Week 7 Session: The Sun's lifetime, calculated

The estimates that rule out chemical burning and gravitational contraction, redone by hand instead of read off the lecture.

30 September 2026

Week 7 said this out loud: you cannot settle whether the Sun burns chemically or gravitationally by arguing about it. You calculate. This session is where you do the calculating, not the lecture.

Before the session#

Bring a calculator, or trust your own arithmetic. You need three numbers, all already given in the lecture: the Sun’s luminosity, about (3.8 \times 10^{26}) joules per second; the Sun’s mass, about (2 \times 10^{30}) kilograms; and a rough chemical energy yield of about (10^7) joules per kilogram, the right order for ordinary combustion.

In the session#

Work the chemical case first. Multiply the Sun’s mass by the chemical energy yield to get the largest energy store generous chemistry could offer, then divide by the luminosity to get a lifetime. Convert seconds into years yourself — do not take the lecture’s answer on trust. You should land somewhere around a couple of thousand years.

Then work gravity. Estimate the gravitational energy released by the Sun having contracted to its current size, using (GM_\odot^2/R_\odot), and divide by the luminosity again. This one takes longer to land, and the answer is a spectacular improvement on chemistry — tens of millions of years — while still falling short of the billions of years radiometric dating gives the Earth and Sun.

Neither estimate needs precision. Get the exponents right and the argument holds; a factor of two either way changes nothing about the verdict.

Afterwards#

Keep your working. It is the same calculation that ends Week 7, done by your own hand instead of read off the page — the moment this course stopped settling anything by arguing.