Astronomy & Celestial Mechanics
the sky, computed
Concept

Lognormal distribution — where it appears

A distribution whose logarithm is normally distributed, describing positive quantities spread by multiplicative effects. The spin parameters of dark-matter haloes follow one, with a median near 0.035 and a spread of about 0.5 in its natural logarithm.

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

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

galaxies · Galaxy spin How much the size difference says about what makes a bar. The ratio of mean disc sizes, barred over unbarred, in haloes of one mass, against the blur w of the spin threshold in ln λ, for disc shares 0.02, 0.03, 0.05. A sharp threshold gives a factor of about two; blurred by w = 0.5 — comparable to the spread of spins itself — the ratio rises to 0.57 at a disc share of 0.03; by w = 1, to 0.70. A measured contrast of 20% would require w ≈ 1.7, a threshold blurred over a factor of 6 in spin: the spin would lean on whether a disc bars rather than decide it.

Barred discs should be the small ones

If a disc bars because its halo spins too slowly, then among galaxies of the same mass the barred ones are those from the slow-spinning haloes — and a slow-spinning halo makes a small disc. A sharp spin threshold would make barred discs half the size of unbarred ones, with no overlap at all. Anything less than a factor of two measures how much else, besides spin, decides whether a disc bars: a contrast of a fifth would mean the spin barely matters.

galaxies · Galaxy spin

Named alongside it

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

Angular momentumDark matter haloDisc scale lengthSpin parameterAdiabatic contractionDisc instabilityDynamical frictionExponential discGalactic barGalaxy formationSpecific angular momentumSurface brightness

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