A wall built by a steep opacity

On a diagram of luminosity against surface temperature, stars that are convective throughout — newly formed stars still contracting toward the main sequence, and old giants swollen to a hundred times the Sun’s radius — lie on nearly vertical lines. A young Sun-like star contracting from a hundred times its present luminosity to one does so at a surface temperature that changes by only a few hundred kelvin, around four thousand. A red giant climbing from ten to a thousand solar luminosities does the same. Their colours hardly change while their sizes change by factors of ten.

Chushiro Hayashi showed in 1961 why. The vertical lines, which carry his name, are not the result of any particular history. They are a boundary: a convective star must lie on one, and no star in hydrostatic equilibrium can lie to the right of it. And the reason the lines are vertical rather than sloped is a single property of one ion — how steeply its opacity rises with temperature.

Two conditions that meet at the surface #

A star whose interior is convective carries its heat by bulk motion of the gas, and efficient convection holds the temperature gradient at the adiabatic value: a parcel of gas that rises and expands without exchanging heat stays at the temperature of its surroundings. Where convection takes over from radiation the whole structure is fixed by the adiabat. For an ideal monatomic gas the pressure and temperature along it are related by $P \propto T^{5/2}$, and the constant in front depends on the star’s mass and radius — for a fully convective star, a polytrope of index three-halves, it goes as $M^{-1/2}R^{-3/2}$. So the interior supplies one relation between the pressure and temperature at the bottom of the photosphere.

The photosphere supplies another. A star’s visible surface is a depth: the level at which the gas above is thin enough for light to escape, where the optical depth measured inward from space is about two-thirds. For an atmosphere in hydrostatic balance that condition becomes $\kappa P = \tfrac{2}{3} g$, where $\kappa$ is the opacity per gram and $g$ the surface gravity. The more opaque the gas, the lower the pressure at which the photosphere sits.

Both relations have to hold at the same point. If the opacity is a power law in pressure and temperature, $\kappa = \kappa_0 P^a T^b$, then substituting the adiabat into the photospheric condition gives an equation for the surface temperature alone, in terms of the mass and radius. Replacing the radius by the luminosity, through $L = 4\pi R^2 \sigma T^4$, gives the line. For the opacity of cool stellar atmospheres, which rises as the square root of the pressure,

$$T_{\rm eff} \propto L^{1/[8(b + 4.25)]} \, M^{7/[4(b + 4.25)]} \, \kappa_0^{-1/(b + 4.25)} .$$

Everything about the Hayashi line is in those exponents. The dependence on luminosity is weak whatever $b$ is, because the adiabat already ties the surface to the interior; the dependence on the opacity’s temperature exponent $b$ decides how weak.

The ion that holds the surface still #

In the atmospheres of stars between about three and seven thousand kelvin, the dominant opacity is the negative hydrogen ion — a hydrogen atom holding a second electron, bound by only three-quarters of an electronvolt. The ion is fragile and scarce: at the Sun’s surface there is about one for every hundred million hydrogen atoms. Its abundance depends on the density of free electrons, and in a cool atmosphere those come from metals, which are partly ionised and whose ionisation climbs steeply with temperature.

The ninth power is what does the work. Suppose a convective star were somehow given a surface cooler than its Hayashi temperature. The photosphere would be much more transparent, light would escape from deeper and hotter layers, and the surface temperature would rise back. Given a surface too hot, the photosphere would be far more opaque, light would escape from higher and cooler layers, and it would fall back. A steep opacity acts as a thermostat on the surface, and the steeper it is, the tighter the thermostat.

The same point, read as a number: the slope of the line is $1/[8(b + 4.25)]$ in logarithmic units.

The consequence can be stated plainly. A young star contracting on a Hayashi line shrinks by a factor of ten in radius and a hundred in luminosity while its colour barely moves; it is not getting cooler as it gets fainter, only smaller. A red giant swelling from ten to a hundred solar radii does the same in reverse, its envelope pushed outward by the shrinking of its core. The temperature of every such star is set by the opacity of one fragile ion and the star’s mass, not by its size.

A temperature that weighs a young star #

If luminosity barely moves the temperature, mass does. The exponent on mass in the formula is fourteen times the exponent on luminosity: doubling a convective star’s mass raises its surface temperature by about ten per cent, while multiplying its luminosity by ten raises it by two. On a Hayashi line, temperature is a measurement of mass.

