I Expansion of a star to become a red giant

[lys04]

TL;DR Summary

Process of star becoming a red giant

How exactly does a star become a red giant, i.e how does its radius expand so much?

When a star runs out of hydrogen to fuse into helium the star's core is essentially all helium, but the core isn't hot enough to start the triple alpha process to start fusing helium yet. So gravity takes over and starts shrinking down the star until to a point where a shell of hydrogen around the core is hot enough to start fusing into helium. Now I get to this point and I'm stuck as to how this leads to the expansion of the star?

 

[Bandersnatch]

Think of a star as composed of two separate parts: the core, and the outer layers. The core (and only the core) collapses, increasing its temperature as it does, which makes it radiate more. The increased radiation pressure from the core raises the outer layers higher.

 

[lys04]

Think of a star as composed of two separate parts: the core, and the outer layers. The core (and only the core) collapses, increasing its temperature as it does, which makes it radiate more. The increased radiation pressure from the core raises the outer layers higher.

But doesn't the radiation pressure need to overcome gravity first in order to raise the outer layers higher?

 

[phinds]

But doesn't the radiation pressure need to overcome gravity first in order to raise the outer layers higher?

And what makes you think the radiation has any problem overcoming gravity? We are not talking about a black hole here.

 

[berkeman]

But doesn't the radiation pressure need to overcome gravity first in order to raise the outer layers higher?

Sure.

Can you please post links to the reading you have been doing about your question, so that we can help you figure out what is confusing you in that reading? Thanks.

 

[Drakkith]

So gravity takes over and starts shrinking down the star until to a point where a shell of hydrogen around the core is hot enough to start fusing into helium. Now I get to this point and I'm stuck as to how this leads to the expansion of the star?

I believe @Ken G addressed this before and said that the shell of fusing hydrogen makes up more volume than the hydrogen core previously did, which means that much more energy is being released during hydrogen-shell fusion vs hydrogen-core fusion, which causes the outer layers to puff outwards under the increased radiation pressure. Correct me if I'm wrong, Ken.

Edit: Hmm. This reference appears to contradicts me, so I am not confident in what I just said. Hopefully someone will clarify.

 

[Astronuc]

A red giant forms after a star has run out of hydrogen fuel for nuclear fusion, and has begun the process of dying.

A star maintains its stability through a fine balance between its own gravity, which holds it together, and the outwards pressure from ongoing thermonuclear fusion processes taking place at its core. Once a star’s core runs out of hydrogen, however, that state of equilibrium is lost and the core begins to collapse. As the core collapses, the shell of plasma surrounding the core becomes hot enough to begin fusing hydrogen itself. As fusion in this shell begins, the extra heat causes the outer layers of the star to expand dramatically, and the surface extends up to several hundred times beyond the former size of the star. The energy at the star’s surface becomes far more dissipated, causing the star's bloated surface to cool, turning from white or yellow to red. A red giant is formed. This process can take hundreds of millions of years and applies to intermediate mass stars (with a mass greater than 80% and less than 800% of the Sun’s mass), which then go on to form planetary nebulae. When a more massive star runs out of hydrogen at its core, it forms a red supergiant instead, prior to exploding as a supernova.

Reference: https://esahubble.org/wordbank/red-giant/

The further out the hydrogen gets, the lower the gravity field, so the less radiation pressure needed to support it. There is both radiation pressure and the thermal (kinetic) energy of the plasma.

See also the discussion here. Halfway down the page is a nice diagram, and further on is a comparison of the sun (Sol) and Betlegeuse
https://pressbooks.online.ucf.edu/a...olution-from-the-main-sequence-to-red-giants/

 

[Ken G]

Apologies for coming into this so late, I wasn't checking for awhile. The question of why red giants expand so much is a very interesting one, but unfortunately there are a lot of misconceptions and half truths associated with it. Maybe those are viewed as minor corrections, or maybe they are promoting important disconnects in the logic, that may depend on your perspective. But here is the key point:

Anything that causes the star's luminosity to increase that much will force its envelope to expand that much, because the envelope will have to be completely convective to carry that much luminosity (if it tried to do it with radiative diffusion, the temperature gradient would go unstable to convective motion, and that's what always causes convection). A convective star with that high of a luminosity will be just like the young protostar Sun was, i.e., very large, it is the solution to the force balance subject to that high of a luminosity.

