Astronomy & Celestial Mechanics
the sky, computed
Concept

Elliptical galaxy — where it appears

A galaxy with a smooth, featureless, roughly ellipsoidal distribution of mostly old stars, supported largely by random motions rather than rotation. Most are thought to be products of mergers, which is why many rotate slowly or not at all while the discs they came from rotated fast.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

galaxies · Velocity dispersion A disc galaxy walked down the plane by three dry mergers. Specific angular momentum against stellar mass, both logarithmic. The solid line is the relation disc galaxies follow, j proportional to M to the two-thirds, normalised to 1000 kpc km/s at 10¹¹ solar masses; the dashed line is the spheroids', a fifth as high. The points follow the median galaxy built from discs that start on the line at 2.51·10¹⁰ solar masses by successive equal-mass mergers, each between two galaxies with the same history as each other, adding spin vectors in random directions and keeping none of the orbital angular momentum. Each merger doubles the mass and moves the point right by 0.3 of a decade; the vector sum keeps on average only two thirds of the specific angular momentum while the line rises by the two-thirds power of two, so the point falls below it: to 0.45 of the line after one merger, 0.18 after two and 0.08 after three. Two dry major mergers carry a disc to where spheroids are measured.

Two spins that cancel, and an orbit that leaves

Disc galaxies and ellipticals of the same stellar mass differ by a factor of five in how much angular momentum each gram of their stars carries. Adding spin vectors in random directions explains most of it — two mergers of equals, with the orbit's angular momentum handed to the dark halo, carry a disc down to where ellipticals sit — and whether a merger makes a spheroid or rebuilds a disc turns on one number the vectors do not supply.

galaxies · Galaxy spin Three initial mass functions, normalised to the same light. The number of stars per logarithmic interval of mass, on a logarithmic axis, for three initial mass functions — Kroupa, Salpeter, bottom-heavy — each normalised to the same number of stars above one solar mass, so that the stars which make an old population's light are the same in each. They differ only below a solar mass. In a 10 billion year old population the stars below half a solar mass hold 62 per cent, 78 per cent, 91 per cent of the living stellar mass, and they give 15.8, 24.2, 41.7 per cent of the light of the main sequence, which is itself a third of the population's. The difference between them is almost entirely mass that emits almost nothing.

Stars that weigh and do not shine

Most of the mass in an old stellar population is in stars too faint to contribute to its light. How many of them there are is set by the mass function stars are born with, and the mass-to-light ratio every dark-matter measurement divides by can be wrong by a factor of two without a single colour changing.

galaxies · Dark matter Four ways for a galaxy to fade with redshift. The dimming of a galaxy's surface brightness, in magnitudes per square arcsecond, against redshift, for the same galaxy placed at greater and greater distance. In an expanding universe the bolometric surface brightness falls as (1+z)⁴: 3.01 magnitudes by z = 1 and 7.8 by z = 5. Measured per unit frequency, as magnitudes in a filter are, one power is absorbed by the stretching of the band and the dimming is (1+z)³. If photons lost energy on a static journey ("tired light"), only one power would remain — 0.75 magnitudes at z = 1. In a static space with no loss there is no dimming at all, because flux and solid angle both fall as the inverse square of distance. At z = 1 expansion and tired light differ by 2.3 magnitudes, a factor of 8 in brightness.

A dimming short by what young stars repay

In an expanding universe a galaxy's surface brightness should fall as the fourth power of one plus its redshift; if light merely tired on its way through static space, as the first power. Measured on elliptical galaxies at redshift 0.85, it falls as about the two-and-a-half power — neither. The verdict on expansion rests on what the galaxies did in the meantime, and the decisive fact is not the size of the shortfall but its sign: expansion needs the galaxies to have been brighter when their stars were younger, and tired light needs them to have been fainter.

cosmology · Expansion

Named alongside it

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

Velocity dispersionAngular-diameter distanceAnisotropyCosmological redshiftDe vaucouleurs lawThe disc–halo degeneracyDistance dualityDry mergerDwarf sensitive featuresDynamical frictionDynamical massEffective radius

All concepts