Brighter for its mass and fainter for its colour

Mass decides almost everything about a star, and the word “almost” carries a second variable. The same mechanics that make luminosity rise as roughly the fourth power of mass also make it depend on what the star is made of — and the dependence is not small. Homology, the scaling argument that gives the shape of the relation, writes the luminosity of a radiative star as

$$L \;\propto\; \frac{\mu^4 M^3}{\bar\kappa},$$

with $\mu$ the mean mass per particle and $\bar\kappa$ the opacity. The relation’s changing slope comes from how the opacity changes with mass. Holding the mass fixed and changing the material instead, and the two factors in front of $M^3$ turn out to say something that sounds contradictory until it is drawn.

A star with a hundredth of the Sun’s iron, carbon and oxygen is, at the same mass, about one and a half times as luminous and several hundred kelvin hotter. Seen at the same temperature as a solar-composition star, it is fainter by half a magnitude. Both statements are true at once, and astronomers named such stars for the second one: subdwarfs, stars that sit below the dwarf sequence. They are not under-luminous. They are over-heated.

What the metals do inside a star #

The metals — everything heavier than helium — make up 1.4 per cent of the Sun by mass. They matter far beyond that share because of how radiation gets out.

In the radiative interior of a star like the Sun the energy diffuses outward as photons absorbed and re-emitted countless times. What absorbs them is mostly two processes. Free–free absorption, in which a free electron absorbs a photon while passing an ion, is supplied by hydrogen and helium and is present in any composition. Bound-free absorption, in which a photon ionises an electron still attached to an ion, needs ions that still hold electrons at millions of kelvin — and at those temperatures hydrogen and helium are stripped bare, so the work is done by the metals. Iron, oxygen and neon, at a combined abundance of about one atom in a thousand, supply a large part of the Sun’s interior opacity.

Remove them and the opacity falls, the radiation leaks out faster, and the star must burn faster to keep up. A star’s luminosity is not set by how fast it burns; it is set by how fast its interior lets energy escape, and the burning adjusts to match. That is the logic behind the relation’s whole shape — the opacity that matters is a harmonic mean dominated by the frequencies where absorption is weakest — and it is why the metals reach the luminosity.

Homology turns that into numbers once the opacity is written down. With bound-free and free–free absorption both following Kramers’s form, $\kappa \propto \rho T^{-7/2}$, the coefficient in front is proportional to $(1+X)$ times a sum of a term for hydrogen and helium and a term proportional to the metal abundance $Z$. The ratio of those two terms at solar composition is the one number the model does not fix; it depends on the detailed atomic physics of iron and oxygen at millions of kelvin, which is precisely what laboratory measurements have been revising. The figures here set it so that metals supply half the opacity of a solar-composition interior, and one figure shows what happens when the share is a third or two thirds.

The relation, shifted #

The model is used differentially, which is the only way homology should be used for quantitative work. Taken absolutely, homology with Kramers opacity predicts luminosity rising as the fifth and a half power of mass, against the measured fourth, because real stars are not exactly homologous — their convective envelopes and their run of opacity with depth change with mass. But the ratio of two stars of the same mass and different composition is much better behaved, because the stars differ only in the factors homology tracks. So the solar-composition curve is the measured relation, from the stars whose masses are known, and composition multiplies it.

The curves for a tenth and a hundredth of the Sun’s metals are nearly parallel to the solar one and above it. Composition multiplies the luminosity at each mass by nearly the same factor, and the factor saturates: going from a tenth to a hundredth of solar metals adds less than going from solar to a tenth, because by a tenth most of the metals’ share of the opacity is already gone. What remains is free–free absorption by hydrogen and helium, which no degree of metal-poverty removes.

Brighter by mass, fainter by temperature #

The paradox lives in the diagram’s geometry. Removing metals moves every star up — more luminous — and to the left — hotter. The hotter part comes from the radius: the same homology says the star contracts slightly, by about seven per cent at a hundredth of solar metals, so the extra light comes out through a smaller surface, which needs a higher temperature. The luminosity rises by about sixty per cent and the temperature by about seventeen.

The main sequence itself runs steeply up and to the left. A shift of seventeen per cent in temperature, along a sequence on which luminosity rises roughly as the sixth or seventh power of temperature, carries a star past solar-composition stars two or three times as luminous as itself. A sixty per cent increase in luminosity does not keep up. So the metal-poor sequence lies to the left of the solar one, and therefore below it at any fixed temperature: at 5,500 K, by about half a magnitude.

Gerard Kuiper named such stars subdwarfs in 1939, from their position in the diagram, before anyone knew why they were there. The name describes the diagram, not the stars: a subdwarf of a given mass outshines a solar-composition dwarf of the same mass.

