Where a shrinking star stops boiling at its centre

A star that has just condensed out of its cloud is large, cool and fully convective, and it shines by contracting. Its surface temperature is pinned near a value set by the opacity of its outer layers — the wall a steep opacity builds on the right of the Hertzsprung–Russell diagram, the Hayashi line — and as it shrinks at that temperature its luminosity falls with its surface area. On the diagram it moves straight down.

Then, for every star above about half a solar mass, the track turns. The luminosity steadies and the surface temperature climbs, and the star moves leftward, nearly horizontally, to its place on the main sequence. That leftward stretch is the Henyey track, and the turn onto it marks a change deep inside: the centre of the star has stopped convecting. A radiative core has formed and is growing outward, and the star’s luminosity, which on the Hayashi track was set by the surface, is now set by how fast radiation can leak through the interior.

The question the turn poses is what decides when it happens. The star is shrinking and heating throughout; at some point the centre crosses a threshold. The threshold is a statement about opacity, and the hero figure draws it.

The gradient radiation would need #

Energy flows outward through a star’s interior either by radiation, diffusing through the gas, or by convection, carried by rising and sinking parcels. Which one does it is decided locally. Radiation can carry a given flux only by maintaining a particular temperature gradient, steeper the more opaque the gas and the larger the flux. If that gradient exceeds the gradient at which a rising parcel stays buoyant — the adiabatic gradient, 0.4 in the logarithmic form used by stellar modellers, for an ideal monatomic gas — the gas boils, and convection takes over. If it does not, the gas stays still and radiation carries the flux.

At the centre of a contracting star the radiative gradient is

$$\nabla_{\rm rad,c} = \frac{3\,\kappa_c P_c}{16\pi a c G\,T_c^4}\,\frac{L}{M},$$

where the luminosity per unit mass, $L/M$, is a fair approximation at the centre for a star whose energy comes from contraction spread through its whole mass. Treating the fully convective star as a polytrope of index 3/2 gives the central density, pressure and temperature in terms of its mass and radius, and on the Hayashi track its luminosity is $4\pi R^2\sigma T_H^4$ at a surface temperature $T_H$ that barely changes.

The hero figure follows the central radiative gradient as four stars, of 0.3, 0.5, 1 and 2 solar masses, shrink. Every curve is a straight line on the logarithmic axes with the same slope: the gradient falls as the 2.5 power of the radius. Where it crosses 0.4 the centre turns radiative. The two-solar-mass star crosses at 7.6 solar radii, the one-solar-mass star at 1.9; the half-solar-mass star would cross only at a radius below its main-sequence size, and the 0.3 never does.

Why the gradient falls #

The slope of 2.5 comes from the opacity. Inside a young star the dominant source of opacity at the centre is the absorption of photons by electrons in the field of ions — bound–free and free–free transitions — and its strength is well described by Kramers’ law, $\kappa \propto \rho T^{-3.5}$: proportional to the density, because more ions mean more absorbers, and falling steeply with temperature, because hotter photons are harder to absorb.

As the star contracts homologously, its central density rises as $R^{-3}$ and its central temperature as $R^{-1}$. The opacity therefore goes as $R^{-3}\cdot R^{3.5} = R^{0.5}$: it falls, slowly, as the star shrinks. The central pressure rises as $R^{-4}$, the luminosity falls as $R^2$, and the fourth power of the temperature rises as $R^{-4}$. Multiplying the pieces, the radiative gradient goes as $R^{0.5}\cdot R^{-4}\cdot R^{2}\cdot R^{4} = R^{2.5}$. The opacity’s fall and the luminosity’s fall together make radiation steadily more able to carry the flux, until it can carry it without help.

The mass dependence follows from the same pieces. At a fixed radius, a heavier star’s centre is hotter — the temperature scales with the mass — and its opacity correspondingly lower, by the 3.5 power of the temperature. Put together, the gradient at fixed radius falls roughly as the mass to the −5.5 power. A star twice the Sun’s mass is transparent enough at its centre to turn radiative while it is still four times larger than a Sun-like star is when it turns.

The steepness of the mass dependence is easiest to see with the masses closer together.

Between 0.7 and 3 solar masses the turning radius grows from about the Sun’s radius to sixteen times it, nearly as the square of the mass. That is the $M^{-5.5}$ dependence at fixed radius, converted by the $R^{2.5}$ slope into a turning radius that scales as roughly $M^{2.2}$. The heavier stars are still distended, cool-surfaced objects when their centres clear; the lighter ones have nearly reached their final size.

