# Why are main sequence stars more massive than the Sun less dense? e.g. Vega, Spica etc

Why do main sequence stars more massive than the Sun have lower densities? e.g. Vega, Spica etc.

• You need to be clearer. Average densities? Core densities? Is there any source that this is typical among more massive stars, or are you just using a sample of 2? – Sir Cumference Oct 31 '16 at 5:05
• yes, fair enough. I'm referring to average density among the most commonly cited examples of main sequence stars where I notice that volume increases much faster than mass. – Arvind H Nov 1 '16 at 4:16

## 1 Answer

Main sequence stars are defined by being hot enough in the core to fuse hydrogen, so their core is at least about 10 million K, and can get up to 20 million K for the more massive ones (because they are more luminous, so their nuclear fusion has to crank itself up a bit more).

The way they get their cores hot enough to fuse hydrogen is via gravitational contraction, but a more massive star does not need to contract as much to reach the necessary temperature. Since it does not need to contract as much, it is at lower density.

To see the scaling approximately, use the virial theorem, which says that the energy per particle in the core (roughly set by the need for fusion) is about $GMm/R$, where $M$ is the mass of the star, and $m$ is the mass of a proton. Expressed in terms of density $\rho$ instead of radius $R$, that's proportional to $M^{\frac{2}{3}}\rho^{\frac{1}{3}}$. Ergo, $\rho$ drops when $M$ rises.

• You might want to add that this is the average density of a star we are talking about. Density is however not distributed uniformly throughout a star but varies quite a lot from the surface (where it might be almost a vacuum) to the core. – Adwaenyth Oct 31 '16 at 15:56
• Yes, good point. We are taking a picture that says as you look at higher mass stars, mostly what changes is their overall scale-- larger radius, lower density, etc., but that the internal structure, like how density decreases with radius over most of the star, does not change much-- so the stars look similar, just with a different overall scale. That's only an approximation, the structure is also changing a bit, because the outer convection zones get thinner, and the core starts to go convective instead, but those are details. – Ken G Oct 31 '16 at 16:00
• Thanks James and Ken. as mass increases fusion rate also has to increase to balance out gravitational contraction 'pressure'. For fusion rate to increase I assume core temperatures and/or densities need to increase. What then is the intuition behind average main sequence stellar density decreasing with mass? – Arvind H Nov 1 '16 at 4:22
• No, there is not an increase in pressure for higher mass stars-- their pressure is actually lower. A lot of places get that wrong. The luminosity sets the fusion rate, not the other way around, because fusion self-regulates to match the luminosity, much moreso than the luminosity self-regulates to match the fusion. The density and pressure must be lower because otherwise the higher mass would lead to too high of a core temperature and too much fusion. – Ken G Nov 1 '16 at 20:28