Astronomy & Celestial Mechanics
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Galaxies

An argument about the innermost kiloparsec

Simulations of cold dark matter have produced the same halo for thirty years — a density rising inward without limit. The rotation curves of the smallest galaxies say the density flattens off. The disagreement is confined to a region a thousandth of the halo's size, and it has not been settled.

Assumes Dark matter and Rotation curves.

16 min read 8 figures Measuring the unreachableScales that defeat intuition

That a galaxy weighs more than its light is not in serious dispute, and neither is the general shape of the thing doing the weighing: a roughly spherical halo, far larger than the visible galaxy, whose density falls outward. What is in dispute is the innermost part of it — a region a thousandth of the halo’s radius across — and the dispute has outlasted three generations of both telescopes and computers.

The reason it matters more than its size suggests is that this is the one place where the standard theory of dark matter makes a sharp, parameter-free prediction about a structure that can be observed. Almost everything else the theory predicts is statistical, or depends on how galaxies form, or can be tuned. The inner slope of a halo is supposed to be a consequence of gravity alone.

A cusp and a core in the same halo: 39 times apart in density, 2.34 in rotation. Two dark-matter density profiles for a halo of 10¹⁰ solar masses, drawn on logarithmic axes so a power law is a straight line and its exponent is a slope. The upper curve is the profile cold dark matter simulations produce: density rising inward without limit, with a logarithmic slope tending to -1.01 — a cusp. It is not a fit to observations; it emerges from simulations run by many groups with different codes, and its robustness is what makes it a prediction worth testing. The lower curve is a cored profile with the same virial mass, flat inside a kiloparsec, slope -0.01. What the rotation curves of gas-rich dwarf galaxies prefer is the second. The disagreement is stark where it is drawn — a factor of 39 in density at 0.10 kiloparsecs — and much less stark in what is actually measured, because a rotation speed depends on the mass enclosed rather than on the local density, and integrating a cusp over a small radius does not accumulate very much. At 1 kiloparsecs the two halos differ by a factor of 2.34 in circular speed — a real difference, and a far smaller one than the 39 in density that produced it. That compression is the whole difficulty of the problem. The observable is an integral of the quantity in dispute; integrating a cusp over a small radius does not accumulate much mass, so the sharpest disagreement lives where the instrument is bluntest. And the innermost points of a rotation curve are also the ones most affected by non-circular motions, by beam smearing, and by the inclination assumed — so the measurement is hardest exactly where it matters most. Whether the resolution is astrophysical — supernova feedback moving gas repeatedly and dragging the dark matter outward with it — or a statement about what dark matter is remains open, and the figure deliberately shows only the alternatives rather than choosing.
Fig. 1 Two dark-matter density profiles for a halo of ten billion solar masses, on logarithmic axes so a power law is a straight line and its exponent is a slope. The upper curve is what cold dark matter simulations produce: a density rising inward with a logarithmic slope tending to −1, a cusp. The lower one is a cored profile of the same virial mass, flat inside a kiloparsec. At a hundred parsecs the two differ by a factor of thirty-nine — and in the quantity actually observed, the circular speed at a kiloparsec, by only a factor of 2.3.

Where the prediction comes from

Nobody derived the cusp. It emerged.

In the 1990s it became possible to simulate the gravitational collapse of a region of an expanding universe filled with collisionless particles, and the halos that formed were found to have a common density profile — one that could be fitted by a two-parameter formula whose inner behaviour was ρ∝r−1\rho \propto r^{-1} and whose outer behaviour was r−3r^{-3}. The parameters were a mass and a concentration; the shape was universal.

What made the result stick was that it kept coming back. Different codes, different resolutions, different cosmological parameters, halos spanning a factor of 10910^{9} in mass — all of them produced approximately the same profile. Later work refined the inner behaviour: the slope does not converge to exactly −1-1 but keeps getting shallower inward, in the way a curve rather than a power law does, which is the Einasto form. But it does not flatten off, and by a hundred parsecs it is still steeper than −0.8-0.8.

