Astronomy & Celestial Mechanics
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The iron clock has no single delay

The α-element knee is drawn as though Type Ia supernovae switched on a billion years after the stars that made them. Measured rates say otherwise — the delays are spread evenly over every decade from forty million years to a Hubble time, as a power law with no timescale in it — and a clock with no timescale bends where a clock with one would break.

Assumes Chemical evolution, Supernovae and Binary stars.

16 min read 6 figures Small perturbations, large consequencesThe sky as a clock

The ratio of α elements to iron in a star’s atmosphere is one of the most useful clocks in astronomy, and it is drawn almost everywhere with a single delay built into it. The argument that introduced the knee went like this: core-collapse supernovae come from massive stars that live a few million years and make oxygen, magnesium and silicon with a little iron; Type Ia supernovae come from white dwarfs, arrive about a billion years later, and make iron with almost no α elements. So a system’s [α/Fe] runs flat at the massive-star value, then bends down at whatever metallicity the system had reached when the first Type Ia exploded.

“About a billion years later” was a simplification, and it was drawn as a step: no Type Ia supernovae before a billion years, all of them after. The actual distribution of delays has been measured, in two independent ways, and it is not a step. It is a power law, with as many explosions in each decade of delay as in any other, from forty million years to the age of the universe.

A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism.
Fig. 1 The fraction of all the Type Ia supernovae from a single burst of star formation that have exploded by a given delay, on a logarithmic time axis, for four distributions: a t−1t^{-1} power law from 40 Myr, a steeper t−1.4t^{-1.4} law, a single delay of 1 Gyr and a Gaussian of 3 ± 1 Gyr. The power law rises as the logarithm of the delay, reaching half by 740 Myr — the geometric mean of its limits — and 55 per cent by 1 Gyr, with the last still exploding after a Hubble time. The steeper law reaches half by 180 Myr.

Two ways to measure a delay

A Type Ia supernova carries no label saying when its progenitor formed. Its delay has to be inferred statistically, by comparing where and when supernovae happen with where and when stars formed.

By host age. Take a large sample of galaxies whose spectra have been decomposed into stellar populations of different ages — how much mass formed less than 400 million years ago, how much between 400 million and 2.4 billion, how much before. Count the supernovae each galaxy has hosted over a survey. The rate in each galaxy is the sum over its age bins of the mass in that bin times the rate per unit mass at that delay, and with thousands of galaxies the rates per unit mass can be solved for. The answer has three features: a large rate at short delays, a rate still well above zero in galaxies with no star formation for billions of years — elliptical galaxies host Type Ia supernovae — and a fall between them close to t−1t^{-1}.

By cosmic history. The volumetric rate of Type Ia supernovae at each redshift is the cosmic star-formation history convolved with the delay-time distribution. The star-formation history is measured independently from the light of galaxies, so the supernova rate at several redshifts constrains the convolution kernel. That kernel also comes out close to t−1t^{-1}.

The two methods share almost nothing — one uses nearby galaxies and population synthesis, the other deep-field supernova searches and a cosmic average — and they agree. The integral of the distribution is measured too: about one Type Ia supernova for every thousand solar masses of stars formed, over a Hubble time.

Why a power law, and why this one

A power law has no characteristic time, which is itself a clue: the mechanism that sets the delay must spread over many orders of magnitude without a preferred scale.

Two classes of progenitor have been proposed. In the single-degenerate picture, a white dwarf accretes from a normal companion star until it approaches the Chandrasekhar mass and ignites; the delay is roughly the companion’s lifetime plus the time it takes to transfer enough mass, and it tends to cluster at particular ages set by which companions can transfer mass stably. In the double-degenerate picture, two white dwarfs in a close binary spiral together by emitting gravitational waves and merge; the delay is the time to form both white dwarfs plus the inspiral time.

The inspiral time for a circular binary goes as the fourth power of the separation:

tGW∝a4m1m2(m1+m2)t_{\rm GW} \propto \frac{a^4}{m_1 m_2 (m_1 + m_2)}

Now suppose white-dwarf binaries emerge from their common-envelope phase with separations spread evenly in logarithm — dN/dln⁡adN/d\ln a constant, which is the distribution of separations observed for binaries of many kinds and has been called Öpik’s law since the 1920s. A distribution flat in ln⁡a\ln a is flat in ln⁡t\ln t, because ln⁡t=4ln⁡a+constant\ln t = 4 \ln a + \text{constant}. And a distribution flat in ln⁡t\ln t is exactly dN/dt∝t−1dN/dt \propto t^{-1}.

