Wrong Explanations for the Luminosity of Main-Sequence Stars

[Ken G]

There is a rather surprising thing afoot in many introductory astronomy texts, and many online web course notes, relating to a commonly seen but completely wrong explanation for why high-mass main-sequence stars have a much higher luminosity than low-mass main-sequence stars. You would really think this would be a basic question, and indeed its answer has been well known for about 100 years, and can still be found in all the high-level texts. But in course notes, and introductory texts, which still ought to know better, you generally find a "truthy" sounding explanation along the lines of, higher mass yields stronger gravity, greater weight, a higher temperature and pressure core, and more rapid fusion as a result.

For example, if I google the question "why are high-mass main-sequence stars more luminous than low-mass main-sequence stars," the first hit does not attempt to answer the question, and the second hit is a Wiki entry that does give the correct answer but it largely plays out in the bibliography to the article, so the reader might go on to the third hit, an Astropedia entry at:
http://m.teachastronomy.com/astropedia/article/Understanding-the-Main-Sequence
which asserts:
" More massive stars have greater gravity that creates higher pressure in the stellar interior. The higher pressure results in higher temperature that causes higher energyoutput by the fusion process, giving both higher luminosity and higher surface temperature."
The next hit does not offer an explanation, it just cites the empirical relationship, but the fifth hit, lecture notes from Lick observatory at http://www.ucolick.org/~mfduran/AY2/lectures/Class-02-21.pdf , repeats verbatim this completely wrong explanation, as though it was cut and pasted from the same source (or written by the same author?). Other websites at least put it into their own words, but they often repeat some version of that false dogma.

So by this point, a student seeking an answer to this question would have found some sites that don't answer it, a Wiki entry that requires some digging to get to the right answer, and two authoritative websites that very clearly "explain" the answer by saying something completely wrong. How can we have this sorry situation in regard to such a simple question?

For those who do not know why that answer is so completely wrong, the correct answer is that high-mass stars do not need to contract as much to get to fusion temperatures in their core, so they have both lower densities, and lower pressures, than lower mass main-sequence stars. That makes them bigger and leakier buckets of light, pure and simple. Yes, that's right, they have lower core pressures. Also, their luminosity is not set by their fusion rates, instead their fusion rates self-regulate to simply replace whatever light is leaking out. The luminosity is set by the rate the light leaks out, which mostly relates to the physics of thermal radiation and how it diffuses, as was known even before fusion was discovered. A somewhat awkward but basically correct version can be found at http://en.wikipedia.org/wiki/Mass–luminosity_relation , which suffices to show that the rate light diffuses out of a star depends almost exclusively on the mass of the star, period, with no reference to fusion, and certainly no false claims that high mass stars have higher core pressures.

So what gives here, why are so many authoritative sources so wrong on this simple issue?

 

[Matterwave]

There is something to be said about the extreme temperature dependence of fusion rates and the fact that high mass stars have much higher temperature cores, up to billions of Kelvin or higher, than lower mass stars though.

Although it is true that a star basically simply has to maintain virial equilibrium to remain on the Main sequence, and the fusion rates are dictated by this virial equilibrium requirement, it is still true that more luminosity = more fusion or more fusion = more luminosity. As you yourself stated, the fusion rate is sufficient to replace the "energy lost" due to luminosity, and thus brighter (main sequence, or more generally any star that is at equilibrium and not evolving towards equilibrium) stars have higher fusion rates at their cores.

 

[Ken G]

There is something to be said about the extreme temperature dependence of fusion rates and the fact that high mass stars have much higher temperature cores, up to billions of Kelvin or higher, than lower mass stars though.

Yes, but the correct cause-and-effect there is usually reversed. What is actually more true is that the reason high-mass stars have higher core temperature is because they have higher luminosity, not the other way around. The way to see this is to take a stellar structure calculation, and just double all the nuclear rates by doubling the cross sections. Will this double the luminosity because fusion physics is so important? No, the luminosity will change hardly at all, what will happen is the core temperature will just adjust itself downward to counter the higher fusion cross sections-- that is the only converged solution you will find. So we see that core temperature self-regulates to match the luminosity, not the other way around as all these incorrect sources suggest.

Although it is true that a star basically simply has to maintain virial equilibrium to remain on the Main sequence, and the fusion rates are dictated by this virial equilibrium requirement, it is still true that more luminosity = more fusion or more fusion = more luminosity. As you yourself stated, the fusion rate is sufficient to replace the "energy lost" due to luminosity, and thus brighter (main sequence, or more generally any star that is at equilibrium and not evolving towards equilibrium) stars have higher fusion rates at their cores.