The physical origin is not fully explained even now, which is worth saying plainly. There are arguments — from the way collisionless material phase-mixes during a merger, from the shape of the initial fluctuation spectrum, from adiabatic invariance during collapse — and none of them predicts the profile from first principles. The prediction is empirical, in the peculiar sense that the experiment is a computation.

The mass of NGC 3198 inside each radius, weighed two ways. The mass inside radius r from the rotation curve, M = v²r/G, against the mass inside r from the light. The luminous curve flattens by about 10 kpc because there is no more disc to enclose; the dynamical curve rises almost linearly, because a flat rotation curve means exactly M ∝ r. At 30 kpc the two differ by a factor of 6.8. The straightness of the upper line is the finding: it is not that a galaxy has more mass than expected, it is that the extra mass keeps arriving at the same rate however far out one looks.
Fig. 2 What a real galaxy’s mass profile looks like when it is decomposed. The stars and gas are measured; the remainder is the halo, and it is defined as a residual. Note where the residual is well constrained: in the outer parts, where it dominates. In the inner parts the baryons contribute most of the mass, so the halo is a difference between two larger numbers — which is a bad way to measure anything, and is the second reason the inner slope is hard.

It is worth being precise about what “universal” means here, because the word does a lot of work. The claim is not that every halo has the same density — obviously not — but that after scaling out one mass and one radius, the profiles lie on top of each other. Two numbers, and the third and fourth and fifth are determined. A galaxy is a far more complicated object than that, and the fact that its dark component is not is one of the strongest indications that the component is genuinely simple: a collisionless fluid with no physics in it beyond gravity would behave exactly this way, and almost nothing else would.

A cusp and a core in the same halo: 279 times apart in density, 6.41 in rotation. Two dark-matter density profiles for a halo of 10¹² solar masses, drawn on logarithmic axes so a power law is a straight line and its exponent is a slope. The upper curve is the profile cold dark matter simulations produce: density rising inward without limit, with a logarithmic slope tending to -1.00 — a cusp. It is not a fit to observations; it emerges from simulations run by many groups with different codes, and its robustness is what makes it a prediction worth testing. The lower curve is a cored profile with the same virial mass, flat inside a kiloparsec, slope -0.00. What the rotation curves of gas-rich dwarf galaxies prefer is the second. The disagreement is stark where it is drawn — a factor of 279 in density at 0.10 kiloparsecs — and much less stark in what is actually measured, because a rotation speed depends on the mass enclosed rather than on the local density, and integrating a cusp over a small radius does not accumulate very much. At 1 kiloparsecs the two halos differ by a factor of 6.41 in circular speed — a real difference, and a far smaller one than the 279 in density that produced it. That compression is the whole difficulty of the problem. The observable is an integral of the quantity in dispute; integrating a cusp over a small radius does not accumulate much mass, so the sharpest disagreement lives where the instrument is bluntest. And the innermost points of a rotation curve are also the ones most affected by non-circular motions, by beam smearing, and by the inclination assumed — so the measurement is hardest exactly where it matters most. Whether the resolution is astrophysical — supernova feedback moving gas repeatedly and dragging the dark matter outward with it — or a statement about what dark matter is remains open, and the figure deliberately shows only the alternatives rather than choosing.
Fig. 3 The same pair of profiles for a halo a hundred times more massive and less concentrated, which is what universality means when it is drawn rather than asserted. The two curves have the same shapes as the dwarf’s and sit at different places, because the only things that changed were a mass and a concentration. The gap between cusp and core at a hundred parsecs is even larger here — a factor of two hundred and seventy-nine — and it is measurable in a large spiral by nobody, because a hundred parsecs from the centre of such a galaxy the mass is overwhelmingly stars. The prediction is sharpest exactly where the observation is worst.

Why a rotation curve is the wrong instrument for the job

The disagreement in density is dramatic. The disagreement in what is observed is much less so, and the compression is not an accident but a consequence of what a rotation curve is.

