Astronomy & Celestial Mechanics
the sky, computed
Concept

Jeans equation — where it appears

The equation of hydrostatic balance for a collisionless system of stars, relating the gradient of their density and velocity dispersion to the gravity holding them. Solved with an assumed anisotropy, it turns a dispersion into a mass.

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

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 Where one dispersion and a resolved profile leave the mass and the orbits. The plane of the mass-sheet parameter λ and the orbital anisotropy β. Light shading: the region within Δχ² = 2.3 of a mock lens with λ = 1 and β = 0.1, given one aperture dispersion measured to five per cent — a long diagonal valley, because a larger λ and more tangential orbits trade against each other. Dark shading: the same, given the dispersion in six rings between a quarter and two Einstein radii, each to three per cent. Marginalised over β with a flat prior, and with λ allowed between 0.8 and 1.15, the aperture measurement gives λ = 0.990 ± 0.091 and the resolved profile λ = 1.000 ± 0.040 — the Hubble constant from this lens to 9 and 4 per cent.

A dispersion that weighs the sheet only if the orbits are known

A lens's images cannot tell how much of its mass is in the galaxy and how much is spread through a sheet around it, and the time delay — and so the Hubble constant — scales with the answer. The stars' velocity dispersion can tell, because orbits feel mass in three dimensions where light only integrates along a line. One dispersion through one aperture weighs the sheet only to sixteen per cent, because the shape of the orbits moves it as much; the same dispersion measured in rings weighs it to four.

galaxies · Strong lensing Haloes that agree on the line of sight, and disagree across the sky. The dispersion of the tangential component of motion on the sky — across the radius vector, as a proper motion measures it — predicted by each of the 13 dark haloes that fit the same flat line-of-sight dispersion of 10 km/s, against projected radius in half-light radii, for stars in a Plummer profile. Coloured by the halo's inner slope; the dashed line is the line-of-sight value all of them reproduce. At the half-light radius the haloes' line-of-sight dispersions agree to 2.0 per cent, and their tangential dispersions range from 8.3 to 10.8 km/s, a factor of 1.29. The anisotropy each halo needed to match the line of sight is written into the motion across it, which no line-of-sight measurement sees.

Motions across the sky measure the orbits first

A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.

galaxies · Velocity dispersion

Named alongside it

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

Orbital anisotropyVelocity dispersionThe core–cusp problemDark matterDwarf spheroidalThe half-light radiusAstrometryDegeneracyEinstein radiusEnclosed massHubble constantIntegral-field spectroscopy

All concepts