Astronomy & Celestial Mechanics
the sky, computed
Concept

Dark matter — where it appears

The mass inferred from motions and from lensing that emits and absorbs no detectable light at any wavelength. It is required by rotation curves, by cluster dynamics, by lensing and by the acoustic peaks, and the four routes agree on an amount five times the ordinary matter.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

gravitation · Virial theorem 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.

galaxies · Rotation curves 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.

galaxies · Dark matter Why an Einstein radius is a mass. The geometry, drawn at an angle some ten thousand times larger than the real one so that anything is visible at all. Light from a source directly behind a lens reaches the observer along every path that passes the lens at the same distance, so the image is a ring rather than a point. The ring's angular radius is θ_E = √(4GM/c² · D_ls/D_l D_s), which for a lens of 1.0×10¹² M☉ at these distances is 2.52 arcseconds and encloses 10 kpc at the lens. Rearranged, it is a mass in terms of an angle and three distances — and the mass so obtained is inside a cylinder rather than a sphere, and assumes nothing whatever about the lens being in equilibrium, which is the assumption every other weighing in this collection makes.

Weighed by the light that bends past it

Every other mass in this collection is measured from something orbiting, which requires the system to have settled down. A gravitational lens weighs whatever is in the way with no such assumption — the light does not care whether the mass is in equilibrium.

galaxies · Strong lensing A cluster weighed three ways, and its stars weighed once. A cluster of galaxies with a velocity dispersion of 1000 km/s, gas at 8 keV and a strong-lensing Einstein radius of 25 arcseconds, each turned into a mass inside 1.5 Mpc by its own relation and nothing else: 7.0, 8.9 and 15.2 × 10¹⁴ M☉. The three assume, respectively, that the galaxies are in equilibrium, that the gas is, and nothing whatever — so their agreement to within a factor of 2.2 is not three restatements of one assumption. The lensing bar is the loosest of the three and is drawn that way deliberately: it measures the mass inside a cylinder of radius 109 kpc, 1.11×10¹⁴ M☉, and carrying that out to 1.5 Mpc as though it were a sphere overstates it. The stars are 2.9 per cent of it.

A cluster weighed three ways

The speeds of a cluster's galaxies, the temperature of its gas and the bending of light behind it are three measurements with almost nothing in common. They agree within a factor of two, and all three exceed the mass of the stars by about a hundred.

galaxies · Clusters The same universe, four times, as a fraction of itself. Each bar is the fractional contribution of the four components to the total density at one epoch, computed from the Planck 2018 parameters by scaling each component from today: radiation as (1+z)⁴, both kinds of matter as (1+z)³, and Λ as a constant. The familiar figure — five per cent baryons, twenty-six dark matter, sixty-nine dark energy — is the top bar and only the top bar. At recombination the same universe is three-quarters dark matter and Λ is one part in ten million; before matter–radiation equality it is mostly radiation. A pie chart of the contents of the universe is therefore a statement about a moment, and the moment is the one it happens to be drawn in.

A budget whose familiar part is five per cent

Five per cent ordinary matter, twenty-six per cent dark matter, sixty-nine per cent dark energy. The figures are quoted everywhere and each one comes from a different measurement, the denominator they are fractions of is itself built out of the expansion rate, and the whole chart is a statement about one instant that was a different chart at every earlier time.

cosmology · Density parameters A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.

A stream is not the orbit it came from

A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

galaxies · Stellar streams 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.

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.

galaxies · Halo profiles A virial mass inflated 23.2-fold by orbits nobody resolved. The factor by which a virial mass is overestimated when the velocity dispersion is measured from single-epoch spectra, against the system's true dispersion, for four numbers of observing epochs. Every star in a binary carries its own orbital velocity, which adds in quadrature to the system's own, and the orbital velocities of ordinary binaries are of order a kilometre a second. A system whose real dispersion is 0.3 km/s therefore measures 1.45, and since a virial mass goes as the square of the dispersion the mass comes out 23.2 times too large. Repeat epochs fix it: the orbital velocities are uncorrelated between visits while the system's own are not, so the binary variance falls as one over the number of epochs and the correction is measured rather than modelled.

A dispersion inflated by orbits nobody resolved

A velocity dispersion measured from single spectra is not the dispersion of the system's centres of mass. Every star in a binary carries its own orbital velocity of a kilometre or so, and for a dwarf galaxy whose real dispersion is smaller than that, the measured value — and the dark-matter content computed from its square — is mostly binaries.

galaxies · Velocity dispersion A lumpy universe makes more of both, and one of them was already too much. The deuterium–lithium plane, with the curve a homogeneous universe traces as its baryon density varies and the points a two-zone universe reaches at a fixed mean density of η₁₀ = 6.13. The dense zone occupies 15 per cent of the volume, and the contrast between the zones runs from 1 — which is the homogeneous case — to 100. Every mixture lies up and to the right of its own homogeneous point, and that is not a modelling choice: an abundance is measured per baryon, so what a telescope averages is η times the abundance — and for both deuterium and lithium that product is a convex function of the density, whose average therefore exceeds its value at the average. The excess is large. At a contrast of 100 the mixture gives D/H = 1.13e-4 against 2.51e-5 smooth, and ⁷Li/H = 1.77e-8 against 4.70e-10, a factor of 37.7. The extra deuterium is exactly what the proposal was for: it lets the mean baryon density be raised while the observed D/H is still matched, which in the 1980s was the one way to make the baryons account for all the matter that dynamics required. The extra lithium is what it costs. The measured abundance was already a factor of 2.9 below the homogeneous prediction, and every step toward the lumpy universe that fixes the density makes that discrepancy worse — which is why the answer to "the baryons are lumpy" turned out to be that the extra matter is not baryons.