When each mass turns #

The ages span three decades. A star of three solar masses spends only twenty thousand years on its Hayashi track before turning, so short a time that such stars are almost never seen there; the Sun spent about two million; a star of 0.7 solar masses spends ten million. Below a threshold mass the turn never comes, because the star reaches its main-sequence radius — where hydrogen burning stops the contraction — before its centre has become transparent enough.

The model places that threshold near 0.55 solar masses. Detailed stellar models, with realistic equations of state, opacities and atmospheres, place it at about 0.35: stars lighter than that are fully convective not only before the main sequence but on it, for their entire hydrogen-burning lives, and they burn almost all their hydrogen rather than a tenth of it because convection mixes the whole star through the core. The polytrope overestimates the threshold because its central temperature for low-mass stars is too high and its treatment of the partly degenerate centre too simple; the existence of the threshold, and its origin in the competition between the opacity’s fall and the contraction’s end, survive the simplification.

The one-solar-mass figure, by contrast, is close to what detailed models give: a radiative core appearing at a little under two solar radii, around two million years. The simple calculation does well where the physics it leaves out — degeneracy, molecules, non-ideal gas — matters least.

Two tracks named for their discoverers, in the wrong order #

The leftward track was found first. In 1955 Louis Henyey and his collaborators computed the contraction of young stars assuming they were radiative throughout, and found them moving nearly horizontally across the diagram onto the main sequence. Six years later Chushiro Hayashi showed that this could not be the whole story: a star whose surface is cool enough has an outer layer so opaque that convection must reach deep, and a fully convective star cannot be cooler than a certain temperature at a given luminosity. Young stars therefore begin on the near-vertical Hayashi track and join the Henyey track only later — when, as the figures show, their centres become transparent.

The order of discovery is a reminder of how the opacity’s role was found. Henyey’s stars were radiative because the calculation assumed it; Hayashi’s insight was that the surface opacity forbade it early on. The turn between the two tracks is the point where the interior opacity, falling as the star contracts, lets the first assumption become true.

Down the line, then across #

On the diagram the turn sorts the stars. The heavier ones leave their convective lines high up, at tens or hundreds of solar luminosities, and cross most of the diagram on radiative tracks, their surfaces heating from five thousand to ten thousand kelvin as they settle. The Sun’s track turns at about one solar luminosity and runs left by a third in temperature. The lightest stars never turn: their tracks run straight down into the main sequence, and the main sequence’s lower end is the place where vertical tracks stop.

The straight dotted lines are a simplification: real Henyey tracks rise slightly in luminosity as the radiative core grows, before settling onto the main sequence. But their starting points are fixed by the opacity criterion, and that is what decides the diagram’s shape for a young cluster. In a cluster a few million years old, the stars above a certain mass have turned and the stars below have not, and the location of that break in the cluster’s sequence is a clock — read, like the turn-off of an old cluster, from the shape of a population rather than from any one star.

Fewer metals, an earlier turn #

The opacity that sets the turn depends on what the star is made of. Bound–free absorption is dominated by the heavier elements — the metals, in astronomers’ usage — because their inner electrons are bound tightly enough to absorb the photons of a stellar interior, and the bound–free opacity is roughly proportional to the metal content.

A metal-poor star’s centre is more transparent at the same temperature and density, so radiation takes over at a larger radius, earlier in the contraction. At a fiftieth of the solar metal content the Sun-like star turns twenty times sooner, and stars down to a quarter of a solar mass grow radiative cores before reaching the main sequence. The same physics changes the Hayashi line itself — metal-poor stars’ outer layers are more transparent too, and their convective tracks lie further to the left — so a metal-poor young cluster’s whole pre-main-sequence diagram is shifted and reshaped.

Lowering the metals moves the whole family of lines down without changing their slope, because the slope comes from the form of Kramers’ law and the metals only change its coefficient. At a tenth of the solar value the Sun-like star turns at four solar radii instead of two, and even a 0.3-solar-mass star grows a radiative core before arriving. Below about a twentieth of solar the bound–free opacity is no longer the largest term — free–free absorption by hydrogen and helium, which does not depend on the metals, takes over — and further lowering the metals changes little.

