<p>The ``mirror principle&quot; is actually that a shell-burning energy source acts as a mirror between what happens in the <strong>core</strong> and the envelope. If the <strong>core</strong> contracts, the envelope expands and vice-versa.</p>
<p>Here is the explanation advanced by Onno Pols in his <a href="https://www.ucolick.org/%7Ewoosley/ay112-14/texts/pols11.pdf" rel="nofollow noreferrer">excellent lecture notes</a>.</p>
<p>Shell-burning sources typically have a very strong temperature dependence and therefore a tendency to stay put in terms of their temperature and radius. If the core contracts beneath the shell, then that necessarily means that the density at the position of the shell-burning region falls. In order to maintain overall mechanical equilibrium, that means the weight of the overlying envelope must also decrease. This can only be achieved by envelope expansion (since gravity decreases as <span class="math-container">$r^{-2}$</span>.)</p>
<p>The opposite change would be that core expansion tries to drive the shell-burning region outwards, but the strong temperature dependence means it maintains its position and instead it gets denser and the pressure increases. For the star to remain in equilibrium, the overlying weight of the envelope must increase, which is achieved by its contraction.</p>
<p>I'm not so keen on thios explanation because there seems to be a disconnect between cause and effect and no clearly identified process that expands the envelope.</p>
<p>Another explanation comes from the lectures of <a href="https://people.ast.cam.ac.uk/%7Epettini/Stellar%20Structure%20Evolution/Lecture12.pdf" rel="nofollow noreferrer">Max Pettini</a>.</p>
<p>If we accept that both energy conservation and the virial theorem must be satisfied, then this implies that both the total gravitational potential energy and the total thermal energy of the star must remain constant separately. Thus if the core contracts, then the envelope must expand to keep the total gravitational potential energy of the star the same. Similarly, if the core becomes hotter, then the envelope must become cooler to keep the total thermal energy constant. Indeed if the rate of energy generation were to increase, as it does when shell burning proceeds, then the envelope must become much cooler.</p>
<p>If you do not find this explanation satisfactory either, I don't blame you. In fact the ``mirror principle&quot; and why stars expand to become red giants appears to be very difficult to explain from first principles - although it happens well eneough in numerical models. Even quite recently there have been new attempts to give such a first principles explanation, though not I think in the simple terms you were hoping for <a href="https://ui.adsabs.harvard.edu/abs/2020MNRAS.492.5940H/abstract" rel="nofollow noreferrer">(Hekker et al. 2020)</a>.</p>

CC BY-SA 4.0
2024-11-20T16:13:13.653Z
Yeah I think right now this is the best explanation, but there really is no distinction between cause and effect in this situation.
Astrovis
CC BY-SA 4.0
2024-11-18T11:44:17.210Z
2024-11-18T11:53:11.967Z
2024-11-18T11:53:11.967Z
ProfRob
<p>I am going to start the answer at the contraction, but I can edit to start it earlier in the processes if that helps.</p>
<p>When the core contracts, it begins to heat up. The physics behind this is a combination of the <a href="https://en.wikipedia.org/wiki/First_law_of_thermodynamics" rel="nofollow noreferrer">first law of thermodynamics</a>  (Can't destroy energy, only transfer or transform it) and the <a href="https://en.wikipedia.org/wiki/Ideal_gas_law" rel="nofollow noreferrer">Ideal Gas Law</a> (As volume decreases, pressure and temperature increase)</p>
<p>This heat is transferred to the next shell via <a href="https://en.wikipedia.org/wiki/Thermal_radiation" rel="nofollow noreferrer">thermal radiation</a> (photons), where some hydrogen remains due to not having been hot enough for fusion. Once this shell heats up hot enough the Hydrogen begins to burn, which continues to heat up the shell.</p>
<p>Again, because of first law of thermodynamics and ideal gas law, we know that the increasing temperature corelates with increasing pressure. The volume does not expand due to pressure because it is temporarily kept in check by the gravity of the inert helium core. Once the pressure increases enough to be stronger than the gravitational force, the envelope of the star expands, cooling in the process due to the same laws.</p>
<p>There are some more complex details at play regarding the role of gravity, and specific entropy gradients which you can read about in <a href="https://arxiv.org/pdf/2001.06064" rel="nofollow noreferrer">this paper</a> that investigates the &quot;red giant bump&quot; related to the short interval in which the increasing luminosity reverses but it has a lot of good info related to the Mirror principal in general.</p>
<p>I don't have a score high enough to just comment yet, or I would have just posted the paper in a comment, so be kind!</p>