Yes, the fusion rate must equal the luminosity, the issue is which is the horse-- and which is the cart. That, and the "truthy" claim that high-mass stars have higher T cores because they have stronger gravity, which exposes a total lack of understanding of their structure. It is a bit counterintuitive, I admit, but I still find it astonishing that so many highly authoritative sources make such a hash of it. They really should know better.

 

[marcus]

... the correct answer is that high-mass stars do not need to contract as much to get to fusion temperatures in their core, so they have both lower densities, and lower pressures, than lower mass main-sequence stars. That makes them bigger and leakier buckets of light, ...

I'd like to have a better intuitive understanding of this, Ken.
Can you break down the pressure for me into matter pressure versus radiation pressure?

Do high-mass and low-mass stars differ as to what fraction of their overall core pressure consists of radiation pressure? And what fraction is mass pressure?

I'm wondering along similar lines about core density.

I don't completely know how to quantify how much something contracts, when you say "do not need to contract as much to get to fusion temperatures. My intuitive handle on how much something contracts is matter density. It would sound like a tautology if I were to say "does not need to contract as much so therefore has lower density". I have the feeling I am missing something.

Could it be something very simple like the surface to volume ratio of the core, which is lower for high mass stars? IOW in a big star there is less surface (per unit core volume) through which light can escape from core. So radiation accumulates more easily in core and becomes a larger component of the overall core density and pressure.

So therefore (hypothetically) the matter density around the core does not have to be as large as it would other wise be (because radiation is doing a larger share of the support job) and since it is matter density that determines "leakiness", a lower matter density means greater leakiness (thinking of scattering, mean free path, random walks here.)

If I want to cultivate some intuition about mass-luminosity and core conditions in high versus low mass stars, does it make sense for me to introduce the surface to volume ratio of the core region, into my thinking? Does it make sense for me to be thinking about how the core pressure breaks down into matter and radiation components?

 

[Matterwave]

Yes, but the correct cause-and-effect there is usually reversed. What is actually more true is that the reason high-mass stars have higher core temperature is because they have higher luminosity, not the other way around. The way to see this is to take a stellar structure calculation, and just double all the nuclear rates by doubling the cross sections. Will this double the luminosity because fusion physics is so important? No, the luminosity will change hardly at all, what will happen is the core temperature will just adjust itself downward to counter the higher fusion cross sections-- that is the only converged solution you will find. So we see that core temperature self-regulates to match the luminosity, not the other way around as all these incorrect sources suggest.
Yes, the fusion rate must equal the luminosity, the issue is which is the horse-- and which is the cart. That, and the "truthy" claim that high-mass stars have higher T cores because they have stronger gravity, which exposes a total lack of understanding of their structure. It is a bit counterintuitive, I admit, but I still find it astonishing that so many highly authoritative sources make such a hash of it. They really should know better.

But in math the equality sign does not have a "cause and effect" relationship. The math comes out to essentially "$\text{higher mass}=\text{higher luminosity}=\text{higher fusion rates}$", I suppose your issue is that the relationship should be explained as "$\text{higher mass}\rightarrow\text{higher luminosity}\rightarrow\text{higher fusion rates}$" rather than "$\text{higher mass}\rightarrow\text{higher fusion rates}\rightarrow\text{higher luminosity}$"? If so, for clarity of presentation I agree; however, I think it's a pretty understandable "mistake" to make.

 

[Ken G]

I'd like to have a better intuitive understanding of this, Ken.
Can you break down the pressure for me into matter pressure versus radiation pressure?

Sure-- radiation pressure is only important for the most massive stars, and that starts to mess with the virial theorem and the simple answers, much like lots of convection messes with the very lowest mass stars. So I really should have said, "for main-sequence stars between about 0.5 to 50 solar masses," so the pressure is nonrelativistic gas pressure and the heat transport is primarily radiative diffusion. Which reminds me, another annoyingly false thing you often find in textbooks is that fusion creates radiation pressure which balances gravity, and that's almost completely false. Except for the highest mass stars (say, above 50 solar masses), radiation pressure never plays any key role, not even in the fusion zone-- fusion just creates heat to replace what is lost, and it's pretty much all regular old gas pressure.

Do high-mass and low-mass stars differ as to what fraction of their overall core pressure consists of radiation pressure? And what fraction is mass pressure?

Yes, there is something known as the "Eddington factor" to establish that ratio, but it is fairly small until mass gets upward of 50 solar. For the Sun, it is about 0.00003, and that's still true in the core.