Thus, the question reduces to, why is a red giant luminosity that high? As mentioned above, it is because hydrogen is fusing in a shell sitting on top of a very dense contracted core (ultimately about the size of the Earth!), and that tends to produce a very high temperature and a very high fusion rate. The fusion is basically going nuts in there, and the only way to regulate it is to lift off weight so as to reduce the density and the amount of gas undergoing fusion (that's what happens in the subgiant phase). Once the star is a red giant, the luminosity is already very high, but it goes even higher as more helium "ash" is added to the degenerate core of the star. That increases the gravity in the shell, raising its luminosity even higher.

So that's more or less what is said above, so what are the misconceptions? The main problem is the reference to "radiation pressure" and the related concept that "fusion causes pressure." (We escaped the equally wrong idea that the core degeneracy causes pressure, but watch out for that one too.) Radiation does cause pressure, but not in any important way in the Sun, so don't worry about radiation pressure in any stage of the Sun's evolution. What about fusion causing pressure? No, the force balance is an essential requirement, it will be there no matter what is happening with fusion or radiation. The star was always in force balance, and always will be, there is no need to find a "cause" for the force balance. The reason the envelope expands is because that is what it needs to do to be in force balance. If it had to shed heat to be in force balance, it would do that, we don't need to look for a "source" of the heat that causes it to expand, it would find the energy somewhere (there is plenty of that in a star). What we need to look for is a reason it needs to expand, and that is because of the rising luminosity (which does indeed come from fusion, but it's not causing the pressure in the star, that was there before fusion and will be there after fusion).

 

[Ken G]

I believe @Ken G addressed this before and said that the shell of fusing hydrogen makes up more volume than the hydrogen core previously did, which means that much more energy is being released during hydrogen-shell fusion vs hydrogen-core fusion, which causes the outer layers to puff outwards under the increased radiation pressure. Correct me if I'm wrong, Ken.

You're right that the key thing is the fusion rate is getting very high in the shell, but not because its volume is large (it's actually quite small), it's because its temperature is high. The solar core today maintains a slow fusion rate by regulating its temperature, but that's what a shell cannot do, the shell in a red giant has to be much hotter because the gravity is so strong (and the geometry of a shell works differently).

Edit: Hmm. This reference appears to contradicts me, so I am not confident in what I just said. Hopefully someone will clarify.

That reference gives a decent account of things, but unfortunately the common misconceptions are showing up (as usual). For example, it says:
"Helium is denser than hydrogen, so the core is actually getting slowly denser with more and more helium being produced."
That's not a physics explanation. Helium is not "denser than hydrogen" in any kind of absolute way, they are both gases and their density will be controlled by a bunch of factors you actually have to figure out. I won't go into detail, but it's quite interesting what makes the second half of the above statement true. (Suffice it to say it is widely known that if the Sun mixed hits helium with its hydrogen, instead of having a mostly helium core and a mostly hydrogen envelope, the helium core would be less dense than the hydrogen core was.)

But I'm being a little picky there, it gets much, much worse:
"Eventually it will get to the point where the contractions will not be able to heat up the interior regions high enough to enable them to produce energy to sustain hydrostatic equilibrium. Even though there is gravity keeping things hot and dense, it won't be enough to help the situation. There is a limit to how tightly you can squeeze stuff and how hot you can get the material."

