Week 7: Why hasn't the Sun burned out?
Chemical burning gives the Sun a few thousand years. Gravity gives it tens of millions. The rocks say billions. This is the week where arguing stops working and you have to calculate.
21 September 2026
Last week ended with the Sun paying almost every energy bill on Earth.
It has been doing that for a very long time.
The oldest rocks on Earth are billions of years old. Life has been here for billions of years too. Whatever keeps the Sun shining has to work on that kind of timescale.
So what is the Sun actually burning?
For centuries the obvious answer was exactly what the word suggests: fuel. Wood burns. Coal burns. Oil burns. Perhaps the Sun was an enormous fire supplied by some enormous chemical store.
We can argue about whether that sounds plausible.
Or we can calculate.
This is the point in the course where those become different things.
First guess: chemical burning#
The Sun radiates energy at roughly
$$ L_\odot \approx 3.8 \times 10^{26}\ \text{joules per second}. $$
That number is its luminosity: how quickly it is losing energy into space.
Suppose, very generously, that the entire Sun were made from material capable of releasing around
$$ 10^7\ \text{joules per kilogram} $$
through chemical reactions. That is the right rough scale for ordinary chemical fuels.
The Sun has a mass of about
$$ M_\odot \approx 2 \times 10^{30}\ \text{kg}. $$
So the largest chemical energy store we could hope for would be roughly
$$ E_{\text{chemical}} \sim 10^7 \times 2 \times 10^{30}#
2 \times 10^{37}\ \text{J}. $$
How long does that last?
There is only one calculation we need:
$$ \text{lifetime}#
\frac{\text{energy available}}{\text{energy used each second}}. $$
So
$$ t \sim \frac{2 \times 10^{37}} {3.8 \times 10^{26}} \approx 5 \times 10^{10}\ \text{s}. $$
Convert that into years and you get only around two thousand.
That answer is catastrophic.
The Earth is not a few thousand years old. The Sun cannot be an ordinary chemical fire.
The calculation has killed the explanation.
Second guess: gravity#
There is another way to release energy without burning anything.
Drop an object and gravity accelerates it. Let an enormous cloud of gas collapse inward and gravitational energy can become heat.
Perhaps the Sun has simply been shrinking.
This idea does much better.
A rough gravitational energy scale for the Sun is
$$ E_{\text{grav}} \sim \frac{GM_\odot^2}{R_\odot}, $$
where (G) measures the strength of gravity and (R_\odot) is the Sun’s radius.
Put in the numbers and the available energy is around
$$ 10^{41}\ \text{J}. $$
Divide by the Sun’s luminosity again:
$$ t \sim \frac{10^{41}} {3.8 \times 10^{26}} \sim 10^{15}\ \text{s}. $$
That is tens of millions of years.
A spectacular improvement.
Still wrong.
Kelvin ran essentially this calculation in the 1860s, and used the result to insist that geologists’ timescales for the Earth were physically impossible. By the nineteenth century, geology was already making it increasingly difficult to fit Earth’s history into such a short lifetime. Modern measurements put the age of the Earth at about 4.5 billion years, and the Sun is roughly the same age.
Gravity buys millions.
We need billions.
There is nowhere left to hide inside the old explanations.
Something much deeper is paying the bill#
The breakthrough is that matter itself contains an energy scale enormously larger than chemistry can reach.
Einstein’s relation is
$$ E = mc^2. $$
The important part here is not memorising the formula. Look at (c^2).
The speed of light is about
$$ c \approx 3 \times 10^8\ \text{m/s}, $$
so
$$ c^2 \approx 9 \times 10^{16}. $$
That multiplier is enormous.
Losing even a tiny amount of mass can release an enormous amount of energy.
That is what happens in the Sun.
Hydrogen nuclei ultimately combine into helium. The helium nucleus has slightly less mass than the ingredients that went into making it. The missing mass has not disappeared.
It has become energy.
For the main hydrogen-to-helium process, roughly (0.7%) of the participating mass is converted.
That sounds tiny.
It is enough.
Only part of the Sun’s mass is available for fusion over its lifetime, but once (E=mc^2) enters the accounting, the available energy is large enough to keep a star like the Sun shining for around ten billion years.
Now the numbers agree with the rocks.
For the first time, we have an energy source with the right scale.
And notice what happened.
Chemical burning was not rejected because nuclear physics sounded more sophisticated. Gravity was not rejected because a newer theory came along.
They were rejected because the numbers failed.
The mathematics did not decorate the explanation.
It chose between explanations.
But the successful answer immediately creates another problem.
The Sun is mostly hydrogen. Hydrogen nuclei are protons, and protons carry positive electric charge.
Bring two positive charges together and they repel.
The closer you push them, the stronger the electrical barrier becomes. Fusion requires nuclei to get extraordinarily close before the strong nuclear interaction can bind them together.
The centre of the Sun is extremely hot.
But for many of its nuclei, it is not classically hot enough.
According to the physics we have used so far, those nuclei should approach, slow down under electrical repulsion, and turn back before they ever get close enough to fuse.
Yet the Sun is shining above you.
So something is getting through a barrier it should not be able to cross.
What we explained#
The Sun cannot be powered by ordinary chemical burning: even an unrealistically generous chemical energy store lasts only thousands of years. Gravitational contraction can extend the lifetime into millions of years, but the Earth and Sun are billions of years old. Nuclear fusion reaches the required timescale because a small loss of mass releases an enormous amount of energy through (E=mc^2).
This is why mathematics arrived this week. The competing explanations sounded plausible until their consequences were calculated. The numbers ruled two of them out.
What became stranger#
How do two nuclei that can’t touch, touch?