I'm wondering along similar lines about core density.

Here is a site where you can calculate the central density in a main-sequence star: http://www.astro.ulb.ac.be/~siess/pmwiki/pmwiki.php/WWWTools/Isochrones
If you check 1 and 7 solar masses, and click the box "extra information", you will see that the central density is about 7 times larger for the lower mass star. This is a natural consequence of the virial theorem, for self-gravitating balls with similar central temperatures.

I don't completely know how to quantify how much something contracts, when you say "do not need to contract as much to get to fusion temperatures. My intuitive handle on how much something contracts is matter density. It would sound like a tautology if I were to say "does not need to contract as much so therefore has lower density". I have the feeling I am missing something.

The point is, it has lower density, and lower pressure. The questionable sources say the opposite-- they think they need an explanation for why there is a faster fusion rate, based on the temperature and pressure in the core, so they imagine the higher mass must lead to higher pressure and density in the core. What it actually does is lead to more light leaking out, so the core needs to self-regulate its temperature to be a little higher, but at lower pressure and density because the latter are consequences of the virial theorem.

Could it be something very simple like the surface to volume ratio of the core, which is lower for high mass stars? IOW in a big star there is less surface (per unit core volume) through which light can escape from core. So radiation accumulates more easily in core and becomes a larger component of the overall core density and pressure.

The key point is, the luminosity is determined by the physics that analyzes how light gathers into thermal pools, and gradually leaks out. That physics never needs to mention fusion, you can get a good understanding of the luminosity of a radiatively diffusing star without even knowing there is any such thing as nuclear fusion (indeed, Russell and Eddington did just that). How you analyze the pooling and escape of the light may be a matter of personal taste, I would point out that the time it takes the light to escape is more or less the same-- higher mass stars are larger so the light has farther to go, but they also have lower density so the light diffuses more easily, so on balance it takes about 100,000 years or so for light to escape any main-sequence star. Given that, the luminosity comes down to how much light is in there at any given time (the "size of the bucket", if you like), and that scales approximately like the size of the star (given that to a very rough approximation, all main-sequence stars have similar core temperatures-- a fact that is due to fusion physics, but it turns out the luminosity would be pretty much the same for any core temperature if the star is virialized, which is why stars tend to both approach, and leave, the main sequence at constant luminosity). So the simplest possible correct answer as to why more massive main-sequence stars are more luminous is "because they are larger buckets of light."

So therefore (hypothetically) the matter density around the core does not have to be as large as it would other wise be (because radiation is doing a larger share of the support job) and since it is matter density that determines "leakiness", a lower matter density means greater leakiness (thinking of scattering, mean free path, random walks here.)

The share that radiation is doing is generally negligible, so I would say the leakiness is of central importance simply because that's what luminosity is. All the same, the radiative force does depend on the luminosity, so there may be a way to frame the situation like that if you have a mind to.

If I want to cultivate some intuition about mass-luminosity and core conditions in high versus low mass stars, does it make sense for me to introduce the surface to volume ratio of the core region, into my thinking? Does it make sense for me to be thinking about how the core pressure breaks down into matter and radiation components?

Yes to the former, and maybe to the latter. But let me just suggest a different way to break it down. Start by saying the star has a characteristic T and R, given its M. Use thermal radiation to say how much light energy is in that bucket at any given time, given T and R. Assert force balance, perhaps by saying the kT of each proton is of order its characteristic gravitational potential energy. Equate the luminosity to the total light energy, divided by the escape time, where the escape time is of order R/v (where here v is the average diffusion speed of light, c/tau for tau the optical depth). If you take a constant opacity to make life simple (say, free-electron opacity), you get that tau ~ rho*R*kappa for constant kappa. Put all those pieces together, and you will see that the luminosity of a virialized radiatively diffusing star with constant opacity per gram scales with (RT)^4 / M, where RT scales with M by the virial theorem. Done-- you get L ~ M^3, and you don't even need to know R and T independently of each other, let alone anything about fusion. This is not a new result, it's what the early astronomers did, I think the name Henyey often gets attached to it.

 

[Ken G]

But in math the equality sign does not have a "cause and effect" relationship.

You are certainly right that at the end of the day, the star has to figure out some way to match its fusion rate to its luminosity, and that will require some iteration in detail, where the fusion physics is going to appear. But there is still conceptual value in attributing cause and effect, and you can see that conceptual value in several ways. One way is to ask, what happened to the luminosity of the Sun when fusion initiated? The answer to that is: "basically nothing at all." Another way is to ask, what happens to the mass-luminosity relationship when we go from the details of p-p fusion to the details of CNO fusion? The answer to that is "not a heck of a lot." So changes in fusion just don't do much. But start monkeying with the opacity in a star, and look out-- the changes in luminosity will be dramatic, and the fusion rates will follow along to match. That's the value in seeing the cause and effect.

