What is the truth about radiation pressure in stars?

[Ken G]

There is a great deal of misconception and misinformation surrounding the topic of radiation pressure in stars. I have found there to be a widespread idea that the cores of stars are held up primarily by radiation pressure, yet this is very rarely true. To see how bad the misinformation there is, consider these quotes:

http://www.astronomydictionary.com/definition/radiation-pressure.html
"Radiation pressure counterbalances the gravitational forces due to the star’s mass
which tend to make it contract. When the star’s energy production ceases and the radiation
pressure is removed, the star will start to collapse. "

The above comes from the very definition of radiation pressure in the online astronomy dictionary, no less, yet it is completely wrong. Not wrong as in some technically detailed sense, but wrong as in the opposite of true. Radiation pressure throughout most stars (especially our Sun) is well known to those who understand stellar interiors to be fairly negligible, even in the core. What's more, the end of fusion most certainly does not bring about the "removal" of the radiation pressure-- the radiation pressure generally increases when fusion ends in a stellar core, but it doesn't matter because it wasn't doing anything important anyway. The only time radiation pressure matters in the core is for stars much more massive than the Sun, like as massive as stars get.

The misinformation is not limited to the online dictionary. Here is a NASA site, no less:
http://imagine.gsfc.nasa.gov/docs/science/know_l2/stars.html
"To add more nucleons to the iron nucleus requires an input of energy,
and so, once the center of the star consists of iron, no more
energy can be extracted. The star's core then has no resistance to the force of gravity,
and once it starts to contract a very rapid collapse will take place."

Here they are talking about a core-collapse supernova, and again we find the patently wrong information that when fusion stops, radiation stops holding up the core of the star. At least they are talking about massive stars here, but most stars that go supernova also do not have cores that are primarily held up by radiation pressure (instead it is usually ideal gas pressure or electron degeneracy pressure). At best they are leaving out most of the story-- what really happens is, fusion ends, hydrostatic equilibrium is just fine for awhile, but gradually the energy losses from the core force the core to (gradually) contract, nothing dramatic there, and little to do with radiation pressure. The dramatic stuff comes later on, generally when additional nuclear "ash" is added to the non-fusing core. Adding mass to the core is the precipitator of all the interesting stuff involving core collapse, and that is true both in a supernova, and at the end of the main sequence when the star begins to expand rapidly.

You can google lots of astronomy course websites and answer-all websites to see this wrong information repeated over and over, I'll just give a few examples of hammering this false theme that the end of fusion causes a loss of radiation pressure which leads to dramatic changes in the core:

http://wind.caspercollege.edu/~marquard/astronomy/sunlike.htm
At some point in time the hydrogen nuclei in the core will run out.
When this occurs there will no longer be a balance between gravitational
pressure inward and photon pressure outward. This is due to the fact that
the photon production ceases when the fusion process runs out of fuel.
At this point the core (almost pure helium) will begin to collapse again.

http://www.ehow.com/about_5438410_life-star-one-solar-mass.html
The main sequence ends when the star's core runs out of hydrogen nuclei. Without the
radiation pressure generated by hydrogen fusion, equilibrium is lost. The star's core,
made up almost entirely of helium now, begins to collapse.

Quite remarkable how consistent is this incorrect theme, once the mainstream understanding of stellar interiors is actually mastered.

 

[Drakkith]

May i ask why this is incorrect? If it is, where can I find correct info?

 

[Ken G]

Elementary physics tells you it is incorrect, and any reasonably complete textbook of stellar interiors will tell you what is correct, or I'll be happy to. The formula for radiation pressure is quite simple, it is of order sigma/c * T⁴. You can easily look this up, and look up sigma (the Stefan-Boltzman constant) and c. Now look at the gas pressure in the core of a star, like the Sun (it is of order GM/R⁴, also from elementary considerations of pressure/gravity balance). You will find that radiation pressure fails by quite a few orders of magnitude there.

What's more, since radiation pressure depends on T, for radiation pressure to be "removed" when fusion stops, the T would need to go down. Do you think the T in the core of a star goes down when fusion stops? It does not, not for a long time anyway. Basically, the claims I criticize above are wrong both in magniture (rad pressure is too weak) and how it changes (it doesn't go away when fusion stops), and even the most elementary understanding of radiation and gas pressure quickly demonstrates that. So the real question is, why are so many seemingly reliable internet sources so completely wrong on this score? I really don't know.