CC BY-SA 4.0
2024-11-18T10:34:53.877Z
I don't think this stacks up as an explanation at all. Why wouldn't a volume expand if you increase the pressure? There is no "time lag" in gas laws. You also haven't explained anything relating to the contraction or expansion of the shell and how it influences (or is influenced by) the  expansion and contraction of the envelope. It is a difficult thing to provide a handwaving explanation about - many have tried.
ProfRob
CC BY-SA 4.0
2024-11-18T10:53:36.050Z
When you say there is no "Time lag" in gas law, I assume you mean there is no lag between temp increase and pressure increase, they increase together essentially (let me know if I misunderstand your comment) - while there is no lag between them, the increase in temperature + pressure together isn't instantaneous, which means there is a time frame in which the force due to pressure outwards is less than the force due to gravity inwards from the helium core.  (which I mentioned) - Volumes don't expand if there is a force acting on it greater than the pressure (ie compressed gas containers)
El Sergio
CC BY-SA 4.0
2024-11-18T11:02:40.653Z
As far as it being difficult thing to provide a handwaving explanation about - I completely agree - but the question asked for an answer suitable to a wide audience, so I figured I would start here, and through discussion and edits, get it to where it needs to be, balancing simplicity and detail as needed. I'm new here, learning the ropes. I appreciate your feedback and comments It helps me learn how to better formulate answers in the future.
El Sergio
CC BY-SA 4.0
2024-11-18T11:54:40.287Z
I've drafted my own attempt, which I think relies on the same sources. I really don't think there is a very good handwaving explanation.
ProfRob
CC BY-SA 4.0
2024-11-18T10:26:57.820Z
2024-11-18T10:26:57.820Z
El Sergio
<p>I understand that stars transition into the red giant phase by obeying the mirror principle, as written <a href="http://low-sky.github.io/stellarev/" rel="noreferrer">here</a>:</p>

<blockquote>
  <p>Shell-burning stars obey something called the mirror principle, which is an observation gleaned from the models that when a shell contracts, the envelope of the star will expand and vice versa.</p>
</blockquote>

<p>However, I read that this information was gleaned from observation. Could anyone explain the reason or the physics behind why this happens?</p>

CC BY-SA 4.0
2019-09-08T11:25:33.353Z
What is the "mirror principle"? I googled for it and did not find anything.
peterh
CC BY-SA 4.0
2019-09-08T12:32:06.627Z
I found it here: http://low-sky.github.io/stellarev/
Vin
CC BY-SA 4.0
2019-09-08T12:32:39.290Z
"Shell-burning stars obey something called the mirror principle, which is an observation gleaned from the models that when a shell contracts, the envelope of the star will expand and vice versa."
Vin
CC BY-SA 4.0
2019-09-08T12:56:27.367Z
Thanks! I understood the post and clarified it.
peterh
CC BY-SA 4.0
2019-09-08T14:44:34.197Z
Section 12.4 of https://www.ast.cam.ac.uk/~pettini/STARS/Lecture12.pdf addresses this, although a proper answer would need more of a dive into the physics, especially the virial theorem, which I have never understood properly.
Steve Linton
CC BY-SA 4.0
2024-07-29T21:25:05.840Z
Maybe it has something to do when the core collapses, it heats up and fuses more, causing the outer layers to expand?
Astrovis
CC BY-SA 4.0
2024-11-17T21:08:34.340Z
temp of the new shell could increase as @Astrovis suggests, but I wonder if there could be a ... focusing effect. imperfect words but,  basically yes net gravity on envelope is through the center of star but some "local" gravity was from surface of the shell. as the shells contract the envelope sees the "shells effect" diminish.  just guessing thanks for the snipe.
tomc
CC BY-SA 4.0
2024-11-18T11:05:53.070Z
I think the Q is a little misleadingly worded. The mirror principle is that when the core contracts, the envelope expands and vice versa. Any shell burning source stays pretty much where it is, or at least at constant temperature.
ProfRob
<star><stellar-evolution><star-formation>
What is the physical reason of the mirror principle in the stellar evolution?
Vin
# What is the physical reason of the mirror principle in the stellar evolution?

