Stars: Structure & Evolution: Balance Sheets of Weight, Heat and Time

In the 1850s and 1860s Hermann von Helmholtz and Lord Kelvin asked where the Sun's energy comes from. Burning coal would last a few thousand years. Their best answer was gravity: let the Sun shrink slowly, and the weight of its outer layers falls inward and turns into heat. It could shine that way for tens of millions of years. Geologists and Darwin wanted hundreds of millions at least, and the argument ran for decades, until nuclear physics in the 1920s and 30s showed a far bigger reservoir.

The lesson is lasting: a star is a bookkeeping problem. Weight pushes in, heat pushes out, and the sums have to balance at every layer, in every moment, for billions of years. Once you see how the books are kept, a surprising amount of stellar life follows from a few lines of arithmetic.

You can see the results with a naked eye. Orion's Betelgeuse is red because it is cool and swollen; Rigel is blue-white because it is hot. The Pleiades are young and blue; the old star clusters are yellow and red. Astronomers read those colours as ages.

Hydrostatic equilibrium: a star that loses heat gets hotter

Cut a slice through a star. The gas below the slice must hold up all the gas above it, so pressure rises toward the centre. The rule is hydrostatic equilibrium: at each layer, the pressure difference across the layer balances the weight of the layer. If it did not, the layer would fall or rise at once. If the balance failed, the Sun would collapse or expand within about an hour, its free-fall time, so any star we see is in this balance.

A first consequence comes from comparing sizes. Gravity's grip on a gas particle grows with the star's mass and shrinks with its radius. For the gas to hold, each particle must be moving fast enough to resist, which means a temperature that scales as mass divided by radius. This is the virial theorem.

Worked for the Sun: with an average particle mass of 0.6 proton masses, this gives 2.8 million K for a uniform Sun. The real core is 15.7 million K, about 5.7 times hotter, because mass piles up toward the centre. Now the strange part. If a star radiates energy and shrinks, R goes down, so T goes up. The star gets hotter as it loses energy. Physicists describe this as negative heat capacity, and it is the reason stars can start fusion at all: a cloud collapsing under its own weight heats until the centre is hot enough to ignite.

Without JavaScript: for the Sun, contraction alone could power it for about 31 million years. The nuclear fuel lasts about 10 billion.

That 31 million years is Kelvin's number. It was close to right for what gravity alone can supply, and it was wrong only about what the Sun uses. In 1920 Arthur Eddington argued that hydrogen turning to helium is the real source, and in 1939 Hans Bethe worked out the reaction chains.

Fusion at fifteen million degrees: tunnelling and a thermostat

Two protons repel each other. To fuse they must approach within about a femtometre, where the electric repulsion is about 1.4 million electron-volts of energy. At the Sun's core the typical thermal energy is 1.35 thousand electron-volts. Classically, the fraction of protons energetic enough is about e⁻¹⁰⁰⁰, around 10⁻⁴⁶⁰. In a Sun of 10⁵⁷ protons, none would ever fuse.

The way out is quantum mechanics (see 34/83). A particle's wave leaks through the barrier. George Gamow worked out the tunnelling in 1928, and Robert Atkinson and Fritz Houtermans applied it to stars in 1929. The reaction rate is the product of two competing things: fewer and fewer particles at high energy, and a tunnelling chance that rises with energy. Their product peaks at a middle energy, the Gamow peak, here about 6 thousand electron-volts. That is about four and a half times the thermal energy, and a few hundred times below the barrier top.

Out of this formula comes a striking result: how fast fusion speeds up as the core gets hotter. For two protons, the rate rises as about the fourth power of temperature. For a proton hitting nitrogen-14 (charge 7), the barrier is 49 times higher in this measure, and the rate rises as about the twentieth power. The Sun mostly uses the proton–proton chain, and the carbon–nitrogen–oxygen (CNO) cycle supplies about 1%, a share the Borexino experiment confirmed by detecting CNO neutrinos in 2020. In stars of about 1.3 Suns and heavier the extreme temperature sensitivity lets CNO take over.