So the measured delay-time distribution is what a population of merging white dwarfs with Öpik separations produces, with no parameters adjusted. That does not prove double-degenerate progenitors — other channels could produce similar shapes, and the minimum delay of about forty million years is set by the lifetime of the most massive stars that make white dwarfs rather than by any inspiral — but it is the most direct connection in the subject between a measured cosmological distribution and a law of gravitational radiation. The same a4a^4 dependence shrinks the orbits of binary pulsars at exactly the rate general relativity predicts.

A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism.
Fig. 2 The t−1t^{-1} law against a steeper t−1.4t^{-1.4} law, both starting at 40 Myr. Half of the power law’s explosions have happened by 740 Myr; half of the steeper law’s by 180 Myr. A steeper law is what separations concentrated at small values would give, or an inspiral whose rate depended more strongly on separation. The measured exponent is about −1.1 with an uncertainty of about 0.1, close enough to −1 that the steeper curve is disfavoured and far enough from it that the question is not closed.

A knee that is a bend

Replace the step in the α-element model with the measured distribution and the knee changes character.

One star-formation history, and a knee at −0.87 or a bend from −1.69 depending on when the white dwarfs explode. [α/Fe] against [Fe/H] for one system forming stars on a 3-Gyr exponential timescale, drawn for different delay-time distributions of its Type Ia supernovae: t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. Everything else — the yields, the plateau at +0.40, the history — is identical. The position where each track first falls 0.05 dex below the plateau is −1.69 for the t⁻¹ from 40 Myr, −1.89 for the t⁻¹·⁴ from 40 Myr, −0.87 for the single delay of 1 Gyr, −0.61 for the Gaussian, 3 ± 1 Gyr, and where it has fallen 0.2 dex is −0.77, −1.21, −0.60, −0.33. A single delay gives the sharp knee that the α-element diagram is usually drawn with, and measured supernova rates rule it out. The power law begins to lower the ratio almost at once, because some of its explosions come within a hundred million years, and takes 0.9 dex of metallicity to fall from 0.05 to 0.2 below the plateau rather than bending at a point. Knees seen in the stars of the Galactic disc look sharper than a one-zone power law makes them, which is either a sign that the delays are less uniform in logarithm than a t⁻¹ law, or — more likely — that the stars at a given metallicity near the Sun are a mixture of populations born at different radii with different histories, and the sharpness is a property of the mixture.
Fig. 3 [α/Fe] against [Fe/H] for one system forming stars on a 3-Gyr exponential timescale, with four delay-time distributions and everything else identical. The single 1-Gyr delay gives the sharp knee: the ratio first falls 0.05 dex below its plateau at [Fe/H] = −0.87. The t−1t^{-1} power law begins to fall at −1.69, nearly a dex earlier, and takes 0.9 dex of metallicity to fall from 0.05 to 0.2 below the plateau. The steeper law begins at −1.89; the 3 ± 1 Gyr Gaussian not until −0.61.

The power law’s prompt explosions — a tenth of them within about seventy million years — begin adding iron while the system is still extremely metal-poor, so the ratio starts to decline almost at once. But they add iron slowly, a constant number per decade of time, so the decline is gradual. By the time a step-function knee would appear, the power-law track has already fallen partway, and it continues to fall for a dex of metallicity further.

That makes the position of the knee a less simple measurement than it was drawn to be. With a single delay the knee is at the metallicity the system reached at that delay, so it measures the star-formation rate directly. With a power law there is no single point, and “the knee” becomes a matter of definition — the first departure from the plateau, or the steepest part of the decline, or the metallicity at which the ratio has fallen halfway — and each definition responds differently to the star-formation history.

