Hydrostatic equilibrium

A star is a self-gravitating sphere of hot plasma that emits radiation because energy is generated in its interior. For most of its life, especially on the main sequence, a star stays about the same size because the inward and outward effects balance.

Hydrostatic equilibrium is a mechanical equilibrium in a fluid body in which the inward gravitational compression is balanced by an outward pressure gradient. In a star, gravity pulls each layer inward. Hot gas pressure and radiation pressure push outward. The surface is not “held up” by anything solid; pressure from hotter, deeper layers supports it.

Thermal pressure is a pressure produced by the random motion of particles in a gas or plasma. Radiation pressure is a pressure produced when photons transfer momentum to matter. In a stellar core, both matter and radiation are energetic, so both can help support the star against gravity.

Why gas laws are useful, but not the whole story

A star is not an ideal gas sitting in a school laboratory, but gas models still give the right first instinct. When a gas cloud contracts, gravitational potential energy is transferred into internal energy, and the temperature rises. A hotter gas has particles with larger average kinetic energy, so for a given density it produces a greater pressure.

That is why the gas laws can model stars: they link pressure, temperature and particle number density. The limitation matters too. Stellar material is an ionized plasma, photons carry energy and momentum, and nuclear reactions change the composition. The model is useful because it captures the pressure–temperature connection, not because a star is literally a simple ideal gas sample.

Equilibrium in a star compared with nuclear stability

The balance in a star is a large-scale equilibrium between gravity and pressure. Nuclear stability is different: a nucleus exists because the short-range strong nuclear interaction can overcome electric repulsion between protons at nuclear separations. If only gravity and electric forces were available, ordinary nuclei could not exist. Gravity between nucleons is far too weak, and the electric force between protons is repulsive. Stars and nuclei both involve competing interactions, but they act on very different length scales and with different dominant forces.

What fusion is

Nuclear fusion is a nuclear reaction where light nuclei join to make a heavier nucleus with a larger binding energy per nucleon. Stars release energy by fusion because the products are more tightly bound than the reactants. The “missing” mass turns up as energy carried by particles and photons.

For energy release calculations, use

In nuclear calculations you may also use

In Sun-like main-sequence stars, the proton–proton chain is the dominant fusion process. Taken as a whole, four hydrogen nuclei become one helium nucleus, with positrons, electron neutrinos and energy produced as well:

A commonly quoted total energy release is about 26.7 MeV per completed helium nucleus, though neutrinos carry away some of that energy.

The proton–proton chain

The usual simplified chain has three stages:

u_e$$: two protons form deuterium, emitting a positron and an electron neutrino. This stage is slow, since it needs a weak-interaction change from proton to neutron.

• ${}^{1}_{1}\mathrm{H} + {}^{2}_{1}\mathrm{H} \to {}^{3}_{2}\mathrm{He} + \gamma$: a proton fuses with deuterium to make helium-3 and a gamma photon.
• ${}^{3}_{2}\mathrm{He} + {}^{3}_{2}\mathrm{He} \to {}^{4}_{2}\mathrm{He} + {}^{1}_{1}\mathrm{H} + {}^{1}_{1}\mathrm{H}$: two helium-3 nuclei make helium-4 and send two protons back into the plasma.

A neutrino is a neutral lepton that interacts only very weakly with matter. Solar neutrinos give direct evidence that fusion is happening in the Sun’s core, since they escape from the core with very little interaction. Historically, detectors found fewer electron neutrinos than expected. The explanation was that neutrinos can change type during flight, so experiments sensitive to only one type measured only part of the flux. It’s a neat example of conservation laws and particle detection pushing the physics into something more subtle.

Energy from binding energy

A binding energy is the energy required to separate a bound system into its individual components. For nuclei, a greater binding energy per nucleon means the nucleus is more stable. In a fusion calculation, compare the total binding energy of the products with the total binding energy of the reactants. If the products have greater total binding energy, the difference is released.

Fusion and fission both release nuclear energy by moving nuclei towards the high-binding-energy region near iron and nickel. The direction is different: fusion combines light nuclei, while fission splits heavy nuclei. Fusion of light nuclei typically gives a large energy release per unit mass because the binding energy per nucleon changes steeply for small nucleon numbers.

