Astronomy & Celestial Mechanics
the sky, computed
Concept

Enclosed mass — where it appears

The mass lying inside a given radius, which for a spherical distribution is the only mass a body at that radius feels. It follows from the shell theorem, and it is why a rotation curve measures where the mass is rather than how much there is in total.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Forty-two minutes, from anywhere to anywhere. Left, the gravitational field inside the Earth: a straight line for a uniform sphere, because the enclosed mass grows as r³ and the field as r, and the PREM curve for the real one, which is nearly flat through the whole mantle at about 9.94 m/s² because the dense core is already all below. Right, what falls through it. A body dropped down a diametric tunnel through a uniform Earth executes simple harmonic motion with ω = √(g/R), reaching the far side in 42.2 minutes; a body dropped down a chord at 0.6 of the radius feels only the component along the tunnel, which is the same ω times a smaller distance, so it arrives in the same time from a shorter trip. The period does not contain the length of the tunnel, its direction, or where the body starts. And 2π/ω is also the period of a circular orbit grazing the surface, 84.3 minutes — the tunnel and the orbit are one ellipse, seen twice. The real Earth is not uniform and gets there in 38.2 minutes instead, which is the honest number and is 9% quicker.

The tunnel that takes the same time from anywhere

Inside a uniform sphere the field grows in proportion to the distance from the centre, which is Hooke's law. A body dropped down any straight tunnel arrives in the same time — and that time is the period of an orbit skimming the surface.

gravitation · Shell 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 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 The pull of a ring, a shell and a point, in the plane. Radial force in the plane of a thin ring of radius 1, against distance from its centre, in units where the same mass at a point would pull with one over the distance squared; positive is outward. Inside a spherical shell of the same radius and mass the force is exactly zero. Inside the ring it is not: it points outward, toward the nearer part of the ring, at 0.34 at half the radius and 2.75 at nine tenths, growing without limit toward the rim. Outside, the ring pulls inward harder than a point would, by 25 per cent at twice its radius and 9 per cent at three times, because the near side of the ring is closer than the centre. The shell is indistinguishable from the point everywhere outside it.

Inside a ring the pull points outward

A hollow sphere exerts no force on anything inside it, and the proof is a pair of cones whose areas grow exactly as fast as gravity weakens. Cut the sphere down to a ring and the arcs grow only as the distance, the nearer side wins, and a planet set in the middle of a rigid ring falls into it within a handful of orbits — which is why Saturn's rings cannot be solid, and why a flat galaxy spins faster than its own enclosed mass allows.

gravitation · Shell theorem Mass profiles that pinch together near the half-light radius. The enclosed mass of each of the 13 haloes that fit the same flat dispersion of 10 km/s, against radius in units of the stars' three-dimensional half-light radius, 391 parsecs. At a tenth of that radius they span a factor of 17, and at four times it a factor of 2.5; they pinch together near the half-light radius, closest at 1.47 of it, where they agree to 14 per cent. The dot is the estimate 3σ²r/G at the radius where the stars' density falls as r⁻³, 0.94 of the half-light radius; there the fitted masses lie between 0.65 and 1.03 times it. Each model trades a wrong guess about the orbits against a wrong guess about the mass, in opposite senses inside and outside.

The one radius where the orbits stop mattering

A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted.

galaxies · Velocity dispersion

Named alongside it

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

Dark matterMass-to-light ratioExponential discRotation curveBaryonBeam smearingCentral condensationChord tunnelCircular velocityThe core–cusp problemCusp core problemDark halo

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