Key Idea: This topic is the physics of stars, from the reaction that powers them to how they live, shine and die.
Inside a star
- p–p chain: 4 ¹H → ⁴He, 26.7 MeV
- Gravity in = pressure out
- Lifetime t = E ÷ L
Measuring stars
- d (pc) = 1/p (arc-seconds)
- b = L/4πd²
- Wien gives T; L = σ4πR²T⁴ gives R
Lives of stars
- HR diagram: hot on the left
- Up to ~8 M☉: white dwarf
- Heavier: supernova, neutron star or black hole
It ties together six ideas: fusion releases energy from a mass defect (E = mc²); the outward pressure it creates balances gravity to keep a star stable; a star has a finite lifetime set by its fuel and its luminosity; we measure stars by their brightness and distance; we read their temperature, size and type from their light (Wien, Stefan-Boltzmann, the H-R diagram); and they end their lives — and forge the heavier elements — according to their mass. It is examined on both papers. Paper 1A is quick multiple-choice — which conditions allow fusion, how brightness changes with distance, a parallax distance, where a star sits on the H-R diagram, the order of a life cycle. Paper 2 is longer structured work — calculate the energy released in MeV, 'show that' a lifetime is about N years, estimate a mass lost or a stellar radius, find a radius ratio from L and T, or outline how an element is confirmed in a star.
Key formulas
Five of these carry the data-booklet badge (look for it). The lifetime relation t = E/L is not printed separately — you build it from 'luminosity = energy used each second', so you remember that one. The mass lost Δm = E/c² and the radius ratio are just E = mc² and Stefan-Boltzmann rearranged.
- energy released by the reaction (J, or MeV)
- mass that disappears — the mass defect Δm (kg, or u)
- speed of light, 3.00 × 10⁸ m s⁻¹ (given constant)
- apparent brightness — power received per unit area at the observer (W m⁻²)
- luminosity — total power the star radiates in all directions (W)
- distance from the star to the observer (m)
- surface area of the sphere the light has spread over (m²)
- parallax angle — half the star's apparent yearly shift, in arc-seconds (″)
- peak wavelength — where the star's radiation is most intense (m)
- surface (absolute) temperature of the star (K)
- Wien's constant, 2.9 × 10⁻³ m K (given)
- Stefan-Boltzmann constant, 5.67 × 10⁻⁸ W m⁻² K⁻⁴ (given)
- surface area of the star; for a sphere A = 4πR² (m²)
Not in the booklet: build them from the ones above.
- main-sequence lifetime (s)
- the Sun's luminosity, radius and surface temperature
The six ideas at a glance
| Idea | Key relationship | What to remember |
|---|---|---|
| Fusion & equilibrium | E = mc² (Δm × 931.5 for MeV) | Light nuclei fuse → product lighter → mass defect becomes energy. Needs high T + high density to beat Coulomb repulsion. Outward pressure (radiation + hot gas) balances gravity → stable radius. |
| Lifetime & mass loss | t = E ÷ L ; Δm = E ÷ c² | Lifetime = fuel's energy ÷ luminosity. Only ~10–12% of mass is core hydrogen, ~0.7% of that becomes energy. Bright stars die young. Mass lost = energy radiated ÷ c². |
| Brightness & distance | b = L/(4π d²) ; d = 1/p | Luminosity L is fixed (total power out); apparent brightness b falls as 1/d². Distance from parallax: d(parsec) = 1/p(arc-second), smaller angle → farther. |
| Wien's law | λₘₐₓ T = 2.9 × 10⁻³ | Read the peak wavelength → get the surface temperature. Shorter (bluer) peak = hotter. λₘₐₓ in metres. |
| Stefan-Boltzmann | L = σ(4πR²)T⁴ → L ∝ R²T⁴ | Links luminosity, radius and temperature. The T⁴ dominates. Radius ratio: RB/RA = √(LB/LA) ÷ (TB/TA)². |
| H-R diagram & evolution | position → type ; mass → fate | Temperature hot on the LEFT, luminosity up. Type from position: main sequence / red giant / white dwarf. Mass decides the ending; massive stars fuse heavier elements up to iron. |
Luminosity vs apparent brightness — the distinction that recurs
| Luminosity L | Apparent brightness b | |
|---|---|---|
| What it is | Total power the star radiates, in all directions | Power you actually receive, per square metre |
| Depends on distance? | No — a fixed property of the star itself | Yes — the same star looks dimmer farther off (b ∝ 1/d²) |
| Unit | watts (W) | watts per square metre (W m⁻²) |
| Where it appears | The vertical axis of the H-R diagram; set by R²T⁴ | What a detector measures; b = L/(4π d²) |
Luminosity is the total power the star pours out (fixed). Apparent brightness is what reaches us per m² (it drops with distance). The H-R diagram and Stefan-Boltzmann use luminosity; a detector reads apparent brightness — read which one the question gives you. And the temperature axis runs backwards on the H-R diagram: hot stars are on the LEFT, cool ones on the right.
Worked exam-style questions
Deep in a star's core, light nuclei fuse into a single heavier nucleus. The combined mass of the original nuclei exceeds the mass of the product by a mass defect Δm = 0.022300 u. Calculate the energy released by this reaction, in MeV. (1 u = 931.5 MeV c⁻².)