A circular speed at radius rr depends on the mass enclosed within rr, not on the density there. Integrating a cusp over a small radius does not accumulate very much mass: ρ∝r−1\rho \propto r^{-1} gives M∝r2M \propto r^{2} and therefore v∝r1/2v \propto r^{1/2}, while a core gives M∝r3M \propto r^{3} and v∝rv \propto r. Both rise from zero; the cusp rises faster; and telling them apart means measuring the shape of the rise over the innermost few resolution elements of an image. Three systematics act on exactly those points and all three push the same way.

Beam smearing. A radio interferometer measures velocity averaged over a beam. If the beam is comparable to the region where the curve is rising, a steeply rising curve is measured as a shallowly rising one — turning a cusp into an apparent core. Early single-dish measurements suffered from this badly, and it was the first objection raised.

Non-circular motions. The method assumes gas moves on circles. A bar, a spiral arm, or a lopsided potential produces streaming motions of tens of kilometres a second, which in the inner regions is a substantial fraction of the whole signal. A bar in particular produces motions that mimic a core.

Inclination. An observed line-of-sight velocity is the circular speed times the sine of the inclination, and a dwarf galaxy’s inclination is not well determined because a dwarf galaxy is not very flat. A ten-degree error near face-on is a large multiplicative error in the derived mass.

The mass of NGC 3198 inside each radius, weighed two ways. The mass inside radius r from the rotation curve, M = v²r/G, against the mass inside r from the light. The luminous curve flattens by about 10 kpc because there is no more disc to enclose; the dynamical curve rises almost linearly, because a flat rotation curve means exactly M ∝ r. At 30 kpc the two differ by a factor of 6.8. The straightness of the upper line is the finding: it is not that a galaxy has more mass than expected, it is that the extra mass keeps arriving at the same rate however far out one looks.
Fig. 4 The same decomposition with the stellar mass-to-light ratio halved. Every point on the stellar curve drops, the halo takes up the difference exactly, and the total is unchanged — because the total is what was measured and the split is what was assumed. That is the fourth systematic, and it is the one the previous three do not include: even a perfectly measured rotation curve gives an inner halo profile only as accurately as the stellar population model that is subtracted from it. In a large spiral this dominates. It is the whole reason the argument is fought in dwarfs, where the curve to subtract is nearly flat at zero.

The three do not average out. Beam smearing always flattens a rise, never steepens one. A bar always adds streaming in the sense that mimics a core. And an inclination underestimated always inflates the derived speed. A field in which every known systematic pushes toward the same conclusion is a field where the conclusion has to be defended carefully, and this is the strongest form of the sceptical case — not that any particular measurement is wrong, but that the errors are not symmetric.

Why dwarf galaxies, and why gas

The measurement is made in the smallest galaxies for two reasons.

The first is that they are dark-matter dominated everywhere, including at the centre. A rotation curve that refuses to fall is evidence for a halo wherever it is seen; in a dwarf the halo is not merely dominant in the outskirts but dominant throughout. A large spiral’s inner kiloparsec is mostly stars, so extracting the halo means subtracting a stellar mass that is known only through a mass-to-light ratio — which is the very quantity that is hardest to pin down. In a gas-rich dwarf the stars contribute a few per cent and the subtraction hardly matters.

The second is that the predicted difference is largest there. Halos of low mass form earlier, when the universe was denser, so they are more concentrated in the appropriate scaled sense and their cusps are relatively more prominent.