 

[Drakkith]

So, you are saying that it is the kinetic energy in the particles that are in the core of the star that hold the star up against gravity? And if the radiation pressure increases when fusion stops, what is giving out that radiation? The heat of the core?

 

[Ken G]

So, you are saying that it is the kinetic energy in the particles that are in the core of the star that hold the star up against gravity?[

Precisely so, that is extremely easy to demonstrate. That is also why the kinetic energy of those particles is of order their escape energy, which is the guts of the virial theorem.

And if the radiation pressure increases when fusion stops, what is giving out that radiation? The heat of the core?

Yes. And to find out if the radiation pressure decreases or increases requires looking at all the possible sources of that heat-- including gravity. After all, gravity is why the core got hot in the first place-- not fusion. There can't even be fusion until there is high temperature. It is certainly true that the absence of fusion is what allows the core to contract, but the contraction does not happen because radiation pressure is dropping, it happens because the mass of the core is rising as more nuclear ash is added to it. Claiming that a drop in radiation pressure is responsible for core contraction is about as wrong as you can get, yet we see it all over the place.

 

[Drakkith]

So the fusion is what releases the energy required to "replenish" the energy lost in the form of light and such? And once fusion stops the core has less and less energy in its particles to hold up against gravity since it is being radiated away?

 

[Ken G]

So the fusion is what releases the energy required to "replenish" the energy lost in the form of light and such?[

Yes, exactly.

And once fusion stops the core has less and less energy in its particles to hold up against gravity since it is being radiated away?

This is much more subtle. When a satellite experiences drag, it loses energy, and its orbit decays. Can we say the satellite loses kinetic energy? No we can not, it gains kinetic energy. It loses total energy, due to the drag. Similarly, all we can say is the particles in the core lose total energy, if it is not being replenished. That does not necessarily mean they lose kinetic energy, or gas pressure, or radiation pressure. Generally, they gain all three of those as the core contracts. But my real point is, it certainly is not a relevant issue if the radiation pressure is dropping, both because it probably isn't, and more importantly, it's usually negligible anyway.

 

[Drakkith]

I see what you are saying. The satellite initially loses kinetic energy to the particles that are causing the drag. If it starts to fall then it starts to lose potential energy as it is converted into kinetic energy of the falling satellite.

 

[Ken G]

I see what you are saying. The satellite initially loses kinetic energy to the particles that are causing the drag. If it starts to fall then it starts to lose potential energy as it is converted into kinetic energy of the falling satellite.

Right, and if there are zillions of satellites doing that all at once, there's never a time where the total kinetic energy is decreasing-- it's always increasing if there is no thrust to counter the drag, or staying constant if there is.

 

[Drakkith]

Right, and if there are zillions of satellites doing that all at once, there's never a time where the total kinetic energy is decreasing-- it's always increasing if there is no thrust to counter the drag, or staying constant if there is.

I guess that's true if the velocity of the satellite never slows down. Due to drag and increasing air resistance that isn't true though.

 

[Ken G]

I guess that's true if the velocity of the satellite never slows down. Due to drag and increasing air resistance that isn't true though.

It's still true of the population of satellites-- not necessarily each one. The kinetic energy of the population is always increasing, because the population is distributed around their orbits so as to satisfy the virial theorem as a whole. The virial theorem says that the average kinetic energy is half the average potential energy (in magnitude), so that is also true of the whole distribution, and both are increasing monotonically. Drag does not reduce the kinetic energy of the population of satellites, just as energy losses from a self-gravitating gas-pressure-supported core does not reduce the temperature of that core. The links above are just plain wrong. The temperature can't drop until degeneracy pressure takes over, much later, and after the contraction has already occured. Above all, what should be understood about stellar cores is that their contraction is generally caused by adding too much non-fusing matter to them, and usually has essentially nothing to do with radiation pressure. Once you understand that, read those links again-- and be amazed.

 

[Drakkith]

Oh my mistake, I forgot you were using the satellite thing as an example for the gas in the core. I think I semi-understand what you are saying. I'll mull it over a bit more and see if I understand it more. Thanks.

 

[Chronos]

Isn't radiation pressure better expressed as a function of fluid kinetic energy using the Bernoulli equation? It appears more relevant to hydrostatic equilibrium in stellar interiors than Stephan's law.