I'm sorry, that is just complete nonsense. It sounds like they are talking about a type Ia supernova, which is the only situation where the gas cannot "heat up enough to sustain hydrostatic equilibrium". But the Sun does not undergo a supernova explosion, and it always maintains hydrostatic equilibrium to a staggeringly good degree. Always, at every single stage of its evolution, it is in force balance, as a whole and in its core, to far more decimal places of accuracy than our simulations of it would ever need to worry about. I do get sick of seeing apparently reputable places claiming that the Sun ever loses hydrostatic equilibrium, but just look at it like this: the sound crossing time for the Sun is less than a day (in the core, less than an hour, in the core of a red giant, less than a minute). The evolutionary timescales are much much longer than that (often thousands, if not millions, of years). That is all you need to know: the Sun is always in a spectacular hydrostatic equilibrium, and any explanation of anything the Sun does that says it happens because it can't find hydrostatic equilibrium is obviously nonsense. And they should know that.

And though it is getting off topic, since that "explanation" has been cited, let's look at another misconception it sows:
"One of the unusual properties of electron degenerate material is that once it is electron degenerate, you can't make it any denser. No matter how hard you squeeze and compress it, it will not get any denser - it will get hotter, but not denser."
Once again, this statement is just plain complete nonsense. It does such a terrible disservice to students trying to understand how gas pressure works! Here's the truth: if you squeeze a gas more than it is already being squeezed, it will contract more than it is already. It won't matter at all if that gas is degenerate or not, that is not at all how degeneracy works. Why people who should know better say these ridiculous things I don't know, but it destroys and chance a student might have of understanding pressure, or degeneracy. It's just obvious, you have an internal pressure, and a force balance, and you squeeze it some more, it contracts some more. You don't need anything but the virial theorem and an adiabatic first law of thermodynamics (can we agree these fundamental laws apply?), and it has nothing whatsoever to do with degeneracy or quantum mechanics, it holds with or without both of those. I cite any text that derives the virial theorem and the first law of thermodynamics (both of which still apply in quantum mechanics and degeneracy). Don't get me started!

Just for the sake of completeness, the blurb talks about the phase of core helium burning, and says:
" The core is still extremely hot, so it is producing a lot of thermal energy as well, which will keep the outer layers puffed up, so the star is still a cool red giant."
This is more of a quibble, because the article is not really about that phase. But for anyone who has read this far, core helium burning is not "no big deal," and the star is not "still a cool red giant." The stars shrinks drastically because the hydrogen shell fusion is no longer sitting on top of a very dense degenerate core. It is now called a "horizontal branch" star, not a red giant, and it is much, much smaller.

Though you might not see it as my place to judge, and that's fine, still I say that on the whole, I think that blurb will help you understand a few things about solar evolution, but at the cost of sowing some crucial misconceptions that might actually make it impossible to figure out what went wrong when you kick the tires of your understanding.

 

[Drakkith]

Thus, the question reduces to, why is a red giant luminosity that high? As mentioned above, it is because hydrogen is fusing in a shell sitting on top of a very dense contracted core (ultimately about the size of the Earth!), and that tends to produce a very high temperature and a very high fusion rate. The fusion is basically going nuts in there, and the only way to regulate it is to lift off weight so as to reduce the density and the amount of gas undergoing fusion (that's what happens in the subgiant phase).

Okay, so the material in the shell pushes back on the material compressing it and this, combined with other effects like radiation pressure and convection, causes the outer layers of the star to expand outwards, relieving the weight on the shell and allowing it to expand and regulate its fusion rate. That about right?

 

[Ken G]

Okay, so the material in the shell pushes back on the material compressing it and this, combined with other effects like radiation pressure and convection, causes the outer layers of the star to expand outwards, relieving the weight on the shell and allowing it to expand and regulate its fusion rate. That about right?

Yes, in the subgiant phase, the expansion of the envelope is required to relieve pressure in the shell (by lifting off the weight). By the time it's a red giant, the weight of the envelope has become irrelevant (in fact everything about the envelope is irrelevant to the core and shell, it really is like two totally separate stars coinhabiting, and the "outer" star is really just like a fully convective protostar with a very significant luminosity being fed in at its lower boundary).