The math comes out to essentially "$\text{higher mass}=\text{higher luminosity}=\text{higher fusion rates}$", I suppose your issue is that the relationship should be explained as "$\text{higher mass}\rightarrow\text{higher luminosity}\rightarrow\text{higher fusion rates}$" rather than "$\text{higher mass}\rightarrow\text{higher fusion rates}\rightarrow\text{higher luminosity}$"?

Yes, that is definitely part of the correction I would offer, an important correction in logical inference.

If so, for clarity of presentation I agree; however, I think it's a pretty understandable "mistake" to make.

And would you also say it is an understandable "mistake" to say that the core pressure and density in higher mass stars is higher? Because that's the real issue here, if one uses the wrong logic, one is led to a very wrong conclusion. It's not just a matter of how we get there, it's where we get in the first place.

 

[Astronuc]

I should be more like

$\text{higher mass}\rightarrow\text{higher gravity}\rightarrow\text{higher density}\rightarrow\text{higher fusion rates}\rightarrow\text{higher temperature}\rightarrow\text{higher luminosity}$

The above line does not convey the feedback effects or interrelationship among density, temperature, and reaction rates. Of course, higher temperature causes higher fusion rates for a given density, but higher temperatures can reduce density at a given pressure.

Fusion reaction rates are a function of nuclear density and temperature.

Pressure is a function of density and temperature.

The higher the fusion rate, the higher the energy density, the higher the temperature, the higher the pressure (transfer of momentum), . . . . , but there is some equilibrium involved. Part of that equilibrium involves loss of energy at the surface of the star.

Temperature is a function of the kinetic energy of the particles (nuclei and electrons), but it is the 'temperature' of the nuclei that governs the fusion reaction rate in conjunction with the particle density. Luminosity is related to the rate at which photons are emitted, and there are several processes involved. Photons may be emitted from atoms, or generated from brehmsstrahlung reactions, or scattered off electrons.

I think one needs to understand stars in terms of equilibrium and the coupled processes, which are described by systems of coupled equations.

Anyone can publish online. I would imagine some sites are published by non-scientists, or science writers, who don't know the details of the science/physics involved. Ideally, folks who publish about science/physics would take care to get the facts or information correct. Too often this is not the case. I wouldn't consider any site authoritative unless it's peer-reviewed by experts in the subject matter.

 

[Ken G]

I should be more like

$\text{higher mass}\rightarrow\text{higher gravity}\rightarrow\text{higher density}\rightarrow\text{higher fusion rates}\rightarrow\text{higher temperature}\rightarrow\text{higher luminosity}$

It should not be like that, that suggests that stars with higher fusion rates have higher density, which would not be correct.

The higher the fusion rate, the higher the energy density, the higher the temperature, the higher the pressure (transfer of momentum), . . . . , but there is some equilibrium involved.

No, the pressure is not higher, it is lower. My whole point is that the only reason anyone would ever think the pressure in a high-mass star would be higher than in a low-mass star, when it is well known that their core temperatures are similar (that's the only place where fusion physics much matters in a main-sequence star, it acts like a core thermostat), is if they are trying to use fusion to explain the higher luminosity, as though the luminosity were being forced to be higher because of the fusion in the core. That's exactly what is not true-- the luminosity is not controlled by fusion physics, see my explanation of that to marcus. All fusion physics does is self-regulate the core T to match the need to replace the luminosity. There can be no understanding of how core fusion is self-regulated if one thinks the luminosity has to carry whatever the core fusion is sending out, which is the way the luminosity is often explained.

I think one needs to understand stars in terms of equilibrium and the coupled processes, which are described by systems of coupled equations.

At the end of the day, any physics question can be answered "that's what happens when you solve the equations." But there is still a place for physical insight, and getting the insight right is important-- it helps us anticipate what would happen if you could double the fusion cross sections, or double the opacity. Improper insights will lead to wrong expectations in situations like that.

Anyone can publish online. I would imagine some sites are published by non-scientists, or science writers, who don't know the details of the science/physics involved.

Yet amazingly, I am talking exclusively about intro textbooks and college course websites. These are sources we would like to regard as authoritative, so when they do have such a large gaffe, it is important to point it out and get it corrected.

