The Sun's core is the power plant. You met fusion in Fission and Fusion: light nuclei joining into heavier ones, with a little mass disappearing as energy. In the Sun the fuel is hydrogen and the product is helium.
Why it works
Four hydrogen nuclei (protons) end up as one helium-4 nucleus, through a series of steps called the proton–proton chain (along the way two protons turn into neutrons, giving off positrons and neutrinos). Weigh both sides with atomic masses: $4 \times 1.007825 = 4.031300$ u before, $4.002603$ u after. The difference, $0.028697$ u, is $0.71\%$ of the starting mass, and it leaves as energy, $E = mc^2$: $0.028697 \times 931.5 = 26.7$ MeV per helium nucleus. Protons repel each other, so fusion needs extreme heat and pressure: about 15 million K and enormous density in the core, where nuclei collide fast enough, with help from quantum tunneling, to fuse.
Key idea
The Sun fuses hydrogen into helium in its core; about $0.7\%$ of the mass becomes energy by $E = mc^2$. To give off $3.83 \times 10^{26}$ W it turns about $4.3 \times 10^{9}$ kg of mass into energy every second.
Worked examples
Worked example: One kilogram of hydrogen
How much energy comes from fusing $1$ kg of hydrogen into helium?
- Mass converted: $0.007 \times 1 = 0.007$ kg.
- $E = mc^2 = 0.007 \times (3.00 \times 10^{8})^2 = 6.3 \times 10^{14}$ J.
- Burning $1$ kg of coal gives about $3 \times 10^{7}$ J: fusion gives about 20 million times as much.
Answer: $6.3 \times 10^{14}$ J.
Worked example: The Sun's mass loss
The Sun gives off $3.83 \times 10^{26}$ W. How much mass does it turn into energy each second, and how much hydrogen does it fuse?
- Mass per second $= \frac{L}{c^2} = \frac{3.83 \times 10^{26}}{9.00 \times 10^{16}} = 4.26 \times 10^{9}$ kg/s, over 4 million tonnes a second.
- That is $0.7\%$ of the hydrogen fused: $\frac{4.26 \times 10^{9}}{0.007} = 6.1 \times 10^{11}$ kg/s, about 600 million tonnes every second.
- Even at that rate, the Sun's $2 \times 10^{30}$ kg lasts billions of years.
Answer: $4.26 \times 10^{9}$ kg/s converted; $6.1 \times 10^{11}$ kg/s of hydrogen fused.
Common mistake
The Sun is not burning like a fire. Burning is a chemical reaction that needs oxygen and releases only about $10^{7}$ J per kilogram; a Sun made of coal would burn out in a few thousand years. Check: Earth's rocks are billions of years old, and only nuclear fusion, millions of times richer than burning, can keep the Sun shining that long.