Astronomy & Celestial Mechanics
the sky, computed
Concept

Optical depth — where it appears

The number of scattering lengths along a line of sight, which decides what fraction of the light gets through unaltered. A depth of one transmits about a third of the light; a depth of ten transmits nothing measurable, which is why a saturated absorption stops responding to what produced it.

Named by 21 essays across 6 fields — each of them below, with the objects they name alongside it.

A wave whose crests count out 35 kilograms per square metre of ring. Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 5.6 km and has shortened to 1.13 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 kilograms per square metre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring.

A ring weighed by the wave crossing it

Saturn's rings are a few tens of metres thick and spread over an area larger than the Earth, made of pieces nobody can resolve, and nothing has ever landed on them. Their mass per unit area is nevertheless known to a few per cent — from the rate at which the crests of a wave crowd together as it travels outwards.

gravitation · Planetary rings Two dips a side, 36.5 seconds apart, symmetric about a body 256 km across. One observer's light curve across a small body with two narrow rings, at a chord 44 km from the centre. The body itself removes the star for 11.2 seconds; the four brief dips, two either side of it, are ring crossings, at 391 km and 405 km from the centre and 7 km and 3 km wide radially. The evidence that they are rings and not two more objects is the symmetry: each pair sits at equal times before and after mid-event, to within 0.07 s here, and two independent bodies have no reason to do that. Each dip is drawn wider than the ring is thick because the chord crosses obliquely — the path length through the material goes as 1/cos, which is also why a ring is deeper near the ansae. Nothing in this light curve was looked for: Chariklo's rings turned up in 2013 in a run recorded to measure a diameter, and every ring system found since has been found the same way.

A star that blinked before it should have

In March 1977 three teams watching a star pass behind Uranus recorded it dimming five times before the planet arrived — and then, symmetrically, five times again on the way out. The symmetry is the whole argument — nothing but a set of rings concentric with the planet produces a mirror image about closest approach.

sky · Occultations Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 10,343 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,400 K, where hydrogen is 78% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.

The surface that is a depth

A star has no surface. What looks like one is the level at which the optical depth reaches about two-thirds — and the edge is sharp only because the opacity climbs so steeply that the transition takes a ten-thousandth of the radius.

starlight · Opacity The curve of growth. The equivalent width of an absorption line against the number of absorbers along the sight line, both logarithmic, computed by integrating a Voigt profile with damping parameter 0.005. Three regimes: the width grows in proportion to the abundance while the line is weak, then almost not at all for two decades once the core saturates, then as the square root once the damping wings dominate. A line measured in the middle stretch carries almost no information about the abundance, and most strong lines in a stellar spectrum are there.

How much of a line is not in its depth

An absorption line stops getting deeper long before it stops getting stronger. What keeps growing is its area, and the way the area depends on the number of absorbers has three distinct regimes over five decades — one of which carries almost no information at all.

starlight · Line formation The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in.

The light that is missing from the edge

The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

starlight · Limb darkening Two means of one opacity, 75 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 6 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 75 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.76 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.13, and then turns up sharply, slope 0.99, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps.

A mean dominated by the gaps

The opacity in the equation of radiative transport is not an average of the opacity. It is a harmonic average weighted by the temperature derivative of the Planck function, which makes it a measurement of the transparent windows between the lines rather than of the lines — and that single fact decides what adding a million spectral lines to a table does.

starlight · Opacity Every shell contributes the same. Concentric shells of equal thickness around an observer, with stars scattered uniformly in volume. The number of stars in a shell grows as its radius squared and the flux from each falls as the radius squared, so the two cancel exactly and every shell delivers the same total light — the drawn counts are 6, 15, 31, 55, 88, in proportion to r³ − r₀³, and the drawn sizes fall as 1/r. That cancellation is the paradox, and it is why no amount of dust helps: dust absorbs the light and then re-radiates it, and in a universe old enough for the sum to converge it would come to the same temperature as the stars. The sum diverges linearly with radius, so something has to stop it — and the only two candidates are that the shells eventually overlap, or that there are no shells beyond a certain distance because there has not been time for their light to arrive.

Why the sky is dark

In an infinite universe of stars every sight line ends on a stellar surface, so the whole sky should be as bright as the Sun's disc. It is not, and the usual answer — that the expansion redshifts the light away — accounts for a factor of six out of a hundred trillion.

cosmology · Olbers's paradox One ionising photon per atom, and where that happens. The number of photons energetic enough to ionise hydrogen, per baryon, against temperature — with temperature falling to the right, so the universe ages to the right. The quantity is the fraction of a Planck distribution above 13.6 eV, integrated numerically, divided by the 6.13·10⁻¹⁰ baryons there are per photon. It crosses one at 5,844 K, which is z = 2143, and it falls by 19 orders of magnitude across the plot because it is the tail of an exponential. That crossing is the answer to why recombination waits until three thousand kelvin. At hydrogen's own ionisation temperature of 157,803 K there are two billion ionising photons per atom and the gas has no chance; the temperature has to fall by a factor of forty before the supply runs out, and it runs out suddenly because an exponential tail does. The number that sets the factor of forty is not an energy at all — it is the photon-to-baryon ratio, which is to say the entropy per baryon, which is to say a number fixed long before any of this and measured today from the second acoustic peak and from deuterium alike.