 

[Ken G]

I'm not sure what alternate suggestion you are making for the formula, other than the one I gave, which is the formula for radiation pressure of a thermal radiation field. Can you be more specific?

 

[Chronos]

Pressure can be defined as a function of energy density, and energy density is easily enough converted to fluid kinetic energy. Stefan's law appears more applicable to the surface of a star, not its interior - where hydrostatic effects dominate. This, however, is not my area of specialty so I am relying on my limited knowledge of fluid dynamics. BTW, I am happy we finally had the opportunity to freely discuss this topic. I find it fascinating.

 

[Ken G]

Stefan's law appears more applicable to the surface of a star, not its interior - where hydrostatic effects dominate. This, however, is not my area of specialty so I am relying on my limited knowledge of fluid dynamics.

I understand where the problem is, there are actually two flavors of the Stefan-Boltzman law-- one that refers to the surface flux, and another, which is more or less just an isotropic version of the first, that gives the energy density instead of the flux (to make the connection, you only have to multiply by 4/c). The pressure of a relativistic fluid, like a thermal radiation field, is 1/3 its energy density. After you have used this to find the radiation pressure, the result can be put into equations, like Bernoulli's, that use pressure.

BTW, I am happy we finally had the opportunity to freely discuss this topic. I find it fascinating.

Me too! I stripped everything that could remotely be considered controversial.

 

[Chronos]

Does the 1/3 energy density field properly account for radiation pressure? I am still confused, I love this stuff. You are the star scientist here so get me where I need to be. I have an abiding interest and wish to learn. You are the only one here who seems interested in talking about stellar dynamics these days.

 

[Ken G]

It's a very general result that if you have an isotropic particle distribution, then there's going to be a direct connection between pressure P and kinetic energy density E/V. All you need is the connection between momentum and energy. If that connection is nonrelativistic, you get P = 2/3 E/V, and if it's highly relativistic, you get P = 1/3 E/V. A thermal radiation field deep in a star is like an isotropic highly relativistic fluid, so obeys the latter relation. Then if you find the E/V using the Planck function integrated over frequency along every ray, and divide by c, you get a Stefan-Boltzmanlike 4*sigma*T⁴/c for E/V, so P = (4/3)*sigma*T⁴/c. That's the analog of the "ideal gas law" for highly relativistic fluids.

Yeah, there does seem to be a general feeling that stars are not interesting, because their basics are so widely understood. One of the purposes of this thread is to show just how untrue that is!

 

[amarante]

Interesting topic. I still have some doubts. I don't know if I missed some point during the discussion.

Well, until now I had this "apparently" misconception about radiation pressure. I had to get the book I used for a Stellar Structure and Evolution course I took. Well the book for reference is: "Stellar Structure and Evolution" R. Kippenhahn and A. Weigert. Well this is a really good book written by people who works with stellar models... anyway. If you can get this book in your library try to follow me.

In Chapter 13 --> They started a discussion about the presure cause by the star gas and they write: "But the pressure in a star is not only given by that of the gas because the photons in the stellar interior can contribute considerably to the pressure." And they find some relations for radiation pressure.

In Chapter 19 --> Studying polytropes they write while explaning a model: "In section X we shall apply this to very massive stars. They are fully convective and dominated by radiation pressure."

And it goes on...
Well my doubt is, did I miss understand what I tried to say, or what you are actually saying that the term "radiation pressure" is not been used correctly?

Because if you say it is wrong, I believe you are saying that a lot, and it is really a lot, of Astronomers are doing it wrong. Mainly because in the stellar structure models they always use the term "radiative core", which is a core dominated by radiation pressure.

Well, and if I missed the point, I still did not get what is the right answer? What supports the stellar from gravity?

Thanks,
João A.

EDIT: and what about the Eddington limit? The maximum luminosity that a central mass M can have and still not have spontaneously eject hydrogen by radiation pressure. You can find more about this in the book: "Radiative process in Astrophysics" - George Ribicki & Alan Lightman

 

[amarante]

Precisely so, that is extremely easy to demonstrate. That is also why the kinetic energy of those particles is of order their escape energy, which is the guts of the virial theorem.

So, if I got it, the kinetic energy of the particles sustain the star.. but what particles?

I mean, in the interior the material is fully ionized, and when astronomer say radiation pressure I think it is the pressure caused by photons that are trapped, not fully, in the high density environment.

