Astronomy & Celestial Mechanics
the sky, computed
Concept

Metallicity — where it appears

The abundance of elements heavier than helium in a star or a gas, usually quoted relative to the Sun's. It is quoted logarithmically as the iron-to-hydrogen ratio relative to the Sun, so a value of −2 means a hundredth of the solar abundance.

Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.

An absorption spectrum at 5772 K. A blackbody continuum at 5772 K with absorption lines cut out of it, each line's depth computed from how much of the gas is in a state that can absorb it. 8 of the 8 lines are strong enough to see at this temperature, which is the whole reason the spectral sequence is a temperature sequence.

Composition, read from what is missing

The dark lines in a stellar spectrum are wavelengths that never arrived. Which ones are absent names the elements present — and the strength of a line says more about temperature than about abundance.

starlight · Spectra The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.

Every distance is measured with the last one

No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

starlight · Distance ladder The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred.

A brightness is a distance only if something is known

The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system.

starlight · Magnitudes A colour term of -0.059 magnitudes per magnitude, and one star that will not obey it. Synthetic photometry of blackbodies from 3,000 to 42,000 K through two V bands: the standard one, and a natural system whose effective wavelength is 9 nm longer and whose width is 1.12 times as large. The vertical axis is the difference between the two magnitudes for the same star — not a constant, because a wider redder band collects a different fraction of a hot spectrum than of a cool one. Fitting a straight line against B − V gives a colour term of -0.0585 magnitudes per magnitude and leaves a residual of 3.1 millimagnitudes, which is why a linear transformation is the standard reduction and why it works. The mark off the line is a cool star with molecular absorption bands in the red, drawn from the same blackbody with three synthetic bites taken out of it: it sits 27 millimagnitudes from the fit, 9 times the blackbody scatter. A colour term knows one number about a star and a spectrum has a shape, and that gap is the reason all-sky photometry stops at a per cent while differential photometry on one field reaches a millimagnitude.

The same star through two telescopes

A magnitude is defined by a response curve, and no two telescopes have the same one. The difference between two observatories' measurements of one star is not a constant to be subtracted but a function of the star's colour — and for a star whose spectrum has structure, not even that.

starlight · Photometric systems Energy generation against core temperature. The proton–proton chain and the CNO cycle, in solar units, against core temperature on logarithmic axes. The CNO curve is far steeper, so the two cross at 18.8 million kelvin — above that temperature a star runs mostly on CNO, and below it mostly on pp.

The furnace that runs cooler than a compost heap

The Sun's core produces 276 watts per cubic metre. A human body produces more. The Sun is bright because it is enormous, and its fusion is astonishingly slow — which is exactly why it lasts.

stars · Fusion The Hertzsprung–Russell diagram. Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.

The main sequence is a place stars sit, not a track they travel

The commonest misreading of the Hertzsprung–Russell diagram is that stars slide down the band as they age. They do not. They sit at one point for ninety percent of their lives and then leave sideways.

stars · The HR diagram Three clusters, three ages, one diagram. Isochrones for populations of 100 Myr, 1 Gyr, 12 Gyr, each drawn as the main sequence up to its own turnoff and then the post-main-sequence track of the turnoff mass. The turnoffs are at 5.52 M☉, 2.20 M☉, 0.94 M☉, from t = 10¹⁰ M/L with this file's own mass–luminosity relation. Nothing here is a track along which a star moves. Every point is a different star of a different mass, all the same age, and the bend is simply where the population runs out of stars that have had time to leave. That is why a cluster has an age and a field star does not: the bend needs a population, and one star is not one. The oldest globular clusters sit near the 12 Gyr line, and in the 1990s the same construction gave them ages of 16 to 18 Gyr against a universe measured at 10 — a two-standard-deviation contradiction that was resolved from the distance side, by Hipparcos, and not from this one.

An age read off a bend

A cluster is a population of one age and many masses, so the place where its stars leave the main sequence is a clock. The bend needs a population — which is why a cluster has an age and a single star does not.

stars · The HR diagram Two populations, and the valley between them. 420 model galaxies, 42 per cent of them drawn from a tight red sequence tilted by -0.08 magnitudes of colour per magnitude of brightness, the rest from a broad blue cloud. Only 3.3 per cent land in the band of width 0.20 magnitudes midway between the two sequences, against 17 per cent in the same band laid on the blue cloud and 27 per cent on the red sequence. The bimodality is the strongest statement the galaxy census makes: a galaxy is usually forming stars or has stopped, rarely in between, and since colour tracks the age of the newest stars, the emptiness of the middle is a statement about a timescale — whatever ends star formation does it in a few hundred million years, not a few billion.

