<p>Whether convection exists depends on whether the interior radiative temperature gradient reaches the adiabatic temperature gradient. </p>

<p>The interior radiative temperature gradient is proportional to the opacity and the outward energy flux, and inversely proportional to <span class="math-container">$T^4$</span>. As the star evolves on the main sequence, the central temperature goes up and the opacity (e.g. Kramer's opacity goes as <span class="math-container">$T^{-7/2}$</span> goes down. You are also removing free electrons (combining them with protons to form He) which reduces the Thomson/Compton scattering opacity. This means the radiative temperature gradient goes down and can fall below the adiabatic gradient, meaning that energy transport reverts to being radiative. </p>

<p>It stays convective right at the centre for longest because that is where the radiative temperature gradient is still largest (driven by the extreme temperature dependence and high outward energy flux of the CNO cycle nuclear reactions).</p>

CC BY-SA 4.0
2019-11-23T12:35:52.497Z
Can you please add that (and why) the adiabatic temperature gradient does *not* change (I myself do not have my old textbooks ready at hand to check it myself). I ask that because I remember that in low mass stars (e.g. our Sun), it's the change in the adiabatic temperature gradient (in contrast to the actual gradient) that causes convection in the envelope.
Hartmut Braun
CC BY-SA 4.0
2019-11-23T15:14:36.253Z
@HartmutBraun The adiabatic temperature does change, but not by the same amount. That is why the connective core shrinks. In the end, the numerical model shows what happens. I offer a rough explanation.
ProfRob
CC BY-SA 4.0
2019-11-23T22:14:32.390Z
@HartmutBraun - The adiabatic gradient doesn't change significantly over the evolution shown above. The core doesn't degenerate so the equation of state is fairly represented by the ideal gas whose adiabatic gradient ∇~0.4. As for low-mass stars, the decrease in the adiabatic gradient is due to the ionization zones in the near the surface. The adiabatic temperature gradient is closely related to the adiabatic exponents whose behaviour in ionization zones can be found here: https://astronomy.stackexchange.com/questions/27769/why-does-the-adiabatic-exponent-decrease-at-the-ionization-zones
Stefano
CC BY-SA 4.0
2019-11-23T11:08:29.973Z
2019-11-23T11:08:29.973Z
ProfRob
<p>The image below shows the evolution of the hydrogen mass fraction profile for a 5 solar mass star in the main sequence. I would expect that the size of the convective core stays roughly constant as the hydrogen is been fused, leading to a final step-like profile when the core hydrogen is exhausted. However, all the literature and simulations show a clear slope in the final step-like profile which results from the shrinking of the convective core.</p>

<p>Does anyone have any insight about the reason behind the shrinking?   </p>

<p><a href="https://i.sstatic.net/scxTz.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/scxTz.gif" alt="Evolution of the hydrogen mass fraction profile for a 5 solar mass star in the main sequence"></a></p>

<p>(source of the image: <a href="http://astro.if.ufrgs.br/evol/evolve/hansen/StellarEvolnDemo/m5z02evoln.html" rel="nofollow noreferrer">http://astro.if.ufrgs.br/evol/evolve/hansen/StellarEvolnDemo/m5z02evoln.html</a>)</p>

<hr>

<p><strong>Edit 1:</strong></p>

<p>Below is the evolution of the temperature profile for the same star. Feel free to comment on it.</p>

<p>Personally, I am surprised how little the temperatures change. Since the main nuclear reaction is the CNO cycle which scale as <span class="math-container">$~T^{16}$</span>, I was expecting a much more violent change. However, the central temperature only increases in about 30% during the whole main sequence. Interesting. </p>

<p><a href="https://i.sstatic.net/fbXZc.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fbXZc.gif" alt="Evolution of the temperature profile for a 5 solar mass star in the main sequence"></a></p>

<p>(Source: <a href="http://astro.if.ufrgs.br/evol/evolve/hansen/StellarEvolnDemo/m5z02evoln.html" rel="nofollow noreferrer">http://astro.if.ufrgs.br/evol/evolve/hansen/StellarEvolnDemo/m5z02evoln.html</a>)</p>

<hr>

<p><strong>Edit 2:</strong></p>

<p>I thought a nice explanation could be in the Ledoux criterion for convection. This criterion states that chemical gradients have a stabilizing effect against convection (i.e., it impedes convection) which would lead to the conclusion that, at the interface between the radiative and convective zone, the radiative transport would take over. However, I simulate the evolution with and without the Ledoux criterion and in both cases the convective core shrinks.</p>

CC BY-SA 4.0
2019-11-23T10:18:52.523Z
It’s more than 20 years ago, since I worked in that area. I vaguely remember that this effect results from the models as a matter of fact. The physical parameters that regulate the onset of convection, the adiabatic and the actual temperature gradient, both depend in a similar way on the changes of properties of stellar matter with radius, and I had no satisfying explanation to why the point of onset moves inwards with time. I’m looking forward to a proper answer!
Hartmut Braun
CC BY-SA 4.0
2019-11-23T10:50:32.037Z
I don't suppose you can plot the central temperature too?
ProfRob
CC BY-SA 4.0
2019-11-23T15:37:07.463Z
@RobJeffries - I added the temperature profile evolution. Please, feel free to comment on it.
Stefano
CC BY-SA 4.0
2019-11-23T20:04:52.530Z
Makes me wonder if anyone's ever gotten numbers on the heat capacities of the various totally ionized nuclei. Differences could easily change convection properties.
Wayfaring Stranger
CC BY-SA 4.0
2019-11-25T20:47:26.527Z
The Ledoux Criterion is only relevant if the core is about to increase in size (instead of shrinking). In that case the lighter layers above heavier material have a stabilizing effect (as the Ledoux criterion predicts) and the core cannot immediately grow. In the convection zone there is no chemical gradient because of convective mixing and the Schwarzschild criterion applies.
Hartmut Braun
<star><stellar-evolution><stellar-astrophysics><main-sequence><modeling>
Why does the convective core in an intermediate- to high-mass star shrink?
Stefano
# Why does the convective core in an intermediate- to high-mass star shrink?

