Explaining the Luminosity-Temperature Relationship for Stars on the Henyey Track

[snorkack]

Why does star luminosity change while on Henyey track?
A lecture:
https://astro.uni-bonn.de/~astolte/StarFormation/Lecture2012_PMS.pdf
Page 31 claims that luminosity changes as 4/5 power of temperature.
I´m not even asking for derivation as to why the power is specifically 4/5, which presumably exists. What I´m asking for is explanation why the power is not 0 and why Eddington is thereby wrong.

 

[Ken G]

Thank you for starting this thread, it gives us a chance to obtain a much better understanding of stars whose interiors are mostly radiative (the "Henyey track"), which is a hugely important stage of the life of stars, including our Sun right now.

The first thing we should all notice, say in the evolutionary tracks here: https://web.njit.edu/~gary/321/iben.gif, is that well before nuclear burning commences, stars have already established what their main-sequence luminosity is going to be-- at least to a reasonable approximation. This can be seen from the horizontal character of most of those tracks (the ones that don't involve degeneracy in their interiors, those pesky red dwarfs). Your question is around why the tracks are not completely horizontal, but won't you first want to know why they are close to horizontal? They shouldn't be exactly horizontal, as these stars are changing radius, density, internal temperature, and surface temperature, all quite significantly. What's more, they are going from not fusing anything, to fusing! So isn't the first question, why are they approximately horizontal, and why does so little happen to the luminosity when fusion commences?

Now, of course you can ask whatever question you want to know about, but surely, any answer of why they are not exactly horizontal must form a contrast between their exact non-horizontal nature, versus their nearly horizontal nature. Is anyone interested in knowing why the tracks are nearly horizontal? If not, I won't force the answer upon you, but if you do want to know that, I'd be happy to explain. It's remarkably simple!

 

[snorkack]

Look at page 38, the same link.
For massive stars, where opacity is due to electron scattering, luminosity should be independent of temperature.
The graphs look closer to horizontal - but still not exactly horizontal.

 

[Ken G]

Look at page 38, the same link.

Yes, that is indeed an insightful page. It derives the relation I have been talking about, in the simplest limit of free electron opacity. You should notice that the luminosity is derived there, as a function of mass, without even mentioning nuclear fusion. This is quite important to understand why that is possible!

For massive stars, where opacity is due to electron scattering, luminosity should be independent of temperature.
The graphs look closer to horizontal - but still not exactly horizontal.

Of course they will not be exactly horizontal, those stars are changing dramatically in radius, core temperature, and surface temperature. So given those radical changes, on the path to the onset of fusion, the interesting fact is that they are reasonably horizontal at all-- not that they are not exactly horizontal. The reason for this is remarkably simple, as long as you avoid regions where degeneracy is important, and as long as you have a good understanding of the sources of opacity. What you don't need is fusion cross sections, not even for the luminosity on the main sequence.

So what I'm saying is, the explanation I was offering is also given in those nice notes you cited. I can put them in words if they are not clear, or you can just read those notes. Indeed this is what I was trying to do in another thread about the luminosity of stars, before it was all deleted. But those notes say it too, so if you don't want my summary, that's fine. The conclusion is simple: an approximate understanding of the luminosity of primarily radiative stars, including main-sequence stars like the Sun, is possible without even knowing if fusion exists, you certainly don't need any fusion cross sections. However, you can't get the answer exactly right unless you include a lot more physics, including fusion physics once those curves reach the main sequence, but also including convection, rotation, magnetism, abundance variations, etc.

Another important point is one you raised already once in our discussion, which is that one thing fusion is quite important for is where do those horizontal curves pause, and hang out at for a very long time (billions of years for the Sun). That's the location in the H-R diagram of the main sequence, which depends mostly on the temperature where fusion turns on in a big way, but also depends on the fusion cross sections, because that's more or less the same issue. However, we already understand the luminosity of the star, before fusion even begins, as those evolutionary diagrams show, and as those notes explain. But it is true that after the H is used up, the evolutionary tracks do not simply continue to the left, and that's because shell fusion changes the internal structure of the star. Shell fusion is the one time that the fusion cross sections become important, because the temperature in the shell cannot be self-regulated (it is controlled by the non-fusing core), so instead it is the mass in the shell that is regulated to match the escaping heat. That changes the luminosity, because it changes the amount of mass that the radiation must diffuse through to escape. This is the crucial difference with the situation when there is not shell fusion.