Without JavaScript: at 15.7 million K the pp rate goes as T^3.8 and CNO as T^19.6. A 10% hotter core makes 44% more pp energy and 6.5 times more CNO energy.

That steep dependence works as a thermostat. Suppose the core makes a bit too much energy. The gas heats and expands, which lowers the temperature, which cuts the fusion rate sharply back. Too little energy, and the core shrinks and heats, and the rate rises. The Sun has held steady for 4.6 billion years this way. The same steepness is why massive stars burn out so fast: a bit more mass means a hotter core and a rate that grows like a high power of temperature.

Getting the energy out: a hundred-thousand-year random walk

The energy made at the Sun's centre must reach the surface. Most of the way it travels as light, but the interior is so dense that a photon cannot go far. It is absorbed and re-emitted within a fraction of a centimetre, in a random direction. A random walk gets out slowly: after N steps of length l you are only about √N × l from the start, so to cover the Sun's radius R takes about (R ÷ l)² steps.

With l about 0.03 cm, a typical value averaged over the Sun, this gives about 170,000 years. Estimates found in the literature run from tens of thousands to millions of years, depending on how the opacity is averaged; 170,000 years is the figure of Mitalas and Sills (1992). Without scattering the light would take 2.3 seconds. The sunshine on your face began its trip long before the first farms. The photons that finally reach you are new ones, made in the last scatterings at the surface.

Without JavaScript: a walker must take about (R ÷ l)² steps, so doubling the distance takes four times as many. For the Sun, that is about 170,000 years.

Deep in the star, in some layers, radiation is not the best way to carry heat; the gas itself rises and sinks, boiling like a pot. This is convection. It takes over where the temperature falls steeply with radius. In the Sun it carries the heat through the outer 30% by radius, which is why the surface is a seething grain of cells. Stars lighter than about a third of the Sun are convective all the way through, which mixes their fuel and lets them burn for trillions of years. In stars heavier than about 1.3 Suns it is the core that boils.

Leaving the main sequence: a cluster is a clock

A star's hydrogen lasts, in round figures, 10 billion years divided by its mass to the power 2.5 (see 34/88). When the core runs out, the core contracts and heats, hydrogen burns in a shell around it, and the outer layers swell and cool. The star leaves the main sequence for the giant branch and gets brighter and redder.

Astronomers use this on clusters: groups of stars born together from one cloud. All the stars have the same age but different masses, so at any time the heavy ones have left the main sequence and the light ones have not. The mass at the point where the main sequence bends off, the turn-off, gives the age of the cluster, because the star at the turn-off is just now running out of fuel. A young cluster like the Pleiades, about 100 million years old, has a main sequence reaching up into hot blue stars. An old globular cluster such as 47 Tucanae, about 12 billion years old, has a main sequence cut off at a star lighter than the Sun.

Without JavaScript: stars of 1 Sun mass leave the main sequence after about 10 billion years in this model, stars of 5 Suns after about 100 million.

The oldest globular clusters come out at roughly 12 to 13 billion years by this method, a little under the 13.8 billion years that cosmology gives for the universe. Two independent clocks that do not contradict each other are a strong check on both.

Endings: what the mass decides

What happens next depends almost entirely on mass. The Sun will swell into a red giant in about 5 billion years, losing a good fraction of its mass in winds. Mercury and Venus will be swallowed; whether Earth survives is borderline in published models. By then the oceans will long since be lost, because the Sun brightens by roughly 10% every billion years. The Sun will shed its outer layers as a glowing shell, a planetary nebula, and its core will remain as a white dwarf of about half a Sun mass, supported by electron degeneracy pressure.

A star above about 8 Suns goes further. It fuses carbon, neon, oxygen and silicon in turn, each stage faster than the last, until an iron core forms. Iron cannot release energy by fusing (see 34/86), so the core collapses in a fraction of a second into a neutron star, and the rebound launches a supernova. The energy is huge: the collapse releases about 3 × 10⁴⁶ joules, of which only about 1% goes into the explosion, 99% leaving as neutrinos. When Supernova 1987A exploded in the Large Magellanic Cloud, about 168,000 light-years away, detectors on Earth caught 25 neutrinos on 23 February 1987 hours before the light arrived: 12 at Kamiokande II, 8 at IMB and 5 at Baksan. The first look inside a collapsing core came from a couple of dozen particles.