That is how the masses of most young stars are estimated. A star in a star-forming region is placed on the diagram from its spectrum and brightness, and its mass is read from the Hayashi line it sits on, its age from how far down the line it has contracted. The second reading is much less secure than the first: luminosity depends on the distance, the extinction and whether the star is a binary, while temperature depends on none of those. For stars of a solar mass and below, the mass read from temperature is the more robust number, and the age the fragile one — which is why the ages of young clusters are argued over far more than the masses of their stars.

The young Sun followed its own line. A million years after it formed it was about twice its present radius and a few times its present luminosity, at a surface temperature near four thousand three hundred kelvin, and it contracted down the line for some ten million years before its core became radiative and it turned onto the horizontal track to the main sequence. Its luminosity during that time is what decided how much water the forming planets could keep, before the habitable zone had settled where it is.

How fast the thermostat acts #

The argument that forbids the right-hand side of the diagram leans on convection being quick, and it is worth seeing how quick. A superadiabatic gradient drives convective motions whose speed grows as the square root of the excess; in the outer layers of a Sun-like star a gradient exceeding the adiabatic value by a part in a million already drives flows of tens of metres per second, enough to overturn a layer a thousand kilometres thick in a day. The imbalance that a star to the right of its line would carry is not a part in a million but tens of per cent, and it is erased on a timescale of years — the time for the convective envelope to redistribute its heat, which is short compared with the star’s evolution by a factor of millions.

That is why the wall is sharp. A star does not drift slowly toward its Hayashi line from the right, as it drifts slowly along it from above; it is either on the line or in a transient state that ends almost at once. The contrast between the two timescales — years to reach the line, millions of years to move along it — is what makes the diagram’s pre-main-sequence region so orderly.

Why nothing can sit to the right #

The line is also a wall. A star lying to the right of its Hayashi line would have to be cooler at its surface than a fully convective star of its mass and luminosity. Working back from that surface, its interior would have to have a temperature gradient steeper than the adiabatic one, because it would need to reach the same central conditions from a cooler starting point. A gradient steeper than adiabatic is unstable to convection, and convection in a stellar interior is so efficient that it erases a superadiabatic gradient in a matter of months to years, a thermal timescale far shorter than any evolution. The star would heat its outer layers and move left until it reached the line.

So the region to the right is not merely unoccupied; it is impossible for a star in hydrostatic and thermal equilibrium. A star can pass through it only briefly — in the very first moments of collapse, before hydrostatic equilibrium is established, or in a violent event — and the pre-main-sequence stars in young clusters, which are the most direct test, are all found on or to the left of the lines their masses predict. The main sequence is a place where stars sit; the Hayashi line is a place they cannot pass.

The giant branch as a metallicity scale #

The formula has a third factor, the opacity’s normalisation $\kappa_0$, and it depends on the star’s composition. The negative hydrogen ion needs free electrons, and in a cool atmosphere they come from metals. A star born from gas with fewer metals has fewer electron donors, a more transparent photosphere at a given temperature, and must therefore be hotter at its surface to bring the opacity up to what the photospheric condition demands. Because the metals are only partly ionised, the electron density goes as the square root of their abundance, and so does the opacity.

That is observed directly. The red giant branches of globular clusters, all old and all of similar stellar mass, differ in colour from one cluster to the next in step with their metallicities: the most metal-poor clusters, with a hundredth of the Sun’s metals, have giant branches hundreds of kelvin hotter than the metal-rich ones near the Galactic bulge. The colour of the giant branch at a fixed brightness was one of the first photometric measures of a cluster’s metallicity, calibrated against spectra, and it is still used for clusters in other galaxies whose stars are too faint for spectroscopy. A colour is a thermometer; on the giant branch it is also, through the opacity of one ion, an assay.

Where the simple version is wrong, and by how much #

The derivation assumes that the adiabat runs unbroken from the centre to the photosphere. It does not. Just below the photosphere there is a thin superadiabatic layer, where convection is inefficient because the gas is thin and radiation leaks out of rising parcels; and deeper down, where hydrogen and helium ionise, the gas’s heat capacity changes and the adiabat bends. Full stellar models treat both, the first through a mixing-length parameter calibrated on the Sun. Their Hayashi lines are shifted by a few hundred kelvin from the power-law version and are not perfectly straight, but they have the same steepness, because the steepness comes from the photospheric opacity, which the models share.

