Astronomy & Celestial Mechanics
the sky, computed
Galaxies

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.

Assumes Star formation, Initial mass function and Stellar evolution.

17 min read 9 figures The sky as a clockScales that defeat intuition

A galaxy makes its own metals. It starts with hydrogen and helium and a trace of lithium, all of it made in the first few minutes of the universe; everything heavier was made in a star, and every star that made it was inside the galaxy.

That closes the accounting, and a closed accounting can be modelled with almost nothing. Suppose a galaxy is a box of gas that turns into stars, that each generation of stars returns a fixed mass of metals per unit mass locked up, and that nothing enters or leaves. Then the metallicity of the gas at any moment is fixed by how much of the gas has been used:

Z  =  yln⁡1μ,Z \;=\; y \ln\frac{1}{\mu},

with μ\mu the fraction of the original gas still gaseous and yy the yield — the mass of new metals produced per unit mass of gas permanently locked into stars. One parameter, and it predicts everything.

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.
Fig. 1 What it predicts, against what is there. The bars are the metallicity distribution of long-lived stars near the Sun — a volume-limited count of dwarfs that live longer than the disc has existed, so none has been removed by dying. The closed box gets the peak roughly right and the tail catastrophically wrong: nine times too many stars below [Fe/H] = −1. That is the G-dwarf problem, and the reason it is an argument rather than a discrepancy is that the yield sets the peak, so moving it to fix the tail moves it away from the data.

The model, and what is buried in it

Three assumptions get the closed box to a single equation and each is worth naming, because two of them are fine and one is the culprit.

Instantaneous recycling. Massive stars are assumed to live no time at all: a generation forms, the massive members explode immediately, and their metals are returned to the gas before the next generation forms. That is a good approximation for the α elements, which come from stars living a few million years against a galaxy’s ten billion, and a bad one for iron.

A constant yield. The mass of metals returned per unit mass locked up is set by the initial mass function and by nucleosynthesis, and both are assumed not to change. This is defensible: the yield is dominated by stars above about ten solar masses, and there is no evidence that their relative numbers vary much.

And a closed box. Nothing enters, nothing leaves.

Why the tail is wrong, and what fixes it

The closed box’s metallicity distribution follows from the equation above by differentiation: the mass of stars formed while the metallicity was below ZZ is proportional to 1−e−Z/y1 - e^{-Z/y}, so the density in ZZ is e−Z/ye^{-Z/y} and in [Fe/H][\mathrm{Fe/H}] carries an extra factor from the change of variable.

The shape has a long low-metallicity tail because, at the start, there is a great deal of gas and it is metal-poor, so a great many stars form out of metal-poor gas. That is the prediction, and it is what fails.

The failure points at the third assumption rather than the first two. If gas keeps arriving — pristine, at roughly the rate the galaxy is consuming it — then the gas mass stays roughly constant while metals accumulate, so the metallicity climbs quickly through the low values and the galaxy spends very little of its history forming stars out of metal-poor material.

The mathematics is a one-line change. With the gas mass held constant by accretion, dZ/ds=(y−Z)/MgdZ/ds = (y - Z)/M_g, so the density of stars in ZZ goes as 1/(y−Z)1/(y-Z) — piling up near the yield rather than spreading below it. No new physics, no new parameter, and the distribution comes out right.

The second clock, built from a ratio

The abundance of one element measures how far enrichment has gone. The ratio of two elements made on different timescales measures how fast.

The α elements — oxygen, magnesium, silicon, calcium — are made in core-collapse supernovae, from stars above about eight solar masses, which live a few million years. Iron is made in those too, but the majority of it comes from Type Ia supernovae, which are white dwarfs and take of order a billion years to arrive after their progenitors formed.

So a system’s [α/Fe][\alpha/\mathrm{Fe}] starts at the value core-collapse supernovae produce, about +0.4+0.4, and stays there for as long as the Type Ia’s have not yet contributed. When they do, iron arrives without α and the ratio falls.