A few more points to stress, perhaps. "Radiation pressure" is not playing any important role (why do a lot of sources like to mention radiation pressure as if it was somehow important when there is fusion?), but the deeper and more important point is that it really doesn't matter at all what is creating the pressure in either a subgiant or a red giant. It suffices that there is pressure balance, the star is always in very good hydrostatic equilibrium. It is also in (fairly good) energy equilibrium, so you could easily understand both the subgiant and the red giant phases without considering any imbalances of any kind. Instead, just ask what is required for the star to be in both force balance and energy equilibrium, and the rest follows directly. The only reason there is expansion of the envelope, or evolution of any kind, is that helium "ash" is building up in the core. This increases the gravity acting on the fusing shell, which requires some adjustment of the stellar structure for everything to stay close to balance.

This is a form of what one might call "equilibrium evolution." You never hear about this concept, which is why so few people (even astronomers) think in these terms (we are too used to understanding motion in terms of unbalanced forces, that's generally just a misconception in stars). Here's a perfect analogy, courtesy of Kirk Korista. When you put air in your car's tires, the car rises, does it not? Is that because the car is going out of force balance, so has to accelerate upward? No, you could do it so the car rises at a constant speed, no force imbalance at all, but that isn't what matters anyway. The car is not going to accelerate at anything close to gravity, so you have a very good force balance there. The tire is holding the car's weight, all that is changing is the pressure in the tire and the surface area where that pressure meets the road. So the tire pressure is going up, but it's not some excess force that explains why the car rises, because the car is always in a good force balance. The car rises because the tire has to put less area on the road to maintain force balance, and that requires the tire has to rise, so the car has to rise. Equilibrium evolution.

It's a little like that for the expanding star, but there's not necessarily a pressure rise (the pressure in the shell drops throughout the subgiant phase, though the envelope is expanding), nor is there any danger of losing force balance, or some new source of pressure kicking in. It's just that if the core has more mass, the shell has to change in some way that requires the envelope to be larger in order to be in both force equilibrium and (near) energy equilibrium. For example, in the red giant phase, the rising core mass forces the luminosity to rise, and a higher luminosity always requires a larger convecting star, just as it did when the Sun was a protostar.

 

[snorkack]

And though it is getting off topic, since that "explanation" has been cited, let's look at another misconception it sows:
"One of the unusual properties of electron degenerate material is that once it is electron degenerate, you can't make it any denser. No matter how hard you squeeze and compress it, it will not get any denser - it will get hotter, but not denser."
Once again, this statement is just plain complete nonsense. It does such a terrible disservice to students trying to understand how gas pressure works! Here's the truth: if you squeeze a gas more than it is already being squeezed, it will contract more than it is already. It won't matter at all if that gas is degenerate or not, that is not at all how degeneracy works. Why people who should know better say these ridiculous things I don't know, but it destroys and chance a student might have of understanding pressure, or degeneracy. It's just obvious, you have an internal pressure, and a force balance, and you squeeze it some more, it contracts some more. You don't need anything but the virial theorem and an adiabatic first law of thermodynamics (can we agree these fundamental laws apply?), and it has nothing whatsoever to do with degeneracy or quantum mechanics, it holds with or without both of those.

Virial theorem is irrelevant here. And so is adiabatic first law of thermodynamics.
Just consider special relativity.
Anything incompressible transmits sound instantly. It also follows that somebody can be incompressible only in one specific inertial frame.

 

[Ken G]

Virial theorem is irrelevant here. And so is adiabatic first law of thermodynamics.

No, that is untrue. Look at the assumptions of both those fundamental laws, they all apply. They are typically not used in elementary texts, but the graduate level ones do. The key assumption of the virial theorem is that the evolutionary timescale is much longer than the sound crossing time (very true here). The key assumption of the adiabatic first law is that energy is conserved, there is no internal energy in the particles being considered, and the compression is coming without heat loss (all valid as well, if one is considering highly degenerate gas that is already at such low entropy that further heat loss is interdicted, the usual assumption about high degeneracy).

Just consider special relativity.

Since we are not talking about type Ia supernova behaviors, relativity is not relevant here.

Anything incompressible transmits sound instantly. It also follows that somebody can be incompressible only in one specific inertial frame.