A cusp and a core in the same halo: 79 times apart in density, 3.36 in rotation. Two dark-matter density profiles for a halo of 10¹⁰ solar masses, drawn on logarithmic axes so a power law is a straight line and its exponent is a slope. The upper curve is the profile cold dark matter simulations produce: density rising inward without limit, with a logarithmic slope tending to -1.02 — a cusp. It is not a fit to observations; it emerges from simulations run by many groups with different codes, and its robustness is what makes it a prediction worth testing. The lower curve is a cored profile with the same virial mass, flat inside a kiloparsec, slope -0.01. What the rotation curves of gas-rich dwarf galaxies prefer is the second. The disagreement is stark where it is drawn — a factor of 79 in density at 0.03 kiloparsecs — and much less stark in what is actually measured, because a rotation speed depends on the mass enclosed rather than on the local density, and integrating a cusp over a small radius does not accumulate very much. At 0.3 kiloparsecs the two halos differ by a factor of 3.36 in circular speed — a real difference, and a far smaller one than the 79 in density that produced it. That compression is the whole difficulty of the problem. The observable is an integral of the quantity in dispute; integrating a cusp over a small radius does not accumulate much mass, so the sharpest disagreement lives where the instrument is bluntest. And the innermost points of a rotation curve are also the ones most affected by non-circular motions, by beam smearing, and by the inclination assumed — so the measurement is hardest exactly where it matters most. Whether the resolution is astrophysical — supernova feedback moving gas repeatedly and dragging the dark matter outward with it — or a statement about what dark matter is remains open, and the figure deliberately shows only the alternatives rather than choosing.
Fig. 5 The same dwarf halo at a concentration of twenty rather than twelve, read at three hundred parsecs rather than a kiloparsec. Concentration is the second of the profile’s two parameters and it is not free: it correlates with mass and with formation epoch, and a low-mass halo that assembled early is more concentrated. So the two effects the paragraph above describes are one effect seen twice, and the figure separates them — the cusp is more prominent in a small halo because the halo is denser everywhere, not because its inner slope is different. The slope is the same. What changed is where the profile turns over.
The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection.
Fig. 6 Why the disagreement is fought in the smallest galaxies. A large spiral’s mass-to-light ratio is near the stellar value in the inner kiloparsec and climbs steeply outward, so extracting a halo there means subtracting a stellar mass known only through a population model. In a gas-rich dwarf the stars contribute a few per cent everywhere and the subtraction hardly matters, which is worth more than any gain in resolution.

What could close the gap

There are three families of answer and they are not mutually exclusive.

The observations are wrong. This was the first response and it is no longer tenable in its strong form. It deserved a hearing: the history of galaxy rotation measurements contains several results that turned out to be about instruments. Two-dimensional velocity fields at arcsecond resolution, from interferometers and from integral-field spectrographs, remove beam smearing and measure the non-circular motions rather than assuming them away. A subset of galaxies still shows cusps, but a substantial subset does not, and the cored ones are not obviously the badly measured ones.

The simulations are incomplete. This is the mainstream position. The simulations that produce cusps contain only dark matter, and real galaxies contain gas that forms stars that explode. Repeated, bursty episodes of star formation drive gas out of the centre and let it fall back; the halo particles feel the changing potential, and — because the changes are neither slow enough to be adiabatic nor fast enough to be impulsive, but resonant with the orbital times — they gain energy and move outward. Simulations that include this do produce cores, and they predict that the effect should switch off below a stellar mass around 10610^{6} solar masses, because below that there are not enough supernovae to do the work.

The feedback account has a testable signature and it is a sharp one. If cores are dug by supernovae, the depth of a core should correlate with how many stars a galaxy has made, and it should vanish in galaxies that made almost none. Halos hosting the very faintest dwarfs — the ultra-faint satellites, with a few thousand stars — should have kept their cusps. Measuring the inner slope of an object with a few thousand stars is at the edge of what is possible, and it is where the argument is currently being fought.

Dark matter is not collisionless. If dark matter particles scatter off each other with a cross-section per unit mass of order a barn per gram, halos thermalise in their dense centres and develop cores of exactly the observed size, while their outer parts are untouched. This is a clean, falsifiable modification with one parameter, and the same parameter is constrained by cluster collisions and by halo shapes.