Most likely those notes go on to talk about that when they get to red giants. The key to the post-main-sequence is the fact that you have an energy source that kicks in outside the core, which breaks the star into two very separate pieces-- a core that is continuing to contract and lose heat, and an envelope that expands. That two-part character of the star is what is different pre- and post-main-sequence, and that's why the star doubles back along its horizontal track. The lower mass stars leave that track because the cores go degenerate, and that changes the internal structure significantly.

The bottom line is, there is something known as the "mass-luminosity" relation which applies pretty much any time you have a primarily radiative star that is not degenerate and is not undergoing shell fusion. In particular, the mass-luminosity relation includes the main sequence of stars other than red dwarfs (which are starting to get degenerate). On that other thread, it was claimed that the mass-luminosity relation is some kind of throwback to before we knew about fusion, but that is simply not correct. The mass-luminosity relation remains the single most important thing to understand about stars, if you want to understand at a fairly simple level, and you realize that it only works in the situations I mentioned-- but it includes our Sun.

 

[snorkack]

Of course they will not be exactly horizontal, those stars are changing dramatically in radius, core temperature, and surface temperature. So given those radical changes, on the path to the onset of fusion, the interesting fact is that they are reasonably horizontal at all-- not that they are not exactly horizontal.

No, it´s interesting that they are not exactly horizontal.

The reason for this is remarkably simple, as long as you avoid regions where degeneracy is important, and as long as you have a good understanding of the sources of opacity.

For example, compare pages 31 with 38.
If opacity is independent of temperature, so is luminosity (page 38). If opacity depends on temperature, luminosity also does (page 31).

What you don't need is fusion cross sections, not even for the luminosity on the main sequence.

But luminosity on main sequence is not luminosity on Henyey track.
Look at the end of Henyey track. All tracks of more massive stars make a zigzag, consistently right before main sequence.
What happens there?

So what I'm saying is, the explanation I was offering is also given in those nice notes you cited. I can put them in words if they are not clear, or you can just read those notes.

Had you read these notes, you´d have seen that the explanation is NOT given there.

The conclusion is simple: an approximate understanding of the luminosity of primarily radiative stars, including main-sequence stars like the Sun, is possible without even knowing if fusion exists, you certainly don't need any fusion cross sections. However, you can't get the answer exactly right unless you include a lot more physics, including fusion physics once those curves reach the main sequence, but also including convection, rotation, magnetism, abundance variations, etc.

True. That´s what I was asking for.

 

[Ken G]

No, it´s interesting that they are not exactly horizontal.

You certainly get to choose what you think is interesting about those curves, but the part that has a simple answer is why they are nearly horizontal, even though a lot is changing in those stars! The only simple answer to why the tracks are not horizontal is that a lot is changing, the simple answer to why they are nearly horizontal involves the physics that is mostly not changing: the radiative diffusion that mostly is what sets the luminosity of those stars (as those notes explain). You can choose if you want to know it or not, as can others reading the thread. But how can you answer your question about why it is not perfectly horizontal, until you understand why it is nearly horizontal?

If opacity is independent of temperature, so is luminosity (page 38). If opacity depends on temperature, luminosity also does (page 31).

Of course, that's why luminosity depends on opacity, as I well know-- it is because the luminosity is set by the escape of the light. What I pointed out is that the luminosity is not particularly sensitive to fusion cross sections, because the luminosity is not determined by the fusion physics. Those notes explain why.

But luminosity on main sequence is not luminosity on Henyey track.

This depends on what one means by the Henyey track. The point is, the derivation of luminosity on the Henyey track works fine for the main sequence, for the simple reason that the luminosity of the star is determined by the same physics: the escape of light. Those notes say the same thing, as does any good derivation of the mass-luminosity relation (which does apply on the main sequence), as was done originally by Eddington.

Look at the end of Henyey track. All tracks of more massive stars make a zigzag, consistently right before main sequence.
What happens there?

Minor adjustments in the stellar interior in response to the onset of fusion. We should certainly not expect zero difference as fusion reconfigures the interior, but what one should get from those tracks, and from those notes, is just how minor those adjustments are. Just look at them-- how reasonable of an approximation can you get for the luminosity by not including those wiggles at all? And if you include other physics not in those sims, like magnetic fields or rotation, don't you think it will have similar effects to what you see there? The point here is not that you don't need simulations to get accuracy, the point is you don't need simulations to get a reasonable approximation. Physics often uses the concept of reasanable approximation in the context of getting understanding, does it not?

Had you read these notes, you´d have seen that the explanation is NOT given there.

I never said the explanation of those minor wiggles is in those notes, I said that you can use those notes to understand the luminosity of main-sequence stars to a reasonable approximation, without any mention of nuclear cross sections. Which is entirely true, and rather significant, but I guess it is only significant to those who wish to have a simple understanding of how to approximate the luminosity of main-sequence stars.

