The star that swells because its centre shrank

A star leaves the main sequence when the hydrogen in its core runs out, and it leaves sideways rather than downwards. What happens next is usually told as a swelling: the star becomes a red giant, and the Sun’s own version of it will reach past the orbit of Mercury.

Told that way it sounds like a relaxation — a star running out of fuel and going slack. It is the reverse. The envelope expands because the centre collapsed, and while the radius climbs by a factor of sixty the density at the centre climbs too.

The two radii are drawn below against the mass of the helium core between them, and nothing about that pair of curves is what a self-gravitating ball of gas is supposed to do.

Two radii that move in opposite directions #

The claim, stated plainly enough to be refused: between the base of the giant branch and its tip, a solar-mass star’s envelope radius and core radius change monotonically in opposite senses, and the central density rises throughout.

The generator refuses it in three independent ways. It walks two hundred sampled core masses and requires the envelope radius to rise and the core radius to fall at every one of them rather than only between the endpoints; it requires the central density to rise at every step; and it recovers the luminosity from the drawn radius and temperature, demanding $L = 4\pi R^2\sigma T^4$ back to one part in $10^{12}$. Any one failing stops the figure being drawn.

Why that is surprising is worth putting crudely. A ball of gas held up by its own pressure has one obvious response to losing its energy source: it contracts, all of it, because gravity is the only thing organising it. Something must sit between the two parts to decouple them.

The source that sits between the two #

What sits between them is a shell of burning hydrogen, thin and very hot. When the core hydrogen is gone, the helium left behind cannot burn at the temperature it finds itself at, so the core has no source and does what a source-less self-gravitating region must: it contracts, and the contraction heats it. The layer immediately outside is still hydrogen, dragged to a temperature the core interior never reached. It ignites, and burns harder than the core ever did. The shell now sets the luminosity, and it sits at a temperature fixed by the core’s mass and radius — and a cold degenerate core’s radius is a function of its mass. So the whole of the star’s output becomes a function of one number, the mass of a core that is not itself burning: the core-mass–luminosity relation, and it is steep, $L \propto M_c^{6}$. The mirror follows from that and one more fact. A deeply convective envelope cannot take any surface temperature it likes; it is pinned near the Hayashi line, a nearly vertical locus in the Hertzsprung–Russell diagram, so $T_{\rm eff}$ varies weakly while $L$ varies by three decades. Stefan–Boltzmann has no freedom left:

$$\frac{R}{R_\odot} = \left(\frac{L}{L_\odot}\right)^{1/2}\left(\frac{T_\odot}{T_{\rm eff}}\right)^{2}.$$

Write $T_{\rm eff} \propto M_c^{q}$ across the branch. With the Hayashi temperature falling from 5,000 K at the base to 3,700 K at the tip while the core grows from 0.149 to 0.48 solar masses, $q = \ln(0.74)/\ln(3.22) = -0.257$, and the exponent the radius inherits is

$$\frac{d\ln R}{d\ln M_c} = \frac{6}{2} - 2q = 3.51,$$

which is the number printed on the hero figure’s envelope curve. The core follows $R_c \propto M_c^{-1/3}$. One exponent is $+3.51$ and the other $-\tfrac13$: the mirror is the sign, and the factor of ten between the magnitudes is why the envelope’s motion is spectacular and the core’s is not.

The centre that ended up denser #

The core’s contraction is modest as a distance and enormous as a density, because density goes as the cube of a length. A core with $R_c \propto M_c^{-1/3}$ has $\rho \propto M_c/R_c^3 \propto M_c^2$, so the rise along the branch is the square of the core-mass ratio: $(0.48/0.149)^2 = 10.4$. The figure’s assertion is not that factor but that the curve rises at every step, which is what the title depends on.

The larger of the two factors is the one the drawing cannot show, and it is the one that changes the physics. The Sun’s centre today is an ordinary hot gas at about 150 g cm⁻³, held up by a pressure that responds to temperature; the base of the giant branch is a thousand times denser and held up by a pressure that has nothing to do with temperature at all. Between those two states the thermostat that keeps a main-sequence star stable is lost, and no part of the curve draws the transition.

Which numbers here are solved, and which are merely stated #

Nothing on any of these drawings was obtained by solving the equations of stellar structure: the sixth-power core-mass–luminosity relation and the Hayashi temperature are results taken from outside this repository and simply stated, and everything else on the figures is exact arithmetic performed on those two statements. A figure implying otherwise would be the worst thing this site could produce — a beautiful drawing that is wrong — so the two claims are worth separating exactly.