I wouldn't consider any site authoritative unless it's peer-reviewed by experts in the subject matter.

In a sense, we are doing the "peer review" now.

 

[Astronuc]

There can be no understanding of how core fusion is self-regulated if one thinks the luminosity has to carry whatever the core fusion is sending out, which is the way the luminosity is often explained.

Luminosity is simply the measure of energy loss from a star. In equilibrium, that energy loss is balanced by the energy transported from the interior of the star, and the source of energy is the fusion processes.

 

[Ken G]

Luminosity is simply the measure of energy loss from a star. In equilibrium, that energy loss is balanced by the energy transported from the interior of the star, and the source of energy is the fusion processes.

Certainly I agree. But the question is, which sets which-- if we want to understand the luminosity, should we expect it to be given to us by the fusion rate, or if we want to understand the fusion rate, should we expect it to be given to us by the luminosity? The latter is much more true, even though certainly any complete solution must balance both. It's like a horse and a cart-- there is no way you can know how fast a horse can pull a cart until you know the details of the cart, but you can still notice that the important player in that relationship is the horse. To see that importance in a star, ask what happens to the luminosity when fusion initiates, which is like asking what happens to how a horse walks when you faster a light cart behind it: not much.

But bear in mind, all this business about the correct logic is just to help unearth what went wrong in the totally incorrect explanation often given. The real point I'm making is that the totally wrong explanation is quite clearly totally wrong: it always includes some version that higher gravity is responsible for the higher temperature, which is wrong, and it generally also says this happens via some imagined higher density and/or pressure in the core, which is also wrong. That's the most egregious part, because it completely misunderstands the physics of stellar structure-- and it is surprisingly common.

 

[Matterwave]

And would you also say it is an understandable "mistake" to say that the core pressure and density in higher mass stars is higher? Because that's the real issue here, if one uses the wrong logic, one is led to a very wrong conclusion. It's not just a matter of how we get there, it's where we get in the first place.

Can you provide me a source that calculates this? I really can't recall this result either way. From the Virial theorem the temperature of a more massive star will certainly be higher since a larger mass equals more potential energy which must be balanced by a higher average kinetic energy. It seems to me that hydrostatic equilibrium would require that more mass = more gravitational force = higher pressure gradient necessary to support this force = higher pressure at the core. Higher pressure and temperature at the core in this elementary analysis leaves the densities indeterminant though since P~rho*T. Is there some nuance that I am missing here? It's been a while since I've done any rigorous stellar structure calculations. Thanks! :)

EDIT: I realized that a much more massive star might be much more puffier than a smaller star, and thereby reduce the gravitational force and therefore the central pressures. Is this a dominant effect?

 

[Ken G]

Can you provide me a source that calculates this?

Certainly, you can use the site I mentioned above, http://www.astro.ulb.ac.be/~siess/pmwiki/pmwiki.php/WWWTools/Isochrones , to calculate it yourself, but it's easier to just use the characteristic force balance or virial theorem, which says that T is proportional to the gravitational energy per particle, so T is proportional to M/R. Since T stays nearly the same as M varies from 0.5 to 50, you can say that R is nearly proportional to M. That means the characteristic density is nearly proportional to M^-2, where note the negative power. So even the most basic possible analysis tells us that any logic that leads us to expect the high luminosity from high-mass stars to stem from high core density is going to be flat wrong.

From the Virial theorem the temperature of a more massive star will certainly be higher since a larger mass equals more potential energy which must be balanced by a higher average kinetic energy.

Be careful, there are two parameters there, T and R. All you know from the virial theorem is that T ~ M/R, but you cannot tell if that will be higher or lower for high-M stars until you understand the physics that sets the luminosity (and that's not fusion, it is radiative diffusion). To make this clear, note that had it worked out that the opacity of high-mass stars was rapidly rising with mass, we would find that the core temperature of high mass stars was lower than low-mass stars, so it's not anything necessarily about the gravity there.

It seems to me that hydrostatic equilibrium would require that more mass = more gravitational force = higher pressure gradient necessary to support this force = higher pressure at the core.

That's the "truthy" sounding answer, but it isn't right. Dig into it-- you will find the flaws, it is actually the opposite of what is true.

Is there some nuance that I am missing here?

Yes-- you are missing the key element to the whole business: the escape of radiation, which is the horse that drives this cart. Before you get to fusion in the core, that "horse" controls the timescale as T rises and R drops, keeping T*R more or less constant as per the virial theorem. When T reaches fusion levels, the process pauses for a long time-- that's all fusion is ever doing in a main-sequence star.

