Astronomy & Celestial Mechanics
the sky, computed
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Velocity dispersion — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Mass from a velocity dispersion, across nine decades. The virial estimate M = 5 σ²R_e/G, drawn for half-light radii from 0.2 to 20 kpc, with four systems placed on it: M32 at σ = 75 km/s and R_e = 0.1 kpc, giving 6.5e+8 M☉; the Milky Way's bulge at σ = 105 km/s and R_e = 1 kpc, giving 1.3e+10 M☉; M87 at σ = 320 km/s and R_e = 7 kpc, giving 8.3e+11 M☉; the Coma cluster at σ = 1000 km/s and R_e = 1500 kpc, giving 1.7e+15 M☉. The same expression spans from a dwarf elliptical to a cluster of galaxies — nine decades of mass — because the virial theorem does not care what the moving objects are. In the last case they are whole galaxies, and the estimate exceeds everything visible in the cluster by a factor of about fifty, which is where the problem was first noticed.

    A galaxy held up by disorder

    An elliptical galaxy barely rotates. What holds it up against its own gravity is the randomness of its stars' motions, and the virial theorem turns that randomness into a mass by exactly the reasoning that turns a rotation speed into one.

    part 1 · galaxies
  2. 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.

    part 2 · galaxies

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