fastest at pericentre, when the tidal radius is smallest, so a stream is made of discrete episodes and has a density structure of its own before any subhalo touches it. Telling that structure apart from perturbations is the difference between counting dark subhaloes and counting the progenitor’s own pericentre passages.