(Asked by: username_0 on 2019-09-08)

I understand that stars transition into the red giant phase by obeying the mirror principle, as written [here](http://low-sky.github.io/stellarev/):

> Shell-burning stars obey something called the mirror principle, which is an observation gleaned from the models that when a shell contracts, the envelope of the star will expand and vice versa.

However, I read that this information was gleaned from observation. Could anyone explain the reason or the physics behind why this happens?

**Tags:** star, stellar-evolution, star-formation

> **Comments:**
> **username_1**: What is the "mirror principle"? I googled for it and did not find anything.  
> **username_0**: I found it here: http://low-sky.github.io/stellarev/  
> **username_0**: "Shell-burning stars obey something called the mirror principle, which is an observation gleaned from the models that when a shell contracts, the envelope of the star will expand and vice versa."  
> **username_1**: Thanks! I understood the post and clarified it.  
> **username_2**: Section 12.4 of https://www.ast.cam.ac.uk/~pettini/STARS/Lecture12.pdf addresses this, although a proper answer would need more of a dive into the physics, especially the virial theorem, which I have never understood properly.  
> **username_3**: Maybe it has something to do when the core collapses, it heats up and fuses more, causing the outer layers to expand?  
> **username_4**: temp of the new shell could increase as @username_3 suggests, but I wonder if there could be a ... focusing effect. imperfect words but,  basically yes net gravity on envelope is through the center of star but some "local" gravity was from surface of the shell. as the shells contract the envelope sees the "shells effect" diminish.  just guessing thanks for the snipe.  
> **username_5**: I think the Q is a little misleadingly worded. The mirror principle is that when the core contracts, the envelope expands and vice versa. Any shell burning source stays pretty much where it is, or at least at constant temperature.  

---

## 2 answers

### Answer by username_5 on 2024-11-18. Score: 3

The ``mirror principle" is actually that a shell-burning energy source acts as a mirror between what happens in the **core** and the envelope. If the **core** contracts, the envelope expands and vice-versa.

Here is the explanation advanced by Onno Pols in his [excellent lecture notes](https://www.ucolick.org/%7Ewoosley/ay112-14/texts/pols11.pdf).

Shell-burning sources typically have a very strong temperature dependence and therefore a tendency to stay put in terms of their temperature and radius. If the core contracts beneath the shell, then that necessarily means that the density at the position of the shell-burning region falls. In order to maintain overall mechanical equilibrium, that means the weight of the overlying envelope must also decrease. This can only be achieved by envelope expansion (since gravity decreases as $r^{-2}$.)

The opposite change would be that core expansion tries to drive the shell-burning region outwards, but the strong temperature dependence means it maintains its position and instead it gets denser and the pressure increases. For the star to remain in equilibrium, the overlying weight of the envelope must increase, which is achieved by its contraction.

I'm not so keen on thios explanation because there seems to be a disconnect between cause and effect and no clearly identified process that expands the envelope.

Another explanation comes from the lectures of [Max Pettini](https://people.ast.cam.ac.uk/%7Epettini/Stellar%20Structure%20Evolution/Lecture12.pdf).