The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 1.1 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 5 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection.
Fig. 7 The mass-to-light ratio the previous section’s subtraction produces, at the lower stellar normalisation. It climbs from a stellar value in the centre to many times that in the outskirts, and the shape of the climb is what the halo is. Note where the curve is flat and where it is steep: the information about the halo lives entirely in the outer part, where the ratio is large and rising, and the inner kiloparsec contributes almost nothing to it. A self-interacting halo would change only the inner region, which is the part of this curve that is closest to the stellar value and hardest to measure — so a diagnostic like this one cannot see the modification at all.

The problem’s relatives

The inner slope is one of a small family of small-scale disagreements, and they are usually discussed together because a single fix would ideally address all of them.

Missing satellites. Simulations predict far more subhalos around a galaxy the size of the Milky Way than there are observed dwarf galaxies. This one has largely dissolved: deep surveys have found many faint satellites, and the physics of how gas is prevented from cooling in the smallest halos accounts for the rest.

Too big to fail. The most massive predicted subhalos are dense enough that they should host visible satellites, and the observed satellites — measured by the disorder of their stars — are less dense than those subhalos. This is the same statement as the cusp–core problem seen in a different population, which is why the two are usually taken to have one cause.

The diversity of rotation curves. Galaxies of the same maximum circular speed have inner rotation curves that differ enormously — some steeply rising, some slowly. Simulations produce a much narrower range. This may be the sharpest version of the problem, because it is about scatter rather than about a mean, and scatter is harder to arrange by tuning. The measurement that would settle the argument is not a better version of any of the above.

What was actually measured

None of the numbers above is a density. What a radio observation produces is a data cube: two sky coordinates and a frequency, with an intensity at every point, where the frequency is the twenty-one-centimetre line Doppler-shifted by the gas’s motion. Extracting a rotation curve from that cube means fitting a set of tilted rings, each with its own centre, inclination, position angle and circular speed, to reproduce the observed velocity field.

That fit has more parameters than it has robust constraints in the inner regions, which is why different groups analysing the same cube have published inner slopes differing by more than their quoted errors. The honest summary of the observational situation is that a majority of well-observed gas-rich dwarfs prefer slopes shallower than −0.5-0.5, that a minority prefer −1-1, and that the scatter between galaxies is larger than the error on any one of them.

For the pressure-supported dwarfs the data are individual stellar velocities, thousands of them, measured with a multi-object spectrograph over several nights. The mass then comes from the virial relation between kinetic and potential energy rather than from a circular orbit, and it is well determined at one radius — roughly the half-light radius — and poorly determined anywhere else. Several galaxies with two distinct stellar populations at different radii have been used to get two such measurements and hence a slope, which is the cleanest version of the measurement anyone has made.

The light of NGC 3198 falls off exponentially. V-band surface brightness against radius for an exponential disc of scale length 2.6 kpc and a stellar mass-to-light ratio of 1.4 in solar units. The vertical axis is magnitudes per square arcsecond and runs the astronomers' way, with brighter upward; because a magnitude is already a logarithm, an exponential disc is a straight line on it, and the slope is one scale length per 1.086 magnitudes. The conventional edge of a galaxy, the 25th magnitude isophote, falls at 12.0 kpc — and the rotation curve is still flat at 30 kpc, 2.5 times further out.
Fig. 8 The quantity that decides where that one well-determined radius falls. A galaxy’s light falls off exponentially, so its half-light radius is fixed by the scale length and by nothing else — and a virial mass from a stellar velocity dispersion is a mass inside that radius, wherever it happens to be. The two-population trick works because a metal-poor and a metal-rich population in the same dwarf have different scale lengths and therefore different half-light radii, so each returns a mass at its own radius and the pair gives a slope. Everything about that measurement is set by this curve rather than by anything about the dark matter.

And it is worth being explicit that a resolution would not be a small result. The inner slope is the only place where the standard dark-matter model makes a parameter-free prediction about an observable structure, so a confirmed core in a galaxy that has made too few stars to have dug one would be evidence about the particle rather than about galaxy formation.