 

[Drakkith]

Amarante, what he is saying is that in stars which do NOT have a very high mass, the radiation pressure from photons is not the dominant force holding the core up against gravity. Instead the dominant force is the actual particles in the core. IE the hydrogen, helium, and electrons that are heated to high temperatures and bouncing around everywhere.

The amount of radiation pressure increases with stellar mass, so high mass stars have much higher amounts of radiation pressure, and after a certain point the radiation pressure can take over as the dominant force holding up the core against gravity. After all, once the core stops fusing and cools down, it stops releasing photons and eventually gravity takes over and crushes a massive star down into a neutron star or a black hole.

 

[amarante]

Amarante, what he is saying is that in stars which do NOT have a very high mass, the radiation pressure from photons is not the dominant force holding the core up against gravity. Instead the dominant force is the actual particles in the core. IE the hydrogen, helium, and electrons that are heated to high temperatures and bouncing around everywhere.

So there is nothing new in what he says. In low mass stars the core is fully radiative and hight mass stars fully convective, so Ok in the core of massive stars the bounce of the particles sustain the star.

The amount of radiation pressure increases with stellar mass, so high mass stars have much higher amounts of radiation pressure, and after a certain point the radiation pressure can take over as the dominant force holding up the core against gravity. After all, once the core stops fusing and cools down, it stops releasing photons and eventually gravity takes over and crushes a massive star down into a neutron star or a black hole

And I think what he said is that when fusion stops the radiative pressure is higher. So why would there be a colapse in low mass stars? So If I got it, what he is saying is only for high mass stars...

 

[Drakkith]

So there is nothing new in what he says. In low mass stars the core is fully radiative and hight mass stars fully convective, so Ok in the core of massive stars the bounce of the particles sustain the star.



And I think what he said is that when fusion stops the radiative pressure is higher. So why would there be a colapse in low mass stars? So If I got it, what he is saying is only for high mass stars...

You have it backwards. In the lower mass stars the core is held up by the particles, in higher mass stars it is held up primarily by radiation pressure.

I think it goes like this:

In the lower mass stars, once fusion stops, the core extra energy from the fusion no longer exists, so the core WOULD start to cool, except for gravity. As fusion begins to cease, the core stats to lose energy and can't hold itself up against gravity as much as it did before. As gravity compresses the core more and more, it gradually heats the core up, which releases MORE radiation pressure since the core is getting hotter and hotter. If there is enough mass in the star, gravity can compress and heat the core so much that Helium fusion eventually starts to occur, and for a time, the star is stable again.

What I got from this, was that the fusing of elements in the star is what provides the energy needed to replace the energy lost from radiation due to gravitational collapse and heating. As long as fusion is going on, the star will be in equilibrium and continue to shine. Once a star like our sun can no longer fuse any elements, it lacks the ability to replace the energy lost and turns into a White Dwarf. These stars still shine, but not due to fusion, it is simply due to how hot they still are. Over time they will lose energy and cool off, gradually releasing less and less light until they turn into Black Dwarfs.

 

[amarante]

You have it backwards. In the lower mass stars the core is held up by the particles, in higher mass stars it is held up primarily by radiation pressure.

Hm.. sorry but, what I know is that low mass stars have a radiative core and convective outer layers, and high mass stars have a convective core and radiative outer layers. I don't know if a radiative core means that is is held up by particles, but what I understand is that radiative core means that it is held by radiation pressure, sorry if I am wrong

As gravity compresses the core more and more, it gradually heats the core up, which releases MORE radiation pressure since the core is getting hotter and hotter.

Hmm.. it stays hotter and hotter, but does it release more radiation pressure? How? There is no burning because the temperature is not enough to break the degeneracy pressure inside the core. If it can breaks the degeneracy pressure than it stars the fusion...

I think what is happening here is a misconception in what we are calling radiation pressure...

 

[Drakkith]

I'm basing my position on what Ken G told me, so I'm far from an expert here.

Hmm.. it stays hotter and hotter, but does it release more radiation pressure? How? There is no burning because the temperature is not enough to break the degeneracy pressure inside the core. If it can breaks the degeneracy pressure than it stars the fusion...