Two colours, and almost nothing between

Plot a few hundred thousand galaxies by colour and brightness and they do not fill the plane. They pile into a tight red sequence and a broad blue cloud with a near-empty gap between, and an empty gap is a statement about how fast something happens.

galaxies · Galaxy populations A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

starlight · Opacity A continuum drawn 4.3 per cent below the real one. A short stretch of spectrum with one strong line in it and 150 weak ones scattered across the same interval. The upper dashed line is the true continuum — the flux the star would emit with no lines at all — and it is not observable. The lower one is what a fit through the highest points of the spectrum returns, which is 4.3 per cent lower, because the weak lines have depressed the gaps between the strong ones. Measuring the strong line's equivalent width against the apparent continuum instead of the real one makes it 11.2 per cent too small. The error has a sign, it is worse in spectra with more lines, and it therefore correlates with metallicity — which is exactly the quantity being measured.

A continuum that was never observed

An equivalent width is an area measured relative to the continuum, and the continuum is not in the data. It is drawn — a curve through the highest points of the spectrum — and in any spectrum with many weak lines those highest points are already below the true continuum, because the weak lines have eaten the gaps.

starlight · Line formation A factor of 2.9, and the density that would remove it. The predicted ⁷Li abundance against the baryon density, with the halo-star measurement as a horizontal band and the microwave background's density as a vertical one. At η₁₀ = 6.13 the network gives 4.70e-10 and the stars show 1.60e-10, a factor of 2.94 — 0.47 in the logarithm, which is the number every proposed resolution has to produce. The obvious cure is drawn as the second marker: ⁷Li rises as η², so the density that would reproduce the observation is η₁₀ = 3.58, a third below the measured one. That is where the cure fails, and it fails on a different element. Deuterium falls as η^(−1.6), so at 3.58 the predicted D/H is 5.946e-5 against the 2.527e-5 measured in quasar absorbers — 2.35 times too much, and 114 times deuterium's own error bar. Helium, meanwhile, moves to 0.2445 and stays inside its measurement, because its curve is flat. The three light elements do not fail together, and that is what rules out a single wrong parameter: whatever is wrong is wrong about mass seven specifically.

A factor of three, and the flatness that prices every cure

The oldest stars in the Galaxy show a third of the lithium the first three minutes should have left. Three kinds of resolution have been offered, and the datum that rules on all three is not how much lithium is missing but how uniformly it is missing.

cosmology · Nucleosynthesis Where a photosphere's free electrons come from. The share of the free electrons donated by metals rather than by hydrogen, against temperature, at a gas pressure of 12000 N/m² and at solar abundance, 1 dex below solar, 2 dex below solar. The electron pressure is not assumed here — it is solved for, as the value at which the ionisation of the gas supplies exactly the electrons the gas contains. Below about 5897 K at solar abundance every free electron in the gas comes from an element present at one part in ten thousand, because hydrogen's 13.6 eV keeps it neutral while magnesium's 7.6 does not. The handover is fast: solar crosses a half at 5897 K, 1 dex down crosses a half at 5088 K, 2 dex down crosses a half at 4467 K. At 5,772 K and this pressure the solved electron pressure is 1.78 N/m².

The continuum is made by one atom in ten thousand

The Sun's light leaves through an ion that exists only because a hydrogen atom will hold a second electron by three-quarters of an electronvolt. The electrons it holds come almost entirely from magnesium, silicon and iron — so the level against which every solar line depth is measured is rationed by elements present at one part in ten thousand.

starlight · Ionisation A carbon-to-oxygen ratio that steps at every front. The carbon-to-oxygen ratio of the gas and of the solids against distance from a 1 solar-mass star, each in units of the star's own ratio. Crossing a condensation front moves one element or both out of the gas and into the solids, so the two curves step in opposite directions at water at 2.7 AU, carbon dioxide at 16.0 AU, carbon monoxide at 196.0 AU. Water takes oxygen and no carbon, so beyond it the gas is carbon-rich — 3.20 times the stellar ratio — and the solids are oxygen-rich. Carbon monoxide takes both, in a ratio of one to one, so beyond that front the gas ratio rises again. A giant planet's atmosphere is made mostly of gas it accreted, so measuring its ratio and inverting this staircase gives a formation radius — and the inversion is not unique, because more than one interval returns the same value once the solids a planet also swallowed are allowed for.

