A shell outside the stream turns nothing

Newton’s shell theorem has two halves, and the familiar one is about the outside: a spherical shell pulls on anything beyond it as if its mass were at its centre. The other half is about the inside: a spherical shell exerts no force at all on anything within it. Both halves survive flattening, provided the shell thins at its poles in exactly the right way — a homoeoid, the layer between two similar, concentric ellipsoids — and the second half is the one this essay is about. A flattened homoeoid exerts no force on anything inside it either.

That essay ended with a question about the Milky Way’s dark halo. Simulations suggest the halo is rounder in its middle, where the disc’s own gravity has pulled it into shape, and more flattened further out, where it still remembers the directions from which its mass arrived. A halo whose layers change shape with radius is not a single nest of similar shells, and its field depends on more than the mass inside each surface. The streams of stars torn from satellites, whose planes turn at a rate set by the halo’s flattening, are the measurement. What the inside half of the theorem says is that each stream measures only part of the halo — and which part.

Why a halo should change shape with radius #

There are two reasons to expect a dark halo’s layers to differ, and they act at opposite ends.

The outer halo remembers how it was assembled. Dark matter falls into a halo along the filaments of the cosmic web, mostly from a few preferred directions set by the tidal field that also spun it, and in simulations of dark matter alone the haloes come out triaxial, elongated along the direction of the filament that fed them, with axis ratios of 0.6 to 0.8 in their outer parts. The outer shells, assembled most recently, carry that elongation most clearly.

The inner halo has been reshaped by the galaxy. Gas cooling into the centre deepens the potential, and the dark matter’s orbits respond: the ones passing through the centre are deflected, box orbits become loops, and the inner halo becomes rounder and, around a disc, slightly oblate in the disc’s plane. Simulations that include the baryons find inner haloes with axis ratios of 0.8 to 0.95 inside about twenty kiloparsecs, turning towards the outer, more elongated shape further out — and, because the contraction that deepens the potential also drags the halo inward, the same process that rounds the inner halo concentrates it.

So the expectation is a halo round inside and flattened or triaxial outside, with the transition somewhere between a few tens of kiloparsecs and the virial radius: exactly the case in which the inside half of the theorem matters most.

A halo built from shells #

Any smooth, axisymmetric halo can be cut into thin layers, each the space between two similar spheroids, and if the halo’s flattening changes with radius, each layer has its own axis ratio $q(a)$. Each layer, being a homoeoid, obeys the theorem: no force inside, and outside a field like that of a flattened body — a monopole plus a quadrupole and higher terms. An orbit at radius $r$ therefore feels the monopoles of every shell inside it, which together make the rotation curve, and the quadrupoles of the same shells, which together make the torque that turns its plane. The shells outside it contribute nothing to either.

The quadrupole of one shell is simple. A homogeneous spheroid of mass $M$ with equatorial radius $a$ and polar radius $c$ has a difference of moments of inertia $C - A = M(a^2 - c^2)/5$; differentiating over a nest of similar spheroids gives the contribution of a thin layer,

$$d(C - A) = \frac{dM\, a^2 (1 - q^2)}{3}.$$

For a halo with a flat rotation curve, whose enclosed mass grows in proportion to radius, the mass in each layer is proportional to its thickness. The quadrupole inside radius $r$ is therefore an integral of $a^2(1-q^2)$ over the shells inside, and the flattening an orbit responds to is the average of $1 - q^2$ over those shells, weighted by $a^2$.

Which shells count #

The weighting has two features, both visible in the figure. Within the orbit, the shells near it dominate: a shell’s quadrupole grows as its size squared, and in a flat rotation curve every shell has the same mass per unit thickness, so the inner half of the halo inside the orbit carries one-eighth of the weight and the outer half seven-eighths. And at the orbit’s radius the weight falls to zero, abruptly, because the shells beyond enclose the orbit and the theorem silences them.

The second feature is the one that matters for interpretation. It means that nothing an orbit does can carry information about the halo outside it. A stream at forty kiloparsecs, however long and however precisely measured, is a measurement of the shells inside forty kiloparsecs, and mostly of those between twenty and forty. The halo beyond could be a pancake or a cigar and the stream would not move.

The hero figure shows what that does to three haloes. A halo flattened to 0.85 everywhere is felt as 0.85 everywhere, because the average of a constant is the constant. A halo round inside and flattened outside is felt as round by every orbit within twenty kiloparsecs — the axis ratio it responds to is 0.988 there — and the flattening it feels lags behind the shells’ at every radius beyond: at sixty kiloparsecs the shells are 0.80 and the orbit feels 0.83, because a quarter of its weight still comes from the rounder shells inside thirty. A halo flattened inside and round outside is felt as flattened well beyond its transition, for the opposite reason. An orbit reports a running average of everything inside it, and nothing at all about what lies outside.