Atomic and nuclear photons

Atomic transitions emit photons when electrons change energy levels in atoms; their energies are usually in the visible, ultraviolet or X-ray range depending on the atom and transition. Nuclear transitions involve changes inside the nucleus and usually emit much higher-energy gamma photons. In both cases the photon is the same kind of particle, but the energy scale and the physical system making the transition are different.

Temperature and density

Fusion in a star needs two conditions at the same time: very high temperature and very high density. Temperature measures the average random kinetic energy of particles in matter. When the core temperature is high, nuclei collide at high speeds and can get close enough for the strong nuclear interaction to act.

Density is mass per unit volume. In a stellar core, high density packs many nuclei into a small volume, so collisions happen often. Temperature makes collisions energetic enough; density makes enough of them happen each second.

Why protons do not fuse easily

Hydrogen nuclei are protons, so they repel each other electrically. For fusion to happen, two protons must come extremely close, within the range of the strong nuclear interaction. Even in the Sun, most proton–proton encounters don't produce fusion. The first step of the proton–proton chain is slow because, while the two particles are briefly close together, one proton must change into a neutron through the weak interaction.

That slow rate is not a defect; it is why Sun-like stars last so long. If proton fusion were easy, main-sequence stars like the Sun would use up their fuel far too quickly for stable planetary systems to develop over billions of years.

Formation of stars from gas clouds

A protostar is a contracting gas object that has not yet settled into stable hydrogen fusion. It starts as a slightly denser region of interstellar gas. Gravity strengthens the density difference, the cloud contracts, and the central temperature rises. Once the core reaches the required temperature and density, sustained fusion begins and the star joins the main sequence.

The first elements, mainly hydrogen and helium with a trace of lithium, formed early in the universe. Heavier elements are made later inside stars and in explosive stellar events. So the short answer to “How are elements created?” is: light elements came from early-universe nucleosynthesis, while many heavier elements are produced by stellar fusion and supernova processes.

Mass controls the whole story

A star’s initial mass does most of the work in setting its lifetime, luminosity and final state. More mass means stronger gravitational compression, so the core has to get hotter to push back with enough outward pressure. Once the core is hotter, the fusion rate rises sharply. That is why massive stars are extremely luminous, but don’t last long.

This often feels backwards at first. A massive star does have more fuel, but it burns through that fuel vastly faster. Low-mass stars live quietly for a very long time; high-mass stars blaze brightly and die violently.

Moderate-mass evolution

A main-sequence star is a star in the long-lived stage during which hydrogen fusion in the core supplies most of its energy. When the core starts to run out of hydrogen, the outward pressure drops. The core contracts and heats, while hydrogen fusion carries on in a shell around it. The outer layers expand and cool, so the star becomes a red giant.

In a moderate-mass star, the core becomes hot enough for helium fusion to produce heavier nuclei such as carbon and oxygen. It does not become hot enough to keep fusion going through many later stages. Eventually, the star expels its outer layers to form a planetary nebula, leaving the hot dense core behind as a white dwarf.

A white dwarf is a compact stellar remnant supported against gravitational collapse by electron degeneracy pressure. Electron degeneracy pressure is a pressure caused by the quantum rule that electrons cannot all occupy the same quantum state. A white dwarf no longer has sustained fusion; it shines because it is hot, then slowly cools.

High-mass evolution

A high-mass star leaves the main sequence with a hotter core. It can fuse heavier elements in a sequence of shells, forming an onion-like internal structure: lighter-element fusion in the outer shells, with heavier products closer to the core. Fusion up to the iron region no longer releases useful energy in the same way, because iron-group nuclei are already near the maximum binding energy per nucleon.

The Chandrasekhar limit is the maximum mass of a white dwarf, about 1.4 solar masses, above which electron degeneracy pressure cannot support the remnant. If the core remnant is above this limit, collapse carries on. Electrons and protons combine to form neutrons and neutrinos, and the outer layers can be thrown out in a supernova.

A neutron star is a compact stellar remnant made mainly of neutrons and supported by neutron degeneracy pressure. A black hole is a region of spacetime from which nothing, not even light, can escape. The Oppenheimer–Volkoff limit is the maximum mass of a neutron star, of order 2–3 solar masses, above which neutron degeneracy pressure cannot prevent collapse to a black hole.

Present observations and future predictions

Astronomers cannot run controlled experiments on stars, so they treat observations of many stars at different stages as a substitute for time-lapse evidence. Since light takes time to travel, looking at distant objects also means looking at earlier states of the universe. From these observations, astronomers build models for stellar evolution and for the universe’s future. The models are not certain on their own; their predictions depend on the quality of the observations and on the assumptions used.