What is incompressible here? Don't tell me you think degeneracy means incompressibility, that's one of the first misconceptions about it that badly needs to be eliminated. The sound speed is the sound speed, we don't want to pretend it's infinite here, and degeneracy does not change it. Degeneracy is a thermodynamic property that relates to the partition of kinetic energy among the particles, but the mechanical properties, such as pressure, depend on the total kinetic energy, not its partition among the particles. The key point is, the pressure is 2/3 the total kinetic energy per volume, like any nonrelativistic gas. That is independent of the thermodynamics, a point that is never stressed enough when talking about degeneracy.

 

[snorkack]

No, that is untrue. Look at the assumptions of both those fundamental laws, they all apply.

They may apply but they are not relevant here, because:

Since we are not talking about type Ia supernova behaviors, relativity is not relevant here.

No, it is:

What is incompressible here? Don't tell me you think degeneracy means incompressibility, that's one of the first misconceptions about it that badly needs to be eliminated. The sound speed is the sound speed, we don't want to pretend it's infinite here, and degeneracy does not change it.

I don´t think degeneracy means incompressibility. The quote you were addressing does:

"One of the unusual properties of electron degenerate material is that once it is electron degenerate, you can't make it any denser. No matter how hard you squeeze and compress it, it will not get any denser - it will get hotter, but not denser."

If anything were incompressible - "you can't make it any denser. No matter how hard you squeeze and compress it, it will not get any denser" - it would have infinite sound speed for the simple reason that if you accelerate one end of it, what is the total volume? The body cannot have undiminished volume unless its other end starts moving exactly when you start pushing it, and that is infinite and therefore faster than light speed of sound. For which special relativity is relevant.
And since the quoted conduct is already addressed by special relativity, laws of thermodynamics are not important here.

 

[Ken G]

I don´t think degeneracy means incompressibility. The quote you were addressing does:

If anything were incompressible - "you can't make it any denser. No matter how hard you squeeze and compress it, it will not get any denser" - it would have infinite sound speed for the simple reason that if you accelerate one end of it, what is the total volume? The body cannot have undiminished volume unless its other end starts moving exactly when you start pushing it, and that is infinite and therefore faster than light speed of sound. For which special relativity is relevant.
And since the quoted conduct is already addressed by special relativity, laws of thermodynamics are not important here.

Are you giving another reason to see that the quote does not make sense? In that case I agree, one of the ways we can know it is nonsensical to say that degenerate gas is incompressible is that it would require an infinite sound speed, which would make it impossible. But I'm not really that interested in the quote that I already know is wrong, I'm interested in what's right. What's right is that special relativity is not important, but the virial theorem and the first law of thermodynamics are.

 

[snorkack]

Yes, in the subgiant phase, the expansion of the envelope is required to relieve pressure in the shell (by lifting off the weight). By the time it's a red giant, the weight of the envelope has become irrelevant (in fact everything about the envelope is irrelevant to the core and shell, it really is like two totally separate stars coinhabiting, and the "outer" star is really just like a fully convective protostar with a very significant luminosity being fed in at its lower boundary).

I don´t see why the envelope would be irrelevant. It seems to me to somehow be essential.
Consider a thin shell of fusible matter on top of a gravitating degenerate core whose compressibility, though not zero, is small.
How can the shell expand? If the shell is gaseous, by heating. But since the shell is thin and in gravity of a passive core, the weight of the shell is not affected by expansion of the shell. If the shell is gaseous, doubling its volume requires doubling its temperature. True, the shell does spend some heat on expansion if it is gaseous, its isobaric heat capacity is bigger than isochoric one, which would not be the case if the shell were also degenerate - but it still has to heat in order to expand.
And since it heats, fusion processes will undergo a thermal runaway till the shell is explosively accelerated away from the degenerate core. Not even force balance.
The one balance state for the thin shell of fusible matter on top of a degenerate core is if the easy conduction of heat through the thin layer and off the surface keeps the temperature of shell low enough that thermonuclear reactions do not happen, only maybe low rate pycnonuclear reactions at the bottom of the shell.
A nova is not in force balance. The state in thermal balance is the internova accumulation of fusible shell, where thermonuclear reactions are chilled down.
We can observe the transitions from red giant to white dwarf. It is the formation of planetary nebulae, whose central stars are young white dwarfs.
What is it that enables the fusing shell of a red giant to stabilize at some luminosity, like novas cannot? At which point does a red giant turning into white dwarf lose the capability for stable shell burning? And what determines the luminosity at which the red giant stabilizes?