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A main-sequence star, Vega-B, has mass M = 2.0 × 10³⁰ kg. About 12% of its mass is hydrogen in the core that can fuse, and 0.70% of that fusible mass is released as energy. Its luminosity is L = 6.0 × 10²⁶ W. Show that its main-sequence lifetime is about 8 × 10⁹ years. (c = 3.00 × 10⁸ m s⁻¹; 1 year ≈ 3.16 × 10⁷ s.)
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A star has luminosity L = 3.2 × 10²⁸ W and is at a distance d = 5.0 × 10¹⁸ m from Earth. (a) Calculate its apparent brightness as seen from Earth. (b) A second, fainter star has a measured parallax angle of p = 0.020 arc-seconds. Find its distance, in parsecs.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
Star X has a black-body peak at λₘₐₓ = 725 nm and Star Y has a peak at λₘₐₓ = 290 nm. (a) Find the surface temperature of each. (b) Star Y is also 9 times as luminous as Star X — find the ratio of Star Y's radius to Star X's radius. (Wien's constant = 2.9 × 10⁻³ m K.)
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
On an H-R diagram, Star P sits at the bottom-left (hot, very dim) and Star Q sits at the top-right (cool, very bright). (a) State the type of each star and which has the larger radius, with a reason. (b) Star Q has a mass of about fifteen times the Sun's. Identify, in order, the stages it will pass through to the end of its life.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Important: Temperature on an HR diagram falls to the right: hot blue stars are on the left. And height means luminosity, not temperature: red giants are high up because they are huge, not hot.
Quick self-check
Tap each card to reveal the answer.
What two conditions let a star's core fuse nuclei, and why? Very high temperature (gives the positive nuclei enough speed to overcome the Coulomb repulsion) AND very high density / pressure (makes those fast collisions frequent enough).
How do you turn a mass defect in u into energy in MeV? Multiply Δm (in u) by 931.5, because 1 u = 931.5 MeV c⁻². (This is just E = mc² with the conversion built in.) For joules, convert Δm to kg first, then × c².
How do you estimate a star's main-sequence lifetime, and the mass it loses? Lifetime t = E ÷ L (fuel's energy ÷ luminosity), then ÷ 3.16 × 10⁷ for years. Only ~10–12% of the mass is core hydrogen and ~0.7% of that becomes energy. Mass lost by radiating: Δm = E ÷ c².
What happens to apparent brightness if a star is three times farther away? It falls to one ninth (b ∝ 1/d², so 1/3² = 1/9). The luminosity is unchanged — only the apparent brightness depends on distance.
Which law gives a star's temperature, and which gives its luminosity/size? Wien (λₘₐₓ T = 2.9 × 10⁻³) turns the peak wavelength into the temperature. Stefan-Boltzmann (L = σ(4πR²)T⁴, so L ∝ R²T⁴) links luminosity, radius and temperature — use the ratio for a radius.
How does a star's mass decide its fate? Low-mass (Sun-like): main sequence → red giant → planetary nebula → white dwarf. High-mass: main sequence → red supergiant → supernova → neutron star or black hole. Only massive stars fuse heavier elements (up to iron).
What is the overall reaction of the proton–proton chain? 4 ¹H + 2 e⁻ → ⁴He + 2ν + γ, releasing about 26.7 MeV.
What are the instability strip and lines of constant radius? The strip: a near-vertical band where stars pulsate. Constant-radius lines slope from upper left to lower right; top right = bigger.
What does a three-solar-mass star end as? A white dwarf: below about 8 solar masses there is no supernova.
Exam tips
Exam tips
- Energy released: find the mass defect FIRST (the product is lighter), then E = mc². In MeV, just multiply Δm (in u) by 931.5; for joules, convert Δm to kg then × c².
- Equilibrium: gravity pulls IN, the PRESSURE from fusion's radiation + hot gas pushes OUT. Say it is the pressure (not the reactions themselves) that balances gravity, and that the balance is self-correcting.
- Lifetime = fuel's energy ÷ luminosity: t = E ÷ L. Cut the fuel down twice (core fraction, then ~0.7% conversion) before E = mc², and convert seconds to years (÷ 3.16 × 10⁷). Mass lost = E ÷ c².
- Keep luminosity (total power out, fixed) and apparent brightness (received per m², falls as 1/d²) apart. Inverse-square: twice as far → a quarter as bright. Equal brightness ⇒ d ∝ √L.
- Parallax: d(parsec) = 1/p, with p in ARC-SECONDS and d straight out in PARSECS. A smaller angle means a farther star.
- Wien: T = 2.9 × 10⁻³ ÷ λₘₐₓ, with λₘₐₓ in METRES (nm → ×10⁻⁹). Stefan-Boltzmann: L = σ(4πR²)T⁴, so L ∝ R²T⁴. For two stars take the RATIO so σ and 4π cancel: RB/RA = √(LB/LA) ÷ (TB/TA)².
- H-R diagram: temperature axis runs BACKWARDS (hot on the LEFT), luminosity UP. Read the type from the position; a cool star can still be bright if it is huge (red giant), a hot one dim if it is tiny (white dwarf).
- Evolution: MASS decides the path — only massive stars go supernova (the Sun ends as a white dwarf). To confirm an element in a star: split its light, find the dark ABSORPTION lines, and match the pattern to that element in the lab.
- A diagram may give the whole angle a star shifts through over six months: halve it to get p, so d = 2/φ.
- Comparing a star with the Sun? Work in solar units: L/L☉ = (R/R☉)²(T/T☉)⁴, and σ and 4π cancel.