Astronomy & Celestial Mechanics
the sky, computed
Concept

Beta model — where it appears

A two-parameter fitting form for the gas density in a cluster of galaxies, falling as (1 + (r/r_c)²) to the power −3β/2. It is empirical rather than derived, and it is used because its line-of-sight integral has a closed form.

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

Gas outweighs stars 6:1, and the baryons come to 15%. Above: enclosed mass against radius for a 7-keV cluster with a beta-model gas profile, β = 0.65 and a 250-kpc core. The total is from hydrostatic equilibrium — the same equation that holds up a star, with the mass following from the density and temperature gradients and from nothing else — and comes to 9.98·10¹⁴ solar masses inside 2 megaparsecs. The gas, integrated from the same profile, is 1.6·10¹⁴; the stars in all the galaxies are 2.5·10¹³, 6 times less. Most of a cluster's ordinary matter is not in anything anybody would call an object. The two total-mass curves differ by the 15 per cent hydrostatic bias: some of the pressure holding the gas up is turbulence left from the last merger rather than heat, and a mass computed from the thermal pressure alone is low by about that — the assumption that makes the measurement possible is the one that biases it. With the correction, the baryons come to 15 per cent of the total, against the 15.7 per cent the microwave background gives for the universe as a whole. A cluster is large enough to have kept everything it started with, so its own accounting is a cosmological measurement. Below: the same gas seen two ways. X-ray surface brightness falls as (1+z)⁻⁴, so a cluster at redshift one is 16 times fainter per unit sky than the same cluster nearby; the Sunyaev–Zel'dovich distortion of the microwave background does not fall at all, because it is a fraction of a background whose own brightness rises by exactly the same factor. That is why a millimetre survey finds clusters at any distance and an X-ray survey finds the near ones.

The baryons that are not in the galaxies

The stars in a cluster are about a seventh of its ordinary matter. The rest is ten-million-kelvin gas that is invisible optically and dominant in X-rays — and weighing it with hydrostatic equilibrium biases the answer low by exactly the amount the assumption is wrong by.

galaxies · Clusters A path length recovered from two integrals of one cluster. The two line-of-sight integrands through an isothermal β = 0.67 cluster, each normalised to its own centre, against distance along the line of sight in core radii. The upper curve is electron density, which the Compton parameter integrates; the lower is density squared, which the X-ray surface brightness integrates. They are integrals of the same gas along the same line and they weight it differently — the half-width is 1.00 core radii for the linear one and 0.64 for the quadratic, and one core radius either side of the centre holds 51 per cent of the pressure signal against 82 per cent of the X-ray. That difference is the whole method. Two integrals with different powers of one unknown density, down one unknown path, are two equations in two unknowns: y₀ = 1.5·10⁻⁴ and a central X-ray surface brightness of 1.63·10⁻⁵ erg cm⁻² s⁻¹ sr⁻¹ give back a central density of 0.006 cm⁻³ and a physical core radius of 0.250 Mpc. Divide that length by the angular core radius the same cluster subtends, 54.2 arcseconds, and the answer is an angular-diameter distance of 952 Mpc — against the 952 Mpc the cluster was built at, which is the round trip this figure exists to close. Nothing in that chain is calibrated on a Cepheid, a supernova or a parallax. It is a length in centimetres measured against an angle.

A length in centimetres, measured against an angle

A cluster's hot gas offers two line integrals of the same electrons — one linear in density, one quadratic. Two equations in two unknowns give back the path length in centimetres, and a length divided by the angle it subtends is a distance with no rung of any ladder beneath it.

cosmology · Sunyaev zeldovich

Named alongside it

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

BremsstrahlungIntracluster mediumThe Sunyaev–Zel'dovich effectAngular-diameter distanceBaryon fractionCluster mass functionCompton parameterCooling flowElectron densityGas clumpingGas fractionHubble constant

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