One star-formation history, and a knee at −0.40 or a bend from −0.96 depending on when the white dwarfs explode. [α/Fe] against [Fe/H] for one system forming stars on a 0.5-Gyr exponential timescale, drawn for different delay-time distributions of its Type Ia supernovae: t⁻¹ from 40 Myr; single delay of 1 Gyr. Everything else — the yields, the plateau at +0.40, the history — is identical. The position where each track first falls 0.05 dex below the plateau is −0.96 for the t⁻¹ from 40 Myr, −0.40 for the single delay of 1 Gyr, and where it has fallen 0.2 dex is −0.31, −0.24. A single delay gives the sharp knee that the α-element diagram is usually drawn with, and measured supernova rates rule it out. The power law begins to lower the ratio almost at once, because some of its explosions come within a hundred million years, and takes 0.6 dex of metallicity to fall from 0.05 to 0.2 below the plateau rather than bending at a point. Knees seen in the stars of the Galactic disc look sharper than a one-zone power law makes them, which is either a sign that the delays are less uniform in logarithm than a t⁻¹ law, or — more likely — that the stars at a given metallicity near the Sun are a mixture of populations born at different radii with different histories, and the sharpness is a property of the mixture.
Fig. 4 A system forming its stars fast, on a 0.5-Gyr timescale, with the power law and the single delay. The single delay’s knee moves to [Fe/H] = −0.40; the power law’s first departure moves to −0.96, and its decline reaches 0.2 dex below the plateau at −0.31. Fast formation moves both to higher metallicity, as the step model said it would — the ordering of systems by star-formation rate survives the change of distribution, while the metallicity at which a given system bends does not.

The ordering survives, and that is the most important thing the figures say. The Galactic bulge, which formed fast, bends at high metallicity; the Milky Way’s thick disc at intermediate metallicity; dwarf spheroidal galaxies, which formed slowly and inefficiently, at [Fe/H] below −1, some below −1.5. Every one of those orderings holds with a power-law distribution, because a faster-forming system reaches any given metallicity sooner and so has had less time for delayed iron whatever the shape of the delay. What changes is the translation from a knee’s metallicity to a star-formation timescale, which becomes a model calculation with the distribution inside it.

The rate across cosmic time

The second way of measuring the distribution is itself a figure worth drawing, because it shows the convolution explicitly.

The Type Ia rate is the star-formation history convolved with a delay, and the delay moves its peak from z = 1.9 to 1.4 or 1.6 or 1.4 or 0.8. The volumetric rate of Type Ia supernovae against redshift, relative to its present value, for the cosmic star-formation history convolved with t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr, beside the star-formation rate itself on the same relative scale. The star-formation history is the Madau–Dickinson fit, peaking near z = 1.9. Every delay pushes the supernova peak later — to lower redshift — and flattens it: the power law peaks at z = 1.4 and is 2.8 times today's rate at z = 1, against 5.6 for star formation. The latest peak belongs to the Gaussian, 3 ± 1 Gyr, at z = 0.8, with a rate at z = 1 of 3.7 times today's. This is the second, independent way the delay-time distribution is measured: from the shape of the rate with redshift rather than from the ages of individual hosts, and the two agree on a power law close to t⁻¹. The rates beyond z ≈ 1.5 come from few supernovae found in deep fields, and at that distance distinguishing the power law from a broad Gaussian depends on exactly the part of the curve that is least well measured.
Fig. 5 The volumetric Type Ia rate against redshift, relative to today, for the cosmic star-formation history convolved with each of the four distributions, beside the star-formation rate itself. Star formation peaks near z = 1.9. The power law moves the supernova peak to z = 1.4 and flattens it: at z = 1 the rate is 2.8 times today’s, against 5.6 for star formation. The steeper law peaks at z = 1.6, the 1-Gyr delay at 1.4, and the 3 ± 1 Gyr Gaussian latest, at z = 0.8.

A convolution with a delay does two things to a history: it shifts the peak to later times, and it smooths the curve by the width of the kernel. The power law does less shifting than a Gaussian at three billion years because so many of its explosions are prompt, and more smoothing than a step because its tail extends across the whole of cosmic time. Today’s Type Ia rate is fed partly by stars formed in the last few hundred million years and partly by stars formed at the peak of cosmic star formation ten billion years ago, and the power law says those contributions are comparable.

The measured Type Ia rate rises by about a factor of two to three from the present to redshift one and then flattens, which is the power-law curve’s shape. Beyond redshift 1.5 the rates come from a few dozen supernovae found in the deepest fields of space telescopes, and the uncertainties there are large enough that a broad Gaussian is not firmly excluded by the cosmic method alone. It is the combination with the host-age method, which constrains short delays well, that pins down the power law.

