You estimated the Sun's lifetime as fuel divided by the rate of use. Binary stars showed how luminosity depends on mass. Put the two together, and you can predict how long any main-sequence star will last.
Why it works
A star's fuel is proportional to its mass $M$ (each burns about the same fraction of its hydrogen on the main sequence). The rate it uses fuel is its luminosity, which grows steeply with mass: roughly $L \propto M^{3.5}$. So the lifetime goes as $$t \propto \frac{M}{L} \propto \frac{M}{M^{3.5}} = M^{-2.5}.$$ Scaling to the Sun's 10 billion years: $t \approx 10\ \text{billion years} \times M^{-2.5}$ (with $M$ in solar masses). A heavier star has more fuel, but its stronger gravity squeezes its core hotter and denser, and fusion speeds up far more than the fuel supply grows.
Key idea
Main-sequence lifetime $\approx 10\ \text{billion years} \times \dfrac{M}{L} \approx 10\ \text{billion years} \times M^{-2.5}$. Massive stars live fast and die young; small stars last for hundreds of billions of years.
| Mass ($M_\odot$) | Luminosity ($L_\odot$, about $M^{3.5}$) | Lifetime (about) |
|---|---|---|
| 0.5 | 0.09 | 57 billion years |
| 1 | 1 | 10 billion years |
| 2 | 11 | 1.8 billion years |
| 5 | 280 | 180 million years |
| 10 | 3200 | 30 million years |
| 25 | 78,000 | a few million years |
Worked examples
Worked example: A 4-solar-mass star
How long does a $4\,M_\odot$ star last on the main sequence?
- $4^{2.5} = 4^2 \times \sqrt{4} = 16 \times 2 = 32$.
- $t \approx \frac{10}{32} = 0.31$ billion years $\approx 310$ million years.
Answer: About $310$ million years.
Worked example: Fuel over rate
A star has $3$ solar masses and $81$ solar luminosities. Estimate its lifetime.
- $t = 10\ \text{billion years} \times \frac{M}{L} = 10 \times \frac{3}{81}$.
- $t \approx 0.37$ billion years $= 370$ million years.
Answer: About $370$ million years.
Common mistake
It seems logical that a big star, with more fuel, would last longer. But it burns that fuel thousands of times faster. A $10\,M_\odot$ star has 10 times the Sun's fuel and about 3000 times its luminosity, so it lasts about $\frac{10}{3000}$ of the Sun's lifetime: some 30 million years. Check: every red dwarf ever born is still shining, because none has had time to run out.