If we accept that both energy conservation and the virial theorem must be satisfied, then this implies that both the total gravitational potential energy and the total thermal energy of the star must remain constant separately. Thus if the core contracts, then the envelope must expand to keep the total gravitational potential energy of the star the same. Similarly, if the core becomes hotter, then the envelope must become cooler to keep the total thermal energy constant. Indeed if the rate of energy generation were to increase, as it does when shell burning proceeds, then the envelope must become much cooler.

If you do not find this explanation satisfactory either, I don't blame you. In fact the ``mirror principle" and why stars expand to become red giants appears to be very difficult to explain from first principles - although it happens well eneough in numerical models. Even quite recently there have been new attempts to give such a first principles explanation, though not I think in the simple terms you were hoping for [(Hekker et al. 2020)](https://ui.adsabs.harvard.edu/abs/2020MNRAS.492.5940H/abstract).

> **Comments:**
> **username_3**: Yeah I think right now this is the best explanation, but there really is no distinction between cause and effect in this situation.  

---

### Answer by username_6 on 2024-11-18. Score: 1

I am going to start the answer at the contraction, but I can edit to start it earlier in the processes if that helps.

When the core contracts, it begins to heat up. The physics behind this is a combination of the [first law of thermodynamics](https://en.wikipedia.org/wiki/First_law_of_thermodynamics) (Can't destroy energy, only transfer or transform it) and the [Ideal Gas Law](https://en.wikipedia.org/wiki/Ideal_gas_law) (As volume decreases, pressure and temperature increase)

This heat is transferred to the next shell via [thermal radiation](https://en.wikipedia.org/wiki/Thermal_radiation) (photons), where some hydrogen remains due to not having been hot enough for fusion. Once this shell heats up hot enough the Hydrogen begins to burn, which continues to heat up the shell.

Again, because of first law of thermodynamics and ideal gas law, we know that the increasing temperature corelates with increasing pressure. The volume does not expand due to pressure because it is temporarily kept in check by the gravity of the inert helium core. Once the pressure increases enough to be stronger than the gravitational force, the envelope of the star expands, cooling in the process due to the same laws.

There are some more complex details at play regarding the role of gravity, and specific entropy gradients which you can read about in [this paper](https://arxiv.org/pdf/2001.06064) that investigates the "red giant bump" related to the short interval in which the increasing luminosity reverses but it has a lot of good info related to the Mirror principal in general.

I don't have a score high enough to just comment yet, or I would have just posted the paper in a comment, so be kind!

> **Comments:**
> **username_5**: I don't think this stacks up as an explanation at all. Why wouldn't a volume expand if you increase the pressure? There is no "time lag" in gas laws. You also haven't explained anything relating to the contraction or expansion of the shell and how it influences (or is influenced by) the  expansion and contraction of the envelope. It is a difficult thing to provide a handwaving explanation about - many have tried.  
> **username_6**: When you say there is no "Time lag" in gas law, I assume you mean there is no lag between temp increase and pressure increase, they increase together essentially (let me know if I misunderstand your comment) - while there is no lag between them, the increase in temperature + pressure together isn't instantaneous, which means there is a time frame in which the force due to pressure outwards is less than the force due to gravity inwards from the helium core.  (which I mentioned) - Volumes don't expand if there is a force acting on it greater than the pressure (ie compressed gas containers)  
> **username_6**: As far as it being difficult thing to provide a handwaving explanation about - I completely agree - but the question asked for an answer suitable to a wide audience, so I figured I would start here, and through discussion and edits, get it to where it needs to be, balancing simplicity and detail as needed. I'm new here, learning the ropes. I appreciate your feedback and comments It helps me learn how to better formulate answers in the future.  
> **username_5**: I've drafted my own attempt, which I think relies on the same sources. I really don't think there is a very good handwaving explanation.  

---