EDIT: I realized that a much more massive star might be much more puffier than a smaller star, and thereby reduce the gravitational force and therefore the central pressures. Is this a dominant effect?

Yes. The key relation is T ~ M/R, so main-sequence stars approximately obey R ~ M because T is fairly constant.

 

[snorkack]

And yet look at a star that leaves main sequence and turns into red giant. Mass is at first constant (later on falls as it is shed), but luminosity changes, and so does radius. Temperature first rises slightly, then drops.

All of it because the properties of fusion change.

 

[Ken G]

And yet look at a star that leaves main sequence and turns into red giant. Mass is at first constant (later on falls as it is shed), but luminosity changes, and so does radius. Temperature first rises slightly, then drops.

All of it because the properties of fusion change.

Yes, the situation is quite a bit different in a red giant. Physically, the reason the details of fusion physics matters to the luminosity of red giants, but not main-sequence stars, is that in red giants, the self-regulation of the fusion rate, there is also self-regulation of the escape process itself, so that's a true give-and-take. That is what does not happen in a main-sequence star, it is handed its T and R from a history of contraction of the star, and when fusion becomes able to replace the luminosity, the contraction stops, end of story-- that's all fusion is doing, no real feedback on anything else. But in a red giant, there is a secularly evolving variable of considerable importance-- the mass of the inert core. That takes away fusion's ability to self-regulate its temperature-- the temperature of a shell-fusing zone, unlike a core-fusing zone, is fixed by the virial temperature of something else, that inert core. So when fusion can no longer self-regulate its T, it must self-regulate something else-- the mass of the shell fusing zone. So what happens in a red giant, is fusion just adjusts the mass of the shell-fusing zone until it supplies the luminosity that diffuses out through that very same shell fusing zone. It does not need to diffuse out of the whole star, just that narrow shell, because the rest of the stellar envelope goes convective. The need to self-regulate that diffusion zone, to match the fusion that happens in that same zone at an externally imposed T, forces fusion and luminosity to feedback significantly into each other.

So in effect, the luminosity of a red giant works like a kind of quasi-main-sequence star where the fusion temperature is handed to you by a secular effect, the buildup of a degenerate core, and what self-regulates to match the luminosity is the amount of mass in the shell-- which also controls the range that the luminosity has to diffuse through. Since that's a whole lot less mass to have to escape through, the light escapes much more easily, you end up with extremely high luminosities as a result.

Don't hold your breath that you will ever see an explanation like that in an intro textbook, if they can't even get the main-sequence stars right! You will need to find the real tomes on the subject, like Kippenhahn. What is really fascinating about red giants is that whole process I just described is happening on a scale of a few 10³ km, deep in a star that is about the size of Earth's orbit-- sitting on top of a core the size of Earth. The intro textbooks can't even be bothered to stress this absolutely stunning fact.

 

[snorkack]

Is the core of a red giant even required to be supported by ideal gas pressure, rather than radiation or degeneracy one?

 

[Ken G]

Usually one would approximate it as pure degeneracy pressure that only needs to support the weight of the core itself (the rest has little weight in comparison), so the core of a red giant is a lot like a white dwarf, but it's a pretty darn hot one so this is only approximate. You can think of a red giant as three independent stars that coexist-- a white dwarf core, a tiny hollow main-sequence star whose T is handed to it by the core and whose M is given by the need to match fusion and radiative diffusion at that T (which is where the fusion physics comes in), and finally a huge envelope that pretty much exists only to convect out that luminosity, created on the scale of a planet, and transported on the scale of a solar system. If there's a more amazing object in all of astronomy than a red giant, I don't know what it is.

 

[Drakkith]

Assume we have two stars of equal mass and luminosity. One is a main sequence star, the other is composed of a non-fusing gas but is identical to the other star in all other aspects. If you add the same amount of mass to both stars over a short amount of time, what will happen?

 

[Ken G]

Assume we have two stars of equal mass and luminosity. One is a main sequence star, the other is composed of a non-fusing gas but is identical to the other star in all other aspects. If you add the same amount of mass to both stars over a short amount of time, what will happen?

Excellent question, because it perfectly illustrates the point I want to make. We must work on timescales shorter than the Kelvin-Helmholtz times such that the two stars remain similar, but that gives us a few million years to play with. We also want to work on timescales longer than the radiative diffusion time, so we are sure the luminosity has time to figure out the changes we are making. So we'd have to add mass on a timescale like a half a million years or so, to really isolate the role of mass over the transient effects. The answer will then be, the two stars will maintain virtually the same luminosity as each other. Meaning, they will both have their fractional luminosity grow like about 3 times their fractional increase in mass (actually it might be closer to 4 times depending on their mass, those are the kinds of details that require a better opacity treatment than a simple analysis uses).