What is stated: the exponent 6; the calibration of that relation to one observed point, the giant-branch tip at 0.48 solar masses of core and 2,500 solar luminosities; the two Hayashi temperatures; the main-sequence and subgiant durations of 10 and 2 billion years; and the degenerate radius normalisation, 0.0125 solar radii at 0.6 solar masses, where the observed white-dwarf mass distribution peaks.

What is computed from those: every radius, by Stefan–Boltzmann; every core radius and central density, from the degenerate relation; the base of the branch, which is not a fourth stated number but the core mass at which the stated relation first delivers the terminal main-sequence luminosity, so the subgiant branch hands over at exactly the luminosity it arrived with; and the time to climb.

The distinction has teeth: the specification this generator was written from quoted a constant that missed the tip it was meant to reproduce by a factor of 2.2, so the calibration was kept and the constant derived from it instead.

The relation is also a clock #

The steep dependence of luminosity on core mass does something a static relation has no business doing: it dates the climb. The shell converts hydrogen to helium at exactly the rate the luminosity demands, so $dM_c/dt = L/(Xq)$, with $X = 0.7$ the hydrogen fraction and $q = 6.4\times10^{14}$ J kg⁻¹ the yield — not a stated constant either, but the 0.7 per cent mass defect of four protons becoming a helium nucleus, times $c^2$. Substituting $L = AM_c^{6}$ gives a closed form,

$$t = \frac{M_{c,0}^{-5} - M_{c,1}^{-5}}{5k},$$

and running it from 0.149 to 0.48 solar masses returns 0.99 billion years — the only duration on the track figure not put there by hand.

A branch that is sparse because it is quick #

The clock turns the track into a statement about how many stars will be seen where, which is this essay’s second argument.

A colour–magnitude diagram records no motion; it records how long stars linger, because a region is populated in proportion to the time spent crossing it. So the numbers on the track figure are a prediction about counts. The giant branch is 0.99 of 12.99 billion years, about 7.6 per cent of a solar-mass star’s post-formation life, while the final doubling of the radius is 4.8 million years, or 0.037 per cent — so in a population of ten thousand such stars, roughly four are in that last doubling at any moment.

A colour–magnitude diagram is therefore a histogram of durations, and the giant branch looks sparse for the same reason a fast-moving object is faint in a long exposure. Its largest stars are its rarest, and the rarity says nothing about how stars form.

What is actually measured about a giant #

Nothing here is observed directly. Everything is inferred from a brightness, a colour, an angle or a frequency, and it is worth naming which.

An angle. A giant’s radius can be had with no stellar model at all, from the baseline at which its fringes vanish: Betelgeuse’s null at about 3.1 metres gives an angular diameter of 0.047 arcseconds, and the radius follows only once a parallax is brought in, so its error is dominated by the distance rather than the angle. That is the one route to a giant’s size that avoids $L = 4\pi R^2\sigma T^4$, which makes it the strongest test these figures admit.

A brightness and a colour. Cluster diagrams give apparent magnitudes and colour indices, which become luminosities and temperatures only after a distance, an extinction and a temperature calibration are supplied. What was measured is the counts; the durations are the occupancy argument applied to them.

A frequency spacing. Red giants oscillate, and two numbers off the mode comb give a mass and a radius with almost no stellar model in the chain: the closest thing to a check on the mirror.

And what is never measured: the core. Its 9,368 kilometres and $1.66\times10^6$ g cm⁻³ are consequences of a stated relation, and the nearest observable is the mass of the white dwarf it becomes once the envelope has gone.

The tip that became a rung of the distance ladder #

The connection that pays for the whole apparatus arrives from an unexpected direction. The core mass at which helium ignites is set by the physics of degenerate matter rather than by the star wrapped around it, so it is nearly the same — close to 0.48 solar masses — for every star below about 1.8 solar masses. Feed that through the core-mass–luminosity relation and the luminosity at the tip is nearly the same too: a quantity calibrated here from one observed point is a near-constant of nature across a wide range of stellar masses.

A near-constant luminosity, recognisable by a sharp cut-off in a colour–magnitude diagram, is a standard candle. The tip of the red-giant branch is one of the ladder’s most-used rungs, at an absolute magnitude near $-4.0$ in the I band once calibrated on globular clusters and Galactic parallaxes, and it reaches ten megaparsecs in a few hours of telescope time. A relation about the interior of a star nobody can see has become a way of measuring the distance to other galaxies.