The surface the background actually is

Hydrogen ionises at 157,800 kelvin and the universe became transparent at 3,000. The factor of fifty between them is not an error, and it is not about energy — it is about there being two billion photons for every atom, so the far tail of the distribution can keep the gas ionised long after the typical photon has become useless.

cosmology · Microwave background A neutral fraction of 2.4·10⁻⁶ is already opaque. The Gunn–Peterson optical depth against the neutral fraction of the intergalactic medium, at z = 3, 5, 6.3, for Ω_b = 0.0493 and h = 0.674. Note the range of the vertical axis. A fully neutral medium at z = 6.3 gives τ = 4.1·10⁵, which is not absorption but extinction of everything; the medium reaches τ = 1 — the point at which it stops transmitting most of the light — at neutral fractions of 6.1·10⁻⁶ at z = 3, 3.3·10⁻⁶ at z = 5, 2.4·10⁻⁶ at z = 6.3. That is why the argument runs from the flux that survives rather than from the flux that does not. A spectrum showing any transmission at all between Lyman α and Lyman β is a measurement that the medium is ionised to better than one part in 164,727, and no fit to any absorption line is needed to establish it.

A trough that proves the forest survived

A uniform neutral medium at redshift six would absorb Lyman alpha with an optical depth of four hundred thousand. So the existence of any transmitted light in a quasar's spectrum is a measurement — of a neutral fraction below one part in ten thousand — made from a detection rather than from an absorption.

cosmology · Reionisation A wind that writes its own terminal speed on the blue edge, at 2017 km/s. A P Cygni profile, computed by integrating a spherical wind on a grid rather than drawn. The wind accelerates outward as one minus the inverse radius to the power 0.8, reaching 2000 kilometres a second, and its density falls as the inverse square of the radius over its own speed; the horizontal axis is the Doppler shift in units of that terminal speed, with blue to the left. The column directly in front of the stellar disc is moving toward the observer at every speed from nearly nothing near the surface up to the terminal value far out, so it takes light out of the beam across that whole range and the absorption trough runs from the rest wavelength to the blue edge at -1.01 of the terminal speed. That edge is the measurement: no model of the star, its distance or its mass enters it, only the geometry of a column seen end-on. The emission comes from everywhere else, where the wind moves across the line of sight and can only add photons; it is Infinity times stronger to the red than to the blue, because the part of the shell receding directly away is hidden behind the star and its blueshifted counterpart is not. Two features, one shell, and between them a velocity and a mass-loss rate.

A speed read off an edge

A hot star drives material off itself at thousands of kilometres a second, and the profile that wind prints on the star's own spectrum has a sharp blue edge. That edge is the terminal velocity, measured with no model of the star, no distance, and no calibration — one of the few numbers in stellar astrophysics obtained from geometry alone.

starlight · Stellar winds Bubbles that meet at z = 5.3, and a scattering depth of 0.047. The fraction of the volume of the universe filled by ionised bubbles, integrated from redshift 20 down to 4.5. The equation has two terms and no others: photons escaping from young galaxies open new volume, at a rate taken from the measured cosmic star formation history with an escape fraction of 0.2; recombinations inside the bubbles close it again, on a timescale that is one over the density times the recombination coefficient times a clumping factor of 3. Early on the density is high and recombination wins almost everything; the curve is nearly flat. As the universe expands the recombination time lengthens as the cube of one plus the redshift while the star formation rate is still rising, the balance tips, and the filling factor runs to one in under half a billion years. It reaches unity at redshift 5.31, which is overlap — the moment the bubbles meet and the last neutral walls between them disappear. The same integration gives an electron-scattering optical depth of 0.0465 for the microwave background, against the 0.054 that is measured, and that agreement is the check: the two observations constrain the same history from opposite ends, one fixing when it finished and the other how long it took.