What an HR diagram shows

A Hertzsprung–Russell diagram is a scatter graph used to classify stars by plotting luminosity against surface temperature. Luminosity means the total power a star radiates. In IB-style HR diagrams, luminosity goes on the vertical axis and temperature goes on the horizontal axis. The odd detail: temperature increases to the left.

For a spherical star, the Stefan–Boltzmann law is

On an HR diagram, stars with constant radius sit on diagonal lines, since luminosity depends on both surface area and temperature.

Main regions of the HR diagram

The main sequence is the diagonal band of stars carrying out core hydrogen fusion. Hot, luminous main-sequence stars sit toward the upper left; cool, dim main-sequence stars sit toward the lower right. The Sun is roughly in the middle of the main sequence.

A red giant is cool but luminous, with a very large radius. It appears toward the upper right of the HR diagram: cool surface, large emitting area, high total luminosity. A supergiant is an extremely large and luminous star found near the top of the HR diagram.

A white dwarf appears in the lower left of the HR diagram. It can be hot but dim because it has a very small surface area. The instability strip is a narrow, nearly vertical region containing stars whose luminosities vary because they pulsate. You need to locate it and interpret it, but detailed Cepheid-variable work is not required.

Surface temperature, colour and spectra

You can estimate a star’s surface temperature from its black-body spectrum: hotter stars peak at shorter wavelengths and look bluer; cooler stars peak at longer wavelengths and look redder. So black-body radiation is not just a thermal physics idea — it is one of the main tools for reading stellar properties from light.

A stellar spectrum shows the intensity of light from a star after it has been separated by wavelength. Absorption and emission lines reveal chemical composition, because atoms and ions absorb or emit photons at characteristic wavelengths. Atomic spectra give information about elements in stellar atmospheres and intervening gas; shifts of those spectral lines can also reveal motion along the line of sight.

Classification as a scientific tool

HR diagrams are useful because they turn a crowd of data points into a pattern. Physics uses classification in many similar ways: scalar and vector quantities, types of radiation, elementary particles in the Standard Model, and regions of the electromagnetic spectrum. Good classification does more than label things; it reveals relationships that help models make predictions.

The distance units

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. You should be able to convert between AU, ly and pc.

Stellar parallax

Stellar parallax measures distance from the apparent shift of a nearby star against distant background stars as Earth orbits the Sun. The full baseline is the diameter of Earth’s orbit, but the parallax angle uses half of that baseline. So, in the simple parsec definition, the distance used is effectively $1\,\text{AU}$.

Use

An arcsecond is $1/3600$ of a degree. A small $p$ gives a large $d$; that’s the practical problem.

Limits and technology

From Earth’s surface, atmospheric turbulence and absorption make precise parallax measurements harder. Space telescopes avoid much of the atmosphere, so they can measure much smaller angular shifts. Missions such as Gaia have transformed stellar astronomy by measuring positions, distances and motions for enormous numbers of stars. This is a good example of technology changing the data available to physics: better instruments don’t just give prettier pictures, they test models more severely.

Radius from luminosity and temperature

To find a stellar radius, start with the Stefan–Boltzmann law. Rearranging $L = 4\pi\sigma R^2T^4$ gives $R = \sqrt{L / 4\pi\sigma T^4}$. You need the star’s luminosity and surface temperature.

It is often neater to compare the star with the Sun:

Rearranging,

Use this form when the luminosity is given in solar units.

Getting luminosity first

A telescope measures apparent brightness; it does not measure luminosity directly. Apparent brightness is the power received per unit area at the observer.

xxxDefinitionEndxxx For nearby stars, the distance may come from parallax. For more distant objects, it may come from other calibrated methods.

Getting temperature from the spectrum

You can estimate the surface temperature from the wavelength where the star’s black-body spectrum peaks.

xxxDefinitionEndxxx The constant has units m K, so T comes out in kelvin.

The working route is: measure parallax to get distance, measure apparent brightness to get luminosity, use the spectrum to get surface temperature, then calculate radius. Each step adds uncertainty, so the final radius is an estimate, not a ruler-length measured directly.

A standard candle is an astronomical object with a known luminosity that can be used to estimate distance from its observed brightness. Standard candles carry the same brightness–distance idea far beyond the range where parallax is practical.