True. That´s what I was asking for.

I know. And as I said, understanding the cause of minor features superimposed on a largely horizontal evolutionary track should begin with an understanding of why the evolutionary track is largely horizontal. If you don't understand that, how can you understand the source of those deviations from horizontal? I cannot tell you the answer to those deviations, but I'm quite sure it is very complicated, depends sensitively on the details of the simulation being done, and likely varies considerably with unconstrained assumptions put into the model. So as I said from the start, I cannot answer your question, but I can tell you why those tracks are nearly horizontal, and I can tell you how to very simply estimate the luminosity of main sequence stars. Those notes tell you these things also. You don't have to care-- that's up to you, it just seems like the proper starting point for understanding the deviations from that starting point. I'd say the person who wrote those notes reasons similarly.

 

[snorkack]

But how can you answer your question about why it is not perfectly horizontal, until you understand why it is nearly horizontal?And as I said, understanding the cause of minor features superimposed on a largely horizontal evolutionary track should begin with an understanding of why the evolutionary track is largely horizontal. If you don't understand that, how can you understand the source of those deviations from horizontal? I cannot tell you the answer to those deviations, but I'm quite sure it is very complicated, depends sensitively on the details of the simulation being done, and likely varies considerably with unconstrained assumptions put into the model.

I'm not so sure it's complicated!
Well - let's try to see the basic derivation for the horizontal.
A) Suppose that the star consists of perfect gas
PV=nRT.
For a self-gravitating sphere of uniform density, P can be derived from V:
If R shrinks 2 times, and M stays unchanged, then surface gravity grows 4 times. The density grows 8 times.
Pressure gradient therefore grows 32 times.
But the radius over which the pressure gradient holds decreases 2 times.
So the central pressure grows 16 times - with inverse 4th power of radius.
Correct so far?

 

[Ken G]

I'm not so sure it's complicated!
Well - let's try to see the basic derivation for the horizontal.
A) Suppose that the star consists of perfect gas
PV=nRT.
For a self-gravitating sphere of uniform density, P can be derived from V:
If R shrinks 2 times, and M stays unchanged, then surface gravity grows 4 times. The density grows 8 times.
Pressure gradient therefore grows 32 times.
But the radius over which the pressure gradient holds decreases 2 times.
So the central pressure grows 16 times - with inverse 4th power of radius.
Correct so far?

Yes.

 

[snorkack]

Now, let's spell out the assumptions I made.
The gas is perfect:
1) P is strictly proportional to T. There is no degeneracy pressure
2) PV is strictly constant. There is no light pressure
and also:
3) n is strictly independent on T. There is no dissociation.
4) The sphere is of uniform density.
Correct?

 

[Ken G]

Actually you did not need PV to be constant to get your conclusion there. Also, your results are scaling laws-- or what can also be called "characteristic numbers" for the star. If you understand those things, you are fine-- the expressions will be of value for a first-brush understanding. Does that make them "correct so far"? That depends on what you are trying to do with them. If you are just trying to understand why characteristic numbers in a star are what they are, you are off to a fine start. If you are trying to demonstrate the internal run with radius, of course you need a pressure gradient. But there's no need for you to try and reinvent the wheel here, regardless of the level of simplicity or complexity you are shooting for. Just look at those notes you found, and see where they show how to estimate the luminosity of a main sequence star. Quite a few approximations are in place throughout those notes, for example on page 26. Are they "correct"? Depends on what you want to do with them. The author of those notes and I both know those are not correct equations in any exact or absolute sense, but they are good enough for the purpose, and I agree. That's always true in science, we never use exact expressions, but we tailor what we use to our needs.

If you continue to follow the explanations in those notes, you can learn an enormous amount about stars. Of course, very few of the expressions there are "correct," but they serve masterfully-- for their purpose. And at one point (page 30), the author does give a rough explanation for why the Henyey tracks leading up to the main sequence are not perfectly horizontal-- as the star loses heat and contracts, its internal temperature rises, causing ionization, and reducing the opacity. This allows freer escape of light. As usual for radiative stars without shell fusion, the key issue in the luminosity is the rate of escape of light. The simplest approximation is to keep the opacity fixed, as for pure free electron opacity, but for the lower mass stars with envelopes that extend out to cooler temperatures, it is necessary to track the degree of ionization, and other possible changes in the opacity. That explains why the Henyey tracks are less horizontal at cooler envelope temperatures, where the surface temperature goes below about 10,000 K. It's all a matter of how much detail one needs, but the salient insight is that luminosity depends on the rate of escape of light-- not on nuclear cross sections, and this is mostly still true even when the main sequence is reached, as those plots show by how little the luminosity changes as fusion ignites.

