Physics · Gravitation and the universe · Year 1

Physics of stars

Where does the Sun stand among the stars, and how will it end?

The situation

Look at the night sky from Maidanak Observatory, high in the mountains of southern Uzbekistan: the stars are different colours — Betelgeuse is reddish, Rigel bluish, Vega pure white. Some are a few light-years away, some hundreds; some are hundreds of thousands of times brighter than the Sun, and some are barely bigger than the Earth. A single diagram, the Hertzsprung–Russell diagram, brings all this variety into order. Where does the Sun stand among the stars, and how will it end?

Topic

Physics and evolution of stars. Hertzsprung–Russell diagram: where the Sun is and how it will end

Key ideas

Hertzsprung–Russell diagram
L = 4πR²σT⁴
Parallax and magnitudes

Formulas

L = 4 π R^{2} σ T^{4}

Stefan–Boltzmann law: luminosity is proportional to the square of the radius and the fourth power of the temperature. The Sun: $L_{⊙} = 3.83 \cdot 10^{26}$ W.

λ_{m a x} = \frac{b}{T}, b = 2.898 \cdot 10^{- 3} m·K

Wien's law: a hot star is blue, a cool star is red.

d [pk] = \frac{1}{p [″]}, m - M = 5 \lg \frac{d}{10 pk}, m_{1} - m_{2} = - 2.5 \lg \frac{E_{1}}{E_{2}}

Distance from parallax (1 pc = 3.26 light-years), the distance modulus and Pogson's formula.

M_{1} + M_{2} = \frac{a^{3}}{P^{2}} (AU, yr, M_{⊙}), \frac{M_{1}}{M_{2}} = \frac{K_{2}}{K_{1}}

The masses of binary stars come from Kepler's third law.

\frac{d P}{d r} = - \frac{G M (r) ρ (r)}{r^{2}}

Hydrostatic equilibrium: in every layer the fall in pressure balances its weight.

Theory

Colour, temperature and spectral classes

A star's surface radiates almost like a black body: by Wien's law a 3 000 K star peaks in the infrared (it looks reddish), a 25 000 K star in the ultraviolet (bluish). The absorption lines in the spectrum also depend on temperature, so stars are divided into spectral classes: O B A F G K M (from hot to cool). The Sun is class G2, 5 772 K. Which atom each line belongs to is explained in the experiment “Atomic structure and spectra”.

Hertzsprung–Russell diagram

Temperature is on the horizontal axis (hot on the left) and luminosity on the vertical. About 90 % of stars lie on the main sequence — stars burning hydrogen in their cores; for them mass decides everything: $L \propto M^{3.5}$ , so the heavy, short-lived O–B stars are at the upper left, and the light, long-lived red dwarfs at the lower right. At the upper right are red giants and supergiants (at the same temperature but hundreds of times larger), and at the lower left white dwarfs (hot, but as small as the Earth). The dotted lines are lines of equal radius: $L \propto R^{2} T^{4}$ .

Distances and stellar magnitudes

The Earth's orbit (1 AU) serves as a baseline: in photographs taken half a year apart a nearby star shifts against the background of distant ones. The distance at which the annual parallax $p$ = 1″ is 1 parsec = 206 265 AU = 3.26 light-years. The parallax of the nearest star, Proxima Centauri, is only 0.768″ — which is why parallax was first measured only in 1838 (by F. Bessel). The Gaia space telescope (2013–2025) measured the parallaxes of about 1.5 billion stars. The apparent magnitude $m$ is how bright a star looks from Earth (Pogson: 5 magnitudes = a factor of 100), and the absolute magnitude $M$ is how it would look from 10 pc. Photometric measurements are covered in the experiment “Luminous flux and illuminance”.

Inside a star: equilibrium and energy

A star is a giant ball of gas: gravity tries to squeeze it, while the pressure of the hot gas inside tries to expand it. In hydrostatic equilibrium they are equal. The conditions at the centre can be estimated roughly: $T_{c} \sim μ m_{H} G M / (k R)$ — for the Sun of the order of 10⁷ K (the standard solar model gives 15.7 million K). At such a temperature protons overcome their repulsion and fuse: the p–p chain $4^{1} H \to^{4} He + 2 e^{+} + 2 ν$ releases 26.7 MeV for each helium nucleus (0.7 % of the mass). In heavier stars the CNO cycle dominates. By mass the Sun is roughly 74 % hydrogen, 25 % helium and 1–2 % heavier elements; inside there is a core (where the energy is produced), a radiative zone and a convective zone. The neutrinos that come out of it are measured on Earth.

Evolution: from birth to death

Stars are born in cold molecular clouds: a clump of the cloud contracts into a protostar. The lifetime on the main sequence is ≈ 10 billion years· ${(M / M_{⊙})}^{- 2.5}$ . When the hydrogen runs out the star becomes a red giant, and helium burns to carbon and oxygen in the core. Stars lighter than ≈ 8 M☉ shed their envelope as a planetary nebula, and the core remains as a white dwarf (≤ 1.4 M☉ — the Chandrasekhar limit). In heavy stars burning goes as far as iron; the iron core collapses — a supernova — leaving a neutron star (pulsar) or a black hole. Elements heavier than iron are made by neutron capture: in AGB stars (the s-process), and in supernovae and mergers of neutron stars (the r-process); a part of the lithium, beryllium and boron is produced by cosmic rays breaking up heavy nuclei. Merging neutron stars and black holes are sources of gravitational waves (first detected in 2015). These elements later go into new stars and planets; the picture of the expanding universe is covered in the experiment “The expanding universe”.

Binary and variable stars; the model

About half of Sun-like stars are in binary or multiple systems. From their orbit (a, P) and radial velocities (Doppler) the masses can be found — the main way of measuring mass directly. If the orbit is seen edge-on (the line of sight lies close to the orbital plane), the stars eclipse each other — eclipsing variables (Algol). In close binary systems matter flows from one star to the other; when a white dwarf collects hydrogen from its companion, a thermonuclear explosion occurs on its surface — a nova. Cepheids pulsate; H. Leavitt (1912) found that the longer the period, the greater the luminosity — this is a “scale” for measuring distance. Eruptive stars (for example flare red dwarfs) suddenly brighten. The model: the tracks roughly match published calculations; orbits are circular; discs are uniformly bright; dust absorption is not taken into account; the sizes of the stars are drawn on a compressed scale.

Did you know?

Maidanak Observatory in the mountains of southern Uzbekistan stands at an altitude of about 2 600 m. The atmosphere there is very calm: the “twinkling” of a star's image is often less than 1″ — one of the best astronomical sites in the world.
Every second the Sun converts about 4 million tonnes of mass into energy: $L_{⊙} / c^{2} \approx 4.3 \cdot 10^{9}$ kg/s. Even so, over 4.6 billion years it has lost only about 0.03 % of its mass.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Newton's cannon: an artificial satellite — Grade 10
Special relativity — Grade 10
Law of universal gravitation — Grade 10
Relativistic dynamics — Grade 10
Kepler's laws — Grade 10
The expanding universe — Year 1
Black holes — Year 4