Three histories, three knees: −0.06, −0.55, −1.10. The ratio of α elements to iron against iron abundance, for three systems that differ in one thing only — how fast they turned their gas into stars. Every track starts flat at +0.40 and then bends downward, and both halves are consequences of stellar lifetimes rather than of chemistry. Core-collapse supernovae come from stars that live a few million years, and they make oxygen, magnesium and silicon along with some iron in a fixed proportion; Type Ia supernovae come from white dwarfs and take about 1 billion years to arrive, and they make iron and almost no α. So for the first billion years the ratio is whatever massive stars produce, flat, and independent of everything. After that iron arrives without α and the ratio falls. The position of the bend is therefore not a measure of time at all — it is a measure of how much iron the system had already made when the clock struck, which is a measure of its star-formation rate. The fast system, forming stars on a 0.5-billion-year timescale, reaches [Fe/H] = -0.06 before its first Ia; the slow one at 12 billion years has only reached -1.10. That is why the Galactic bulge, the thin disc and the Sagittarius dwarf have knees a dex apart while being made of the same elements by the same stars — and why two galaxies of identical mass and colour can be told apart by the ratio of two lines in a spectrum.
Fig. 2 The knee, for three star-formation histories differing in one thing only: how fast the gas is being turned into stars. Every track starts flat and bends downward, and the position of the bend is not a measure of time — it is a measure of how much iron the system had already made when the Type Ia clock struck, which is a measure of its star-formation rate. The fast system reaches nearly solar iron before its first Ia; the slow one has only reached a tenth of it. Same elements, same stars, same nucleosynthesis; a dex of difference in where the bend falls.

The consequence is a diagnostic that works on entire galaxies. The Galactic bulge knees near solar iron, so it formed fast. The thin disc knees around −0.5-0.5. The Sagittarius dwarf knees near −1.2-1.2 and is α-poor at metallicities where the disc is α-rich, so it made its stars slowly — and stars in the halo with that chemical signature are recognisable as accreted from a dwarf, torn out by the same tidal field that makes a tail rather than formed in place.

Two galaxies of identical stellar mass and colour are told apart by the ratio of two elements.

A one-parameter model, and 6 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: 9.9 per cent of its stars fall below [Fe/H] = -1 against an observed 1.6 per cent, a factor of 6. 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 2.3 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.
Fig. 3 The same comparison at a higher yield. A closed box predicts a definite distribution of stellar metallicities with the yield as its only parameter, and raising it shifts the whole distribution up without changing its shape — so no yield reproduces both the position of the peak and the scarcity of metal-poor stars at once. Below about 0.9 the generator refuses to draw the comparison at all, because there the infall model stops being an improvement on the box it is supposed to fix.

What was actually measured

Every number above comes from spectra, and the chain from a spectrum to an abundance has more model in it than any other measurement in this essay.

An abundance is derived by computing a synthetic spectrum from a model atmosphere with an assumed temperature, gravity, microturbulence and composition, and adjusting the composition until the computed line strengths match the measured ones. The uncertainties are dominated by the model atmosphere rather than by the photon noise: a hundred-kelvin error in the assumed temperature moves an iron abundance by several hundredths of a dex, and the temperature is itself inferred. The G-dwarf sample itself is the other half. It has to be volume-limited, complete, and free of the selection that would arise from choosing stars by their metallicity or their kinematics. Early samples were none of those; the deficiency of metal-poor stars survived every improvement, and the modern versions — from surveys of hundreds of thousands of stars with spectroscopic abundances — put the fraction below [Fe/H]=−1[\mathrm{Fe/H}] = -1 at about two per cent locally.