What the figure cannot show

The hero figure draws two profiles and lets them be compared. It does not, and cannot, show the thing that makes the comparison hard, which is that neither curve is observed. What is observed is a set of velocities at a set of radii with error bars, and both curves are fitted to them through a model that includes the stars, the gas, the inclination and the beam.

Nor does it show the halo’s shape. Everything above assumes spherical symmetry, and simulated halos are triaxial — typically with axis ratios around 2:1 in the inner regions. A triaxial halo viewed at an unlucky angle produces a rotation curve that a spherical fit reads as cored, and the effect is of the right size to matter.

And it does not show time. Both profiles are drawn as though a halo had a fixed density distribution, and a real halo is being continually stirred by infalling subhalos, by its own baryons and by whatever the last merger left behind. The relaxation time of a dark halo is far longer than the age of the universe, so it does not forget — but it is not in equilibrium either. A profile is a snapshot of something still settling.

The obstacle is not the quality of the spectroscopy, and it has not been for a decade.

The measurement that would settle it

Every mass in this argument comes from one line-of-sight velocity component, and that is the root of the ambiguity rather than a detail of it.

A dwarf spheroidal’s mass is inferred from the dispersion of its stars’ radial velocities, and the same dispersion is produced by a shallow potential with radial orbits or by a deeper one with tangential orbits. That is the mass–anisotropy degeneracy, and it is not a small effect: it is comfortably large enough to convert a measured cusp into a measured core and back.

Nothing about better radial velocities helps. The degeneracy is between two things that a single velocity component cannot separate, so a thousand more stars measured the same way tightens the error bars on a quantity that is still ambiguous. What breaks it is the other two components — the proper motions, which say how the stars move across the sky.

The numbers involved are brutal. A star in a satellite at a hundred kiloparsecs moving at ten kilometres a second has a proper motion of about twenty microarcseconds a year, so measuring a dispersion of that size means measuring individual stellar motions at the microarcsecond level. That is below what any survey delivers for stars this faint, and the route being taken instead is a long baseline: images of the same field separated by fifteen or twenty years, from the same instrument, where a small annual motion has accumulated into a measurable displacement.

Two such measurements now exist for the nearest classical dwarfs, and they are the reason this problem is expected to move rather than remain open indefinitely. With all three velocity components the anisotropy is measured rather than assumed, the mass profile follows without the degeneracy, and the inner slope becomes a number with an error bar rather than a preference expressed by a fit.

The pattern is the one this whole subject runs on: an unresolvable ambiguity in one projection, resolved not by measuring the projection better but by obtaining a different one. It is the same move as adding an interferometric angle to a Doppler track, or a second stellar population at a second radius, and it is more valuable than any amount of precision in the component already in hand.

Where the ladder goes

The next rungs on this anchor go outward and inward. Outward: the halo’s outer profile and its concentration, which are measured by the shear of the galaxies behind it and by strong lensing rather than by orbits and are in much better agreement with the simulations. Inward: what happens where the halo meets a central black hole, and whether the cusp is steepened there into something detectable.

There is also a thread that is not about galaxies at all. The same simulations that predict a cusp predict a halo mass function extending to Earth masses, and the smallest predicted halos are far too faint to see by any means — except by what they do to a stellar stream passing through them. That is currently the most promising route to the question underneath all of this, which is not what the inner slope is, but what the particle is.

Where this goes next

The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection. The mass that is not the light A galaxy's mass-to-light ratio is not a number, it is a curve, and it rises without turning over. What stellar populations can plausibly weigh sets a ceiling; the dynamics sit far above it, and the gap is a shape rather than a discrepancy. The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not. A rotation curve that refuses to fall Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure. An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation. An average that weighs what cannot be watched A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Beam smearingCusp core problemDark matterDwarf galaxyEnclosed massHalo concentrationInner slopeNfw profileNon-circular motionsSelf interacting dark matterStellar feedback