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

<p>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.</p>

<p>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.</p>

<p>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.</p>

<p>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.</p>

<p>As to the last part of your question, yes there are <a href="https://en.m.wikipedia.org/wiki/Astrophysical_maser" rel="nofollow">astrophysical masers</a> 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.</p>

CC BY-SA 3.0
2015-07-12T19:17:28.250Z
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
CC BY-SA 3.0
2015-07-12T19:22:46.960Z
@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
CC BY-SA 3.0
2015-07-12T19:31:37.253Z
@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
CC BY-SA 3.0
2015-07-12T19:43:34.753Z
@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
CC BY-SA 3.0
2015-07-12T20:04:44.437Z
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
CC BY-SA 3.0
2015-07-12T20:05:16.677Z
@supercat I think I read the question differently. My answer is how hot stars in general can become. The answer is different if you ask how hot a star of a given mass can become. Fusion reactions act as a thermostat and the high temperature dependence means that the interior will be only a little bit hotter than the threshold temperature at which a particular reaction becomes significant.
ProfRob
CC BY-SA 3.0
2015-07-12T20:16:32.753Z
@supercat The premise of your last comment *is* incorrect. In an ideal gas $P \propto \rho T$. The fusion reaction rate is $\propto \rho^{\alpha} T^{\beta}$. For the pp chain $\alpha=1$, $\beta=4$. For the triple alpha process $\alpha=2$, but $\beta=40$ !. Stars live a long time because energy transfer processes act quickly enough to prevent the core heating up -a star's luminosity is the energy released by nuclear reactions. If the core did heat up, the pressure would increase (though not in a degenerate gas), the core would expand and *cool*. It's the decreased T that matters.
ProfRob
CC BY-SA 3.0
2015-07-12T07:57:44.740Z
2015-07-12T19:21:11.737Z
2015-07-12T19:21:11.737Z
ProfRob
<p>I think that <em>size</em> and <em>mass</em> do not correlate to <em>temperature</em>, but then again these factors contribute to the internal pressure. </p>

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

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

CC BY-SA 3.0
2015-07-12T06:29:07.757Z
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
<star><temperature>
Is there a limit to how hot a star can be?
user6760
# Is there a limit to how hot a star can be?

(Asked by: username_0 on 2015-07-12)

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?

**Tags:** star, temperature
> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2015-07-12. Score: 9 | Accepted 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](https://en.m.wikipedia.org/wiki/Astrophysical_maser) 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.

> **Comments:**
> **username_3**: 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.  
> **username_2**: @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.  
> **username_3**: @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.  
> **username_2**: @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.  
> **username_3**: 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?  
> **username_2**: @supercat I think I read the question differently. My answer is how hot stars in general can become. The answer is different if you ask how hot a star of a given mass can become. Fusion reactions act as a thermostat and the high temperature dependence means that the interior will be only a little bit hotter than the threshold temperature at which a particular reaction becomes significant.  
> **username_2**: @supercat The premise of your last comment *is* incorrect. In an ideal gas $P \propto \rho T$. The fusion reaction rate is $\propto \rho^{\alpha} T^{\beta}$. For the pp chain $\alpha=1$, $\beta=4$. For the triple alpha process $\alpha=2$, but $\beta=40$ !. Stars live a long time because energy transfer processes act quickly enough to prevent the core heating up -a star's luminosity is the energy released by nuclear reactions. If the core did heat up, the pressure would increase (though not in a degenerate gas), the core would expand and *cool*. It's the decreased T that matters.  

---