From what I understand, the hotter the core gets the more radiation it releases, just like anything else does. (IE Blackbody Radiation)

 

[amarante]

From what I understand, the hotter the core gets the more radiation it releases, just like anything else does. (IE Blackbody Radiation)

It gets hotter, but we are talking about a high density environment and a very high pressure! At this pressure the matter is degenerated. Its not an ideal gas anymore, it contracts, it heats up but it does not expand. It keeps contracting until you reach the temperature limit to break the degeneracy. There will be no photons going out because they are trapped.

 

[Drakkith]

It gets hotter, but we are talking about a high density environment and a very high pressure! At this pressure the matter is degenerated. Its not an ideal gas anymore, it contracts, it heats up but it does not expand. It keeps contracting until you reach the temperature limit to break the degeneracy.

I don't remember expansion being part of this. The radiation pressure can increase without the core expanding.

 

[Ken G]

Hello amarante, and thank you for your interest in this thread. You are giving me a chance to show how widespread the misconceptions are!

Well the book for reference is: "Stellar Structure and Evolution" R. Kippenhahn and A. Weigert. Well this is a really good book written by people who works with stellar models... anyway. If you can get this book in your library try to follow me.

Yes, that's the book I use when I teach it, it's excellent.

In Chapter 13 --> They started a discussion about the presure cause by the star gas and they write: "But the pressure in a star is not only given by that of the gas because the photons in the stellar interior can contribute considerably to the pressure." And they find some relations for radiation pressure.

Yes, you can see where I give those relations in the thread above.

In Chapter 19 --> Studying polytropes they write while explaning a model: "In section X we shall apply this to very massive stars. They are fully convective and dominated by radiation pressure."

Also mentioned above, those are stars with mass approaching or exceeding 100 times that of the Sun!

Well my doubt is, did I miss understand what I tried to say, or what you are actually saying that the term "radiation pressure" is not been used correctly?

To recap, there are two common mistakes around radiation pressure in stars near the end of their main-sequence lives. These mistakes appear in lots of places (but of course not Kippenhahn and Wiegert, that's practically the bible here). The errors are:
1) that most stars (including our Sun) have important radiation pressure in their cores, and
2) radiation pressure in the core drops as the star runs out of nuclear fuel.
As I explained above, those are both wrong. It's not some technical thing like the words are being used awkwardly, it is flatly, completely, and totally wrong, and for reasons that are accessible at the undergraduate level, they don't even require a graduate-level text like Kippenhahn.

Because if you say it is wrong, I believe you are saying that a lot, and it is really a lot, of Astronomers are doing it wrong. Mainly because in the stellar structure models they always use the term "radiative core", which is a core dominated by radiation pressure.

Certainly not. "Radiative core" refers to the fact that that the energy transport is via radiation, not the momentum transport. There's a rather big difference there. And as far as I know, no astronomers are doing it wrong, but a lot of educational websites sure are.

Well, and if I missed the point, I still did not get what is the right answer? What supports the stellar from gravity?

Generally, gas pressure.

EDIT: and what about the Eddington limit? The maximum luminosity that a central mass M can have and still not have spontaneously eject hydrogen by radiation pressure. You can find more about this in the book: "Radiative process in Astrophysics" - George Ribicki & Alan Lightman

Don't forget to check the kinds of mass that stars approaching that limit have, it is another way to prove my point.

ETA: the thread hasn't been closed, so I removed the comment about that.

 

[Ken G]

I think what is happening here is a misconception in what we are calling radiation pressure...

Radiation pressure is a very straightforward concept, there's no problem with it. Like all pressure, radiation pressure is momentum flux density. That means, if you imagine a plane like a window, and track the rate of momentum going through from one direction, and add it to the rate of momentum going through the other way (with a minus sign to the latter, going the other way as it does), and take the dot product with the area normal, you get a scalar called pressure. This is also the scalar whose gradient gives the force per unit volume. It's just like gas pressure, except it's 1/3 the energy density (being relativistic), rather than 2/3 like nonrelativistic gases.

 

[Ken G]

You have it backwards. In the lower mass stars the core is held up by the particles, in higher mass stars it is held up primarily by radiation pressure.

That's correct. Higher mass stars have convective cores, so the energy transport is by the gas, but the pressure is by the radiation. Pressure has to do with momentum transport, and energy density, but not energy transport. Lower mass stars have radiative cores, so the energy transport is by the photons, but the pressure is just by the protons and electrons, which is also where the thermal energy is. That is indeed pretty confusing, and maybe this is the reason we see it wrong in so many places.

To be clear, I am referring to main-sequence stars, or stars just ending the main sequence-- not degenerate cores, that's another matter.

