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Stars are where A-level physics measures things nobody can touch. A star’s colour gives its temperature, its temperature and luminosity give its size, and its apparent brightness gives its distance. Examiners test this with short chained calculations, H–R diagram interpretation, and a 6-mark account of how a star evolves.
By the end you’ll be able to
Use Wien’s displacement law λ_max T = 2.9 × 10⁻³ m K and Stefan’s law L = 4πr²σT⁴
Use intensity I = L/4πd² and standard candles to find distances; define the parsec and light-year and use parallax
Use magnitude scales and m − M = 5 log(d/10) where required (AQA)
Classify stars by spectral class (O B A F G K M) and relate absorption lines to temperature
Place main-sequence stars, red giants, white dwarfs and supergiants on an H–R diagram and describe evolutionary paths, supernovae, neutron stars and black holes
What the exam asks
Black-body calculations: temperature from peak wavelength (Wien), then luminosity or radius (Stefan), often chained together.
Distances: the inverse-square law with standard candles, parallax and the parsec, and (AQA only) magnitudes.
Spectra: how absorption lines form and (AQA) why the hydrogen Balmer lines depend on temperature.
The H–R diagram: its axes and regions, comparing radii, and evolutionary paths.
End states: white dwarfs, neutron stars, black holes and supernovae.
Core ideas
Luminosity, intensity and the inverse-square law
LuminosityL is the total power a star radiates. IntensityI is the power received per unit area at distance d (Cambridge calls it radiant flux intensity, F):
I
vii.Check your understanding
3 questions on stars: luminosity, temperature, classification and evolution. Every option is explained once you answer.
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PromptCard 1 of 3
State Wien’s displacement law.
+15
Rs≈2GM/c2
P=σAT4
Edexcel 9PH0 (Paper 2) and IAL (Unit 5): I=L/4πd2, standard candles, parallax, L=σAT4, Wien’s law with 2.898×10−3 m K, and the H–R diagram with the life cycle of stars.
OCR A (5.5.1–5.5.2): star formation and evolution, electron degeneracy pressure, the Chandrasekhar limit, and line spectra. The data sheet gives λmax∝1/T with no constant (compare with the Sun or use the value you are given) and L=4πr2σT4.
Cambridge 9702 (25.1–25.2, Paper 4): F=L/4πd2 (radiant flux intensity) and λmax∝1/T, both to be recalled, plus the Stefan–Boltzmann law to estimate radii. There is no H–R diagram or magnitude scale.
Eduqas (Component 2) and WJEC (Unit 1): stellar spectra, black bodies, Wien, Stefan and the inverse-square law, and multiwavelength astronomy. Magnitudes are not required.
CCEA (AS 2): the same calculations. Check your specification for which star topics it lists.
=
4πd2L
This assumes the star radiates equally in all directions and that nothing absorbs the radiation on the way. A standard candle is an object of known luminosity, such as a Cepheid variable or a type Ia supernova. Measure its intensity and d=L/4πI.
Distance units and parallax
Unit
Definition
Value
astronomical unit (AU)
mean distance from the Earth to the Sun
1.50×1011 m
light-year (ly)
distance light travels in one year
9.46×1015 m
parsec (pc)
distance at which 1 AU subtends 1 arcsecond
3.08×1016 m = 3.26 ly
As the Earth orbits the Sun, a nearby star appears to shift against the distant background stars. The parallax anglep is half the total shift over six months. With p in arcseconds, d=1/p in parsecs. The method only works to a few hundred parsecs; further away, the angles are too small to measure.
Magnitudes (AQA only)
Apparent magnitudem measures how bright a star looks. On the Hipparcos scale the brightest stars are about 1 and the faintest stars visible to the naked eye are 6. Brighter means smaller. A difference of 5 magnitudes is an intensity ratio of 100, so 1 magnitude is a ratio of 1001/5=2.51.
Absolute magnitudeM is the apparent magnitude the star would have at 10 pc, so it measures luminosity:
m−M=5log10d(d in parsecs)
Black-body radiation: Wien and Stefan
A black body absorbs all radiation that falls on it and emits a continuous spectrum that depends only on its temperature. Stars are close to black bodies. A hotter body’s curve is higher at every wavelength, and its peak is at a shorter wavelength.
Wien’s law: λmaxT=2.9×10−3 m K, so hot stars are blue-white and cool stars are red.
Stefan’s law: L=σAT4=4πr2σT4, with σ=5.67×10−8W m−2K and T in kelvin.