How fast the planes turn #

The quadrupole’s effect on an orbit is to make the vertical oscillation slightly faster than the orbital motion, so the point where the orbit rises through the halo’s equatorial plane falls back a little each time round and the plane turns. For a nearly circular orbit the regression per orbit is $3\pi G Q(<r)/(r^3 V^2)$, where $Q(<r)$ is the quadrupole of the shells inside. For a constant flattening in a flat rotation curve this is $\pi(1 - q^2)/3$ radians per orbit, the same at every radius — the quadrupole and the mass both grow in proportion to radius, and their ratio, which sets the turning per orbit, does not change. A flattening of 0.85 turns a plane by about seventeen degrees per orbit at every distance from the centre.

Per unit of time the turning slows outward, because outer orbits take longer. At twenty kiloparsecs, where an orbit takes six hundred million years, the plane turns by twenty-seven degrees per billion years; at eighty, by seven. That is one reason the streams that measure the halo best are not the most distant ones: a stream at a hundred kiloparsecs has completed only a few orbits in the age of the Galaxy, and its plane has had little time to turn, whatever the halo’s shape.

The same formula is the one that reads the Earth’s shape off a satellite’s node, applied at the other end of the scale of masses. For a satellite above a planet the quadrupole belongs to a body entirely inside the orbit, and the regression measures that body’s flattening completely. For a stream inside a halo the quadrupole belongs to the part of the halo inside the orbit, and the regression measures that part and not the rest.

Three streams in one halo #

Three streams in the halo that is round inside tell three different stories. The stream at fifteen kiloparsecs has turned eleven degrees in eight billion years, consistent with a nearly round halo, and it is right about the halo inside it. The stream at thirty-five has turned sixty-four degrees; at seventy, seventy-three. Fitted one at a time with a single flattening, each would return a different halo, and fitted together with one flattening they would return a compromise that describes none of the shells — a halo too flat for the inner stream and too round for the outer.

This is not a hypothetical difficulty. The Milky Way’s streams have been fitted for two decades with halo shapes that disagreed: oblate from one stream, prolate from another, triaxial with an orientation no formation model expected from the longest of them, nearly spherical from the thin streams of globular clusters within twenty kiloparsecs. Some of the disagreement was the analysis — a stream is not the orbit it came from, and treating it as one biases the fit — and some was a perturbation the models left out, which the last figure of this essay addresses. But some of it may simply be the halo’s layers having different shapes, read by streams at different radii as running averages of different parts of it.

The streams that exist sample the halo unevenly. The thin streams from globular clusters — GD-1 at about fifteen kiloparsecs, Palomar 5 at about twenty — are long, cold and old, and their planes have turned the least; they constrain the inner halo, and they say it is close to round. Their cold, thin structure also makes them sensitive to small passing masses, which punch gaps whose size measures mass times time and which a fit of the halo’s shape has to treat as noise. The streams from dwarf galaxies are broader and reach further — the Orphan stream to about sixty kiloparsecs, Sagittarius to a hundred — and they carry the information about the outer shells, together with most of the perturbation from the Cloud.

A ring over the poles reads the same average #

Streams are not the only orbits whose planes can be watched turning. A few per cent of lenticular and early-type galaxies carry polar rings — rings of gas and young stars orbiting over the poles of the central galaxy, at several times its radius, probably captured from a passing companion or accreted along a filament. A ring at a fixed radius is a single orbit made visible, and if the halo inside it were flattened the ring would turn and twist, its inner and outer edges precessing at different rates until it smeared into a disc.

That the rings survive at all says the halo inside them is close to round, or that the ring’s own gravity holds it together against the differential turning. Fits of their rotation, which give the potential in the ring’s plane, compared with the central galaxy’s rotation, which gives it in the equatorial plane, have found haloes ranging from flattened along the ring to flattened along the host. Each ring is the same measurement as a stream: a constraint on the shells inside its radius, weighted towards its radius, and silent about everything beyond it. Most rings lie at fifteen to thirty kiloparsecs, inside the region where the halo is expected to have been rounded by the galaxy it surrounds.

Reading the shells back out #

If each stream measures an average of the shells inside it, the shells themselves can in principle be recovered: the average out to one radius, multiplied by the volume, minus the same at the next radius in, is the contribution of the shells between them.

The recovery works — it finds the transition at thirty kiloparsecs and the flattening of 0.8 beyond — and it shows what it costs. Each recovered shell is the difference of two cumulative quantities, both large, and the difference is small; the measurement errors on the two do not cancel. Because the cumulative quantity grows as the cube of the radius, the same precision in the turning of each stream gives a precision in the recovered shells that worsens with distance: a hundredth in axis ratio near the centre, four hundredths in the outermost interval drawn. That is with a degree per orbit of precision in each stream’s turning, which is better than any stream’s turning is known.