 

[Ken G]

I don´t see why the envelope would be irrelevant. It seems to me to somehow be essential.

It's not irrelevant to the star as a whole (it kind of is the star as a whole), it is irrelevant to the core and shell that are the engine of the star.

Consider a thin shell of fusible matter on top of a gravitating degenerate core whose compressibility, though not zero, is small.

I don't know what you mean by its compressibility being small. Its pressure is nothing special for gas in a star. I guess you mean it is small compared to the radius of the whole star, that is certainly true.

How can the shell expand? If the shell is gaseous, by heating. But since the shell is thin and in gravity of a passive core, the weight of the shell is not affected by expansion of the shell.

Correct, this is the essential feature of shell fusion. But it ceases to hold if the width of the shell starts to exceed the radius of the core. Thus the other essential feature of shell fusion is that its width is pegged to the radius of the core. Also, its width is essentially its isothermal scale height, because the fusion shell is essentially an atmosphere on top of the core. Hence, its temperature is handed to it by the core mass and radius. This is what matters about the shell, the envelope plays no key role in any of that. (The Kippenhahn and Wiegert book explains this well, apparently it's from a paper by Wiegert.)

If the shell is gaseous, doubling its volume requires doubling its temperature.

Yes, that's because it is essentially an atmosphere on top of the core.

And since it heats, fusion processes will undergo a thermal runaway till the shell is explosively accelerated away from the degenerate core. Not even force balance.

No, you are quoting what happens for shell fusion late in the life of an AGB star, which holds only because the shell is narrow compared to its radius. The shell fusion in a red giant has similar width to the core radius, that is how it does not run away. It is more like core fusion in this regard.

The one balance state for the thin shell of fusible matter on top of a degenerate core is if the easy conduction of heat through the thin layer and off the surface keeps the temperature of shell low enough that thermonuclear reactions do not happen, only maybe low rate pycnonuclear reactions at the bottom of the shell.

No, energy transport through the shell is by radiative diffusion, not convection. The convection in the envelope is how the energy is transported after it has left the shell, and really doesn't matter any more. It's almost like you are arguing that the convection in the shell pulls heat away faster than if the envelope wasn't there at all, when actually it pulls it away at the same rate as if it wasn't there at all, which is another reason why the envelope isn't doing anything important.

A nova is not in force balance. The state in thermal balance is the internova accumulation of fusible shell, where thermonuclear reactions are chilled down.

The fusion in a nova is not like the fusion in a red giant, it's a much thinner shell so it tends to run away once it starts.

We can observe the transitions from red giant to white dwarf. It is the formation of planetary nebulae, whose central stars are young white dwarfs.

Yes, that does certainly happen.

What is it that enables the fusing shell of a red giant to stabilize at some luminosity, like novas cannot?

The width of the shell. The weight of the envelope is negligible, the shell pressure gradient is balanced by gravity.

At which point does a red giant turning into white dwarf lose the capability for stable shell burning?

When the shells are far out, and much thinner than their radius.

And what determines the luminosity at which the red giant stabilizes?

The mass of the core. This is the whole point, the physics of the core completely determines everything in a red giant. You start with the (evolving) mass of the core. That gives you a radius and a gravity. The shell will conform to that radius, under the influence of that gravity. That gives you the temperature of the shell, and it is very hot, so the luminosity is very high even though the density is low. The density is self-regulated to match the fusion rate to the radiative diffusion rate through the shell. That much would be essentially no different if the envelope wasn't even there. The envelope doesn't even alter the luminosity, all it does is present a larger cooler surface, making the star red instead of being an X-ray object.