The Type Ia rate is the star-formation history convolved with a delay, and the delay moves its peak from z = 1.9 to 1.4 or 0.8. The volumetric rate of Type Ia supernovae against redshift, relative to its present value, for the cosmic star-formation history convolved with t⁻¹ from 40 Myr; Gaussian, 3 ± 1 Gyr, beside the star-formation rate itself on the same relative scale. The star-formation history is the Madau–Dickinson fit, peaking near z = 1.9. Every delay pushes the supernova peak later — to lower redshift — and flattens it: the power law peaks at z = 1.4 and is 2.8 times today's rate at z = 1, against 5.8 for star formation. The latest peak belongs to the Gaussian, 3 ± 1 Gyr, at z = 0.8, with a rate at z = 1 of 3.7 times today's. This is the second, independent way the delay-time distribution is measured: from the shape of the rate with redshift rather than from the ages of individual hosts, and the two agree on a power law close to t⁻¹. The rates beyond z ≈ 1.5 come from few supernovae found in deep fields, and at that distance distinguishing the power law from a broad Gaussian depends on exactly the part of the curve that is least well measured.
Fig. 6 The power law and the 3 ± 1 Gyr Gaussian alone, to z = 2, where the measurements are. The power law is 2.8 times today’s rate at z = 1 and the Gaussian 3.7 times; star formation is 5.8 times. The two supernova curves diverge most above z ≈ 1, where the Gaussian has already peaked and is falling while the power law is still rising — exactly the range in which a handful of distant supernovae carry the whole comparison.

Iron as a supernova’s signature

A Type Ia supernova — the explosion told apart from a collapsing core by the lines its spectrum lacks — makes about six-tenths to seven-tenths of a solar mass of iron-peak elements, mostly as radioactive nickel-56 that decays through cobalt to iron; a core-collapse supernova makes about a tenth of that. The light curve of a Type Ia is powered by exactly that decay, and its peak brightness measures the nickel mass, which is why these explosions can be standardised at all. The iron that eventually ends up in stars’ atmospheres is, for most elements of the iron peak, mainly the ash of these explosions.

That is what makes the delay-time distribution a chemical quantity. The rate per unit mass sets how fast iron accumulates; the delay sets when. Over a Hubble time about one Type Ia per thousand solar masses of stars formed, each making about 0.7 solar masses of iron, gives roughly 0.7 × 10⁻³ of all the mass that formed stars returned as iron — comparable to what the ten or so core-collapse supernovae per thousand solar masses of the same stars supply. Delayed iron therefore roughly doubles what the prompt explosions made, which is why the α-to-iron ratio of the Sun is lower than that of the oldest stars by a factor of two to three, and why that factor is set as much by the normalisation of the distribution as by its shape.

Iron where no stars are forming

The long tail of the distribution has consequences that are easy to overlook because they happen in galaxies that have stopped doing anything else.

An elliptical galaxy — a member of the red sequence that has almost nothing between it and the blue cloud — that finished forming stars eight billion years ago still hosts Type Ia supernovae, at a rate per unit stellar mass a few times lower than a star-forming spiral’s. On the power law that is exactly as expected: the stars formed at the start are now at delays of eight to ten billion years, and a t−1t^{-1} distribution still delivers about a twentieth of its explosions per factor of 1.3 in delay there. Those supernovae explode into hot, thin gas rather than into a star-forming disc, and their iron has nowhere to go but the galaxy’s halo — or, in a cluster, the intracluster gas that holds most of the cluster’s ordinary matter.

The iron in that gas is measured from X-ray lines, at about a third of the solar abundance, spread over hundreds of kiloparsecs, and it is a problem. Adding up the iron that the cluster’s stars could have made — core-collapse supernovae at their birth plus Type Ia supernovae with the field galaxies’ delay-time distribution and normalisation — falls short of the iron observed, by a factor of two or more. Either clusters’ stars produced more Type Ia supernovae per unit mass than stars elsewhere, which the supernova rates measured in clusters do suggest, or the initial mass function was different, or more stars were stripped into intracluster light than are counted. The integral of the distribution, in other words, may not be universal, and the most iron-rich places in the universe are where it is least likely to be.