There is an additional effect this model leaves out, and it works in the same direction. The diagram is drawn against temperature, but observers measure colour, and colour depends on composition too: the metals’ absorption lines crowd the blue and ultraviolet, so a metal-poor atmosphere of a given temperature is bluer than a solar one — the ultraviolet excess that was the first photometric sign of metal-poverty. Plotted against observed colour, a subdwarf is shifted further to the blue and lies further below the main sequence than it does against temperature. The offsets quoted for subdwarfs in colour–magnitude diagrams, up to a magnitude or more, are this interior effect and that atmospheric one added together.

Where the extra light comes from #

The homology lets the effect be taken apart, by switching on one change at a time. A metal-poor star differs from the Sun in three ways at once: fewer metals, slightly less helium — because helium, like metals, is made in stars and accumulates in the gas each generation forms from — and correspondingly slightly more hydrogen.

Opacity is the large term. At a hundredth of solar metals, opacity alone would double the luminosity at fixed mass.

The mean mass per particle takes some of it back. A fully ionised gas of hydrogen contributes two particles — a proton and an electron — per atomic mass unit, and helium three particles per four units, so a gas with less helium has more particles per gram and a lower $\mu$. Lower $\mu$ means a lower central temperature is enough to support the star, through the same hydrostatic balance that sets the temperature before fusion does, and luminosity in homology goes as a high power of $\mu$ — the fourth in the simplest form, and higher once Kramers opacity’s own temperature dependence is included. The few hundredths of helium a metal-poor star lacks cost it about a fifth of its luminosity.

The hydrogen fraction enters through the energy generation, since the proton–proton chain’s rate goes as the square of the hydrogen abundance. More hydrogen lets the star meet its luminosity at a slightly lower central temperature, which changes the radius a little and the luminosity hardly at all.

The decomposition also says where the model’s sensitivity lies. Helium enters as such a high power that a star with substantially more helium than its neighbours is noticeably brighter and hotter at the same mass. Several globular clusters contain stars born with helium mass fractions up to about 0.35 or more, against the 0.25 of their first generation, and in the best-measured of them the main sequence is split in two, with the helium-rich sequence bluer. The homology exaggerates the size of that split, because the low-mass stars involved have deep convective envelopes it does not describe, but it gets its direction right, and the direction is the observation.

A mass read off the wrong relation #

The practical consequence is the most direct. Masses of single stars are almost never measured. They are inferred from a luminosity, or from a temperature and a luminosity, through a relation calibrated on binaries — and the binaries that calibrate it are overwhelmingly nearby disc stars of roughly solar composition.

A metal-poor star read off that relation is assigned the mass of a solar-composition star of the same luminosity, which is a heavier star. Because the relation goes as the fourth power of mass in this range, the error in mass is the fourth root of the luminosity factor: about thirteen per cent too much at a hundredth of solar metals, if the metals supply half the solar opacity, and between five and twenty-six per cent across the plausible range of that share. The sign never changes. The best eclipsing binaries now give masses good to one to three per cent, so this is a systematic error several times larger than the measurement errors of the calibration it is read from.

The same error reaches every quantity derived from a mass: the size of a planet from the wobble it raises, the age of a star from its position on a track, the mass budget of a stellar population. A planet orbiting a metal-poor star whose mass was read off the solar relation has its own mass overestimated by two thirds of the same fraction, since the reflex velocity scales as the two-thirds power of the star’s mass.

The turnoff names a mass only once the composition is known #

A star’s main-sequence lifetime is its fuel divided by the rate it burns it, so a star that shines brighter for its mass lives less long. A 0.8 solar-mass star of solar composition would stay on the main sequence for nearly twenty billion years in this estimate; with a hundredth of the metals it lasts twelve.

That moves the turnoff, the point where a cluster’s stars leave the main sequence and the feature from which a cluster’s age is read. At an age of twelve billion years a solar-composition cluster would be losing stars of 0.94 solar masses; a metal-poor cluster of the same age is losing stars of 0.80, which is close to the turnoff mass that full stellar models give for the old globular clusters of the Galactic halo. Read with solar-composition stars, those clusters’ turnoffs would be taken for heavier stars burning more slowly for their light: a turnoff at the luminosity of the metal-poor 0.80 solar-mass star belongs, at solar composition, to a star of about 0.9 solar masses that lives more than thirteen and a half billion years, and the cluster would come out more than a billion years too old.

The cluster age depends on the composition twice: once through the luminosity of the stars at the turnoff, which is what this figure draws, and once through the distance, because the distance to a globular cluster is often found by fitting its main sequence to nearby subdwarfs of the same composition with measured parallaxes. A main sequence fitted to solar-composition stars assumes the cluster’s dwarfs are as luminous as solar dwarfs of their colour, which they are not, and so puts the cluster too far away by the half-magnitude offset in the diagram above — making its turnoff look intrinsically brighter and therefore younger. The two errors partly cancel, which is the reason the composition must be measured and not assumed: a partial cancellation is not a guarantee.