The opacity itself is a power law only over a limited range. Below about three thousand kelvin, molecules form in the atmosphere — water, titanium oxide, and in the coolest stars dust — and the opacity no longer follows the negative hydrogen ion’s ninth power. The Hayashi lines of the lowest-mass stars and brown dwarfs bend accordingly, and the line for half a solar mass drawn here extends into a temperature range where the simple law is no longer the whole story.

The same bending carries the argument below the hydrogen-burning limit. A brown dwarf contracting from its birth is fully convective like a young star, and it too descends a nearly vertical line at first, near two and a half thousand kelvin for the heaviest. But it never reaches a main sequence; it keeps contracting until electron degeneracy stops it, and then it cools at nearly constant radius, sliding to the right across the diagram through temperatures a fully convective star in equilibrium could not have. That is not a violation of the wall. The wall belongs to stars whose interiors are ideal gases on an adiabat, and a degenerate interior is neither, so its surface temperature is no longer tied to its luminosity by the same pair of conditions. The Hayashi line ends where degeneracy begins.

And the normalisation is set, not derived: the lines are anchored so that a solar-mass, solar-metallicity star at one solar luminosity has a surface temperature of four thousand kelvin, close to what the full models give. The exponents — how the temperature scales with luminosity, mass and metallicity — are what the argument derives, and they are what the figures claim.

The Sun itself is the calibration point for all of this, and it is worth being explicit about what it fixes. The Sun is not on a Hayashi line; its interior is radiative out to seven-tenths of its radius. But its outer convection zone is described by the same mixing-length treatment, with one free parameter, and that parameter is chosen so that a model of one solar mass reaches the Sun’s measured radius and temperature at its measured age. The same parameter is then used for every other star, convective or not. Every Hayashi line in every published set of stellar tracks inherits its position from that one calibration, and the observed spread of red giant branch colours at fixed metallicity is one of the few checks that it transfers from the Sun to stars with deep convective envelopes.

What was measured #

The Hayashi lines are tested wherever convective stars of known mass can be placed on the diagram. Young clusters are the cleanest case: their pre-main-sequence stars are a population of one age, spread along lines of their masses, and their positions agree with the lines to within the uncertainties of converting colours to temperatures. Eclipsing binaries among young stars give masses, radii and temperatures independently, and they sit on the predicted lines to a few per cent in temperature — with systematic departures for the most magnetically active stars, whose starspots and suppressed convection make them larger and cooler than the models, the same discrepancy that appears in the darkness a field pays for.

Red giant branches are tested against the metallicity dependence, and the sensitivity cuts both ways. The same factor that makes the giant branch colour a metallicity indicator makes it an opacity test: if the negative hydrogen ion’s opacity, or the free electrons from metals, were wrong in the models, the giant branches of clusters of known metallicity would sit at the wrong colours. They sit at the right ones to within about a hundred kelvin.

A surface fixed by the rarest ion #

The pattern recurs throughout stellar physics: a large-scale property of a star decided by a small-scale detail of its gas. The mean opacity of a stellar interior is dominated by the gaps between lines, not by the lines; the continuum of the Sun is made by an ion present at one part in a hundred million; and the colour of every convective star in the Galaxy — every star in the youngest clusters and every red giant — is fixed by how steeply that ion’s abundance rises with temperature. The luminosity of such a star can vary by a factor of ten thousand while its colour changes by less than an astronomer’s error bar on it a century ago, and the reason is the ninth power of a temperature in an ionisation equilibrium.

Still open: where a convective star leaves its line #

The Hayashi line is a boundary, and contracting young stars follow it only while they are fully convective. At some point a radiative core develops — earlier for heavier stars — and the star leaves the line, turning left along a nearly horizontal track toward the main sequence. When that happens is set by the opacity of the deep interior, not the photosphere: the core becomes radiative when its opacity has fallen enough that radiation can carry the luminosity without a superadiabatic gradient. Whether young stars’ radiative cores appear when standard models say, or later, is tested by the depletion of lithium, which burns at a temperature reached only at the base of a deep convection zone. The lithium measured in young clusters disagrees with the models in a way that points to inflated, magnetically active stars, and whether the fix belongs in the convection, the magnetic field or the interior opacity is not settled.