Three histories, three knees: −0.02, −0.41, −0.93. The ratio of α elements to iron against iron abundance, for three systems that differ in one thing only — how fast they turned their gas into stars. Every track starts flat at +0.40 and then bends downward, and both halves are consequences of stellar lifetimes rather than of chemistry. Core-collapse supernovae come from stars that live a few million years, and they make oxygen, magnesium and silicon along with some iron in a fixed proportion; Type Ia supernovae come from white dwarfs and take about 1.5 billion years to arrive, and they make iron and almost no α. So for the first billion years the ratio is whatever massive stars produce, flat, and independent of everything. After that iron arrives without α and the ratio falls. The position of the bend is therefore not a measure of time at all — it is a measure of how much iron the system had already made when the clock struck, which is a measure of its star-formation rate. The fast system, forming stars on a 0.5-billion-year timescale, reaches [Fe/H] = -0.02 before its first Ia; the slow one at 12 billion years has only reached -0.93. That is why the Galactic bulge, the thin disc and the Sagittarius dwarf have knees a dex apart while being made of the same elements by the same stars — and why two galaxies of identical mass and colour can be told apart by the ratio of two lines in a spectrum.
Fig. 4 And the second clock with the supernova delay stretched from one billion years to one and a half. Alpha elements come from massive stars within a few million years and iron largely from supernovae much later, so the ratio falls at a metallicity that depends on both the delay and how fast the system was forming stars — the knee is a clock reading the product. Lengthening the delay moves every knee to higher metallicity, which is why the delay-time distribution has to be measured separately before a knee is an age.

The third measurement: what leaves

The closed box fails in the other direction too, and the evidence is a straight line.

Rearranged, the closed-box relation says that every system should lie on one line of ZZ against ln⁡(1/μ)\ln(1/\mu), with slope equal to the yield and with no dependence on mass, size, age or history. Measuring gas fractions and gas-phase metallicities for a range of systems tests that directly.

One line for a closed box, and an effective yield 2.8 times lower in the small systems. Gas-phase metallicity against the logarithm of the inverse gas fraction, for six kinds of star-forming system. A closed box has an exact prediction here and it is the simplest one in this collection: Z = y ln(1/μ), a straight line through the origin whose slope is the yield, with no dependence on the mass, the size, the age or the history of the system. Everything down to about a hundred kilometres a second of rotation sits on it, and the slope those systems share — 0.58 in solar units — is the only fitted number in the panel. Everything below that sits under it, and further under it the smaller the system: the effective yield — the slope each point implies on its own — falls from 0.56 for the large systems to 0.20 for the dwarfs, a factor of 2.8. There is only one way to be below this line. The yield is a property of stars and does not know what galaxy it is in, so a system that has made metals and does not have them has lost them, and lost them selectively: what leaves is the hot, freshly enriched gas from the supernovae themselves, driven out of a potential well too shallow to hold it. The trend with rotation speed is the argument, because the depth of that well is the one thing that changes along it. What the picture cannot show is where the metals went — the diffuse hot halo that should hold them is the hardest thing in this subject to observe, and the accounting has never been closed.
Fig. 5 The test, and the result. Everything down to about a hundred kilometres a second of rotation lies on the line. Everything below it lies under it, and further under it the smaller the system: the effective yield each point implies falls by a factor of nearly three across the sequence. There is only one way to be below this line. The yield is a property of stars and does not know what galaxy it is in — so a system that has made metals and does not have them has lost them, and the trend with rotation speed identifies the depth of the potential well as the thing that decides.

What leaves is not the galaxy’s gas in general. It is the hot, freshly enriched material from the supernovae themselves, driven out along the path of least resistance before it has had time to mix. That distinction matters: an outflow that carried away gas of the average composition would reduce the amount of star formation without changing the metallicity at a given gas fraction, and would leave the system on the line.

A third production site, arriving late

The two-site picture — core collapse fast, thermonuclear slow — accounts for the α elements and iron, and it leaves out roughly half the periodic table above iron.

Elements heavier than the iron peak cannot be built by fusion, because fusion past iron absorbs energy rather than releasing it. They are built by neutron capture, and neutron capture comes in two regimes distinguished by whether captures are faster or slower than the beta decays that compete with them. The slow route operates in the helium-burning shells of evolved intermediate-mass stars over thousands of years and produces barium, strontium and lead. The rapid route requires a neutron flux enormous enough to run a nucleus far from stability before it can decay, and it produces europium, gold, platinum and the actinides.