A standard candle with an age in it

The delay-time distribution also reaches into cosmology, by a route that runs through the brightness that has to be standardised before a Type Ia supernova is a distance.

Standardisation corrects each explosion’s peak brightness by its light-curve width and colour. After that correction a residual remains that correlates with the host galaxy: supernovae in massive or old hosts are slightly brighter, by a few hundredths of a magnitude, than those in young ones. If that residual is a property of the progenitor’s age — if white dwarfs that explode at long delays produce slightly different explosions from those at short delays — then the delay-time distribution matters directly for distances.

The reason is that the mix of delays changes with redshift. A supernova at redshift one exploded when the universe was six billion years old, so it cannot have had a delay of ten billion years; the power law’s long tail is truncated at high redshift and fully present today. A population of supernovae whose standardised brightness depends on delay therefore has a mean brightness that drifts with redshift, and a drift of a few hundredths of a magnitude between redshift zero and one is the size of the signal that distinguishes a cosmological constant from a slowly varying dark energy. Whether the host correlation is an age effect, a dust effect or a metallicity effect is currently argued, and the answer determines whether the distribution drawn at the top of this page is a nuisance parameter in the equation of state of the universe.

Where the delay matters for a disc

In a disc whose gradient is built ring by ring, each ring has its own history, and each ring’s iron arrives with the same distribution of delays after its own star formation. Inner rings that formed stars fast bend at high metallicity and outer rings that formed slowly bend low, so a disc does not have one α-element knee: it has a family, one per radius.

Stars near the Sun, which are a mixture of stars born across the disc, therefore do not trace any single track. The two sequences seen in the solar neighbourhood’s [α/Fe] against [Fe/H] — a high-α sequence and a low-α sequence, separated by a gap — have been explained by two distinct episodes of gas accretion, by a thick disc formed fast and a thin disc formed slowly, and by the superposition of many rings mixed together by migration. The delay-time distribution enters all three explanations identically; what separates them is the history of star formation and the movement of stars, which is why the gap is still argued about.

What the figures cannot show

The chemistry is a single-zone calculation with instantaneous mixing: every supernova’s iron is assumed to be spread through the whole system immediately. In a real galaxy iron from a single explosion is mixed into a few hundred thousand solar masses of gas over tens of millions of years, and in the earliest, most metal-poor stars the abundance pattern of an individual supernova can show through. The figures also treat the yields as fixed, and Type Ia yields depend on the explosion mechanism — whether the white dwarf is at the Chandrasekhar mass or well below it — which changes manganese and nickel much more than iron, and is one of the few chemical handles on which progenitor channel dominates.

And the distribution is taken to be universal. It may not be: the fraction of stars in close binaries, the efficiency of common-envelope ejection and the white-dwarf mass distribution all depend on metallicity, and a distribution measured in present-day galaxies of near-solar metallicity is being applied to systems a hundred times poorer in metals.

A distribution with no timescale draws no feature of its own

A single delay is a timescale, and a timescale produces a feature — a knee, a break, an edge — at the moment it is reached. A distribution with no timescale produces no feature of its own; any sharp feature in the output must come from somewhere else. When a measured knee is sharp and the delay distribution is a power law, the sharpness is information about the star-formation history or about mixing, not about the supernovae.

The same reading applies wherever a response is a convolution. A disc’s gradient history seen in old stars is the birth gradient convolved with migration, and the flatness of the result is a property of the kernel. A cosmic supernova rate is a star-formation history convolved with a delay. In both cases the kernel can be measured independently, and in both the temptation is to read a feature of the output as a feature of the input.

Still open: the first stars’ yields

The earliest stars formed from gas with no metals at all, and their supernovae — possibly including pair-instability explosions of stars hundreds of times the Sun’s mass, which leave no remnant and make a distinctive abundance pattern — enriched the gas that the oldest surviving stars formed from. No such first-generation star has been found, and perhaps none survives. What is left is the abundance pattern of the most metal-poor stars known, a few with iron abundances below a millionth of the Sun’s, whose ratios of carbon to iron and of odd to even elements are the only evidence there is about a population nobody has observed.

