Understanding the Luminosity of Radiative Stars

In summary, the conversation discusses the idea that nuclear fusion is not the only factor that determines the luminosity of main-sequence stars. The Wikipedia entry on the mass-luminosity relation is used as a starting point for the discussion, which shows that a basic understanding of a star's luminosity can be obtained without referencing nuclear fusion. This is because a star's luminosity is determined by factors such as temperature, density, and radius. The conversation also mentions the assumptions and simplifications needed to derive the mass-luminosity relation, and how fusion plays a role in the feedback mechanism that sets both luminosity and fusion rate. The conclusion is that the idea of nuclear fusion solely determining a star's luminosity is incorrect.
  • #1
Ken G
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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.
 
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  • #2
Ken G said:
What all this means is, everyone who says that nuclear fusion sets the luminosity of a main-sequence star is simply incorrect.

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).
 
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  • #3
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.
 
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  • #4
Ken G said:
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.
...
...

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.

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
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
Ken G said:
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.

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?

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.

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
Drakkith said:
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.

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
Matterwave said:
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.
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
Drakkith said:
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).
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.
Also, why would we not be able to determine the temperature of the star without knowing the luminosity?
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.
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.
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.
 
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  • #10
Ken G said:
...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..

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
Ken G said:
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.

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

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.

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 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.

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
Ken G said:
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.
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
davenn said:
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
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
Drakkith said:
Okay, by luminosity do you mean the amount of energy being brought to the surface of the star from the core?
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.

I'd measure the 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.

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.
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.
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?)
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
snorkack said:
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 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).

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.
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.
 
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  • #16
Ken G said:
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.

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.

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!

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
Drakkith said:
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.
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?
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.
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?
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.
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
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
Ken G said:
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?

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.

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.

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.

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.

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
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.
 
  • #21
Chronos said:
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.

which is why I mentioned spectrum many posts ago

thanks for the backup Chronos :)

Dave
 
  • #22
Apparently I am not making myself clear. I am not talking about how to measure the temperature, or the luminosity, of a star. I am talking about how to know why the luminosity is what it is (and then the surface temperature). I'm talking about how to calculate it if you only know the laws of physics, but have never seen a star in your life. Understanding the luminosity, not observing the luminosity. Like what the Wiki is trying to do, I don't see spectra there.
 
  • #23
Ken G said:
Apparently I am not making myself clear. I am not talking about how to measure the temperature, or the luminosity, of a star. I am talking about how to know why the luminosity is what it is (and then the surface temperature). I'm talking about how to calculate it if you only know the laws of physics, but have never seen a star in your life. Understanding the luminosity, not observing the luminosity. Like what the Wiki is trying to do, I don't see spectra there.

You've made yourself perfectly clear to me. I just don't agree with you.
 
  • #24
Do you agree with the Wiki derivation?
 
