Understanding the Luminosity of Radiative Stars

1. Apr 8, 2014

Ken G

In the past I have tried to start threads that provide surprising insights into the processes that set the luminosity of main-sequence stars that have radiative diffusion as their primary energy transport mechanism (as opposed to fully convective stars). However I failed to include professional references to support my argument, so my arguments were easily mistaken as non-mainstream and that does not serve this forum. So I will rectify that here, by basing the argument on a reliable reference as a starting point for discussion. I would note that when a mainstream idea appears non-mainstream to casual inspection, that is about as "surprising" a result as we ever encounter, so is worth a second look.

To use the Wikipedia entry on http://en.wikipedia.org/wiki/Mass–luminosity_relation as a starting point, because it is excellent (except for the early part that equates the surface T to the average T, that's just not something you want to do with a star it is conceptual kryptonite), note that it says: "Deriving a theoretically exact mass/luminosity relation requires finding the energy generation equation and building a thermodynamic model of the inside of a star. However, the basic relation L ∝ M3 can be derived using some basic physics and simplifying assumptions. The first such derivation was performed by astrophysicist Arthur Eddington in 1924."

Of particular significance in this quote is the well-known fact that in 1924 Eddington did not have the slightest clue about the existence of nuclear fusion. So we immediately see the fact that we can get a basic semi-quantitative understanding of why main-sequence stars have the luminosity they do, without referencing nuclear fusion in any way. Indeed, the simple fact here is that you can indeed get a fairly decent working understanding of the luminosity of a main-sequence star and know absolutely nothing about nuclear fusion. You can also get a decent working understanding of the radius of a main-sequence star if you know only one thing about nuclear fusion: that it is very temperature sensitive, and kicks in for hydrogen in a big way at T around 10 million K.

The Wiki derivation shows all these facts, but if you want the Cliff notes on it, recognize the following:
1) a star is a big leaky bucket of light, and its luminosity is set by how much light is in there, and how long it takes to leak out. Those are simply not issues that directly involve nuclear fusion, they involve the temperature, density, and radius of the star. In other words, a snapshot of the thermodynamical structure of a star tell you its luminosity, and if you use characteristic scaling laws to understand that structure, then knowledge of the mass and radius is all you need to get the luminosity.
2) in a surprising flourish, the luminosity ends up not depending on the radius after all, so you only need the mass to get the luminosity, which is why there is a mass-luminosity relationship for stars that have a simple internal thermodynamical structure and transport energy by radiative diffusion.
3) if you know the T at which fusion initiates (about 10 million K), and you know it is highly temperature sensitive so acts like a thermostat around this T, then you know the radius at which a star will cease contracting and enter the main sequence. This is all you need to know about fusion, to get this basic understanding-- all other details are only required for better quantitative results.

To get this result, the only simplifying assumptions you need are as follows, as you can see from the Wiki derivation:
1) the star has to be "all one thing", in the sense that characteristic numbers like its internal T, its radius R, mass M, and their connection to density, must all be interrelated by the standard simple scaling laws. In particular, you cannot have shell fusion, because shell fusion tends to break the star into a core and envelope in a way that puffs out the envelope and essentially turns the star into "two different things", which means that the characteristic scaling relations between temperature, mass, radius, and density, no longer apply in a global way.
2) the energy transport must be radiative diffusion, not convection, so stars near the Hayashi track are not applicable (including protostars and red giants).
3) the opacity that restricts radiative diffusion must be treated in some simple way, for example we can assume the cross sections per gram are constant. This is rather rough, but if you want a full simulation of a star, you cannot use simple conceptual insights.
4) to get L scaling like M3, you also need the gas pressure to exceed the radiation pressure. This is a standard property of all but the highest-mass main sequence stars. But the mass-luminosity relation can also be extended to those very high-mass stars, you just get a transition to L scaling in proportion to M. You still don't need to know anything about nuclear fusion to get that result, it is called "the Eddington limit" and does not refer to fusion.

What all this means is, everyone who says that nuclear fusion sets the luminosity of a main-sequence star is simply incorrect. A good semi-quantitative understanding of L can be obtained without knowing anything about fusion, a good semi-quantitative understanding of R can be obtained knowing only the characteristic T of fusion, and a complete detailed quantification requires a self-consistent calculation that involves both fusion and radiative diffusion. In none of those cases does fusion set the luminosity-- the rough relation is that luminosity sets the fusion rate, and the precise relation is that the two achieve a feedback mechanism that sets both of them.

