# NUCLEAR FUSION IN STARS

### RETURN OF THE H-R DIAGRAM

• One more fact and we can understand the Main Sequence in the H-R Diagram (!)

Q.   A more massive star has a higher or lower central temperature?
A.   Higher of course - hydrostatic equilibrium demands this.

The P-P Chain reaction rate increases (steeply) with temperature. A higher central temperature means the P-P rate is higher and more energy is released - this means a larger luminosity.

So more massive stars have higher central temperatures which results in higher P-P rates which releases more energy and causes the star to have a higher luminosity. So for stars on the main sequence, more massive stars = higher luminosities

• Pay attention now - this explains why the main sequence exhibits a Luminosity-Temperature relation, and why as you move to higher temperature and luminosity along the main sequence you are also moving along a mass sequence.

### NUCLEAR FUSION AND THE LIFETIMES OF STARS

• The Universe is changing in one direction -   H He .

• The Sun, and other stars too, will eventually run out of fuel.

We calculated the lifetime of the Sun as an H-fusion-powered object, but this assumed the entire Sun was pure hydrogen and that ALL the hydrogen could be fused to helium. In fact, by the time that about 10% of the hydrogen is fused to helium, the Sun will change its structure and will no longer be a main-sequence star.

• We can already figure out the rough main-sequence lifetimes of stars of different masses.

A 10 M has a central temperature of 30,000,000 K and a luminosity of 40,000 L .

We know that the fuel of stars is mass and that the rate of fuel consumption is the luminosity. So a 10 M star would be expected to live 10 times longer based on its extra fuel, but, 1/40000 less long based on its fuel consumption:

 Lifespan ( 10 M ) = 10 x Lifespan ( 1 M ) = 0.00025 x Lifespan ( 1 M ) 40,000

Similarly a 0.3 M star with 0.01 L would have a main-sequence lifetime that was:

 Lifespan ( 0.3 M ) = 0.3 x Lifespan ( 1 M ) = 30 x Lifespan ( 1 M ) 1/100

So even though more massive stars have more fuel (mass), they use it up faster (higher luminosity) and so have shorter main sequence lifetimes than the less massive stars.