  • #25
Drakkith, I somehow missed your post before, I did not ignore your specific points. Let me address them in detail, all will become clear if you bear with the argument:
Drakkith said:
A star's surface emits radiation because it is hot.
Yes, that is absolutely true-- if we look at the microphysics and ask why does it emit light, it is because of its temperature. I am not talking about why the surface emits light, I'm talking about how to understand how much light it will emit. If you had some way to understand the surface T, then that would be a valid position, but here's the point: there is absolutely no way to understand the surface T of a star until you understand what its luminosity is. Note again that I am saying "understand" in the meaning of "can derive it from first principles", not can measure it using observational diagnostics like spectral information. Surely, we can agree that understanding in physics equates to derivations from first principles?
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.
The stove is a faulty analogy to the surface of a star for the express reason that a stove has a thermostat that allows you to set the temperature, so the stove automatically generates whatever internal heat is needed to maintain that temperature. That is just precisely the opposite of what is happening to the surface of a star: there it is the rate that internal energy is welling up that is determined (and not by fusion, at least not for the simplest semi-quantitative understanding: as per the Wiki), and the temperature comes to whatever it needs to in order to carry that luminosity. This is the point.
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.
What I mean by the luminosity of a star is the net rate the star is emitting heat (which is where the planetary analogy fails and that analogy is probably where this binary star issue came up, it's a bit of a red herring for understanding the luminosity of a star so I probably should not have brought up planets). If you shine a light on the surface of a star, it's net luminosity will not change, but that is what we are trying to understand here. You may regard this as a semantic difference, but my point is, it's easy to change the surface T, and it's easy to change the amount of light something emits by shining light on it, but neither of those things will tell you what is the net rate that it is losing energy. You just can't that via your method, but the Wiki works quite well. Please do not that the Wiki never mentions the surface temperature, there is a very good reason for that-- you can never get the luminosity of a star by first deriving, from first principles, the surface temperature. Instead, you will derive the luminosity from first principles, and get the surface temperature from that. Try it any other way (without cheating and looking at the star)!
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.
No, it's exactly the other way around, see my discussion of what is different about a stove.
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!
Yes, exactly. Now, can we do it the other way-- can we derive the surface temperature without first determining the luminosity?
I can find the power in a circuit without knowing the voltage, but that doesn't mean that power doesn't depend on voltage.
Whether you can do that or not depends on the physical situation, that's why we need to understand that situation. This is an important point, we have equations that connect power, voltage, and resistance. Is there no causation implied there? An equation implies no causation, necessarily, but the physical situation does. This is a powerful insight in physics. Give me the equation P = V2/R, and I can design a physical situation where P is determined by V and R (it's the usual situation with a battery and a resistor). Now write the equation V = Sqrt[PR], and I can find a physical situation where V is determined by P and R, but it won't be the same situation as one where P is determined by V and R! The equations hold in all cases, of course, but the causation does not. To make V = Sqrt[PR] look like the causative relation in that physical situation, I must build a circuit that has a specific I or P, and then the V comes to whatever it needs. That's not going to be a battery.
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!).
(ETA) I had to edit this part, because I did not at first realize that the Wiki connected the good second part of the derivation to the awful first part! Oh my, that's horrendous. There is absolutely no need to even mention the surface T, or the Stefan-Boltzmann law, in that second derivation, they just did a derivation in terms of a totally wrong argument (that first part is totally wrong, as you can see if you plug in numbers the result has no relation to the luminosity of a real star, and the scaling law they get is clearly completely coincidental if its numerical result is totally wrong), and then expressed the second part in terms of the wrong first part, in such a way that the errors in the first part don't matter! How did I not realize that, I just looked at it and said, yes, they are using the average in internal T, and the diffusion physics, and getting L. To connect that to the surface T is utterly unnecessary, and conceptually terrible. Wiki, what have you done to me!?

Now I have to fix it by explaining the logic of what they really do, as you can see. They look at how the mass and radius set the internal T, then find the radiative diffusion rate. Look at what you now have: you have an internal energy in the radiation field (the T gives you a radiative energy density, yes?), and you have diffusion physics to tell you how long it takes to diffuse out, so you have a luminosity. Correct? Done, that's the luminosity! They then use L to calculate the surface T, which has no bearing on L, but since they did a wrong calculation first, they felt the need to insert the right calculation via essentially a correction factor to the wrong calculation! Think about that for a moment-- it would be like calculating the temperature of the Sun by first calculating the temperature of your stove, and then inserting a correction factor that turns the temperature of your stove into the temperature of the Sun! Note the proof of what I say-- their ultimate answer depends only on M, the surface T never needs to appear anywhere. It may now be hopeless for me to get you to see this is what they've done, I must seek a different reference that has some sense to it.
 
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  • #26
Here's one that does it much better: http://www.astro.virginia.edu/~jh8h/Foundations/chapter5/box5b.html
I'm not enthralled with their approach, because I feel (subjectively, here) that if you are going to do diffusion physics it's easier to just think in terms of how long it takes the photons to get out, rather than use the diffusion equation, but their approach is still completely valid, and quite good. (By the way, to completely follow you would need their equation 5.10, but that's going to be some version of the virial theorem-- it's just the statement that you have an ideal gas that is gravitationally bound, so the average T, not the surface T (!), is going to be proportional to M/R.)