Where this goes next

A one-parameter model, and 9 times too many metal-poor stars. The metallicity distribution of long-lived stars near the Sun — the bars — against two models of how a galaxy enriches itself. The closed box is as simple as a model of a galaxy can be: gas turns into stars, stars make metals and return them, nothing enters and nothing leaves. It has exactly one parameter, the yield, and it predicts the whole curve. It gets the peak roughly right and the tail catastrophically wrong: 13.7 per cent of its stars fall below [Fe/H] = -1 against an observed 1.6 per cent, a factor of 9. This is the G-dwarf problem, and the reason it is an argument rather than a discrepancy is that the failure cannot be fixed by changing the yield: the yield sets where the peak is, and moving the peak to fix the tail moves it away from the data. What is wrong is a boundary condition. Let gas keep arriving — pristine, at roughly the rate it is being consumed — and the gas is never both abundant and metal-poor for long, so few stars form while it is. That is the second curve, with 3.2 per cent below -1, and it needed no new nucleosynthesis and no new parameter beyond the fact of accretion. The picture cannot show the thing that would settle it directly, which is the infall itself: the gas arriving on the disc now is a few solar masses a year spread over twenty kiloparsecs, and it has never been securely detected. A histogram that says the box was not closed The simplest model of a galaxy enriching itself has exactly one parameter and predicts the whole metallicity distribution of its surviving stars. The solar neighbourhood's disagrees, in a specific direction — and the failure is not in the nucleosynthesis but in a boundary condition. Four kinds of explosion, told apart by which line is missing. Spectra of 4 supernova types — Ia, II, Ib, Ic — near maximum light, stacked, over the same wavelength range and drawn with the same photospheric expansion speed of 10,000 kilometres per second. Every feature is a P Cygni profile: a trough blueshifted by the material expanding towards the observer and a peak at rest wavelength from the material moving across the line of sight, and the width of both is the expansion speed. Reading the silicon trough of the type Ia off the drawing puts it at 614.0 nanometres, which recovers 10,131 km/s from a rest wavelength of 635.5. The classification is a decision tree on absences. Hydrogen present makes it a type II; no hydrogen but silicon makes it a Ia; no hydrogen and no silicon but helium makes it a Ib; none of the three makes it a Ic. That tree was written down before anybody knew what these objects were, and it separates a detonating white dwarf from a collapsing massive core almost perfectly — because what the letters are actually reading is how much of the star's outer envelope was still there when it exploded. The type II kept its hydrogen, the Ib had lost it and shows the helium beneath, the Ic had lost that too. Two explosions told apart by a missing line The classification of supernovae is a decision tree on absences — is hydrogen there, is silicon there, is helium there — and it was drawn up decades before anybody knew what any of these events were. It nevertheless separates a detonating white dwarf from a collapsing stellar core almost perfectly, and the reason it does is worth the essay. A disc built from the inside out, with a gradient of −0.066 dex per kiloparsec at 12 Gyr. The metallicity of the gas, in solar units on a logarithmic scale, against galactocentric radius, at ages of 2, 6, 12 Gyr, for a disc in which every ring is its own box with infall: pristine gas arrives on a timescale that grows with radius, from 1 Gyr in the centre to 7 Gyr at 8 kpc, turns into stars on the depletion time of a Kennicutt law, which is shorter where the gas is denser and much longer below a threshold of 7 M☉ pc⁻², and keeps everything it makes. The slopes fitted between 4 and 14 kpc: −0.259 dex/kpc at 2 Gyr, −0.124 dex/kpc at 6 Gyr, −0.066 dex/kpc at 12 Gyr. The inner disc has had its gas early and turned it over many times, so it is near the yield; the outer disc is still accreting and forming stars slowly, so its gas is diluted and young in the chemical sense. At 8 kpc the model's present abundance is 1.16 of the yield. Every ring is independent here: no gas flows between them and no star moves, which are the two processes that real discs add and which both act to flatten what is drawn. A gradient the old stars have walked away from The gas in a disc galaxy is richer in metals near the centre than at the edge, by about six-hundredths of a dex per kiloparsec in the Milky Way. Two ingredients of disc growth make that slope, a disc that grows from the inside out makes it flatten with time — and the old stars that should carry the steeper history have moved several kiloparsecs from where they were born.

The objects this essay names

Each one links to every other essay that touches it.

Alpha enhancementAlpha kneeChemical evolutionConvolutionCosmic star formation historyDelay time distributionDouble degenerateGravitational wavesSingle degenerateType ia supernovaeWhite dwarf