What happens to the R and the core T is interesting as well. The fusing star will maintain its core temperature, because fusion is an excellent thermostat, so its fractional increase in radius will match its fractional increase in mass (as per the virial theorem). The non-fusing star will have its core T adjust to match the internal energy of however much mass you add, but let's say you drop it in from afar, so it has kind of a "normal" amount of energy in it, and it falls in deep enough that it doesn't just radiate it away more quickly than the normal diffusion time. In that case, the R will do pretty much the same thing, and the two stars will be more or less indistinguishable until the K-H time reveals that only one of them has an internal energy source that can keep it from gradually contracting.

 

[Drakkith]

For those who do not know why that answer is so completely wrong, the correct answer is that high-mass stars do not need to contract as much to get to fusion temperatures in their core, so they have both lower densities, and lower pressures, than lower mass main-sequence stars. That makes them bigger and leakier buckets of light, pure and simple. Yes, that's right, they have lower core pressures.

What do you mean by 'contract as much'? If we have two gas clouds, the first with 1 solar mass and the second with 5 solar masses, each identical in volume prior collapse, what's going on? Why does the latter contract less?

 

[Ken G]

What do you mean by 'contract as much'?

I mean contract to as high a density-- that's why higher-mass main-sequence stars are lower-density stars.

If we have two gas clouds, the first with 1 solar mass and the second with 5 solar masses, each identical in volume prior collapse, what's going on? Why does the latter contract less?

Because of the virial theorem: T is proportional to M/R, or if you like, density is proportional to T³/M². So since all stars have to contract to roughly the same T to initiate fusion, that means a star 5 times more massive only needs to reach a significantly lower density to initiate fusion. This is what the elementary textbooks should be saying, not that higher mass leads to higher pressure which leads to more fusion, which is 100% baloney.

 

[TEFLing]

found the following link online

http://www2.astro.psu.edu/users/alex/astro497_6.pdf

 

[TEFLing]

from Wikipedia

radiation power density

 

[TEFLing]

from Wikipedia

radiation power density

P ~ n² T^1/2​

assuming comparable temperatures, total luminosity

L ~ VP ~ V²P/V ~ M²/R³​

according to the virial theorem,

T ~ M/R​

if that is constant, then L ~ 1/R which is not the case

 

[TEFLing]

found the following link online

http://www2.astro.psu.edu/users/alex/astro497_6.pdf

if I understand correctly

power density ~ n² T⁴

luminosity ~ power density x volume

L_core ~ M² T⁴ / R³

L_surface ~ T⁴ R²

from which follows

M^2 ~ R^5

 

[TEFLing]

the above is a crude order of magnitude estimate least wrong for low mass stars generating energy via Ppp

 

[TEFLing]

for high mass stars, we observe roughly M ~ R^3/2

working backwards, and using the virial theorem

that implies a T^8 dependence for CNO cycle stars

 

[Ken G]

if I understand correctly

power density ~ n² T⁴

Yes, that is the p-p fusion rate, approximately, for low-mass stars. For stars like the Sun, it's more like n²T⁵, and any more massive, the T dependence goes way up, due to CNO cycle fusion. But this is a reasonable expression for stars with masses in the range 0.5 to 1, though if you want to talk about the range I mentioned, 0.5 to 50 solar masses, a steeper T dependence is required, making the T fairly thermostatic.

luminosity ~ power density x volume

This is also true as an equation, but it is important to recognize this is not the physics that sets the luminosity, which is my point. We already have a constraint on the luminosity, which comes from radiative diffusion, and it tells us that luminosity is roughly proportional to M³ (if we take a simple constant opacity idealization), without knowing anything at all about fusion or even if there is any fusion going on. So since we already know the luminosity, your expression is now used to constrain the combination of T and R, given M, which is the only thing we need fusion for.

L_core ~ M² T⁴ / R³

L_surface ~ T⁴ R²

from which follows

M^2 ~ R^5

That's not a very good constraint, because you have equated surface and core T, but they are very different animals-- not just in magnitude, but also in physical meaning. The only thing worse than thinking a star determines its luminosity from its core temperature is thinking the star determines its luminosity from its surface temperature! The luminosity is already determined before you know either of those.

