Evolution on the Main Sequence

By and large, once a star is on the Main Sequence it stays on the Main Sequence until it runs out of fuel (hydrogen--A star burns roughly 10 % of its mass while on the Main Sequence). However, a star does change a little as it burns hydrogen into helium in its core. The reason a star evolves while it is on the Main Sequence is quite easy to understand.

For stars like the Sun, the internal pressure is generated through ordinary gas pressure. The pressure is determined by how fast the gas particles move (as measured by the temperature of the gas, T) and by how many particles that there are in the gas volume. Quantitatively, we have that
- P = N k T
Now, make the obvious observation that during the fusion process, the number of particles in the core of a star gets smaller -- 4 hydrogen nuclei are fused into 1 helium nucleus ===> if the temperature remains the same, the pressure should go down.
This, in fact, does occur and the core of the star reacts by shrinking slowly which does 2 things:
- it increases the density of the core (increases N)
- it increases the temperature of the core (due to the compression)

Both of the above effects cause the nuclear reaction rates to increase; the former by increasing the rate of collisions and the latter by making each collision more energetic. The upshot of this type of evolution is that the luminosity of a Main Sequence star will increase while it is on the Main Sequence

This is a pretty substantial effect for the Sun.

Over the last 4.6 billion years, the luminosity of the Sun has increased by 30 %. In terms of the average temperature (equilibrium temperature) of the Earth, if nothing else happened, the average temperature would have increased by ~ 10 % over the lifetime of the Earth.

Show that a temperature increase of ~ 10 % is expected if the Earth is in thermal equilibrium (assume that the input of energy is due to the absorption of solar radiation).

Interestingly enough, the Earth has not been heating. It, in fact, is fairly certain that the Earth was warmer in the past than it is today. That is, when the Sun was fainter the Earth was warmer. This is obviously saying that we really do not understand how the climate of the Earth reacts to changes in the amount of energy received from the Sun.