Astronomy & Celestial Mechanics
the sky, computed
Concept

Intracluster medium — where it appears

The ten-million-kelvin plasma filling a galaxy cluster, which outweighs all the stars in all its galaxies by a factor of six or seven. It is invisible optically and dominant in X-rays, and its density and temperature profiles are what a cluster's mass is measured from.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

A cluster weighed three ways, and its stars weighed once. A cluster of galaxies with a velocity dispersion of 1000 km/s, gas at 8 keV and a strong-lensing Einstein radius of 25 arcseconds, each turned into a mass inside 1.5 Mpc by its own relation and nothing else: 7.0, 8.9 and 15.2 × 10¹⁴ M☉. The three assume, respectively, that the galaxies are in equilibrium, that the gas is, and nothing whatever — so their agreement to within a factor of 2.2 is not three restatements of one assumption. The lensing bar is the loosest of the three and is drawn that way deliberately: it measures the mass inside a cylinder of radius 109 kpc, 1.11×10¹⁴ M☉, and carrying that out to 1.5 Mpc as though it were a sphere overstates it. The stars are 2.9 per cent of it.

A cluster weighed three ways

The speeds of a cluster's galaxies, the temperature of its gas and the bending of light behind it are three measurements with almost nothing in common. They agree within a factor of two, and all three exceed the mass of the stars by about a hundred.

galaxies · Clusters 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 One cluster moved to z = 1.8: one signal unchanged, the other 57 times fainter. The same cluster of galaxies placed at a series of redshifts, with two ways of detecting it compared. The Sunyaev–Zel'dovich decrement is flat, because it is a fraction of the microwave background and the microwave background has the same surface brightness at every redshift a cluster can sit at — moving the cluster further away shrinks it on the sky but does not make its shadow shallower. The X-ray surface brightness of the identical object falls as (1+z)⁻⁴, the dimming every surface brightness suffers in an expanding universe, and by z = 1.8 it is 57 times below where it started. This is why the cluster surveys that reach the early universe are millimetre surveys: an X-ray telescope's cluster catalogue thins out with distance and a Sunyaev–Zel'dovich catalogue is limited only by how large the cluster looks and by how faint a fractional distortion the instrument can measure. What the effect cannot supply on its own is a distance — the signal that does not know how far away the cluster is also cannot say — so every one of these clusters still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it.

A shadow that does not get fainter with distance

Every other way of finding a cluster of galaxies gets harder the further away the cluster is. One does not. A cluster's hot gas scatters about one microwave background photon in a hundred to a higher frequency, and because the result is a fraction of a background rather than a flux from a source, the same cluster is exactly as detectable at ten billion light years as at one.

cosmology · Sunyaev zeldovich Gas kept inside 3.6 to 6.8 kiloparsecs, depending only on the speed of the fall. The pressure holding a disc galaxy's gas down, against radius, with the ram pressure of the cluster gas it is falling through drawn as three levels. The restoring pressure is two pi times the gravitational constant times the product of the stellar and gaseous surface densities, both exponential with scale lengths of 3 and 5 kiloparsecs, so it falls steeply outward; the ram pressure is the intracluster density times the square of the infall speed and does not depend on radius at all. Where the level crosses the curve, the gas goes. A galaxy entering at 700 kilometres a second keeps everything inside 6.8 kiloparsecs; at 1600 it keeps only 3.6. Nothing in this touches the stars, which are not a fluid and feel no pressure at all, so the galaxy emerges with its stellar disc and its rotation curve intact and its star formation stopped from the outside in. That is the mechanism behind the most conspicuous fact about clusters: their spirals are red, gas-poor and still spiral-shaped, which no process acting on the stars could produce.

Red, gas-poor, and still spiral-shaped

A galaxy falling into a cluster meets a headwind of hot gas at two thousand kilometres a second. Where that ram pressure exceeds the disc's own grip on its gas, the gas goes; the stars, which are not a fluid and feel no pressure, do not. The result is a spiral with no fuel, and it is the commonest kind of galaxy in a cluster.

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 that leans on no other distance measurement.

cosmology · Sunyaev zeldovich A cluster moving at +500 km/s, and the frequency where only the motion is left. The two distortions one cluster imprints on the microwave background, against observing frequency, scaled to the largest excursion drawn. The thermal effect is from the random motion of electrons at 8 keV, with a central Compton parameter of 10⁻⁴; the kinematic effect is from the bulk motion of the same gas at 500 km/s along the line of sight, positive meaning receding, through an optical depth of 0.00639, which is the Compton parameter divided by kT/mₑc². A bulk velocity shifts every scattered photon by a common Doppler factor, and a blackbody shifted by a common factor is a blackbody at another temperature — so the kinematic distortion has exactly the shape of a temperature change, ΔT/T = −τv/c, which here is −29.0 µK at every frequency. In intensity that shape is the derivative of the Planck spectrum, and its largest value falls at 217.5 GHz, the same frequency at which the thermal distortion crosses zero, 217.5 GHz: both conditions reduce to x coth(x/2) = 4. At 150 GHz the thermal decrement is −260 µK, so the motion is 11.2 per cent of it there and all of the signal at the null. What the kinematic spectrum cannot be told apart from is the primary microwave background itself, which is also a temperature change with this shape — so the frequency that isolates the velocity from the gas is no help at all against the sky behind it.