Last edited: Apr 8, 2014
2. Apr 8, 2014

Fantasist

Why would someone say this? According to the Stefan-Boltzmann law (which is the basis of the derivation you quoted), the luminosity of a star only depends on its surface temperature (not the interior temperature), so you would get the same luminosity e.g. for an isothermal star (which obviously would imply the absence of nuclear fusion at the center).

Last edited: Apr 8, 2014
3. Apr 8, 2014

Ken G

The issue of the surface temperature is another interesting case. The way to understand the surface temperature of a star is to recognize that it must be set by the luminosity, not the other way around. The way to see that is to consider a snapshot of the internal structure of the star, which has some history that has made it that way, but seeing how the light is diffusing out through that structure will tell you its luminosity. When that light gets to the surface, it will heat the surface until the surface reaches the right temperature to radiate that light, as has been going on throughout the history of that object. So you'd never have any way to know what the surface temperature of a star will be without first understanding what its luminosity is, which requires understanding that diffusion physics (if it is radiative and not convective). In particular, you will not need to know if there is any fusable fuel in its core, that will only tell you (to first order anyway) the timescale for evolving from one internal thermodynamic structure to the next.

A rather analogous situation appears if you want to understand why planets have the surface temperatures they do. Planets also radiate light according to the Stefan-Boltzmann law, so we could just as easily say that the rate a planet radiates heat is determined by its surface temperature. But then we see it as some kind of amazing coincidence that the rate it emits heat balances the rate it absorbs heat from the Sun! The resolution is clear-- the rate it absorbs heat from the Sun sets the rate it must emit heat, and the rate it must emit heat sets its surface temperature, not the other way around.

Last edited: Apr 8, 2014
4. Apr 8, 2014

Fantasist

You shouldn't forget about gravity here. The gravitational potential energy of an atom at the surface of the sun corresponds to a kinetic energy of several million degrees (kT = GMm/3R as stated in the Wikipedia article). The radiation temperature is very much negligible compared to this. This is quite different for planets, where the 'gravitational temperature' is negligible, not only because of the much smaller mass M, but also because they are solid/fluid and thus the mechanical equilibrium is maintained by static pressure, not a dynamic pressure as for gaseous objects.

5. Apr 8, 2014

Ken G

I agree that gravity plays a key role in the temperature of stars, it is best understood via the "virial theorem" whereby we can know that the average gravitational potential energy is, in magnitude, twice the average kinetic energy of the gas (for the stars that are not the very high-mass stars that I mentioned work a bit differently). However, once one understands the role that gravity plays in setting the temperature of the bulk of the star, one can then determine the luminosity, and then the luminosity sets the surface temperature as I described. Gravity does not play a direct role in the surface temperature, but does have that key indirect role in everything that is happening to a star.

So what we can see is that the luminosity of the star is set neither by the surface temperature, nor by the fusion rate, because both of those are set by the luminosity (to first order). The luminosity is actually set by the structure of the star, which is constrained primarily by its mass, and the star moves through different stages of that structure over timescales that fusion does strongly affect.

6. Apr 8, 2014

Staff: Mentor

I'm not following you. I'm under the impression that temperature of the outer surface of the star sets the amount of energy that is radiated per unit of surface area, and combined with the size of the star determines the luminosity. Both of these are of course set by the mass, structure, and composition of the star (and whatever other properties I may have forgotten).

Also, why would we not be able to determine the temperature of the star without knowing the luminosity?

I don't see how you're concluding that the rate at which the surface emits radiation depends directly on the absorbed radiation. While the absorbed radiation will obviously affect the surface temperature, the amount and distribution of emitted radiation depends solely on the temperature of the object, not on the absorbed radiation.

7. Apr 8, 2014

Matterwave

Ken G has made one implicit assumption in that statement, which is that the temperature of the planet is constant. This means that the rate of energy absorption must equal the rate of energy emission. Or else the planet would either be heating up or cooling down.

This is not true for planets in very elliptical orbits which may not be in a nice thermal equilibrium.