Two elements in a ratio, and a birthplace read off it

Carbon and oxygen freeze out at different places, so the gas between the fronts is carbon-rich and the solids are oxygen-rich. A giant planet is made mostly of gas it accreted, so measuring the ratio in its atmosphere and inverting the staircase should give the radius it formed at — and the inversion turns out not to be unique.

exoplanets · The snow line Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the V light reddens the index by 0.093 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5331 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.005 magnitudes against 0.405 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5744 K, which is 28 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.

Three shifts larger than the error bar, and two that cancel

An unresolved companion, a reddening and a metallicity each move a colour index by more than any modern photometer's precision. Two of them move it in opposite directions, so the three together can return the right temperature by cancellation rather than by accuracy.

starlight · Stellar colour A push that depends on how fast the wind is already going. The force multiplier — the radiative push on a line-driven wind as a multiple of the push on its free electrons — against t, the electron-scattering optical depth across one thermal Doppler width, which is small where the wind accelerates steeply. The solid curve is an ensemble of spectral lines whose strengths follow a power law, the number of lines stronger than a given strength falling as that strength to the power 1 − α, with α = 0.6. If every line were optically thin the push would be the flat line at 2000, independent of the flow; if every line were thick it would fall as 1/t, set only by the velocity gradient. The ensemble lies between: over most of the range it falls as t to the power −0.59, because as t rises lines cross from thin to thick one after another. That single exponent is the whole of the theory — it fixes the terminal speed, the mass-loss rate and how both scale with the star — and it is a property of the atomic physics of the ions in the wind, not of the star.

A rate the wind has no choice about

A hot star's wind is driven by starlight caught in hundreds of thousands of spectral lines, and the way those lines are distributed in strength fixes everything else. The mass-loss rate turns out not to be a free parameter at all but the one value for which a smooth flow can leave the surface — and multiplying it by the wind's speed and the square root of the star's radius cancels the star's mass, leaving a relation with the luminosity alone.

starlight · Stellar winds How planet occurrence rises with the host star's iron, by kind of planet. Planet occurrence relative to its value at solar metallicity, on a logarithmic axis, against the host star's iron abundance [Fe/H] (zero is the Sun; one unit is a factor of ten). Each class is a power law, occurrence ∝ 10^(β[Fe/H]). Giant planets within a few astronomical units follow β = 2, the relation found in the first large radial-velocity survey to measure it: 12 per cent of stars at [Fe/H] = +0.3 have one, against 3 per cent at solar and under 0.3 per cent at −0.5 — a factor of 158 across the range drawn. The small planets found by transits rise far more gently; the slopes drawn, β = 0.5 for sub-Neptunes and 0.2 for super-Earths, are representative of published fits, which range from nearly flat to about a doubling across this interval depending on orbital period.

A threshold that makes iron matter only to giants

A star with twice the Sun's iron is four times as likely to have a giant planet, and barely more likely to have a small one. The steep law and the flat one look like two mechanisms. A core racing its disc, run over a spread of discs, produces both — steep where winning is rare, flat where it is common.

exoplanets · Occurrence rates Hayashi lines: a luminosity that moves and a temperature that does not. The effective temperature a fully convective star must have, against its luminosity, for 0.5, 1, 2 solar masses, drawn as a Hertzsprung–Russell diagram with temperature falling to the right. The temperature follows from matching the star's convective adiabat to a photosphere whose opacity, from the negative hydrogen ion, rises as the ninth power of temperature. Over a factor of thirty thousand in luminosity each line moves by only 363 K, 398 K, 436 K respectively: the lines are nearly vertical, at 3650 K, 4000 K, 4383 K for one solar luminosity. A contracting young star, or a swelling giant, moves up or down its line almost without changing colour. To the right of the lines no star in hydrostatic equilibrium can sit.