This matters for reading ages from young clusters, and more for reading them from the first stars. The oldest stellar populations formed from gas with almost no metals, and their young stars would have spent much less time fully convective. Whether that affected how they spun down, how their magnetic fields formed, or how their surface lithium survived — all processes tied to the depth of the convection zone — is a question about stars no one can now observe young.

What the turn does to a young star’s surface #

The turn is invisible inside the star and conspicuous outside it. While the star is fully convective its outer convection zone reaches to the centre, and anything at the surface is carried down to the hot interior and back on a timescale of months. Lithium, which is destroyed by protons above about two and a half million kelvin, is burned wherever the convection carries it that hot, so a young star’s surface lithium is a record of how deep its convection has reached and how hot the bottom was.

When the radiative core appears, the base of the convection zone begins to retreat outward and to cool. In stars above about a solar mass it retreats fast enough that the bottom of the convection zone drops below the lithium-burning temperature before much lithium is lost, and the surface keeps most of what it was born with. In the lightest stars, which never turn, the convection reaches the centre throughout the contraction, the centre passes the burning temperature, and all the lithium is destroyed within a few tens of millions of years. The mass at which a young cluster’s stars switch from having lithium to having none — the lithium depletion boundary — moves to lower masses as the cluster ages, and is one of the most precise clocks for clusters between twenty and two hundred million years old.

The convection zone’s depth also governs how a young star loses its spin. A deep convection zone couples to the star’s magnetic field and its wind, and the clock that starts by forgetting — the spin-down that eventually lets a star’s rotation date it — begins in earnest only after the core has decoupled from the envelope. Stars that never turn spin down differently, keeping their rapid rotation for billions of years and then braking abruptly.

What the opacity is doing #

The pattern in all three figures is that the energy source never enters the decision. The star is shining by contraction throughout; nothing about its power supply changes at the turn. What changes is whether the interior can transmit that power without boiling, and that is a property of the gas’s opacity at the temperature and density the contraction has reached.

It is the same lesson the surface that is a depth teaches at the outer edge: where a star’s structure changes, the change is usually written in its opacity. The Hayashi line is a surface opacity rising steeply with temperature, the Henyey turn is a central opacity falling with temperature, and between them they draw the two straight segments of every young star’s path. The main sequence is a place, not a track; the tracks that lead to it are drawn by the opacity, from the outside in.

What the model leaves out #

The star is a polytrope of index 3/2 throughout. That is exact for a fully convective ideal-gas star and approximate as the centre approaches the turn, where the structure begins to change. It sets the central conditions to within tens of per cent for Sun-like stars and worse for the lightest, whose centres are partly degenerate.

The Hayashi temperatures are interpolated from a table. Real Hayashi tracks shift with mass, age and metal content by a few hundred kelvin, which moves the luminosity at a given radius and so the gradient; the turn’s radius moves by a corresponding few per cent.

The opacity is Kramers’ law plus electron scattering. At the centres of young stars, a few million kelvin, Kramers’ law is a fair approximation to the tabulated opacities; it is not accurate to better than a factor of two, and the threshold mass is sensitive to that factor.

The luminosity is spread uniformly by mass. Contraction releases energy everywhere, but deuterium burning, which briefly halts the contraction of young low-mass stars, concentrates it at the centre for a time and delays the turn slightly.

The Henyey track is drawn as a straight line. Its real shape — a slight rise and a hook onto the main sequence — depends on the growth of the radiative core and the onset of nuclear burning, which the criterion here does not follow.

Still open: whether magnetic fields move the threshold #

The fully convective threshold at about 0.35 solar masses is visible in the properties of red dwarfs on the main sequence. Their rotation and magnetic activity change character there, because a star without a radiative core has no boundary layer between core and envelope where an ordinary solar-type dynamo is thought to operate. And there is a narrow gap in the density of stars on the main sequence near that mass, measured in the space astrometry catalogues, attributed to an instability as a star’s tiny radiative core comes and goes while its centre’s helium-3 burning settles.

What is not known is whether magnetic fields, strong in young low-mass stars, move the threshold. A strong field inhibits convection, which is one proposed reason why red dwarfs come out several per cent larger and cooler than models predict. If the field also shifts where the radiative gradient crosses the adiabatic one, the masses at which young stars turn — and the mass below which they never do — would depend on how magnetic they were in their first few million years, and the clock read from a young cluster’s turn would run at a rate set partly by its stars’ fields.