To find a radius, get T from Wien’s law and L from the inverse-square law, then use r=L/4πσT4. For ratios, use L∝r2T4.
Spectra and spectral classes
The hot, dense surface of a star emits a continuous spectrum. Cooler, low-density gas in its atmosphere absorbs photons whose energy hf matches a gap between its energy levels, then re-emits that energy in all directions. Less light of that wavelength reaches us, so a dark absorption line appears. Each element has a unique set of lines, which reveals the atmosphere’s composition.
Class
Colour
Temperature / K
Prominent absorption lines (AQA)
O
blue
25 000–50 000
He⁺, He, H
B
blue
11 000–25 000
He, H
A
blue-white
7500–11 000
H (strongest), ionised metals
F
white
6000–7500
ionised metals
G
yellow-white
5000–6000
ionised and neutral metals
K
orange
3500–5000
neutral metals
M
red
below 3500
neutral atoms, TiO
The visible Balmer lines need hydrogen atoms whose electrons are already in n=2. In cool stars almost all hydrogen is in the ground state, and in very hot stars most of it is ionised. Around 10 000 K (class A) the n=2 population is largest, so the Balmer lines are strongest.
The Hertzsprung–Russell diagram
The H–R diagram plots luminosity (log scale, or absolute magnitude with −10 at the top) against surface temperature, which decreases to the right.
Main sequence: a diagonal band from hot and luminous (top left) to cool and dim (bottom right). These stars fuse hydrogen in their cores, and about 90% of stars are here.
Red giants and supergiants (top right) are cool yet very luminous, so they must be huge.
White dwarfs (bottom left) are hot yet dim, so they must be tiny, about the size of the Earth.
For two stars at the same temperature, r1/r2=L1/L2.
Life cycle of stars
Formation: a nebula of gas and dust collapses under gravity, and the lost gravitational potential energy heats it into a protostar. At about 107 K, hydrogen fusion starts in the core.
Main sequence: the outward pressure of gas and radiation balances gravity. The Sun stays here for about 1010 years, but massive stars use up their fuel in a few million years.
Low-mass stars (up to about 10 solar masses in OCR’s scheme): when the core runs out of hydrogen, it contracts and hydrogen fuses in a shell around it. The outer layers expand and cool, and the star becomes a red giant (it moves up and to the right). Helium fuses into carbon and oxygen. The outer layers then drift off as a planetary nebula, leaving a white dwarf (bottom left). A white dwarf has no fusion. Electron degeneracy pressure (no two electrons can share a quantum state) holds it up, provided its mass is below the Chandrasekhar limit of 1.44 solar masses.
Massive stars become red supergiants and fuse successively heavier elements, up to iron. Iron fusion releases no energy, so the core collapses and the outer layers are blasted away in a supernova. What remains is a neutron star (10–20 km across, density about 1017kg m−3), or a black hole if the core is more than about 3 solar masses.
AQA extras. In a type Ia supernova, a white dwarf pulls matter from a binary companion until it passes the Chandrasekhar limit. They all peak at about the same absolute magnitude, −19.3, so they are standard candles. Their light curve rises to a sharp peak in about 20 days, then fades over months. A black hole’s event horizon has the Schwarzschild radiusRs≈2GM/c2, where the escape velocity equals c. Supermassive black holes sit at the centres of most galaxies. Gamma-ray bursts come from supergiants collapsing, and in seconds they release about as much energy as the Sun will emit in its lifetime.
Worked examples
Exam technique
Chains run Wien (temperature), then inverse square (luminosity), then Stefan (radius). Keep unrounded values in your calculator.
Ratios need no constants: write L∝r2T4 and substitute the factors.
H–R sketches: label both axes with scales, with temperature decreasing to the right. Mark the Sun at about 5800 K and 1L⊙.
Evolution 6-markers: take the stages in order. For each, name the physics (gravity against pressure, what fuses, what supports the remnant) and the star’s movement on the H–R diagram.
Common mistakes
Quick recap
I=L/4πd2, and standard candles give distance from measured intensity.
d(pc)=1/p(arcsec), and 1 pc =3.08×1016 m =3.26 ly.
λmaxT=2.9×10−3 m K and together give stellar radii.
AQA: 5 magnitudes is a factor of 100, and m−M=5log(d/10).
Balmer lines are strongest in class A stars, because the n=2 population peaks near 10 000 K.
A Sun-like star becomes a red giant, then a white dwarf. A massive star becomes a red supergiant, then a supernova, then a neutron star or black hole.