In the book "A Really Short History Of Nearly Everything", I read that the larger the star is, the faster it burns itself. Whys that? Wouldn't there be more energy to burn if it's larger, and just be brighter? Our sun can burn for billions of years, but a planet bigger than our sun will only burn for millions of years. Why is that?
Nuclear fusion rates in the core of a star have very non-linear and strong dependences on temperature, pressure and density (for temperature it's like T^40 for some processes - http://www.astro.soton.ac.uk/~pac/PH112/notes/notes/node117.html). So, as the star gets bigger, denser and hotter at the centre, the rates of fusion rise much faster and the star loses energy quicker, hence living a shorter life.
Note that the star tries to balance itself to maintain stability, so the actual dependence is not that high, and ends up being related to the mass luminosity relationship, where the dependence is not that steep.
The mass luminosity relation is non-linear (L = M^3.5 for the main sequence) so more massive stars burn at a much faster rate. This is a non-linear relationship because the fusion is caused by pressure in the core from the star trying to gravitationally collapse on itself.
It's easy: the biggest the star is the more matter it has to burn but also:
- It has bigger burning zones, so it burns at higher rates.
- As the star is bigger it reaches to higher pressures and temperatures at it's core, so the star can start heavier elements (than H) burning earlier than smaller and cooler stars This creates an onion like structure where different elements are fused at each layer ending in a core of inert Ni and Fe that can't be fused to create energy.
We could compare a big star to a small one like a Formula 1 car to a simple car: the F1 is faster, but it runs out of fuel earlier.
The easiest way to think of it is terms of the cube square law coming into play.
The speed at which a star can radiate heat is mostly dependent on its surface area. but it's ability to generate heat is dependent on its volume. Surface area increases with the square of the increase in radius but volume increases with cube of the radius. As a star grows in a diameter it's inside grows faster than its outside.
Even if all other factors stayed the same, the star would grow hotter because it has less surface area to radiate from.
Since gravity is dependent on volume, the gravitational compression (simplistically) scales with the cube as well. So, a larger star compresses its interior more per unit of mass than does a smaller star.
As Joan.bdm pointed out, stars have an interior volume, the burning zone down towards the core in which the pressures are high enough for fusion to take place. Probably should call it the burning volume
The burning volume is the volume in which fusion can occur. Fusion occurs once a specific threshold of heat and pressure have been reached. Below that threshold, fuel consumption is zero.
The cube square law means that the relative size of the burning volume to total volume increases faster than the diameter. E.g If a small star has ten percent of its mass inside the burning zone, a star with twice the diameter will have 20% of its total volume inside the burning zone. (actual ratios invented for demonstration purposes.)
A higher percentage of the larger star's total mass is inside the burning zone compared to a smaller star so a higher percentage of the star's mass is subject to fusion at any given time, so a higher percentage of the star's total "fuel" is being "burned" at any one time than in a smaller star.
The addition of more fusible isotopes also does not scale linearly. Fusing heavier elements does not release as much energy per unit of mass as lighter ones and they require higher temperatures to start i.e. heavier elements are a poorer fuel. The heavier elements are also produced by fusing lighter ones so the star has to burn fuel to make fuel. Since heavier elements require higher temperatures to fuse, the burning volume for them is smaller than that for lighter elements. Stars usually can't burn more than small percentage of the elements isotopes they produce so the notional increase in fuel supply doesn't compensate for the higher burn rate of lighter elements.
It's like the bard said, "The candle that burns twice as bright, burns half as long."