The universe that was lumpy at one second

If the baryons were unevenly spread when the network fired, each region ran its own nucleosynthesis and what is observed is an average. For a decade that was the one way to make ordinary matter account for all the matter — and the reason it fails is a theorem about convex curves.

cosmology · Nucleosynthesis Two galaxies that stopped moving apart, and the mass that did it. The separation of two galaxies on a radial orbit that began together at the big bang, against cosmic time, solved so that after 13.797 Gyr they are 770 kpc apart and approaching at 110 km/s — the present separation and approach speed of the Milky Way and the Andromeda galaxy. The curve is a cycloid, r = A(1 − cos θ) and t = B(θ − sin θ), and only one cycloid passes through that point with that slope. It rose to 1037 kpc, turned round when the universe was 8.5 Gyr old, and on this purely radial orbit the two meet 3.3 Gyr from now. Its period fixes the mass: A³/(GB²) = 4.2·10¹² solar masses. The dashed curve is the same calculation with the cosmological constant's outward push included, integrated rather than solved; to arrive at the same place at the same speed against that push it needs 4.76·10¹² solar masses, 13 per cent more. Far more than the stars of the two galaxies, it is the timing argument's measurement of the Local Group's dark matter.

The age of the universe weighs the Local Group

The Andromeda galaxy is approaching the Milky Way, and in an expanding universe that means the two once moved apart, stopped and turned round. One radial orbit passes through their present separation with their present speed after exactly the age of the universe, and its period fixes the mass that turned them — four trillion suns, twenty-five times what their stars can account for.

cosmology · Expansion Five speeds from the same galaxies, and the relation each one gives. The exponent and the scatter of the baryonic Tully–Fisher relation fitted to the same 90 model galaxies with five different speeds, fitted as speed on mass. outer speed: exponent 3.98, scatter 0.080 dex in mass; peak speed: exponent 3.86, scatter 0.110 dex in mass; at 2.2 scale lengths: exponent 3.29, scatter 0.167 dex in mass; W50 ÷ 2: exponent 3.73, scatter 0.119 dex in mass; W20 ÷ 2: exponent 4.05, scatter 0.118 dex in mass. The galaxies were built with the relation in their outer halo speed, and the other four speeds each lose some of it: the ones read from the inner curve inherit how concentrated each galaxy's stars are, and the line widths add the turbulence of the gas, which matters most in the slowest galaxies.

The speed a line width stands in for

A galaxy does not have a rotation speed. It has a rotation curve, rising in the smallest galaxies and peaking early in the largest, and the Tully–Fisher relation is fitted to whichever single number is read off it. Build galaxies with the relation placed in their outer speed and read four other speeds from the same curves, and each gives a shallower or looser relation — which is why the choice of speed is a statement about where the relation lives.

galaxies · Tully–Fisher How the scatter depends on how much dispersion is added to rotation. The scatter of the baryonic Tully–Fisher relation, in dex of mass, when the same 80 turbulent model discs are measured with S = √(K Vᵣₒₜ² + σ²), against the weight K given to the rotation, for rotation measured at 1 and 2.2 disc scale lengths. Measured at 1 scale length the scatter is smallest, 0.076 dex, at K = 0.55, and is 0.076 dex at K = 0.5; measured at 2.2 scale lengths the scatter is smallest, 0.077 dex, at K = 0.23, and is 0.134 dex at K = 0.5. For an exponential disc with constant dispersion the pressure correction is exactly K = one over twice the radius in scale lengths, so the best weight depends on where the rotation is measured. The widely used K = 0.5 is the value that makes a pure rotator and a pure isothermal sphere of the same mass agree; it is also the pressure correction for rotation measured at one scale length, and not at any other.

A disc that turns slower than its mass requires

The star-forming discs of ten billion years ago were not the thin, cold, orderly discs of today. Their gas moved randomly at tens of kilometres a second, and that motion is a pressure that holds up part of each disc, so it turns more slowly than its mass alone would require. Read those rotation speeds as if rotation did all the work and the Tully–Fisher relation tilts and scatters as if galaxies had evolved, when what differs is how much of their weight is carried by disorder.

galaxies · Tully–Fisher

Named alongside it

The objects these essays reach for when they reach for this one.

Velocity dispersionMass-to-light ratioEnclosed massRotation curveBaryon densityThe baryonic Tully–Fisher relationCritical densityDynamical massGravitational lensingTully fisher relationVirial massVirial theorem

All concepts