The general shape of the problem is familiar. A quantity integrated over a region from the centre outward can be differentiated to give the profile, and differentiation amplifies noise; a core weighed by something that never went in faced the same trade, with the exterior gravity field of a planet in place of streams. The useful habit is to say which quantity a measurement actually constrains — here, a running average — and to quote that, rather than the profile, unless the profile’s errors are quoted with it.

A lump the theorem does not silence #

The theorem’s silence applies to homoeoids, and to nothing else. Mass outside an orbit that is not arranged in similar flattened shells does exert a force inside — inside a ring the pull even points outward — and a tidal torque.

A single massive satellite outside an orbit is the extreme case. Its tidal field at the orbit grows as the orbit’s radius squared over the cube of the satellite’s distance, and it turns the orbit’s plane about the line to the satellite rather than about the halo’s axis. The Large Magellanic Cloud, now known to be about a tenth of the Milky Way’s mass and fifty kiloparsecs away, turns the plane of an orbit at twenty kiloparsecs as fast as a flattening of a tenth would, and orbits further out faster still. It has been close for only the last few hundred million years — it is on its first passage — so the accumulated turning is limited, but it acts on the outer streams more strongly than any plausible flattening of the halo while it is near.

That is the perturbation that reconciled much of the disagreement about the Sagittarius stream. Models that include the Cloud’s pull, and the reflex motion it induces in the Milky Way’s own centre, describe the stream with a halo much closer to spherical than models without it. The theorem explains why the Cloud could not be absorbed into a shape: no arrangement of homoeoids outside an orbit can act on it, so a torque from outside has to come from something that is not a homoeoid, and fitting it with a halo shape assigns an exterior lump’s effect to interior shells.

What the calculation assumes #

The halo has a flat rotation curve at every radius and its layers are axisymmetric spheroids sharing one axis. Real haloes are triaxial, and their axes twist with radius as well as their axis ratios change; the shells inside an orbit then contribute quadrupoles about different axes, and the orbit’s plane turns about an axis that belongs to none of them. The averaging argument survives — the shells outside still contribute nothing — but the average is over tensors rather than numbers, and the turning is not a regression about a fixed axis but a slow wobble.

The orbits are nearly circular and slightly inclined. Streams are made by satellites on eccentric orbits, which sample a range of radii and respond to a range of running averages; the precession of an eccentric orbit is roughly that of a circular orbit at a typical radius, weighted by the time spent there, so the eccentric orbit’s apocentre region matters most. And the disc’s own quadrupole, inside every stream, adds a flattening that is not the halo’s at all; for streams within about twenty kiloparsecs it is comparable to the halo’s and has to be subtracted first.

A shell of a real halo is also not quite a homoeoid. The theorem needs the thickness to vary in proportion to the distance from the centre along each direction; a layer of a halo whose shape changes with radius is the space between two spheroids of slightly different axis ratios, and it does exert a small force inside, proportional to how fast the axis ratio changes across it. The calculation here treats each thin layer as a homoeoid of its own axis ratio, which is exact in the limit of thin layers when the change of shape is gradual, and an approximation where it is abrupt.

And the halo is treated as fixed. A real halo grows: its outer shells were added over the last several billion years, so a stream whose satellite was disrupted eight billion years ago orbited, for much of its life, in a halo that ended closer to its orbit than the halo does now. That makes no difference to the inside half of the theorem — shells added outside an orbit exert no force on it whenever they arrive — but it matters for the inner shells, which were themselves being reshaped as the galaxy grew, so the stream’s turning is a time-average of a running average.

Still open: the shape at each radius #

The streams of the Milky Way are now numerous enough — dozens of thin streams, mapped by astrometric surveys in position, distance and motion — that they sample many radii, and the questions asked of them have shifted from the halo’s shape to its shape as a function of radius. What the inside half of the theorem says is that the measurement at each radius is cumulative and the outer halo is reached only by the outer streams, which are the fewest, the oldest and the slowest to turn.

The cleanest test of whether the halo’s layers differ is therefore a comparison of streams that are close in radius but differently oriented, rather than of streams at very different radii. Two streams at the same distance, inclined differently to the halo’s axes, sample the same shells, and their turning rates should agree on one flattening; if they do not, the halo inside that radius is not axisymmetric. Combined with the Cloud’s pull, whose effect is largest on the outer streams and calculable from its now well-measured mass and orbit, that comparison will say whether the Milky Way’s halo is the nest of similar shells the simplest models assume, or a structure whose layers record different chapters of its assembly.