8. Apr 8, 2014

Ken G

Right, although no planets are in an orbit so elliptical that what I said is not basically correct. Still, your point is key: the surface of the planet is taken to be in radiative energy balance. That's what makes it analogous to the surface of a star-- those are also in radiative energy balance, to a reasonable degree of accuracy (nothing is ever exact, of course).

9. Apr 8, 2014

Ken G

The logic isn't quite right there-- the mass, structure (including radius), and composition don't set the surface T, which then sets the luminosity, they set the luminosity, which then sets the surface T. Of course they all have to be determined self-consistently with each other if you want an exact answer, but for a good approximation (which is what understanding is always about), we can notice that if I somehow changed the rate that radiation was diffusing up from the depths of the Sun, the surface T would quickly respond to that, so the luminosity would quickly adjust to match what was diffusing up from depth. This makes it clear that the surface T is the slave to the luminosity, not the other way around, when you consider the overall physical situation-- not just the requisite equations.
Tell me how you would determine the surface temperature of a star if I tell you the mass and radius of the star, but I don't tell you the rate the light can diffuse out (which means, telling you the opacity coefficients). You can't, you'd have no idea in fact. But if I tell you those opacity coefficients so you can calculate the diffusion rate, you can figure out the surface T in a few seconds.
Matterwave answered this, but if you want more, we must probe into the difference between "depends on" and "is determined by." The distinction is subtle, but quite important, and indeed we invoke it all the time when we try to calculate stuff from a conceptual (rather than plug in all the equations and let the computer find the solution) perspective.

I grant you that there is an equality between a function of the surface T, and the radiative flux it emits. That equality allows us to say the radiative flux "depends on" the surface T. But to say it is "determined by", I am taking the meaning that we are trying to figure out both the rate that the Earth emits heat, and its surface T, and I'm asking which of those do we need to know first, in order to get the second, as a solution strategy in an actual physical situation. You need to know the distance of the Earth to the Sun, so you can know the rate the Earth absorbs sunlight. You also need to know the surface of the Earth will reach radiative energy balance where it re-emits what it absorbs. If you don't know those two things, you are dead in the water, you can't get the answer at all. But if you do know those two things, then the calculation proceeds that you use the energy balance to tell you the rate the surface must emit heat, and you use that answer to tell you the temperature. Ergo, the rate it absorbs heat controls the rate it must emit heat, and that determines the temperature, just look at the order in which you know those quantities in the actual process of logical analysis.

Last edited: Apr 8, 2014
10. Apr 8, 2014

davenn

I don't need to know its mass, radius or rate of light diffusion

2 common ways of measuring star surface temperature is by its colour and its spectrum

http://zebu.uoregon.edu/~soper/Stars/color.html

Dave

11. Apr 9, 2014

Staff: Mentor

Okay, by luminosity do you mean the amount of energy being brought to the surface of the star from the core?

I'd measure the temperature. But that's when I was thinking luminosity meant the radiation emitted from the surface of the star into space. I'm not sure that's how you were using it.

I think I can see where you're going, but that ignores the fact that the Earth (and other planets) have internal heat which raise the temperature of the surface higher than when you just factor in the absorbed radiation. Perhaps you were ignoring this since we were talking about stars? (But even then wouldn't you need to factor this effect in for certain stars like close binaries?)

12. Apr 9, 2014

snorkack

If you changed the opacity coefficients near the surface of the Sun but left the core and luminosity unchanged (which is easy to do, most of the interior, luminosity and mass of Sun is stagnant - added metals would be mixed into a very small mass of convective atmosphere, IIRC about 0,02 solar masses, so the pressure on the interior of Sun would be little changed and thus little effect on core fusion and luminosity), you would not only change the surface temperature. You would also change the radius - without changing the mass. OR the actual average interior temperature.
That´s the difference between a dwarf and subdwarf - for equal mass, the luminosity should be about equal but the subdwarf is appreciably smaller and hotter.

13. Apr 9, 2014

Ken G

You are talking about measuring the temperature, you are right that can be done in several ways. I'm talking about how to calculate, a priori knowing only physics, what the temperature of a star will be. That's what I mean be "determines" the temperature, what process decides what the temperature must be. The purpose is to understand the surface temperature, and the luminosity.