Indeed, the change in luminosity during the main sequence phase is fairly weak, even though there are dramatic changes going on to the fusion cross sections per gram (the H is turning into He!). So we can say we owe our existence on this planet to the fact that solar luminosity is not particularly sensitive to fusion cross sections!

 

[snorkack]

Actually you did not need PV to be constant to get your conclusion there.

How about, needing the assumption 2), of no light pressure?

Just look at those notes you found, and see where they show how to estimate the luminosity of a main sequence star. Quite a few approximations are in place throughout those notes, for example on page 26.

In place, but not spelt out.
Now, return to my assumption 4):
The sphere is of uniform density.
One which is manifestly false at all times. The density increases inwards, a lot.
But now let's replace assumption 4) with another one:
4B) Whatever the radial distribution of density is, as the star contracts, the density distribution remains unchanged.
On that assumption, the previous conclusion:
* The central pressure increases with inverse fourth power of radius
also holds for such unchanging radial density distributions.
Correct so far?

 

[Ken G]

How about, needing the assumption 2), of no light pressure?

Eddington found a very simple way to include light pressure, but it only matters for high-mass stars. So again, it's a question of what you are trying to do. An even simpler, and more approximate way (there's always that tradeoff, is there not?) to account for light pressure is to simply let the mass-luminosity relation you get without accounting for light pressure saturate to the Eddington luminosity when the luminosity would otherwise exceed it.

Now, return to my assumption 4):
The sphere is of uniform density.

Again, that works fine for an understanding of the characteristic density in a star, which is more like the average density. Real stars have density gradients, which can then be understood via various levels of approximation. A simple approach that is not "correct", but is still useful, is to assume a polytrope, for example. But even if you do that, the characteristic densities you get will still obey the scaling laws you just gave. Those relations are certainly not meaningless, even if you don't regard them as "correct" for your purposes.

One which is manifestly false at all times.

Again, only if you don't understand what you are doing with those expressions.

The density increases inwards, a lot.

I'm well aware of that, anyone who even knows what a "Henyey track" is would certainly know that. Nevertheless, such a person can still fathom a concept of "characteristic density" of a star. Do you know what it is for the Sun? Interestingly, about the same as water. Can you get that from your expressions above? Yes, to some useful level of approximation. You simply have to know what you are doing.

But now let's replace assumption 4) with another one:
4B) Whatever the radial distribution of density is, as the star contracts, the density distribution remains unchanged.
On that assumption, the previous conclusion:
* The central pressure increases with inverse fourth power of radius
also holds for such unchanging radial density distributions.
Correct so far?

Yes, that's what is done with polytrope models, which are one way of assuming the radial distributions don't change, and just looking at the scaling laws as the radius changes (for example). I'm not sure where you are going with this though-- these kinds of arguments are all in those notes, and done quite well there.

 

[snorkack]

Again, that works fine for an understanding of the characteristic density in a star, which is more like the average density.

To what purposes?

I'm not sure where you are going with this though-- these kinds of arguments are all in those notes, and done quite well there.

Applied there, but not spelt out.
Now, go to the next conclusion.
Since PV=nRT (assumptions 1 and 2)
and n is independent on T (assumption 3)
and V varies with 3rd power of R (assumption 4B), while P varies with 4th inverse power of R (also assumption 4B)
T must vary with inverse 1st power of R.
Correct?

 

[Ken G]

To what purposes?

To the purpose of understanding the characteristic density, pressure, and radius of stars like our Sun. Some like to be able to understand why those values are what they are, in simple if approximate terms. I guess others don't!

Since PV=nRT (assumptions 1 and 2)
and n is independent on T (assumption 3)
and V varies with 3rd power of R (assumption 4B), while P varies with 4th inverse power of R (also assumption 4B)
T must vary with inverse 1st power of R.
Correct?

Yes, though of course once again we must understand the meaning of this T-- it is the characteristic temperature of the gas in the star. There are simpler ways to get it, by the way-- I like the virial theorem, which essentially involves setting the characteristic velocity of the protons to the escape speed. So by self-consistent application of these simple formulae, we can understand why the Sun has the radius it has (approximately), and why the escape speed is what it is (approximately), from knowledge of the T where fusion kicks in in a big way. We can set T to the fusion T because that is characteristic of the gas in the Sun. We must certainly avoid associating T with the surface T of the Sun, that T only applies over a very tiny fraction of the gas, and is set by the radius (which we have just determined) and the luminosity (which we determine from the photon escape rate, just as done in those notes, though we haven't gotten there yet in this simple calculation).