A velocity that has the colour of the sky

A cluster moving through the microwave background shifts the light it scatters by a common Doppler factor, which leaves a spectrum shaped exactly like a change of temperature. That shape is loudest precisely where the hot gas falls silent — and it is the one shape the background itself already has.

cosmology · Sunyaev zeldovich At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral.

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

cosmology · Sunyaev zeldovich Two lobes, one halo, and two different paths out. A double radio source drawn in the plane containing the line of sight, the observer far to the right. The host galaxy sits at the centre of a hot halo — the rings mark 20, 40, 60, 100 kpc, in a β-model of core radius 40 kpc — and its two lobes lie 60 kpc out along an axis 45° from the line of sight. Light from the near lobe leaves the halo through its outskirts; light from the far lobe has to cross the halo's middle. With a tangled field of 3 µG at the centre in cells of 5 kpc, the rotation measure in front of the near lobe varies across the beam with a dispersion of 9 rad m⁻² and in front of the far lobe 28 rad m⁻². On the sky the two lobes are the same distance from the core, 42 kpc, and nothing in a total-intensity image says which is which.

The fainter polarisation belongs to the lobe further away

The two lobes of a radio galaxy lie at the same distance from the core on the sky, and nothing in a picture of them says which is nearer. Their polarisation does. Light from the far lobe crosses the magnetised hot halo of the host on its way out and light from the near lobe mostly does not, so at long wavelengths the far lobe loses its polarisation first — and the one-sided jet, it turned out, points at the lobe that keeps it.

starlight · Faraday rotation A lobe, the shell it pushed out, and two sight lines through it. A radio lobe of radius 20 kpc drawn in the plane containing the line of sight, the observer to the right, 60 kpc from the host in the same halo as before. Growing at Mach 2, it swept the gas that filled its volume into a shell compressed 2.29 times and 2.9 kpc thick, and the field in that shell was amplified along the boundary. A sight line through the lobe's centre crosses the shell head-on, where the amplified field lies across the line of sight and contributes nothing: the shell there turns the light by about 6 rad m⁻². A sight line at 85 per cent of the radius crosses it obliquely, along 4.8 kpc of shell with the amplified field partly along the path, and the shell turns it by about 18 rad m⁻². The rotation the halo model attributes to the gas in front of the lobe is partly written by the lobe itself, and it is written most strongly around its edge.

A rotation measure the lobe writes on its own boundary

The magnetic fields of hot group and cluster gas are mostly measured through the Faraday rotation of the radio galaxies that live in it, on the assumption that a lobe is a passive lamp behind a screen. But a lobe made its own room. The gas that filled it is now a shell around it, compressed if the lobe grew faster than sound, and the shell rotates the light too — so the inferred field comes out too low for a lobe that grew gently and up to three times too high for one that drove a shock. Only sources behind the halo, which carry no shell, can say which.

starlight · Faraday rotation The middle of a cluster cools faster than the universe ages. The time the gas of a cool-core cluster, of central electron density 0.05 per cubic centimetre, would take to radiate its heat at constant pressure, against radius, both logarithmic. Inside 76 kiloparsecs it is less than 7.7 billion years, the time since a redshift of about one when the cluster's core is thought to have settled, and at 10 kiloparsecs it is 0.8 billion years. Beyond a few hundred kiloparsecs it is longer than the age of the universe, and the gas is simply hot.

A cooling flow that never arrives

The gas at the centre of a rich cluster radiates its heat in under a billion years, so it should be sinking inward and turning into stars at hundreds of solar masses a year. For two decades that was the accepted picture and it was treated as a measurement. Then X-ray spectra looked for the gas on its way down and found that it stops at a third of the temperature it began with, while the galaxy in the middle forms stars a hundred times too slowly. Something pays the radiative bill, and bubbles of the right size are visible in the X-ray images.

galaxies · Clusters

Named alongside it

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

The Sunyaev–Zel'dovich effectBremsstrahlungCompton y parameterThomson scatteringAngular-diameter distanceBaryon fractionBeta modelCluster mass functionCooling flowCosmic microwave backgroundDepolarisationFaraday rotation

All concepts