It ends when the walls meet

Reionisation is usually described as the universe becoming transparent, which makes it sound like a change of state. It is not. Each young galaxy opens a bubble of ionised gas around itself and recombination closes it again, and what ends the epoch is geometric — the bubbles meet, and the last neutral walls between them disappear.

cosmology · Reionisation A spot is dark because it is squeezed. Pressure against depth below the quiet photosphere's optical surface, drawn as a ratio to the pressure there. The rising curve is the surrounding gas, which grows exponentially with a 140-kilometre scale height because that is what hydrostatic equilibrium in an ideal gas produces. The flat pair of bands is the spot's own budget: a magnetic pressure of 3.58·10⁵ dyn/cm² from a 3000-gauss field, which is 25 per cent of the total, plus the gas pressure left over. Horizontal balance requires the two columns to reach the same total at the same geometric level, and the level at which they do is 348 kilometres below the quiet surface — so the spot's own optical surface sits in a hollow. Measured Wilson depressions, obtained from the foreshortening of a spot near the limb, are four to six hundred kilometres. Nothing about the darkness was assumed: a field strength read off a Zeeman splitting fixes the magnetic share, the share fixes the depression, and the depression fixes the temperature the deeper layer must have to carry the reduced flux.

The darkness a field pays for

A sunspot is not cool because something is missing. It is cool because a three-thousand-gauss field supplies part of the pressure that holds the column up, the gas therefore supplies less, and the level at which that gas becomes opaque sits some hundreds of kilometres deeper than the surface around it.

stars · Plasma beta Below 0.46 microns a grain is not in orbit at all. The ratio of the radiation force to the gravitational force on a dust grain, against the grain's radius, for three densities. Every line has slope exactly −1 because gravity acts on the mass and radiation on the cross-section, and the ratio of a volume to an area is a length. Two horizontal lines matter and they are different statements. At β = 1 the star does not attract the grain at all. At β = 1/2 a grain released at rest from a circular orbit is already unbound, because it keeps the speed appropriate to the full stellar mass while feeling only half of it — and since dust is made by breaking up larger bodies that were on circular orbits, the lower line is the one that applies. For rock at 2500 kilograms a cubic metre that is 0.46 microns; for ice it is 1.15, and for iron 0.15. A collisional cascade that grinds material finer runs into this floor and stops, and the material that would have been finer leaves the system on a hyperbola.

The drag that sorts a disc by size

Starlight does three different things to a dust grain depending on how big it is — blows it out of the system, drags it inward over millennia, or ignores it entirely. Which one happens is decided by a single length, and whether it happens at all is decided by how crowded the disc is.

orbits · Collisional cascade A 5 per cent continuum error, and an optical depth wrong by 1.1. The mean transmitted flux of the Lyman-alpha forest against redshift, with a 5 per cent uncertainty in the quasar continuum drawn as a band. The continuum is not observed: at these redshifts every part of the spectrum blueward of the emission line is absorbed, so the level has to be extrapolated from the red side across a region where the quasar's own spectrum has structure. A fractional error in that level is a fractional error in the flux, and since the optical depth is minus the logarithm of the flux, the resulting error in the optical depth is the fractional error divided by the flux — which grows without bound as the forest goes black. At z = 2 it is 0.06; at z = 6.2 it is 1.1. That is why measurements of when reionisation ended are quoted as limits rather than values above about redshift six.

A forest with no continuum left

Measuring how much neutral hydrogen sits between here and a distant quasar means measuring the fraction of its light that survives, which means knowing how much light there was. At high redshift nothing survives at the wavelengths that would show it, so the level is extrapolated across the region being measured — and the optical depth is the logarithm of a number divided by a guess.

cosmology · Reionisation One curve the temperature fixes, and one line the polarisation adds. The plane of the optical depth to reionisation against the amplitude of the primordial fluctuations. The temperature power spectrum of the microwave background measures the product of the amplitude and the exponential of minus twice the optical depth, so it constrains a curve rather than a point: more electrons scattering the photons out is indistinguishable from fewer fluctuations to begin with, and the two trade along the drawn locus across a factor of 1.22 in amplitude. What breaks it is the polarisation at the largest angular scales, where rescattered photons regenerate a signal whose amplitude is proportional to the optical depth itself rather than to its exponential. That constraint is nearly vertical here, it comes from a handful of multipoles at the very largest scales, and it is the single hardest measurement the microwave background has demanded — because at those scales the Galaxy's own polarised emission is larger than the signal.

An amplitude and a depth that arrive multiplied

The microwave background's temperature fluctuations are the primordial ones damped by everything that scattered them since. The damping is uniform, so a smaller starting amplitude and more scattering produce identical maps — and separating them requires a signal from a handful of the largest angular scales, where the Galaxy's own emission is larger than what is being measured.

cosmology · Reionisation The colour of the zenith at twilight, with and without ozone. The colour of the zenith sky relative to sunlight — the 450 nm brightness against the 650 nm brightness, in magnitudes, bluer upward — against the Sun's depression, computed by single scattering with a 300 Dobson-unit ozone layer and again with none. Scattering alone favours blue by λ⁻⁴, which would make the sky 1.60 magnitudes bluer than sunlight if nothing were removed on the way; that is the dotted line. But after sunset every ray has travelled a long grazing path, and Rayleigh scattering removes blue from that path faster than red, so the two effects fight. Without ozone they very nearly cancel: the zenith is 0.01 magnitudes from sunlight's own colour at sunset and 0.06 at 6° — a pale, colourless sky — and only turns bluer, −0.13 at 10°, once the lit layer has climbed above most of the air the grazing ray used to cross. With ozone the zenith is −0.34 at sunset, −0.67 at 6° and −0.76 at 10°: bluer than without by 0.72 magnitudes at 6°, because the grazing ray also crosses the ozone layer near its tangent point, and ozone's Chappuis band absorbs orange and red rather than blue. The ozone cross-sections are approximate, and the conclusion does not depend on them to better than a factor of two.