The same mirror in a cluster of stars #

The structure — a contracting inner region, an expanding outer one and an energy flow between them — is not peculiar to stars.

A globular cluster does exactly this, with stars in place of gas. Two-body encounters carry energy outwards from the dense core; the core loses it, contracts and heats, while the outer parts absorb it and expand. The result is a runaway rather than an equilibrium — the gravothermal catastrophe — and it produces the collapsed cores seen in about a fifth of the Galaxy’s globular clusters, in a system whose support comes from disorder rather than pressure.

So “a self-gravitating body contracts when it loses energy” is a statement about a body with no internal structure. Give it two regions and a source between them, and the sign of the response can differ across them.

The mirror running the other way #

If a contracting core forces an expanding envelope, the converse should hold: an expanding core should force a contracting envelope. It does, and watching it happen is how the rest of a low-mass star’s life is read.

When the degenerate helium core finally ignites, the energy released lifts the degeneracy and the core expands. The shell source above it, which had been squeezed between a shrinking core and the envelope, finds itself in a less compressed layer at a lower temperature, and its output falls. The envelope, no longer being pushed by so much flux from below, contracts — and a contracting envelope is a hotter, smaller, bluer star.

So the ascent of the giant branch reverses. The star leaves the tip, drops in luminosity by a factor of several, moves left across the diagram at roughly constant brightness, and settles onto the horizontal branch, where it burns helium in its core for perhaps a hundred million years.

The whole excursion is the same mirror, with the sign of the core’s motion flipped, and the fact that it works in both directions is the strongest argument that the mechanism is the shell rather than anything about the specific fuel.

The horizontal branch’s shape then carries information the ascent could not. A cluster’s horizontal-branch stars all have nearly the same core mass, because they all ignited helium at the same core mass — so their positions along the branch are set by how much envelope each retained, which is set by how much each lost during the ascent. A cluster whose branch extends far to the blue lost more envelope than one whose stars cluster to the red.

That makes the branch a record of mass loss, measured on a population rather than on an individual — and mass loss on the giant branch is precisely the quantity that no theory predicts and that the initial–final mass relation has to be calibrated around.

The branch’s colour distribution is also sensitive to composition and to age, and separating those three influences is the long-standing second-parameter problem — a case where one observable carries three causes and the population has to be large before any of them can be read.

What the picture cannot show #

The relations are stated, and saying so twice is deliberate. The core-mass–luminosity law and the Hayashi temperature come from outside this repository; no equation of stellar structure was integrated to make any figure here. What the drawings guarantee is internal exactness — Stefan–Boltzmann to machine precision among the quantities drawn, and a clock integrated rather than asserted — which is strictly weaker than having computed a star. A real calculation produces loops, a bump where the shell crosses a composition discontinuity, and a dependence on convection theory that none of these curves has.

The core is drawn cold. The degenerate radius used is the zero-temperature one, so the drawn radius is a lower bound and the density an upper one, loosest at the left-hand end where a real core is only partly degenerate.

The branch ends where the drawing ends, and the star does not. At the tip helium ignites, and in a degenerate core it ignites as a flash rather than a settling; the star then drops to the horizontal branch and later climbs the asymptotic giant branch to a larger radius than anything here. The 122 solar radii on the hero figure is the red-giant tip, not the biggest the star ever gets.

The horizontal axis is a core mass, not a time. Equal intervals on the mirror and density figures are wildly unequal intervals of time, and only the track figure carries the clock that says so. Nor is there mass loss anywhere: a real giant sheds much of its envelope in a wind.

One more mass shows that the mirror is not a property of a solar-type star.

Where the ladder goes next #

Later rungs on this anchor: the subgiant branch and the Hertzsprung gap; the helium flash and what lifting the degeneracy costs; the horizontal branch and the red clump as a second standard candle; thermal pulses and dredge-up; mass loss and the planetary-nebula phase; the initial–final mass relation that connects a star to the white dwarf it leaves; and the same sequence for a star massive enough that its core never becomes degenerate, where the mirror still operates and the clock runs a thousand times faster.

The mirror was not discovered by looking at stars. It emerged from the first numerical evolution calculations of the 1950s and 1960s as a regularity nobody had asked for — every model with a shell source expanding its envelope as its core contracted — and the account of why came afterwards, which in this subject is the usual order.