Where the rapid route happens was argued about for sixty years, and the argument was settled by a chemical-evolution measurement of exactly the kind this essay is about. Plot europium against iron for stars in the Galactic halo and disc, and the ratio behaves like a delayed product: high at low iron in some stars and not others, with an enormous star-to-star scatter at the lowest metallicities that narrows as the metallicity rises.

That scatter is the signature. A product made in every core-collapse supernova would be well mixed with iron from the start and would show the small scatter the α elements show. A product made in a rare event that yields a great deal at once produces a scatter that shrinks only as the number of contributing events grows — so a large scatter at low metallicity is a direct measurement that the site is rare and prolific rather than common and modest.

The rate and yield implied point at compact-object mergers rather than at ordinary supernovae, and the direct confirmation arrived when a neutron-star merger detected in gravitational waves was followed by an optical transient whose spectrum reddened over days in the way an expanding cloud of freshly synthesised heavy elements should. A chemical argument made from the abundances of old stars in this Galaxy predicted the site, and an entirely different instrument found it.

Its consequence for the models here is that a third delay time has to be carried, longer than the α elements’ and comparable with or longer than the Type Ia’s, with a rate low enough that early enrichment is stochastic rather than smooth. One-zone models with instantaneous mixing cannot represent that at all.

Where the model stops

Instantaneous recycling is wrong for iron by construction, which is why the α-knee exists at all — so any model that uses the simple closed-box equation for iron is using an approximation whose failure is the second half of this essay.

A galaxy is not one box. Metallicity varies with radius in every disc, typically falling outwards by 0.05 dex per kiloparsec, so a one-zone model describes an average of things with different histories. Radial gas flows move material between the zones, and they are neither measured nor negligible.

The yield is not measured independently. It is computed from nucleosynthesis calculations and an assumed initial mass function, and the computed value depends on the treatment of mass loss, rotation and explosion energy in massive stars — factors of two are live. Every statement about an effective yield being low is therefore relative to a plateau fitted to the data rather than to a theoretical number.

And the picture cannot show where the metals went. The material expelled from dwarf galaxies should be somewhere, and the diffuse hot halo that ought to hold it is one of the hardest things in the subject to observe. Estimates of the metal content of the circumgalactic medium have moved by factors of several within the last decade, and the cosmic metal budget has never been closed.

What the fix implies, and how it was checked

Saying “gas kept arriving” is easy and would be worth little if nothing else depended on it. Three independent things do.

The gas is there. High-velocity clouds of neutral hydrogen fall towards the Galactic disc at a hundred kilometres a second and more, and several of them have measured metallicities well below solar — which is what a model of infall by pristine gas requires and what a model of material recycled from the disc would not give.

The rate is right in order of magnitude. The disc forms stars at one to two solar masses a year and its gas would be exhausted in a couple of billion years without resupply; the accretion rate estimated from the clouds is of the same order, though with a factor-of-several uncertainty that nobody has removed.

And the same requirement appears at other masses. A galaxy’s stellar mass and its gas content, followed over cosmic time, cannot be reconciled with a closed system at any mass: the observed population converts gas to stars faster than it could have started with. The G-dwarf distribution is the local, stellar-archaeological version of a constraint that also comes from surveys at redshift two.

What none of that establishes is the form of the infall. The extreme model used above holds the gas mass exactly constant, which is a limiting case chosen for having a closed-form solution rather than for being right. Any accretion history that keeps the gas from being both abundant and metal-poor for long produces a comparable distribution, so the observation constrains the class of models rather than the member.

Reading a birth certificate

The essay so far has treated abundances as a bulk property of a galaxy. There is a much more ambitious use of the same data, and it is worth stating because it is the direction the subject has been moving and because its difficulties are instructive.