Ozone keeps the twilight zenith blue

After sunset the sky overhead turns a deep blue, and scattering alone cannot explain it. The light that reaches the zenith has first grazed hundreds of kilometres of air, which strips blue out of the sunlight as fast as scattering puts it back, and the two very nearly cancel. What tips the balance is a gas that makes up a few parts in ten million of the atmosphere and absorbs the orange and red end of the spectrum.

sky · Twilight How far the Sun has to sink before the sky darkens, on four worlds. The dimming of the zenith sky relative to sunset, in magnitudes, against the Sun's depression, for single scattering in an isothermal atmosphere with the stated scale height and vertical optical depth, scattering isotropically: the Earth (R 6371 km, H 8.5 km, τ 0.097); Mars (dust) (R 3389.5 km, H 11.1 km, τ 0.5); Titan (haze) (R 2574.7 km, H 50 km, τ 4); Pluto (haze) (R 1188 km, H 50 km, τ 0.02). These are representative values for the layer that does the scattering — air on the Earth, dust on Mars, haze on Titan and Pluto — not full models of those atmospheres. The angle over which a sky darkens is set by the height of the scatterers against the size of the planet, √(2H/R), because that is the depression at which the shadow over the zenith has risen one scale height. The Earth dims by ten magnitudes at 8.6° of depression, which at the equator of a world whose solar day is 24 hours takes the Sun 35 minutes. Mars (dust) dims by ten magnitudes at 13.4° of depression, which at the equator of a world whose solar day is 24.66 hours takes the Sun 55 minutes. Titan (haze) dims by ten magnitudes at 29.8° of depression, which at the equator of a world whose solar day is 382.7 hours takes the Sun 31.7 hours. Pluto (haze) dims by ten magnitudes at 43.9° of depression, which at the equator of a world whose solar day is 153.3 hours takes the Sun 18.7 hours.

The air's height against the planet sets the twilight

On the Earth the sky overhead fades over about nine degrees of the Sun's descent. That angle is not a property of air or of sunlight. It is the square root of twice the height of the scattering layer divided by the radius of the planet, and on a small world with a tall haze it grows to tens of degrees — so that twilight on Titan lasts more than a day.

sky · Twilight The sky's darkness, measured as an opacity. Optical depth to electron–positron pair production against gamma-ray energy, for sources at redshifts 0.03, 0.1, 0.3, 1, both axes logarithmic. A gamma ray of energy E is absorbed most readily by background photons near twice the square of the electron rest energy divided by E: at 1 TeV that is 2.37 µm and at 100 GeV it is 0.24 µm, so the energy axis is a wavelength axis for the background light, running backwards — and the background it is evaluated against is the same two-component spectrum the star formation history produced, not a flat number. Above the marked τ = 1 the universe is opaque. The depth is computed in the delta-function approximation, the cross-section replaced by 0.2 of the Thomson value over a bandwidth of order the energy, with the background's comoving density evolving as (1+z)^1.2. That is a factor-of-two calculation and the shape is what it gets right: the horizon closes from z = 0.59 at 100 GeV to z = 0.16 at 1 TeV. The measurement runs the other way. A blazar's spectrum is observed, the absorbed part is the difference between it and the spectrum the source is believed to have emitted, and that difference gives the background — in the near infrared, where no direct measurement can subtract the zodiacal light well enough to compete.

A background weighed by what it stops

The faintest light in the universe cannot be photographed from inside the Solar System, because the zodiacal foreground is a hundred times brighter. It can be weighed instead, by the bite it takes out of a blazar at a trillion electronvolts.

cosmology · Olbers's paradox Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one.

Two skies where the paradox comes out right

Olbers argued that every sight line should end on a source and the sky should blaze. In neutrinos and in gravitational waves it does — the walls are at one second and at no time at all — and both of those skies have now been detected.

cosmology · Olbers's paradox 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

Named alongside it

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

Thomson scatteringMean free pathThe Gunn–Peterson troughOpacityRecombinationReionisationSaturationCompton y parameterCosmological dimmingEarth shadowExtragalactic background lightIntergalactic medium

All concepts