As long as fusion is going on, the star will be in equilibrium and continue to shine. Once a star like our sun can no longer fuse any elements, it lacks the ability to replace the energy lost and turns into a White Dwarf. These stars still shine, but not due to fusion, it is simply due to how hot they still are. Over time they will lose energy and cool off, gradually releasing less and less light until they turn into Black Dwarfs.

Yes, that's true.

 

[amarante]

Thanks Ken G. Now I got it. ;)

 

[Ken G]

Excellent. To be honest, I had not realized how confusing the term "radiative core" is, this goes a long way toward helping me to understand where the confusion is coming from!

 

[amarante]

Excellent. To be honest, I had not realized how confusing the term "radiative core" is, this goes a long way toward helping me to understand where the confusion is coming from!

Actually, I think I was confused about the energy transport and momentum transfer. Now it is all clear.

 

[Pigkappa]

Here they are talking about a core-collapse supernova, and again we find the patently wrong information that when fusion stops, radiation stops holding up the core of the star. At least they are talking about massive stars here, but most stars that go supernova also do not have cores that are primarily held up by radiation pressure (instead it is usually ideal gas pressure or electron degeneracy pressure). At best they are leaving out most of the story-- what really happens is, fusion ends, hydrostatic equilibrium is just fine for awhile, but gradually the energy losses from the core force the core to (gradually) contract, nothing dramatic there, and little to do with radiation pressure.

I 'm not sure it works this way. It is true (I think) that most stars that go supernova do not have cores that are primarily held up by radiation pressure and "gradually the energy losses from the core force the core to (gradually) contract", but it isn't true that radiation pressure has little to do with it.

I think that this is what happens. When the fusion stops, no more photons are produced. Then the radiation pressure drops and the core begins to contract. This is a very slow contraction because the radiation pressure wasn't too important, but it's the only pressure which can be reduced (the electron, nuclei and nucleons gas pressure depends on the chemical composition, T and density of the core and they don't change just because the fusion stopped).
Since the core has started to contract, its temperature and density rise and this changes the equilibrium between free $$\alpha$$ particles and Fe nuclei and the reaction $$Fe + \gamma \rightarrow 13 \alpha + 4n$$ happens, some of the $$\alpha$$ particles break up releasing free protons, and some of the protons capture some electrons thus reducing their pressure contribute.
The radiation pressure variation is important because it allows the contraction to start.

I'm facing this topic for my 3rd-year thesis (a minor presentation we have to make which is kinda pointless) and the Stellar Physics course is at the 4th year, so I hope I didn't mess up anything.

 

[Ken G]

I 'm not sure it works this way. It is true (I think) that most stars that go supernova do not have cores that are primarily held up by radiation pressure and "gradually the energy losses from the core force the core to (gradually) contract", but it isn't true that radiation pressure has little to do with it.

Most stars that go supernova do not have cores that are primarily held up by radiation pressure, and radiation pressure plays little role anywhere in the supernova process.

I think that this is what happens. When the fusion stops, no more photons are produced.

Right away this is incorrect. Photons are not just produced by fusion, they are produced by hot gas. If you don't want to make photons, you'd have to instantly cool the gas to absolute zero. Generally, what instead happens is the gas begins to heat up, thereby producing more photons. Your understanding is exactly the misconception I am pointing to, so I'm glad that you are willing to share it, because it supports my claim that this misconception is quite widespread.

Then the radiation pressure drops and the core begins to contract.

Again, no, radiation pressure depends only on one thing in the core: its temperature. Period. So if you think the radiation pressure drops, you are claiming the core temperature drops. That is not usually true.

This is a very slow contraction because the radiation pressure wasn't too important, but it's the only pressure which can be reduced (the electron, nuclei and nucleons gas pressure depends on the chemical composition, T and density of the core and they don't change just because the fusion stopped).

And right there is your core misconception: radiation pressure is not fundamentally different from other forms of pressure, they all depend (in different ways) on the local plasma properties like T and density and composition, it's just that radiation pressure is particularly simple because it only depends on T (we are talking about a thermal radiation field here, the equations for it can be found anywhere).

I'm facing this topic for my 3rd-year thesis (a minor presentation we have to make which is kinda pointless) and the Stellar Physics course is at the 4th year, so I hope I didn't mess up anything.

I'm glad we had this talk!