A star’s photosphere is a sample of the gas it formed from, preserved unchanged for as long as it lives. If the gas in a star-forming cloud is well mixed, every star born in that cloud has the same composition — not merely the same iron abundance, but the same pattern across twenty or thirty elements. And if different clouds have different patterns, then the pattern is a label: stars sharing one are siblings, and a dissolved cluster could in principle be reassembled from a survey of a million field stars long after its members have scattered across the Galaxy.

That is chemical tagging, and its appeal is that it would supply what Galactic archaeology otherwise lacks. Kinematics are erased — orbits mix, and the phase-space structure of a dissolved cluster fades within a few orbital times — while a composition is not erased by anything except the star’s own internal mixing.

The difficulty is dimensionality. Tagging requires the abundance patterns of different clusters to be genuinely distinct in enough independent directions to separate thousands of them, and what the surveys have found is that they are not. Most of the variation across stars lies along one or two directions — essentially the overall metallicity and the α-to-iron ratio, which are the two quantities this essay has already been using — and the residual scatter in the remaining elements is comparable with the measurement precision.

So the label has fewer bits in it than the problem requires. Chemical tagging works in the strong sense for a handful of chemically peculiar groups and works in a weak sense everywhere: a star’s position in the two-dimensional abundance plane says which broad population it belongs to, which is how accreted halo stars are separated from ones formed in place.

The failure is informative in its own right. If clusters were chemically distinctive, star formation would have to be poorly mixed on the scale of a cloud; that they are not says the interstellar medium mixes efficiently before it collapses, which is a constraint on turbulent mixing obtained from stellar spectra rather than from any fluid measurement.

Both of the model’s free parameters can be moved, and the shape of the disagreement with the data is what says which of them is wrong rather than merely mis-set.

A one-parameter model, and 5 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: 7.7 per cent of its stars fall below [Fe/H] = -1 against an observed 1.6 per cent, a factor of 5. 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 1.7 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.
Fig. 6 The closed box with the yield raised by eighty per cent. The whole distribution shifts to higher metallicity and the shape of the excess at the metal-poor end is unchanged, which is why no choice of yield removes the problem: the disagreement is in the shape and the yield only moves the mean.
Three histories, three knees: −0.00, −0.20, −0.81. The ratio of α elements to iron against iron abundance, for three systems that differ in one thing only — how fast they turned their gas into stars. Every track starts flat at +0.40 and then bends downward, and both halves are consequences of stellar lifetimes rather than of chemistry. Core-collapse supernovae come from stars that live a few million years, and they make oxygen, magnesium and silicon along with some iron in a fixed proportion; Type Ia supernovae come from white dwarfs and take about 1 billion years to arrive, and they make iron and almost no α. So for the first billion years the ratio is whatever massive stars produce, flat, and independent of everything. After that iron arrives without α and the ratio falls. The position of the bend is therefore not a measure of time at all — it is a measure of how much iron the system had already made when the clock struck, which is a measure of its star-formation rate. The fast system, forming stars on a 0.2-billion-year timescale, reaches [Fe/H] = -0.00 before its first Ia; the slow one at 6 billion years has only reached -0.81. That is why the Galactic bulge, the thin disc and the Sagittarius dwarf have knees a dex apart while being made of the same elements by the same stars — and why two galaxies of identical mass and colour can be told apart by the ratio of two lines in a spectrum.
Fig. 7 Three star-formation histories a factor of two faster than those above. The knee moves to higher metallicity in each case, because a faster history reaches a given metallicity sooner and the delayed-detonation clock has less time to run — which is why the knee’s position is a star-formation timescale rather than a composition.

The generalisation

This is a case where the failure of the simplest model is more informative than a model that fits, and the reason is structural.

The closed box has one parameter and predicts a whole distribution. That is an enormous amount of prediction for one number, and it means the model is falsifiable in a way a flexible one is not — a five-parameter model of the same data would have fitted, and would have said nothing. The discrepancy is large, specific, and in a direction that identifies which assumption is wrong: not the nucleosynthesis, not the stellar lifetimes, not the mass function, but the boundary condition.