A wall built by a steep opacity

A star that is convective all the way to its surface cannot choose its surface temperature. The interior fixes a relation between pressure and temperature, the photosphere fixes another through its opacity, and the negative hydrogen ion's opacity rises so steeply with temperature that the two meet at nearly the same temperature whatever the star's size. Young stars contracting and old stars swelling slide up and down a line in the diagram almost without changing colour — and nothing in equilibrium can sit to its right.

starlight · Opacity Brighter by mass, fainter by temperature. Main sequences from 0.5 to 1.3 solar masses in luminosity against effective temperature, hotter to the left, for solar composition, [Fe/H] = −1.5, from homology applied to the solar-composition relation. The dots join stars of the same mass: lowering the metals moves each one up and to the left, so a 0.8 solar-mass star goes from 0.41 to 0.66 solar luminosities and from 5049 to 5879 K. The main sequence is steep, so a move of that size to the left carries a star past solar stars much brighter than itself: at a fixed temperature of 5800 K the metal-poor sequences lie 0.59 magnitudes below the solar one. A metal-poor star is brighter than a solar one of the same mass and fainter than a solar one of the same temperature — which is why such stars were called subdwarfs.

Brighter for its mass and fainter for its colour

Two stars of the same mass are almost the same star only if they are made of the same material. Take most of the metals out and the opacity falls, the radiation leaks out faster and the star settles brighter and much hotter — so far hotter that, at a given colour, it sits below the ordinary main sequence. The relation between mass and light has composition inside it, and every mass, age and distance read off it inherits that.

stars · The mass–luminosity relation Atomic gas that stops growing: a cloud holds at most 10, 35, 104 M☉ pc⁻² of HI. The surface density of atomic hydrogen in a slab lit on both faces, against the slab's total gas surface density, both logarithmic, for metallicities of solar, 0.3 solar, 0.1 solar. A thin slab is atomic throughout and sits on the one-to-one line (grey). A thick one shields its interior and turns it molecular, so every gram added beyond the transition is a gram of H₂ and the atomic column stops at 10.1 M☉ pc⁻² for solar, 34.7 M☉ pc⁻² for 0.3 solar, 103.7 M☉ pc⁻² for 0.1 solar. The saturation is the same thing the 21-centimetre maps of nearby galaxies show: atomic gas surface densities that rarely exceed about ten solar masses per square parsec in metal-rich discs, however much gas there is, and that climb in metal-poor dwarfs.

An atomic ceiling set by the dust

Pile more gas into a galaxy's disc and the atomic hydrogen stops growing at about ten solar masses per square parsec; everything beyond that is molecular. The ceiling is the column a cloud needs to shield its interior from ultraviolet light, and because dust both makes the molecules and does the final shielding, it scales as the inverse of the metallicity — and hardly at all with the starlight, since the cold gas thickens in step with the field that would break it apart.

galaxies · Interstellar medium Isochrones that crowd on the main sequence and fan out above it. Isochrones for stars of solar metallicity at 1, 2, 4, 6, 8, 10, 12 billion years, luminosity against temperature with temperature falling to the right. On the main sequence they lie almost on top of one another — a star brightens and barely changes colour as it ages — and only where they turn off and cross towards the giants do they separate. Two boxes show the measurement errors of a star with a good parallax and spectrum (twice 70 K and 3 per cent): one around a Sun-like star of 4.6 billion years, whose age they allow anywhere in 4.53 (3.38–5.63) billion years; one around a subgiant of the same mass at 10.5, allowed only 10.5 (10.3–10.7).

A lone star is dated only as it leaves

A star in a cluster is dated by the bend its neighbours trace. A star alone has one position on the Hertzsprung–Russell diagram, and an age for it means finding which isochrone passes through that point. On the main sequence they almost all do: a Sun-like star measured as well as a parallax and a spectrum allow is anywhere from three and a half to five and a half billion years old. A star the same mass on its way off the main sequence is dated to two per cent.

stars · The HR diagram The carbon a star makes, and the carbon it needs. For a 30 solar-mass star born with no carbon, two quantities against its central temperature, on logarithmic scales: the carbon fraction its core needs for the CNO cycle to carry its luminosity (falling steeply, because a hotter core needs less catalyst), and the carbon the triple-alpha process makes in the hundred thousand years the core takes to contract (rising even more steeply). They meet at 108 million kelvin and a carbon fraction of 1.7 × 10⁻¹⁰: 1.7 parts in ten billion by mass, made by the star itself, are enough to switch on the cycle that runs every massive star of the present day. The contraction stops there.

