# Is there a limit to how hot a star can be?

I think that size and mass do not correlate to temperature, but then again these factors contribute to the internal pressure.

I would like to know if there is a limit to how hot a star can get and what mechanism(s) could drive a star to get unusually hot.

I also know that negative temperature occurs in laser is hotter than a positive temperature, and can a star produce negative temperature?

• Core or surface? Stable or during collapse? I think, during the collapse and formation of Neutron stars, the core will reach over a trillion degrees, but once formed, the Neutron Star cools off fairly quickly. – userLTK Jul 12 '15 at 6:29

## 1 Answer

Yes, there is a limit. If the radiation pressure gradient exceeds the local density multiplied by the local gravity, then no equilibrium is possible.

Radiation pressure depends on the fourth power of temperature. Radiation pressure gradient therefore depends on the third power of temperature multiplied by the temperature gradient.

Hence for stability $$T^3 \frac{dT}{dr} \leq \alpha \rho g,$$ where $\rho$ is the density, $g$ is the local gravity and $\alpha$ is some collection of physical constants, including how opaque the material is to radiation. Because there must be a temperature gradient in stars (they are hotter on the inside than the outside) this effectively puts an upper limit to the temperature. It is this that sets an upper limit of around 60,000-70,000 K to the surface temperature of the most massive stars, which are dominated by radiation pressure.

In regions of higher density or higher gravity, radiation pressure is not such an issue and temperatures can be much higher. The surface temperatures of White dwarf stars (high density and gravity) can be 100,000 K, the surfaces of neutron stars may exceed a million K.

Of course stellar interiors are much denser and consequently can be much hotter. The maximum temperatures there are controlled by how quickly heat can be moved outwards by radiation or convection. The very highest temperatures of $\sim 10^{11}$ K are reached at the centres of core-collapse supernovae. Ordinarily, these temperatures are unattainable in a star because cooling by neutrinos can carry energy away highly effectively. In the final seconds of a CCSn, the density gets high enough that neutrinos become trapped and so the gravitational potential energy released by the collapse cannot escape freely - hence the high temperatures.

As to the last part of your question, yes there are astrophysical masers found in the envelopes of some evolved stars. The pumping mechanism is still debated. The brightness temperatures of such masers can be much higher than anything discussed above.

• According to The Disappearing Spoon, the rate at which fusion occurs in the core of a star diminishes with temperature, so would seem to limit temperatures in stars whose primary heat source is nuclear fusion. When stars collapse and generate heat from converted potential energy rather than fusion, such limits go out the window, but for "stable" stars I would think they'd be the primary limiting factor. – supercat Jul 12 '15 at 19:17
• @supercat I do not know what Disappearing Spoon is, but it's wrong. As you may judge from the fact that massive stars with higher interior temperatures are orders of magnitude more luminous. – ProfRob Jul 12 '15 at 19:22
• @RobJeffries: It's a book. It doesn't say that all stars have the same equilibrium temperature (they clearly don't), but that for a given level of pressure the fusion rate drops off with temperature. Stars which are more massive can achieve higher pressures, and thus have higher equilibrium temperatures, but for a star with some particular amount of mass, the temperatures that fusion can reach will be limited by the aforementioned feedback. – supercat Jul 12 '15 at 19:31
• @supercat So you (or the book) are saying that if $\rho T$ is a constant, then as you increase $T$ fusion reactions decrease. Seems incorrect to me. The $T$-dependence of fusion reactions is far steeper than the $\rho$ dependence. In fact the central density and pressure of higher mass main-sequence stars is lower.. The limiting factor is radiation pressure in the most massive stars. Central temperatures in less massive stars are lower, because they don't need to be as high. – ProfRob Jul 12 '15 at 19:43
• My understanding of what the book is saying is that at a given pressure, increasing temperatures will reduce the density of stellar matter sufficiently to reduce the rate at which it fuses. If increasing temperatures don't reduce the rate of fusion, why would stars be able to last for millions of years? – supercat Jul 12 '15 at 20:04