That pattern recurs whenever a system is described by a conservation law with a term left out. A rotation curve that refuses to fall is Newtonian gravity with a mass term missing. A cluster whose galaxies move too fast is the virial theorem with the same term missing. Here it is mass conservation with an inflow and an outflow missing, and in each case the diagnostic value comes from the shape of the discrepancy rather than its size.

A model simple enough to be wrong in a describable way is worth more than one flexible enough to be right.

And the same knee at a shorter delay time for the second production site, which is the parameter the nuclear physics is supposed to supply.

Three histories, three knees: −0.16, −0.74, −1.31. The ratio of α elements to iron against iron abundance, for three systems that differ in one thing only — how fast they turned their gas into stars. Every track starts flat at +0.40 and then bends downward, and both halves are consequences of stellar lifetimes rather than of chemistry. Core-collapse supernovae come from stars that live a few million years, and they make oxygen, magnesium and silicon along with some iron in a fixed proportion; Type Ia supernovae come from white dwarfs and take about 0.6 billion years to arrive, and they make iron and almost no α. So for the first billion years the ratio is whatever massive stars produce, flat, and independent of everything. After that iron arrives without α and the ratio falls. The position of the bend is therefore not a measure of time at all — it is a measure of how much iron the system had already made when the clock struck, which is a measure of its star-formation rate. The fast system, forming stars on a 0.5-billion-year timescale, reaches [Fe/H] = -0.16 before its first Ia; the slow one at 12 billion years has only reached -1.31. That is why the Galactic bulge, the thin disc and the Sagittarius dwarf have knees a dex apart while being made of the same elements by the same stars — and why two galaxies of identical mass and colour can be told apart by the ratio of two lines in a spectrum.
Fig. 8 The same three histories with the type Ia delay shortened to six hundred million years. Every knee moves to lower metallicity, and by an amount comparable with the difference between the histories — so the delay time and the star-formation timescale are partly degenerate, and separating them needs a population where one of the two is known independently.

One more reading shows what an effective yield below the true one is actually measuring.

One line for a closed box, and an effective yield 2.8 times lower in the small systems. Gas-phase metallicity against the logarithm of the inverse gas fraction, for six kinds of star-forming system. A closed box has an exact prediction here and it is the simplest one in this collection: Z = y ln(1/μ), a straight line through the origin whose slope is the yield, with no dependence on the mass, the size, the age or the history of the system. Everything down to about a hundred kilometres a second of rotation sits on it, and the slope those systems share — 0.4 in solar units — is the only fitted number in the panel. Everything below that sits under it, and further under it the smaller the system: the effective yield — the slope each point implies on its own — falls from 0.56 for the large systems to 0.20 for the dwarfs, a factor of 2.8. There is only one way to be below this line. The yield is a property of stars and does not know what galaxy it is in, so a system that has made metals and does not have them has lost them, and lost them selectively: what leaves is the hot, freshly enriched gas from the supernovae themselves, driven out of a potential well too shallow to hold it. The trend with rotation speed is the argument, because the depth of that well is the one thing that changes along it. What the picture cannot show is where the metals went — the diffuse hot halo that should hold them is the hardest thing in this subject to observe, and the accounting has never been closed.
Fig. 9 The closed-box line against an effective yield reduced by outflow. A galaxy losing metals looks like a closed box with a smaller yield, and the ratio between the two is the fraction of the produced metals that left — which is how outflows are weighed without being seen.

Where this ladder goes next

Later rungs on this anchor: the mass–metallicity relation, which is the effective-yield trend restated as an observable and extends to redshift three; radial abundance gradients and what they say about how discs assemble; the detailed abundance patterns of individual elements, which fingerprint individual nucleosynthetic sites and are the basis of chemical tagging; the delay-time distribution of Type Ia supernovae, which is measured from the knee and independently from supernova rates and is the least certain ingredient in every model here; and the first stars, whose yields are recorded in the abundance patterns of the most metal-poor stars known and are the only evidence there is about a population nobody has observed.