Now, please do notice the following points:
1) Not only do they not need the surface T, they approximate it as zero! (They could of course go back and put in the surface T that the L they derive would produce, as I've said, and then redo their calculation with that new surface T, iterating to an answer, but that would be silly-- the answer would not change essentially at all, and certainly not to within the errors they are already accepting with their useful simplifications.)
2) The resulting L depends only on M, not even on R and certainly not on anything that has anything to do with fusion.

The bottom line is, some places on the web understand the luminosities of stars (really, stars that are mostly radiative throughout), others do not. Any place that tries to get it using the fusion rate invariably massacres the true physics at some point along the way (believe me, I've seen all kinds of things on seemingly authoritative online sources, from "higher mass stars have higher pressure in the core, which yields faster fusion" to "higher mass stars produce higher temperatures when they contract owing to their stronger gravity, which yields faster fusion", both of which are really staggeringly false, not even consistent with the first things we should know about stars). But I must say, the Wiki that I inadvertantly cited sets a new standard in bizarre ways to arrive at L: they do it right ultimately, but by first doing it completely wrong, and then inserting a huge correction factor to make it right! Believe me, when you understand what they really did there, you just have to laugh, but I'll give them this: it's not wrong.
 
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  • #27
Ken G said:
Yes, that is absolutely true-- if we look at the microphysics and ask why does it emit light, it is because of its temperature. I am not talking about why the surface emits light, I'm talking about how to understand how much light it will emit. If you had some way to understand the surface T, then that would be a valid position, but here's the point: there is absolutely no way to understand the surface T of a star until you understand what its luminosity is. Note again that I am saying "understand" in the meaning of "can derive it from first principles", not can measure it using observational diagnostics like spectral information. Surely, we can agree that understanding in physics equates to derivations from first principles?

I understand your position, but I simply don't agree. Without knowing that the surface emits radiation because it is hot, you will not understand why the star has the luminosity that it has. You are literally missing a key step. Again, I see the mass-luminosity relation as a shortcut. You can use it to determine the luminosity of the star without actually knowing or calculating the surface temperature only because we already know and understand how matter emits thermal radiation.

The stove is a faulty analogy to the surface of a star for the express reason that a stove has a thermostat that allows you to set the temperature, so the stove automatically generates whatever internal heat is needed to maintain that temperature. That is just precisely the opposite of what is happening to the surface of a star: there it is the rate that internal energy is welling up that is determined (and not by fusion, at least not for the simplest semi-quantitative understanding: as per the Wiki), and the temperature comes to whatever it needs to in order to carry that luminosity. This is the point.

Sorry, by "stove" I meant the heating element on top. AKA the burners. Imagine one of the old style heating elements where the dial determines the current through it and there is no temperature control. In that case the internal generation of energy through ohmic heating is constant, and the temperature of the element is determined by the amount of heat radiated away from the element (Again, assuming the element is in a vacuum). And before criticizing the analogy, remember that it's just an analogy. Of course it won't match the exact properties of a star.

What I mean by the luminosity of a star is the net rate the star is emitting heat (which is where the planetary analogy fails and that analogy is probably where this binary star issue came up, it's a bit of a red herring for understanding the luminosity of a star so I probably should not have brought up planets). If you shine a light on the surface of a star, it's net luminosity will not change, but that is what we are trying to understand here.

(underlined by me)
And this is wrong. The luminosity MUST change because the surface is heated by the light. It's just like heating up something with a laser. The temperature and luminosity both increase because you have another source of energy for the star. I don't know offhand how the internal areas of the star would be affected, if at all, but the surface would most definitely heat up and become more luminous. I don't see how you can deny this, it's basic thermodynamics.

You may regard this as a semantic difference, but my point is, it's easy to change the surface T, and it's easy to change the amount of light something emits by shining light on it, but neither of those things will tell you what is the net rate that it is losing energy.