 

[snorkack]

Yes, though of course once again we must understand the meaning of this T-- it is the characteristic temperature of the gas in the star.

No.
If we take the assumption 4B) that it is the whole density distribution that remains unchanged, and that assumptions 1-3) hold throughout the distribution, then it is the whole radial distribution of temperature which remains unchanged as T increases with inverse R. Exactly over the whole distribution - leaving the matter moot as to which point of distribution to pick as "characteristic".

There are simpler ways to get it, by the way-- I like the virial theorem, which essentially involves setting the characteristic velocity of the protons to the escape speed.

I dislike it. It fails to spell out the assumptions. Like 1) (no degeneracy).

We can set T to the fusion T because that is characteristic of the gas in the Sun. We must certainly avoid associating T with the surface T of the Sun, that T only applies over a very tiny fraction of the gas,

Just how do you back up the assumption that fusion T is somehow "characteristic", rather than apply only over a very tiny fraction of the gas?
Now, return to enumerating assumptions:
5) The heat conductivity of gas is completely independent on density, temperature and composition. dL/dS=k*(dT/dR) everywhere.
Correct?

 

[Ken G]

No.
If we take the assumption 4B) that it is the whole density distribution that remains unchanged, and that assumptions 1-3) hold throughout the distribution, then it is the whole radial distribution of temperature which remains unchanged as T increases with inverse R.

How does that lead to your saying "no"? Sounds like "yes" to me, because this is just the meaning of a "scaling law." This is all quite simple, I'm not sure what point you are trying to make. If you assume you have what is known as a "homology", such as a polytrope model of given index, what it means is that you have a run of all the parameters, like density and temperature, as a function of radius, that you are keeping fixed, except for overall scale. Then all that changes are the characteristic T, and characteristic density, and so on, which can be thought of as average values or just loosely as estimates that characterize the general state. Scaling laws and homologies are elementary applications of partial differential equations, they are a common technique for achieving simple and highly comprehensible results. They are also approximate, because homologies cannot in general be strictly enforced. What are you trying to tell me that I don't already know?

Exactly over the whole distribution - leaving the matter moot as to which point of distribution to pick as "characteristic".

Again, that simply doesn't follow. There is no difficulty choosing the meaning of a "characteristic" value, we do it all the time! Even the H-R diagram does this-- it plots luminosity against surface temperature, which are, of course, characteristic values of those quantities for each star!

I dislike it. It fails to spell out the assumptions. Like 1) (no degeneracy).

Again I have no idea what you mean here-- I spelled out the assumption of no degeneracy from the very start. Are you complaining that not all physics theories spell out all their assumptions? It is normal to have assumptions that are taken for granted, though I have no objection to spelling them out. I really have no idea what you are trying to say, we seem to have gotten very far from understanding the luminosity of stars!

Just how do you back up the assumption that fusion T is somehow "characteristic", rather than apply only over a very tiny fraction of the gas?

Oh, that's very easy. The fusion temperature of H is in the vicinity of 10 million K. The mass-weighted average temperature in the Sun is also in the vicinity of 10 million K. The thermal speed of protons at 10 million K is also characteristic of the escape speed from any part of the Sun. All of these are elementary applications of the concept of characteristic values, I'm not sure the issue you have with this but it is a completely standard way to carry out an approximate analysis!

Now, return to enumerating assumptions:
5) The heat conductivity of gas is completely independent on density, temperature and composition. dL/dS=k*(dT/dR) everywhere.
Correct?

Again, you seem to be saying that the expressions used in all physics theories are not exact. We should all be already aware that the expressions used in physics theories are not exact! Can you give me an example of any physics analysis you have ever seen that used "correct" expressions, in the sense you seem to mean? What they actually use are expressions that serve their purpose, just like the ones you are now talking about. It's all a questions of purpose. Many people actually do have the purpose of wanting to have a simple, albeit approximate, understanding of the characteristic values of stars, like radius, internal temperature, and the luminosity that results. They really do! In fact, all simulations do exactly that-- all that differs is the degree of precision in the characteristic values. For example, most simulations employ the concept of a "grid", within which are values of various parameters. Those values are characteristic of the gridzones-- but they are certainly not "correct", unless all you mean is that they serve their purpose. Right?

Put differently, some of the expressions you have written could be regarded as "one-point quadrature values" for the physics theories being employed. This is a routine thing to do, in the limit of greatest simplicity, and least accuracy-- yet they still work to provide characteristic values for the star, if you know what that means. The notes you cited do exactly this, is that not clear? Are you objecting to the notes you cited?