A star that has to make its own catalyst

Every massive star now forming burns hydrogen through a cycle of reactions that carbon, nitrogen and oxygen catalyse, and the first stars had none of those elements. A massive star made of hydrogen and helium alone contracts and heats until its core reaches a hundred million kelvin, where helium begins to fuse into carbon; a few atoms of carbon in ten billion are enough to start the cycle, and the star settles there — three times hotter at its centre than any star of the same mass today, and smaller, and bluer.

stars · The mass–luminosity relation Molecules take longer to form than a cloud takes to fall. Three timescales for a shielded cloud of 100 hydrogen atoms per cubic centimetre, against its metallicity relative to the Sun's on a logarithmic scale, all in years on a logarithmic scale. Molecular hydrogen forms on dust grains, and the time to turn the cloud's atoms into molecules is inversely proportional to the dust, so to the metallicity: 5.3 million years at solar, 53 million years at a tenth, 530 million years at a hundredth. The cloud's free-fall time, 4.3 million years, does not depend on its metals. The two cross near solar metallicity: above it the molecules form as fast as the cloud can collapse, below it the cloud collapses before most of its hydrogen has become molecular. The dashed line is a galaxy's molecular depletion time, two billion years — the timescale that equilibrium arguments implicitly assume is short compared with everything else.

A molecule that forms too slowly to wait for

The atomic hydrogen in a galaxy's disc turns molecular wherever a cloud is thick enough to shield itself from starlight, and stars form where it does. That is an equilibrium statement, and equilibrium takes time. Molecular hydrogen forms on dust grains, so in gas with a tenth of the Sun's metals it forms ten times more slowly — slowly enough that a cloud can finish collapsing before most of its hydrogen has become molecular, and form its stars from atoms.

galaxies · Interstellar medium A peak that follows the snow line inwards. Giant planets per star per unit of ln a against orbital distance, for stars of 0.3, 0.6, 1, 1.5 solar masses, if the peak's position scales as the 1 power of the star's mass — snow line, young star's luminosity. The shape is the broken power law measured round Sun-like stars, rising as a⁰·⁸ and falling as a⁻¹, with its height scaled in proportion to the star's mass as the giants' overall rate is. The peaks sit at 0.9 AU, 1.8 AU, 3.0 AU, 4.5 AU: a 0.3-solar-mass star would have its giants piled up 3.3 times closer in than the Sun's. The dots mark each peak.

A peak that should move with the star

Round Sun-like stars, giant planets are commonest at about three astronomical units, close to where water first froze in the disc. If the snow line put them there, the peak should sit closer in round smaller stars — three times closer round a star of a third of the Sun's mass. Every hypothesis agrees about the Sun, which is why the Sun-like stars cannot test them; and one of them makes a Jupiter at the peak wobble its star by the same angle whatever the star's mass.

exoplanets · Occurrence rates How far a wrong metallicity term moves the Hubble constant. The fractional change in the Hubble constant against the error in the adopted metallicity term, in magnitudes per dex, for supernova hosts whose Cepheids are 0.1, 0.2, 0.4 dex richer in metals than the geometric calibrators'. A term too weak (too close to zero) by 0.2 mag/dex with hosts 0.2 dex richer moves H₀ by 1.9%. The shaded band is the ±8% disagreement between the local measurement and the microwave background's prediction: reaching it with a 0.2-dex difference needs the term wrong by 0.9 mag/dex.

A metallicity term too small to close a gap

Cepheids of the same period are slightly brighter when they are richer in metals, by about a fifth of a magnitude per factor of ten in metal content. The galaxies whose supernovae calibrate the Hubble constant hold Cepheids a little richer than the galaxies whose distances are known geometrically, so the term enters the answer. How much it can move it is simple to compute: a few per cent at most for any plausible error — and the eight per cent separating the local Hubble constant from the early universe's would need the term to be wrong by four times its own size.

stars · Variable stars

Named alongside it

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

Main-sequenceCore accretionDistance modulusIsochroneMain-sequence turnoffOpacitySystematic errorAstronomical unit (AU)Bound-free absorptionCNO cycleColour indexConvection

All concepts