Topic 5.6: E = mc²: The Energy in a Star
In Topic 3.7 we met Einstein's equation. Now we use it.
E = m c squared
- E is energy in joules (J).
- m is mass in kilograms.
- c is the speed of light, 300,000,000 m/s, so c squared is 9 x 10^16.
That huge number says that a very small amount of mass is equal to a very large amount of energy.
How Much Energy?
One kilogram of matter: E = 1 x (3 x 10^8) squared = 9 x 10^16 joules. That is about the energy of a 21-megaton nuclear bomb, or enough electricity to power a large city for months. A kilogram of coal burned in a power plant gives 3 x 10^7 J, about three billion times less.
Where the Missing Mass Goes
When four hydrogen nuclei fuse into one helium nucleus, the helium weighs slightly less than the four protons added together:
| Mass (atomic mass units) | |
|---|---|
| 4 hydrogen nuclei | 4 x 1.00783 = 4.0313 |
| 1 helium nucleus | 4.0026 |
| Mass lost | 0.0287 (about 0.71%) |
That 0.71% turns into energy, mostly gamma rays and the motion of particles. For each helium nucleus formed, 26.7 million electron volts of energy are released. That is a tiny amount, but the Sun makes about 10^38 helium nuclei every second.
The Sun's Power Bill
The Sun's power output (its luminosity) is 3.8 x 10^26 watts. Divide by c squared to find how much mass becomes energy each second:
m = 3.8 x 10^26 / 9 x 10^16 = 4.2 x 10^9 kg per second, about 4 million tonnes every second.
That sounds like a lot, but the Sun has 2 x 10^30 kg. It has lost only about 0.03% of its mass to fusion over its whole life so far.
How Long Can It Last?
Only about 10% of the Sun's mass is in the core where fusion can happen, and only 0.71% of that fuel's mass turns into energy.
- Mass converted: 0.10 x 0.0071 x 2 x 10^30 kg = 1.4 x 10^27 kg
- Energy available: 1.4 x 10^27 x 9 x 10^16 = 1.3 x 10^44 J
- Time at 3.8 x 10^26 watts: 1.3 x 10^44 / 3.8 x 10^26 = 3.3 x 10^17 seconds = about 10 billion years
That is exactly what other methods give.
Who Figured This Out?
- Einstein published E = mc squared in 1905.
- In 1920 Arthur Eddington (the same man who tested relativity at the 1919 eclipse) suggested that stars shine by turning hydrogen into helium.
- In 1938-39 Hans Bethe worked out the detailed reactions, the CNO cycle and the pp-chain, and won a Nobel Prize for it in 1967.
(A note on timelines: the lifetimes and ages in this lesson come from the standard model of how stars work, which is tested by physics we can measure today. Many Christians who study science accept those ages; others, including creation scientists, hold to a much younger universe. How a star shines, how bright it is and how hot it is do not depend on that debate. We give the numbers because they appear in textbooks and tests, and so you can recognize them when you meet them.)
Worth a Pause
Colossians says that in Christ "all things consist" (Colossians 1:17), that is, they hold together. Physicists use a related word for something similar, that mass and energy are conserved, one turning into the other without anything being lost. A star is the loudest example: it spends four million tonnes of itself every second and does not run down for ages. It is worth being a little amazed that the same few numbers (c, G, the mass of a proton) are the same in every star we can see.
Try It: Mass Matters. Slide a star's mass and watch its color, brightness, size, lifetime and fate change. Play Mass Matters »
Practice Problems
- Write Einstein's equation and say what each symbol means.
- How much energy is in 1 kg of matter? (c = 3 x 10^8 m/s)
- How much mass does the Sun convert to energy each second? (L = 3.8 x 10^26 W)
- What fraction of the mass of four hydrogen nuclei is lost when they fuse into helium?
- Why does the Sun not lose a noticeable fraction of its mass even after billions of years?
- Who first suggested that stars shine by fusing hydrogen into helium, and who worked out the details?
- Compare the energy released by burning 1 kg of coal (3 x 10^7 J) with converting 1 kg of mass completely into energy.