# Burning Out Stars

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?

• How is it that my comment has been deleted here? o.O – Py-ser Jul 3 '14 at 5:01
• What comment? You never did. – ʇolɐǝz ǝɥʇ qoq Jul 3 '14 at 5:01
• Uhm, yes I did, at the very beginning. – Py-ser Jul 3 '14 at 5:06
• Maybe it got flagged and removed. – ʇolɐǝz ǝɥʇ qoq Jul 3 '14 at 5:33

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.

• "tries" is probably a bit too metaphorical here. Can you describe how the process is balanced? – naught101 Jul 3 '14 at 2:13
• It has to maintain hydrostatic equilibrium (HSE) to be stable (or else the net force will drive it towards HSE, and hence stability). If it compresses too much, the core temperature increases, causing the nuclear reaction rates to increase steeply, which creates enough thermal/radiation pressure to push the star outwards and balance the compression. If it expands too much, it cools, lowers the thermal pressure, allowing gravity to pull the star inwards again. That's what I meant by 'tries to balance', as often it can't, for example in the instability strip containing variable/pulsating stars. – Takku Jul 3 '14 at 8:53
• More massive stars have less dense cores. The density dependence of the nuclear burning rate, it is all about the central temperature. – ProfRob Dec 20 '14 at 14:23

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.

• But doesn't big stars have more fuel? – ʇolɐǝz ǝɥʇ qoq Jun 27 '14 at 17:16
• Yes, they do have more fuel, but the additional gravity pushes that fuel into the centre which burns it brighter and faster. – Carl Jun 30 '14 at 3:36
• That's why they are bigger :) there are known star up to 265 solar masses en.wikipedia.org/wiki/R136a1 – Joan.bdm Jun 30 '14 at 6:12
• I don't see how a bigger surface means burning faster. Stars consume their matter by fusing Hydrogen and Helium in their core. This phenomena does not occur on their surface. Please correct that. – harogaston Jul 2 '14 at 22:41
• I'm sorry if I didn't explain well. By surface I meant the burning zones. These zones are not exactly at the core but at a shell surrounding it. At the exact core there is a sphere of inert elements that can't be fused to get energy (in big start these are Ni and Fe (nickel & iron)). These zones also create an onion like structure where different elements are fused in each layer. upload.wikimedia.org/wikipedia/commons/3/37/… So basically you have higher fusion rates than in smaller stars. – Joan.bdm Jul 3 '14 at 6:30

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."

• The rate at which a star radiates is mostly dependent on its photospheric temperature. "A larger star compresses its interior more..." - incorrect, central density of main sequence stars decreases with increasing mass. – ProfRob Dec 20 '14 at 14:22
• Gravity is not dependent on volume. It is dependent on mass, energy, and momentum. – Sir Cumference Nov 20 '15 at 4:12