(Asked by: username_0 on 2019-11-23)

The image below shows the evolution of the hydrogen mass fraction profile for a 5 solar mass star in the main sequence. I would expect that the size of the convective core stays roughly constant as the hydrogen is been fused, leading to a final step-like profile when the core hydrogen is exhausted. However, all the literature and simulations show a clear slope in the final step-like profile which results from the shrinking of the convective core.

Does anyone have any insight about the reason behind the shrinking?

[![Evolution of the hydrogen mass fraction profile for a 5 solar mass star in the main sequence](https://i.sstatic.net/scxTz.gif)](https://i.sstatic.net/scxTz.gif)

(source of the image: <http://astro.if.ufrgs.br/evol/evolve/hansen/StellarEvolnDemo/m5z02evoln.html>)

---

**Edit 1:**

Below is the evolution of the temperature profile for the same star. Feel free to comment on it.

Personally, I am surprised how little the temperatures change. Since the main nuclear reaction is the CNO cycle which scale as $~T^{16}$, I was expecting a much more violent change. However, the central temperature only increases in about 30% during the whole main sequence. Interesting.

[![Evolution of the temperature profile for a 5 solar mass star in the main sequence](https://i.sstatic.net/fbXZc.gif)](https://i.sstatic.net/fbXZc.gif)

(Source: <http://astro.if.ufrgs.br/evol/evolve/hansen/StellarEvolnDemo/m5z02evoln.html>)

---

**Edit 2:**

I thought a nice explanation could be in the Ledoux criterion for convection. This criterion states that chemical gradients have a stabilizing effect against convection (i.e., it impedes convection) which would lead to the conclusion that, at the interface between the radiative and convective zone, the radiative transport would take over. However, I simulate the evolution with and without the Ledoux criterion and in both cases the convective core shrinks.

**Tags:** star, stellar-evolution, stellar-astrophysics, main-sequence, modeling
> **Comments:**
> **username_1**: It’s more than 20 years ago, since I worked in that area. I vaguely remember that this effect results from the models as a matter of fact. The physical parameters that regulate the onset of convection, the adiabatic and the actual temperature gradient, both depend in a similar way on the changes of properties of stellar matter with radius, and I had no satisfying explanation to why the point of onset moves inwards with time. I’m looking forward to a proper answer!  
> **username_2**: I don't suppose you can plot the central temperature too?  
> **username_0**: @RobJeffries - I added the temperature profile evolution. Please, feel free to comment on it.  
> **username_3**: Makes me wonder if anyone's ever gotten numbers on the heat capacities of the various totally ionized nuclei. Differences could easily change convection properties.  
> **username_1**: The Ledoux Criterion is only relevant if the core is about to increase in size (instead of shrinking). In that case the lighter layers above heavier material have a stabilizing effect (as the Ledoux criterion predicts) and the core cannot immediately grow. In the convection zone there is no chemical gradient because of convective mixing and the Schwarzschild criterion applies.  

---

## 1 answers

### Answer by username_2 on 2019-11-23. Score: 6 | Accepted answer

Whether convection exists depends on whether the interior radiative temperature gradient reaches the adiabatic temperature gradient.

The interior radiative temperature gradient is proportional to the opacity and the outward energy flux, and inversely proportional to $T^4$. As the star evolves on the main sequence, the central temperature goes up and the opacity (e.g. Kramer's opacity goes as $T^{-7/2}$ goes down. You are also removing free electrons (combining them with protons to form He) which reduces the Thomson/Compton scattering opacity. This means the radiative temperature gradient goes down and can fall below the adiabatic gradient, meaning that energy transport reverts to being radiative.

It stays convective right at the centre for longest because that is where the radiative temperature gradient is still largest (driven by the extreme temperature dependence and high outward energy flux of the CNO cycle nuclear reactions).

> **Comments:**
> **username_1**: Can you please add that (and why) the adiabatic temperature gradient does *not* change (I myself do not have my old textbooks ready at hand to check it myself). I ask that because I remember that in low mass stars (e.g. our Sun), it's the change in the adiabatic temperature gradient (in contrast to the actual gradient) that causes convection in the envelope.  
> **username_2**: @HartmutBraun The adiabatic temperature does change, but not by the same amount. That is why the connective core shrinks. In the end, the numerical model shows what happens. I offer a rough explanation.  
> **username_0**: @HartmutBraun - The adiabatic gradient doesn't change significantly over the evolution shown above. The core doesn't degenerate so the equation of state is fairly represented by the ideal gas whose adiabatic gradient ∇~0.4. As for low-mass stars, the decrease in the adiabatic gradient is due to the ionization zones in the near the surface. The adiabatic temperature gradient is closely related to the adiabatic exponents whose behaviour in ionization zones can be found here: https://astronomy.stackexchange.com/questions/27769/why-does-the-adiabatic-exponent-decrease-at-the-ionization-zones  

---