 

[snorkack]

Do you agree to enumerating assumption 5) - that the heat conductivity of gas is completely independent on density, temperature and composition. dL/dS=k*(dT/dR) everywhere.

 

[Ken G]

Do you agree to enumerating assumption 5) - that the heat conductivity of gas is completely independent on density, temperature and composition. dL/dS=k*(dT/dR) everywhere.

Are you asking me if I think that equation is ever exactly true? No, I don't think that equation is ever exactly true. Do I think it could be useful for certain purposes? Sure, it depends on the purpose though. Since I don't know what your purpose is, I cannot answer your question. If your purpose is to get an approximate but simple and useful understanding of stars, then there are certainly ways to simplify the concept of "heat conductivity," but your way doesn't look like a reasonable approach to me.

The notes you cited would have used an approximate radiative transfer equation to do that, as you should be able to see, because heat conduction in the stars of interest is primarily by radiative diffusion (this is actually the most crucial factor for understanding the luminosity of these stars). It appears that you mean dL/dS as luminosity per surface area, also known as flux density. If so, then your expression is not very good-- you should instead look at the expressions used on p. 22 in the notes you cite. The usual way this is done is to say that the radial gradient in T⁴ times the photon mean-free-path is proportional to the radiative energy flux density (per area), which is like applying the Stefan-Boltzmann law at two facing surfaces on opposite sides of the photon mean-free path. This is the radiative diffusion equivalent of thermal conduction, those notes do assume a lot of physics is already known so are not necessarily perfect for introductory material. But the point is, it is the gradient in T⁴, not T, that you should focus on. Also, the opacity appears, and does not have to be constant, though constant opacity (per gram) is certainly a good starting point for comparison.

 

[snorkack]

If your purpose is to get an approximate but simple and useful understanding of stars, then there are certainly ways to simplify the concept of "heat conductivity," but your way doesn't look like a reasonable approach to me.

The notes you cited would have used an approximate radiative transfer equation to do that, as you should be able to see, because heat conduction in the stars of interest is primarily by radiative diffusion (this is actually the most crucial factor for understanding the luminosity of these stars). It appears that you mean dL/dS as luminosity per surface area, also known as flux density. If so, then your expression is not very good-- you should instead look at the expressions used on p. 22 in the notes you cite. The usual way this is done is to say that the radial gradient in T⁴ times the photon mean-free-path is proportional to the radiative energy flux density (per area), which is like applying the Stefan-Boltzmann law at two facing surfaces on opposite sides of the photon mean-free path. This is the radiative diffusion equivalent of thermal conduction, those notes do assume a lot of physics is already known so are not necessarily perfect for introductory material. But the point is, it is the gradient in T⁴, not T, that you should focus on.

That much is a good insight to remember!
Now, note that d/dR(T⁴)=4*T³*(dT/dR)
As the star contracts with the aforesaid unchanging radial density and temperature distribution, density increases with 1/R³, and the factor of 4T³ also increases with 1/R³ - so these cancel out, and heat conductivity is constant after all:
5) dL/dS=k*(dT/dR) everwhere
but nice to remember that heat conductivity is independent on density and temperature just because these cancel out.
Next assumption:
6) Heat is transferred outwards only by conduction, in form aforesaid - no convection anywhere.
Correct?

 

[Ken G]

That much is a good insight to remember!
Now, note that d/dR(T⁴)=4*T³*(dT/dR)
As the star contracts with the aforesaid unchanging radial density and temperature distribution, density increases with 1/R³, and the factor of 4T³ also increases with 1/R³ - so these cancel out, and heat conductivity is constant after all:
5) dL/dS=k*(dT/dR) everwhere
but nice to remember that heat conductivity is independent on density and temperature just because these cancel out.

Good point, the ratio of T³/rho is constant with depth, so the coefficient in the heat conduction does act approximately like a constant within a given star. However, it will not be the same constant from star to star, instead it will depend on mass. I can show you why if you are not aware of this.

Next assumption:
6) Heat is transferred outwards only by conduction, in form aforesaid - no convection anywhere.
Correct?

Yes, that's all fine-- so long as you recognize that the homology argument does not mean k cannot depend on M. (In fact, it will.)

 

[snorkack]

Then we can notice that:
T increases with 1/R;
R also decreases;
therefore dT/dR increases with 1/R²;
but therefore it also is dS/dL that increases with 1/R²;
But S itself decreases with R²!
Therefore L remains constant with changing R! QED.
But now remember the assumptions I spelt out and enumerated as 1 to 6. And it´s worth to also spell out 7), which is related to 4B):
7) The radial distribution of L also remains unchanged.