Iron is not the metal that matters #

Every curve here is drawn against [Fe/H], the logarithm of a star’s iron abundance relative to the Sun’s, because iron is what a spectrum measures best: it has thousands of lines across the visible, and the abundance comes from their depths. But the opacity in a star’s interior is not mostly iron. Oxygen, neon and the other light metals made by massive stars contribute as much or more, and their abundance relative to iron is not fixed.

In the old stars of the Galactic halo it is not solar. Those stars formed from gas enriched only by the short-lived massive stars that explode as core-collapse supernovae, which make oxygen and the other light metals in abundance and relatively little iron; the type Ia supernovae that supply most of the Galaxy’s iron arrived later, with no single delay. So a halo star’s oxygen-to-iron ratio is typically two and a half times the Sun’s.

For the luminosity that changes the arithmetic. The quantity the opacity responds to is the total metal content, and with the light metals enhanced by that factor a star at [Fe/H] = −2 has about twice the total metals that its iron alone suggests — the equivalent, for its interior, of a star at about −1.7 with solar proportions. On the flat part of the curves, below a tenth of solar metals, that makes little difference to the luminosity. Between a tenth and a third of solar — the composition of the thick disc and of the most metal-rich globular clusters — it moves the luminosity by several per cent and the inferred mass by about one, which is comparable to the precision of the best binaries.

A test in the clusters themselves #

The model can be checked where it matters most. Detached eclipsing binaries give masses and radii directly, and a few have been found in globular clusters, near the turnoff, where their masses are the turnoff mass and their luminosities fix the cluster’s composition-dependent relation at one point.

In the globular cluster 47 Tucanae, with about a fifth of the Sun’s iron, a pair of stars just below the turnoff has masses of about 0.86 and 0.88 solar masses measured to about one per cent. A solar-composition relation reading their luminosities would give larger masses; the metal-poor models that fit their masses and radii together give the cluster an age of about twelve billion years. Similar measurements in more metal-poor clusters are harder, because the stars are fainter and the binaries rarer, and they are the observations that would test the flat part of the curves directly.

The field supplies the other test. Nearby subdwarfs with parallaxes are bright enough for their luminosities to be measured well, and those in binaries give masses. Their position below the main sequence, at the size the composition predicts once both the interior and the atmospheric effects are included, is what makes main-sequence fitting possible at all: the method works because the offset is understood well enough to be corrected.

Where the model stops #

Homology. The model treats the composition change as a rescaling of a star whose structure is otherwise unchanged. Real stars of different composition have different depths of convective envelope, and below about 0.8 solar masses that difference dominates the surface temperature. The shifts drawn are the right size for radiative interiors and directionally right below.

Two opacity sources. The opacity is free–free plus bound-free in Kramers’s form. Electron scattering, important in hotter stars, and the negative hydrogen ion, which dominates in cool envelopes, are omitted.

Fixed helium enrichment. Helium falls with the metals along one assumed law, $Y = 0.247 + 1.36Z$. A population with its own helium history departs from it.

Zero-age stars. The stars are on the main sequence with their initial composition. As a star burns, its core’s $\mu$ rises and its luminosity with it, so the relation drifts through the main-sequence lifetime — a separate problem, and one in which composition also enters.

A relation is a function of everything held fixed #

The mass–luminosity relation is usually stated as if it had one variable, and for stars of the Sun’s neighbourhood that is close enough: their compositions differ by factors of two, which move the luminosity by a few tens of per cent at most. The relation was calibrated on those stars and works for them.

A relation calibrated on one population carries that population’s composition inside it, as a hidden constant. Applied to a population with a different composition it returns answers with a systematic error of a definite sign, and the error is invisible in any check made within the calibrating sample. The stars of the Galactic halo, a hundredth as metal-rich as the Sun, are where that constant stops being constant.

Still open: the stars with no metals at all #

The first stars formed from gas left by the Big Bang, with no metals whatever. For them bound-free absorption by metals is absent entirely, and there is no carbon, nitrogen or oxygen to catalyse hydrogen burning in massive stars, so the CNO cycle cannot start until the star has made its own carbon by burning helium. Models say such stars were hotter and more compact than any star now forming, and that the first generation was weighted towards high masses. None has been observed. The most metal-poor stars known have iron abundances below a ten-millionth of the Sun’s, but some metals nonetheless, so they formed from gas already enriched by a previous generation. Whether any metal-free stars of low mass formed — which would still be on the main sequence today — and where the mass–luminosity relation of that first generation lies, are open.