14. Apr 9, 2014

Ken G

I mean the rate that the surface emits heat, but yes, because the surface is in radiative energy balance, that must equal the rate it is being brought up from the interior. That fact is what sets the surface temperature.

Spoken like a true observer! But what if all you have are the laws of physics, and you are not actually looking at any stars? How do you know what the luminosity of a star will be then? This is required if you wish to understand the luminosity of a star.

Yes, I am neglecting that, but for most planets (including Earth), it is rather insignificant. Any time we attempt to achieve understanding, we must idealize and simplify, the real universe does not fit in our brains.
In close binaries, there is a "reflection effect", which can measurably raise the surface temperatures of both stars on their facing sides, but it will not measurably change the luminosity of either star. Thank you for that additional way to show that surface temperature is not the process that sets the luminosity!

15. Apr 9, 2014

Ken G

That's an informative case for demonstrating my point. If you only change the opacity right at the surface, it will not have any effect on either the radius or the luminosity of the star. If you change the opacity over a significant part of the envelope, it will change the rate that light can diffuse out, so that affects the calculation I described (and you can see where it would alter the Wiki calculation). If you don't think these things are true, then find in that Wiki calculation where they needed to specify the opacity at the surface (and you are right that the surface has very different physical conditions than the bulk of the star, so if the surface opacity mattered, the Wiki would be completely wrong to not specify what the surface opacity is).

The Wiki on cool subdwarfs says "The explanation of their underluminosity lies in their low metallicity: these stars are unenriched in elements heavier than helium. The lower metallicity decreases the opacity of their outer layers and decreases the radiation pressure, resulting in a smaller, hotter star for a given mass." Unfortunately, that is of course completely ridiculous, radiation pressure cannot possibly play any role in the case of the cooler stars they were talking about. This is the problem with online sources-- sometimes they are quite good (like the Wiki I cited), and other times they are just baloney, so you have to analyze critically if it makes any sense.

ETA: I think I understand what is going on with cool subdwarfs (hot subdwarfs are electron degenerate white dwarfs that are still cooling down, so are very different animals because their temperature is strongly affected by their closeness to their quantum mechanical ground state, so don't have a high internal temperature like main-sequence stars and the stars the Wiki derivation, and this thread, are about). It is not quite accurate to characterize cool subdwarfs as underluminous, simply because they lie below the main sequence on an H-R diagram. Since the argument of this thread is that the mass sets the luminosity of any star in radiative energy balance throughout its interior (whether you even know if it is undergoing fusion or not), the proper comparison to make for the cool subdwarfs is with regular main-sequence stars of the same mass. I believe what is happening is that the low metallicities of the cool subdwarfs are causing them to look a bit bluer than normal (stars are not perfect blackbodies, so even the concept of "surface temperature" is an inexact idealization). If you take a normal main-sequence star, change neither its mass nor its luminosity but make it look bluer, it falls under the main sequence in an H-R diagram. That's my bet, so if I'm right, then it's mostly a spectroscopic phenomenon, not much related to the luminosity.

Last edited: Apr 9, 2014
16. Apr 9, 2014

Staff: Mentor

Well then I don't agree at all. Your argument is that the luminosity of the star sets the temperature of the surface. But, similar to your previous thread that got locked, both the luminosity and temperature are related and depend on one another. Luminosity depends on temperature, and temperature depends on the rate of energy being radiated away with the rate of energy coming up from the inside of the star. I don't see how you can separate the two. In fact, it's even more complicated because the luminosity and temperature relationship can change slightly depending on the makeup of the outer layers of the star.

Except that it appears to me that it is. The stars will be hotter than they would otherwise, as you stated, which would require that their luminosity rise, if only a small amount.