 

[Ken G]

Then we can notice that:
T increases with 1/R;
R also decreases;
therefore dT/dR increases with 1/R²;
but therefore it also is dS/dL that increases with 1/R²;
But S itself decreases with R²!
Therefore L remains constant with changing R! QED.

Exactly. This is the easy way to see that radiatively diffusive stars have a given mass-luminosity relationship, regardless of their state of contraction. Then we have only to notice that when fusion begins, mostly what it does is self-regulate to replace the heat that is being lost, stopping the contraction. Indeed, many textbooks make great hay of the stability of the self-regulation of fusion, but don't seem to recognize this is why fusion doesn't alter the internal structure very much! It is the extreme temperature sensitivity of fusion that is crucial-- it means small adjustments in T are all you need, which have little effect on the heat conduction! Thus, the same argument gives us the luminosity of main-sequence stars-- with no knowledge of fusion cross sections. We should say that the fusion rate does not control the luminosity, the luminosity controls the fusion rate (mostly).

But now remember the assumptions I spelt out and enumerated as 1 to 6. And it´s worth to also spell out 7), which is related to 4B):
7) The radial distribution of L also remains unchanged.

This is why there is some small restructuring of the interior distribution when fusion begins, but it is not very much-- as can be seen from the H-R diagram, the onset of fusion does not change L much. Just look at the evolutionary tracks above, you see that little squiggle as fusion begins-- little impact on L at all! Indeed, most of the change in L during fusion happens much later, as H is converted to He, which has an impact on the opacity and so the constant k in your expression for the heat conduction. There is also a change in mean molecular weight, that enters the force balance. Neither of those have anything to do with fusion cross sections!

Bottom line: if we are not concerned about small changes in the run with radius, we can treat the entire evolution of a star throughout its radiative phase, including the onset of fusion, with the above analysis, deriving a useful approximation for the stellar luminosity, called the "mass-luminosity relation", which does apply on the main sequence, and does not require knowing fusion cross sections. Of course, to add the next layer of detail and accuracy, one will need to include how fusion alters (somewhat) the internal structure. However, there are many other effects that can alter internal structure that might not even be in a detailed simulation, such as rotation, differential rotation, magnetism, abundance variations, and an a priori model of convection. So one always has to choose one's goals in any calculation!

 

[davenn]

Hi @Ken G

have to thank you for you excellent posts. I was totally unaware of the Henyey track before reading this thread
Has been lots of new learning regards
Dave

 

[Ken G]

Yes, the Hayashi tracks get all the press, but Henyey tracks are actually easier to understand-- and more relevant to stars the way we normally think about them! It also let's you understand red giants by combining Henyey and Hayashi physics: you start with the mass of the degenerate core as the external input, which sets the temperature in the fusion shell. The fusion shell then self-regulates the amount of mass in the shell, to create a kind of mini-me Henyey star, which is just a shell! The fusion in the shell self-regulates, by regulating the mass in the shell at the externally imposed temperature (controlled by the amount of mass in the core), such that fusion just replaces the heat that is radiatively diffusing out of that shell. This sets the luminosity, which is then the input to the Hayashi envelope. Normally we think of Hayashi stars as having an imposed radius and a resultant luminosity, but in red giants, it's the other way around! The fact that the mini "Henyey shell" is narrow means you get a lot more rapid escape, and the T gets pretty high in there as the core builds up mass, so the combination of those two facts is why the luminosity goes way up. The reason they all get to about the same place in the red giant branch is because that's where the core gets enough mass to get hot enough to start fusing He-- regardless of the mass of the rest of the star!

A warning-- once you understand these things, you will have a hard time reading elementary textbooks-- they make such a horrible mess of it!

 

[Drakkith]

A warning-- once you understand these things, you will have a hard time reading elementary textbooks-- they make such a horrible mess of it!

Indeed. I groan every time I hear or read that it is fusion that causes stars to shine... I guess everyone just forgot about black body radiation and don't realize that a non-fusing ball of gas at 6,000 K would shine with approximately the same spectrum as the Sun.

 

[snorkack]

It is the extreme temperature sensitivity of fusion that is crucial-- it means small adjustments in T are all you need, which have little effect on the heat conduction! Thus, the same argument gives us the luminosity of main-sequence stars-- with no knowledge of fusion cross sections. We should say that the fusion rate does not control the luminosity, the luminosity controls the fusion rate (mostly). This is why there is some small restructuring of the interior distribution when fusion begins, but it is not very much-- as can be seen from the H-R diagram, the onset of fusion does not change L much. Just look at the evolutionary tracks above, you see that little squiggle as fusion begins-- little impact on L at all!

