The big idea: A full petrol tank lasts far longer in a frugal car than in a gas-guzzler — how long it lasts is just the fuel ÷ how fast you burn it. A star's lifetime works exactly the same way.
Its fuel is the fusible hydrogen in its core, and its burn rate is its luminosity — so its lifetime is the energy the fuel is worth ÷ how fast the star spends it.
And because that energy comes from converting mass (E = mc²), the star slowly gets lighter as it shines.
What runs the star
- Its fuel = the hydrogen in the core it can actually fuse
- Its burn rate = the luminosity L (energy radiated each second)
- Each kilogram of fuel is worth a fixed amount of energy (E = mc²)
How long it lasts
- Total energy available = fuel turned into energy
- Lifetime = energy available ÷ how fast it is used
- So a bright, hungry star burns out faster than a dim one
New words, plainly: Main sequence = the long, stable middle of a star's life, when it is steadily fusing hydrogen into helium.
Luminosity (L) = the total energy a star radiates every second — its power output, in watts (W = J s⁻¹).
Fusible hydrogen = only the hydrogen in the hot, dense core can fuse, and only a small fraction of its mass is released as energy. So the usable fuel is far less than the star's whole mass.
| You need | What it is | Symbol / unit |
|---|---|---|
| The fuel's energy | energy the fusible hydrogen releases when it fuses | E (J) |
| The burn rate | luminosity — energy the star radiates per second | L (W = J s⁻¹) |
| The lifetime | how long the fuel lasts at that burn rate | t (s, then years) |
Why a brighter star dies younger: A bright star has a huge luminosity — it burns through its fuel fast.
A dim star sips its fuel slowly, so it lasts far longer.
That is why the most luminous stars live only millions of years, while faint ones can outlast the Universe so far.
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Two steps. First find the energy the fusible hydrogen can release, using E = mc². Then divide by the luminosity (the energy used each second) to get the lifetime in seconds — and convert to years.
- energy released as the fusible hydrogen fuses (J)
- mass that is actually converted to energy (kg)
- speed of light, 3.00 × 10⁸ m s⁻¹ (given constant)
- main-sequence lifetime — how long the star keeps fusing hydrogen (s)
- total energy the fusible hydrogen can release (J)
- luminosity — the energy the star radiates each second (W = J s⁻¹)
Only a sliver of the mass is fuel: Two cuts shrink the usable fuel right down:
- Only the core's hydrogen fuses — typically about 10–12% of the star's mass. - Of that mass, only about 0.7% is actually released as energy when hydrogen turns into helium.
So multiply the star's mass by both fractions before using E = mc².
A Sun-like star, Helios-B, has mass M = 2.4 × 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 = 5.0 × 10²⁶ W. Show that it stays on the main sequence for about 1.1 × 10¹⁰ years. (c = 3.00 × 10⁸ m s⁻¹; 1 year ≈ 3.16 × 10⁷ s.)
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How this is tested — stellar lifetime and mass loss is a Paper 2 topic that comes in three linked parts:
Paper 2 — the lifetime
- Show that a star lives ≈ N years: find the fusible-H energy with E = mc², then t = E ÷ L, then convert seconds to years.
Paper 2 — the follow-ups
- State one assumption behind the estimate (e.g. luminosity stays constant; only core hydrogen fuses).
- Estimate the mass the star loses by radiating: Δm = E ÷ c².
The classic trap: Using the star's whole mass as fuel. Only the core's hydrogen fuses, and only ~0.7% of that mass becomes energy — multiply by both fractions first.
| The question type | What you do |
|---|---|
| Show the lifetime ≈ N years | energy of fusible H ÷ luminosity, then ÷ (3.16 × 10⁷) to get years |
| State an assumption | e.g. luminosity stays constant; only core hydrogen fuses; fixed fusible fraction |
| Estimate the mass lost | Δm = E ÷ c², where E is the total energy radiated |
A hot, bright main-sequence star, Lyra-A, has mass M = 4.0 × 10³⁰ kg. About 10% of its mass is fusible core hydrogen, and 0.70% of that fusible mass is released as energy. Its luminosity is L = 3.0 × 10²⁷ W. Show that its main-sequence lifetime is about 2.7 × 10⁹ years. (c = 3.00 × 10⁸ m s⁻¹; 1 year ≈ 3.16 × 10⁷ s.)
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State one assumption made when estimating a star's main-sequence lifetime this way.
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Over its whole main-sequence life, Helios-B radiates a total energy of about E = 1.8 × 10⁴⁴ J. Estimate the total mass it loses by radiating this energy. (c = 3.00 × 10⁸ m s⁻¹.)
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