Where this goes next

The star-formation law, and the clock inside it. 46 model discs on the Kennicutt–Schmidt plane, built from Σ_SFR = 2.5e-4 Σ_gas^1.4 with 0.28 dex of scatter, with the exponent then fitted back off the drawn points at 1.42. The diagonal dashed lines are constant depletion times: gas divided by the rate consuming it. Every disc on this plane consumes its gas within a few gigayears, and a galaxy that has been forming stars for ten of them therefore cannot have been working from the gas it started with. The slope steeper than one is what makes the depletion time shorter where the gas is richer, which is the opposite of the intuition that a full tank lasts longer. The gas runs out before the galaxy does Divide a disc's gas by the rate it is turning that gas into stars and the answer is about two billion years — a seventh of the age of the disc. Every spiral still forming stars is therefore being fed from outside, and the star-formation law says the richer the disc, the sooner it starves. One population, three integrals, two ends. The same mass function weighted three ways, each normalised so that the area under it is one, against mass on a logarithmic axis — so a share of the page is a share of the total. The light is a zero-age population's, every star still on the main sequence, which is the only age at which the top of the range is present at all. The steep curve on the left is the number of stars, which peaks at 0.08 M☉ and falls away because the function is steeper than m⁻¹; the middle one is the mass they carry; the one on the right is the light they emit, computed from this file's own main-sequence relation and peaking at 55.0 M☉ — 682 times further up the axis. The population that is counted and the population that is seen are two different populations. Above 8 M☉ there are 0.63% of the stars, 21% of the mass and 99% of the light. The slopes are α = 1.3 above 0.08 M☉, α = 2.3 above 0.5 M☉, and the low-mass end is where the honesty runs out: it has never been measured in another galaxy, and every mass inferred from a luminosity assumes it. Most stars are small, and most of the light is not The mass function is steep and the mass–luminosity relation is steeper, so counting stars and measuring their output are integrals of the same function dominated by opposite ends of it. Every mass inferred from a brightness passes through that mismatch. Four abundances, one free parameter. The abundances big-bang nucleosynthesis predicts, against the one number it is free to choose: η₁₀, the ratio of baryons to photons in units of 10⁻¹⁰. Four curves spanning nine decades, from a helium mass fraction of about a quarter down to a lithium abundance of one atom in ten billion, and they are not four independent predictions — they all come out of the same reaction network run at the same density. The horizontal bands are what is measured in the sky, each at its published one sigma. The measurement that matters is deuterium, because its curve is the steep one: inverting the drawn curve at D/H = 2.527e-5 gives η₁₀ = 6.11, and the ends of the observed interval give 6.06 to 6.15. The vertical band is what the microwave background gives, 6.13 ± 0.04, from the height of the second acoustic peak relative to the first. Those two agree to 0.4 per cent, and they have nothing whatever in common: one is a nuclear-reaction network run in the first three minutes and read off a quasar absorption line, the other is a fluid oscillation at four hundred thousand years read off a sky map. Lithium is the exception and it is not a small one — the network predicts 4.70e-10 at the microwave background's density and the halo stars show 1.60e-10, a factor of 2.9 too much, which is unresolved. Four abundances and one free parameter In the first three minutes the universe ran a nuclear reaction network with exactly one adjustable number in it. That number predicts four abundances spanning nine orders of magnitude, three of them are observed and match, and the fourth is wrong by a factor of three and has been for twenty-five years.

What this makes readable

Essays that name this one as a prerequisite.

What links here

The 8 of 9 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Alpha enhancementAlpha kneeChemical evolutionClosed box modelEffective yieldG-dwarf problemGalactic outflowGas infallInstantaneous recyclingMetallicity distributionType ia supernovaeYield