Yes, but that´s hindsight...
Step 1 was to spell out the assumptions that come into the derivation of the constancy of L.
Now that we are expressly aware of all the assumptions, when any of these are violated, we can expect some deviations from the result.
Step 2 should be to estimate what direction of effect could be expected when a deviation occurs.
And Step 3 should be to estimate the size of the deviation.
Now, the assumption
4B) The radial distribution of density, temperature and luminosity remains unchanged
is actually a vital one.
Look at the example of a red giant. Its radius is blowing up. So its escape speed is decreasing. Therefore its "characteristic" density and temperature should be decreasing, right?
Yet the central density, pressure and temperature of a red giant are increasing, a lot. And so is luminosity. Because the assumption 4B), of unchanged radial distribution of density, is being deviated from.
Now, when a star leaves Henyey track and enters into main sequence, the radial distribution of luminosity certainly changes. A star heated by contraction is releasing heat throughout its interior - its luminosity is small near the centre and increases far out.
Whereas, because of the stated "extreme temperature sensitivity of fusion", fusion only happens at the highest temperature part of star - a small mass of gas nearest the centre, otherwise not characteristic of the star. The luminosity is concentrated near the centre, where dL/dS acquires high values. Right?
This necessarily also breaks down assumption 4B).
As it happens, the change in luminosity on reaching main sequence is relatively small, compared to change of luminosity on leaving main sequence. But that´s Step 3, estimating actual size of the deviation. We are not at Step 2 yet.
Do you agree with the point that a change in radial distribution of L is likely to change L itself?

 

[Ken G]

Yes, but that´s hindsight...

Science is always full of hindsight. How did we know the Sun underwent fusion at all? Hindsight! What matters is that both models, and observations, verify that the onset of fusion does not significantly change the internal structure, or luminosity, of a star.

Look at the example of a red giant.

As I said above, shell fusion is not at all like core fusion-- shell fusion rips the star in half, creating two completely different pieces-- an inert core that is just piling up mass and getting hotter, and a convective envelope obeying completely different physics, reminiscent of the protostar phase. Nevertheless, we can still understand the luminosity of a red giant if we recognize that the shell fusion zone is a kind of mini Henyey object, that is, radiatively diffusive. But yes, shell fusion completely changes the internal structure of a star, core fusion does not. These are very important things to know about stars, if one will have a simple understanding of how they work. I recommend Kippenhahn and Wiegert to see all this. Of course, it is up to every individual if they wish to have intellectual understanding of how stars work-- one can always just take a "black box" approach, and say, "it's in the simulation somewhere." Personally, I like to have my own understanding, even if it requires some hindsight and approximation!

Its radius is blowing up. So its escape speed is decreasing. Therefore its "characteristic" density and temperature should be decreasing, right?

As I said above, to use concepts like "characteristic values", you do still have to know what you are doing. In a main sequence star, the internal structure is very much as though the star is "all one thing", wearing a cool coat at the interface with deep space. Hence, for main sequence stars, "characteristic values" characterize the star as a whole. But red giants aren't like that at all-- they are very much several different objects, a degenerate core, a shell fusion zone, and a Hayashi envelope, all coexisting. But once you understand that, it just means you need to understand which characteristic value is needed to understand each of those parts. Once you do, you can actually understand red giants quite well-- without solving a full-blown simulation, it fits right in your own head. But again, it's a personal matter if people wish to have that kind of understanding, or not.

Whereas, because of the stated "extreme temperature sensitivity of fusion", fusion only happens at the highest temperature part of star - a small mass of gas nearest the centre, otherwise not characteristic of the star. The luminosity is concentrated near the centre, where dL/dS acquires high values. Right?

All I can tell you is, I stand by everything I have said. Read Kippenhahn and Wiegert, it is an advanced graduate level stellar astrophysics text, and everything I've said is in there too.

Do you agree with the point that a change in radial distribution of L is likely to change L itself?

No, not significantly-- if the change is not significant. Look at the H-R diagram for that answer.

 

[Ken G]

Indeed. I groan every time I hear or read that it is fusion that causes stars to shine... I guess everyone just forgot about black body radiation and don't realize that a non-fusing ball of gas at 6,000 K would shine with approximately the same spectrum as the Sun.

My favorite is the claim that higher-mass stars are more luminous because they have higher pressure cores, so fuse faster! Baloney.