A car's range is the fuel in its tank divided by how fast it burns fuel. The Sun is the same. With fusion as the fuel and its luminosity as the burn rate, scientific notation turns a big question into a short calculation: how long can the Sun shine?
Why it works
Lifetime $=$ energy available $\div$ power. Only the core gets hot enough to fuse, and by the time the core has turned about a tenth of the Sun's hydrogen into helium, the Sun begins to change. So the energy available is about $0.1 \times 0.007 \times M_\odot c^2$. Divide by $L_\odot$, and the answer is about 10 billion years. Alternatives fail this test badly: a Sun that burned chemically would last a few thousand years, and one that shone only by slowly shrinking (gravity heating it) would last about 30 million years. Rocks and meteorites show the solar system is 4.6 billion years old, so only fusion fits.
Key idea
Main-sequence lifetime $\approx \dfrac{\text{fuel}}{\text{rate}} = \dfrac{0.1 \times 0.007\,M_\odot c^2}{L_\odot} \approx 10$ billion years for the Sun. It is about 4.6 billion years old, so about halfway through.
Worked examples
Worked example: Fuel divided by rate
Estimate the Sun's main-sequence lifetime. Use $M_\odot = 1.99 \times 10^{30}$ kg, $L_\odot = 3.83 \times 10^{26}$ W, $10\%$ of the mass fused, $0.7\%$ of that converted.
- Mass converted: $0.1 \times 0.007 \times 1.99 \times 10^{30} = 1.39 \times 10^{27}$ kg.
- Energy: $1.39 \times 10^{27} \times 9.00 \times 10^{16} = 1.25 \times 10^{44}$ J.
- Time: $\frac{1.25 \times 10^{44}}{3.83 \times 10^{26}} = 3.27 \times 10^{17}$ s; divide by $3.156 \times 10^{7}$ s per year: $1.0 \times 10^{10}$ years.
Answer: About 10 billion years.
Worked example: A coal Sun
If the Sun were coal, releasing $3 \times 10^{7}$ J per kilogram, how long could it shine?
- Energy: $1.99 \times 10^{30} \times 3 \times 10^{7} = 6.0 \times 10^{37}$ J.
- Time: $\frac{6.0 \times 10^{37}}{3.83 \times 10^{26}} = 1.6 \times 10^{11}$ s, about $5000$ years.
- Recorded history alone is longer than that, so the Sun cannot be burning.
Answer: About $5000$ years.
Common mistake
Two opposite myths: that the Sun will burn out within a few thousand years, and that it will end in a supernova. It has about 5 billion years of hydrogen left in its core. And it is far too light to explode: only stars of more than about 8 solar masses end as supernovae. The Sun will swell into a red giant, puff off its outer layers, and leave a white dwarf. Check: a supernova needs a heavy iron core collapsing, and the Sun will never make iron.