Highest Star Core Temperature: Does Size Matter?

[goldsax]

does it follow that the larger the star the higher the core temperature?
if so what would be the highest temperature attained.
thanks

 

[QuArK21343]

Yes, the larger the star the highest the core temperature must be. This is because in main-sequence stars a state of hydrostatic equilibrium is attained: the gravitational forces that tend to make the star collapse on itself and the forces due to radiation pressure that tend to make the star expand, on each infinitesimal volume, balanced themselves. The greater the mass, the greater the pressure gradient has to be to make the star stable. Temperatures inside a star are of the order of 10-15 million kelvin at least. The pressure gradient is due to the processes of nuclear fusion that occur in the stellar plasmas (protons need huge kinetic energies to overcome the potential barrier due to electrostatic repulsion and to actually fuse into helium).

I hope it is clear.

 

[goldsax]

Thanks for the clear explanation ..
I should have realized that the only way to counteract the mass would need an increase in heat(kinetic energy).
Just as a matter of interest .. What would be the core temp in the largest stars? Thanks again

 

[Nabeshin]

Given that we know massive stars fuse up to iron, which requires core temperatures of ~billion K, this is at least a pretty decent lower limit on the hottest core temperature achieved in the most massive stars.

 

[QuArK21343]

There are at least three known mechanisms for nuclear fusion.

Stars with mass comparable to that of the sun rely almost completely on the proton-proton chain. This requires temperatures of about 15 million kelvins. From a classical point of view, even such temperatures would not be sufficient to overcome the coulomb potential between protons in the stellar plasma. Only quantum mechanics, and in particular so called tunnel effect, fully accounts for the entire process. Also electroweak forces intervene: they are responsible for the process that takes a proton into a neutron and an electric neutrino. This stage produces deuterium; deuterium can then fuse to produce helium-3 particles and finally, alpha-particles.

For stars with about 10 solar masses, carbon cycle is the dominant mechanism.

For even more massive stars (red giants and supergiants) with temperature above 100 million kelvins, helium can fuse directly to form beryllium and then carbon (triple alpha process).

With still more massive and older star, hydrogen fuel begins to run out. Neon and oxygen burning take place. Then, the star approaches the peak of the binding curve, the so called iron peak: when iron is produced the star is doomed, since this process is endothermic and so takes energy from the outside: the star will collapse and finally explode in a type II supernova. A very brief phase before this moment is the so called silicon burning process: here massive stars can reach temperatures of even 3.5 billions K!

 

[goldsax]

3+billion k ... Wow ..
Thanks again for the explanation

 

[Drakkith]

Does the core temperature vary with the type of fuel being burnt at the time? Or does it simply depend on the mass? Are the most massive stars burning Hydrogen at a temperature of 3 billion k or more?

 

[QuArK21343]

A very rough approximation of the temperature of a stellar core is given by

$$T=\frac{GMm_p}{RK_{B}}$$

where $M$ is the mass of the star, $R$ its radius and $m_p$ tha mass of the proton. This is an order of magnitude relation and can be derived from a condition of hydrostatic equilibrium (just equate the force due to pressure gradient to the gravitational force and use the equation of state for a perfect gas to get $\partial p/\partial r=-GM_r\rho_r/r^2$). So you can see that the greater the mass the higher the temperature and the more compact the star is the higher the temperature. Of course, the type of fuel being burnt determines the temperature. No star burns hydrogen at temperature of billions of degrees. Only stars with masses greater than 10 solar masses and approaching the iron peak can reach such temperatures (if hydrogen runs out, gravitational collapse sets in and the temperature increases till fusion of silicon can take place).

 

[Drakkith]

Alright, so when Hydrogen burning stars a more massive star simply burns a lot more of it in a bigger core than a star like the Sun?