Without JavaScript: below about 0.075 Suns, brown dwarf; to about 8 Suns, white dwarf; to roughly 20–25 Suns, usually a neutron star; above that, often a black hole.

White dwarfs cannot exceed about 1.4 Suns, the Chandrasekhar limit. A white dwarf in a binary that gains matter can approach it and explode as a type Ia supernova, with a fairly standard brightness. That is why these explosions work as distance markers across the universe (see 34/88). What they leave, and how neutron stars and black holes behave, is the work of Compact Objects & High-Energy Astrophysics (36/43).

Stars make the elements

The Big Bang made hydrogen, helium and a trace of lithium. Everything heavier was made in stars or in the events that end them. The oxygen you breathe and the calcium in your bones were made in massive stars, and carried out by supernovae and winds into the gas clouds that later formed the Sun. In 1957 Margaret and Geoffrey Burbidge, William Fowler and Fred Hoyle published the foundational paper on how it all works (usually called B²FH), after Hoyle had predicted in 1953 the special energy level in carbon that makes the triple-alpha reaction possible.

Iron is the end of the line for fusion, and heavier elements need neutrons. Slow neutron capture in giant stars builds barium and lead. Rapid capture needs a flood of neutrons for a second or less, and for gold that source was long a mystery. In August 2017 the LIGO and Virgo detectors recorded gravitational waves from two neutron stars merging (GW170817), and telescopes then watched the afterglow, a kilonova, whose light matched freshly made heavy elements. So mergers make r-process elements. Whether they make most of them is not settled.

Without JavaScript: hydrogen, helium and some lithium from the Big Bang; carbon in giant stars; oxygen and much iron in supernovae; barium in giant stars; gold in neutron-star mergers and possibly other events.

What a graduate just sees

• A star is a balance of weight and pressure at every layer. Lose energy and it contracts and heats.
• Temperature scales as mass over radius, so a collapsing cloud always finds the temperature to start fusion if it has enough mass.
• Fusion works by tunnelling, and its rate climbs as a steep power of temperature, which makes the Sun a thermostat.
• Light takes about a hundred thousand years to leave the Sun, but eight minutes to cross space afterwards.
• A cluster is one age and many masses, so the turn-off reads its age.
• Mass is destiny: brown dwarf, white dwarf, neutron star or black hole, with a thin fuzzy border between them.
• Most of the elements heavier than helium were forged in stars and scattered by their deaths.

Where it shows up next

• Compact Objects & High-Energy Astrophysics (36/43): white dwarfs, neutron stars and black holes are the remnants this piece stops at.
• Galaxies & the Milky Way (36/42): a galaxy's colour and history are the sum of its stars' lifetimes, and the elements stars make enrich each generation.
• Planetary Science & Exoplanets (36/40): a star's brightening over billions of years moves its habitable zone outward.
• In the world: solar neutrino detectors, fusion energy research (which copies the Sun's tricks at lower density and higher temperature), and stellar models behind the age of the oldest known stars.

If you want to go deeper

• 📘 Textbook. Bradley Carroll and Dale Ostlie, An Introduction to Modern Astrophysics (Cambridge University Press, 2017 reissue of the 2nd edition), for the standard undergraduate treatment. Rudolf Kippenhahn, Alfred Weigert and Achim Weiss, Stellar Structure and Evolution (Springer, 2nd edition 2012), for the full structure equations.
• 📘 Gentler. James Kaler, Stars and Their Spectra (Cambridge University Press, 2nd edition 2011), a readable tour of the stars and what their light says.
• 🌐 Free tool. MESA, the open-source stellar evolution code used in research, with its documentation and tutorials.
• 🌐 Free course. MIT OpenCourseWare 8.901 Astrophysics I (Spring 2006), covering stellar structure and evolution.