Here is what you want to do. Use radiative diffusion to say that L ~ R⁴ T⁴/M, as is done in a lot of places that do understand stellar structure (by that I mean, not the intro textbooks that say L is higher because core temperature and pressure are higher). Then combine this with the virial theorem, T ~ M/R, to get L ~ M³. Then we are done with luminosity, we know the luminosity without even saying if there is any fusion going on (this is sometimes called the "Henyey track"). But what is missing is we do not know T and R, for that we will need fusion physics. So that's what we can use your luminosity ~ power density * volume expression, and we then say the fusion rate per volume is n² Tⁿ, where you have n=4 for fairly coolish p-p fusion, but can be as high as n=18 for CNO. Put these all together and you find that this last expression is only needed to get R(M) and T(M), and they come out R ~ M^(n-1)/(n+3) and T ~ M^4/(n+3). So when n=18, R is nearly proportional to M, and the core T scales like M to about the 1/5 power, a very weak scaling that allows us to imagine the core T is thermostatic. But again, this only matters for understanding R, it has little to do with L unless we are being very exact-- and if we are doing that, we will need to include mixing length theory and convection and varying chemical composition and all that jazz.

 

[TEFLing]

...Here is what you want to do. Use radiative diffusion to say that L ~ R⁴ T⁴/M

...Then combine this with the virial theorem, T ~ M/R, to get L ~ M³...Put these all together and you find that this last expression is only needed to get R(M) and T(M), and they come out R ~ M^(n-1)/(n+3) and T ~ M^4/(n+3). So when n=18, R is nearly proportional to M, and the core T scales like M to about the 1/5 power...

L ~ R²T_surface ⁴

So

R²T_surface⁴ ~ R⁴T_core⁴ / M

( T_core / T_surface ) ~ M/R²

So low mass stars, having higher surface gravities, are relatively hotter inside, as compared to high mass stars

When n=4,

( T_core / T_surface ) ~ M^1/7 ~ constant

When n=18,

( T_core / T_surface ) ~ M^-13/21 ~ M^-3/5

 

[Ken G]

L ~ R²T_surface ⁴

So

R²T_surface⁴ ~ R⁴T_core⁴ / M

( T_core / T_surface ) ~ M/R²

So low mass stars, having higher surface gravities, are relatively hotter inside, as compared to high mass stars

Yes that does follow, at least when n=4 (main-sequence stars a bit less massive than the Sun). Note that this is not a constraint on the core temperature, it is a constraint on the surface temperature. The surface temperature is quite easily adjusted by the star, and this is the constraint it must follow when it does that.

When n=4,

( T_core / T_surface ) ~ M^1/7 ~ constant

When n=18,

( T_core / T_surface ) ~ M^-13/21 ~ M^-3/5

Yes, those are interesting scaling relations for the surface temperature, they explain why higher-mass stars have hotter surfaces, especially for CNO-cycle stars.

 

[TEFLing]

...

( T_core / T_surface ) ~ M/R²

...

From VT,

T_core ~ M/R

So

T_surface ~ R

I think that is qualitatively correct, and maybe even semi quantitatively so

 

[Ken G]

From VT,

T_core ~ M/R

So

T_surface ~ R

I think that is qualitatively correct, and maybe even semi quantitatively so

Yes, that logic works for me. It overestimates a little how much the surface T varies, but that's not surprising given that the opacity is assumed constant and convection is neglected. If anyone is just now coming into the thread, notice we are strictly talking about ideal-gas-pressure stars that transport energy by radiative diffusion, so these expressions do not apply to giants or white dwarfs, or even main-sequence stars at very low or very high mass. But it should all give reasonably semi-quantitative results for main-sequence stars between about 0.5 and 50 solar masses.

 

[TEFLing]

In QM, you can calculate the Bohr radius of H by writing the equation for electron energy, and differentiating to find the dx = h/dp which minimizes the total energy

Does something similar work for stars as well?

U ~ -2/5 GM^2/R
K ~ -1/2 U

?

 

[Ken G]

In QM, you can calculate the Bohr radius of H by writing the equation for electron energy, and differentiating to find the dx = h/dp which minimizes the total energy

Does something similar work for stars as well?

U ~ -2/5 GM^2/R
K ~ -1/2 U

?

Only for white dwarfs, which are the stellar equivalent of a multiple-particle Pauli-excluded Bohr atom.

 

[TEFLing]

Maybe for NS also?

I have a question about radiative equilibrium...

Outside the stellar core, shouldn't L(r) remain constant? Otherwise energy would be accumulating or draining from material in some spherical shell

If so, then

4pi r^2 T(r)^4 = L(R) = constant

T ~ r^-0.5