17. Apr 9, 2014

Ken G

This is very much a sidelight from the main thrust here (which is what sets the luminosity, not what sets the surface T), but there is no question whatsoever that the luminosity of the star sets its surface T. Just look at the Wiki derivation I started with, which derives the luminosity of the star without ever mentioning the surface T! Note that it would then be trivial to use that derived luminosity to derive the necessary surface T. You will never in a million years be able to do the opposite-- derive what the surface T must be without knowing the luminosity, and use that to find the luminosity. Do you see the distinction?
Yes, a lot of things can affect the result slightly, the point is to understand what mostly matters for understanding the luminosity, and then the surface T, of a star. The exact result is vastly complex and requires a black-box computer calculation, the details of which could never fit in our heads, so cannot convey understanding to us. What conveys understanding is idealization and simplification, while preserving the key physics. Don't you ever do that in other things you think about?
Let me ask you this then. Do you think the stellar luminosities will change a lot more, a lot less, or about the same amount, as the surface temperatures in the close binary? Because if you can see that the answer is the luminosities will change a lot less than the surface temperatures, that should tell you which is the dog and which is the tail. The luminosities are very hard to affect, as they result from diffusion over the entire star. Anything you do at the surface, like shine light on it, will have very little effect on the luminosity, but can have all kinds of effects on the surface properties.

We saw something similar in the discussion about changing the opacity at the surface of a cool subdwarf, that will also not change the luminosity much at all, unless you change the opacity at the surface so much you are affecting the optical depth of the star as a whole, and hence its diffusion physics. This is the point-- a star is a big leaky bucket of light, and no one will ever understand the first thing about its luminosity if they think they are going to get it by first deriving the fusion rate, or by first deriving the surface T, because both those things are set by the rate of that leaking bucket. This is unquestionably true, but don't take my word for it, just either follow the Wiki derivation, or find a flaw in it. Note that I'm not basing any of this on the Wiki, it's all things I already know, but it is understandable for people to want to see it from an accepted authority before they should believe it is worth their careful consideration.

18. Apr 10, 2014

Chronos

Ken, the spectrum of a star tells you everything you need to know about its temperature. Boltzmann figured this out in the mid 19th century. As far as I know, that rule is still good. If you have any examples to the contrary, please cite them.

19. Apr 10, 2014

Staff: Mentor

No, not at all. Here's how I see it. A star's surface emits radiation because it is hot. If we increase the temperature of the surface, then it emits more radiation, just like my stove emits radiation when I run current through the heating element. The luminosity of both the star and my stove heating element depends almost entirely on what the temperature is (assuming my stove is in a vaccum). The rate at which energy is radiated (luminosity) from either the star or my stove's heating element, compared to the energy generated internally, determines what the temperature will balance out at.

In short, the temperature determines the luminosity and the luminosity determines what the temperature balances out at. I see no way around this conclusion.

I don't see how you're getting this. If the temperature of the star's surface changes then luminosity must change accordingly (If all else is the same). Shine a light on the surface of the star and the star's luminosity will increase to balance out the absorbed light.

The rate that light is leaking out of this leaky bucket is directly affected by the temperature of the star's surface, just like my example using my stove. You can't just separate the two and say one causes the other. That's my key point here, that changing either one necessarily affects the other.

Also, just because you can derive the luminosity without knowing the temperature only means that we know enough about stars to develop a way of determining the luminosity without being required to input the temperature! I can find the power in a circuit without knowing the voltage, but that doesn't mean that power doesn't depend on voltage. Besides, the only reason we can determine the luminosity using the mass is because it's already been derived using the Stefan-Boltzmann law, which requires a temperature (This is in the wiki page you linked!). The mass-luminosity relation is merely a shortcut to determine the luminosity using the mass, nothing more.

20. Apr 10, 2014

snorkack

I should say that the temperature on the surface depends, above all, on the complex dependence of opacity on temperature as various atoms get progressively ionized.
Look at what happens with a sunlike star as the luminosity grows with nearly constant mass. Faint young Sun was appreciably colder but also smaller than Sun now. As the fusion rate slowly grows, Sun has been growing hotter but also bigger.
Sun is still very slightly heating, but mainly growing. Sun is predicted to reach maximum temperature very slightly above present while being much brighter and bigger. Then Sun would go on brightening but with only very slight cooling through the subgiant period - then while still continuing to brighten cool down drastically and expand drastically as it becomes a red giant.
There is nothing universal about this pattern! Massive stars are in no way required to become red supergiants because they often become blue supergiants or yellow supergiants or luminous blue variables instead.
Red dwarfs have never been seen to evolve, but it is predicted that dwarfs below about 0,15 solar masses should NOT swell into red giants, ever, because they would heat up as subgiants until they go out, without expanding so as to cool.
In short - temperature does not determine luminosity because the star will shrink or swell so as to match its temperature and area to its luminosity.