It appears that you are ignoring the fact that the light is providing more energy to the star which must be radiated away.

You just can't that via your method, but the Wiki works quite well. Please do not that the Wiki never mentions the surface temperature, there is a very good reason for that-- you can never get the luminosity of a star by first deriving, from first principles, the surface temperature. Instead, you will derive the luminosity from first principles, and get the surface temperature from that. Try it any other way (without cheating and looking at the star)!

(Bolded and underlined by me)
Yes it does. It mentions it at least once and refers to the Stefan-Boltzmann law several times.

Yes, exactly. Now, can we do it the other way-- can we derive the surface temperature without first determining the luminosity?

No, and neither can you properly derive the luminosity without knowing the surface temperature. (As BOTH of your links show, which I will explain if more detail below)

Whether you can do that or not depends on the physical situation, that's why we need to understand that situation. This is an important point, we have equations that connect power, voltage, and resistance. Is there no causation implied there? An equation implies no causation, necessarily, but the physical situation does.

I agree. And this is exactly my argument. The math is (apparently) telling you that the luminosity determines the temperature but that the temperature doesn't determine the luminosity. Looking at the physical situation I see that as incorrect.

(ETA) I had to edit this part, because I did not at first realize that the Wiki connected the good second part of the derivation to the awful first part! Oh my, that's horrendous. There is absolutely no need to even mention the surface T, or the Stefan-Boltzmann law...

I'll answer this below.

Now I have to fix it by explaining the logic of what they really do, as you can see.

Your entire explanation after this requires knowing the Stefan-Boltzmann law in order to find the amount of radiation emitted inside the star. It's right there in BOTH of of your links. (More explanation below)

Now, please do notice the following points:
1) Not only do they not need the surface T, they approximate it as zero! (They could of course go back and put in the surface T that the L they derive would produce, as I've said, and then redo their calculation with that new surface T, iterating to an answer, but that would be silly-- the answer would not change essentially at all, and certainly not to within the errors they are already accepting with their useful simplifications.)

Let's look at what they actually say: We next need to relate the energy density of the photons to one of our stellar variables. To a good approximation, we can use the blackbody-radiation rules given in Chapter 4; the energy density of photons, which is immediately related to the intensity of the light, is thus proportional to the fourth power of the temperature, tex2html_wrap_inline30 . The rate at which the photons leak out will be determined, then, by the change in the fourth power of temperature with radius. Since the temperature goes down as one moves out through the star, and the photons move toward the cooler regions, the radiation will eventually leak out of the star.

And in chapter 4: Blackbody radiation is a very specific type of spectrum that corresponds to photons in equilibrium. This radiation is completely characterized by one parameter, the temperature of the emitter.

And: Next we approximate the temperature difference in equation (5.13) by the difference between the core temperature and the surface temperature, and we will make the approximation that the surface temperature is zero. This may seem drastic, but the surface temperature of a star is very much lower than its core temperature, so it is actually not a bad first estimate.

So let's see... They determine the energy flux (aka the luminosity as you've said) from the core to the surface by first finding the temperature of the core, and by saying that this energy is radiated outwards. The surface temperature is approximated as zero in order to simplify things to get an approximation of the gradient of the photon energy density.

I'm sorry, Ken, but, as you can see, this entire derivation is utterly dependent on temperature. Just finding the energy radiated away from the core requires that you use the Stefan-Boltzmann law.

2) The resulting L depends only on M, not even on R and certainly not on anything that has anything to do with fusion.

Only because we know that R and T are set by M. So knowing M you automatically know L, R, and T.

Now, before you reply, there's a few things I'd like to point out. Things that I believe both you and I agree with. I'd like to point them out now so that neither of us try explaining things the other already agrees with.

1. The energy radiated from the star is equal to the energy generated internally by the star. (Ignoring external energy sources like light)
2. The mass of the star is by far the biggest factor in determining its radius, temperature, and luminosity. By that I mean that the mass is the one variable we can freely change in our examples without having to worry about magic pixie dust or something. The other three are intimately related (what's the correct term for this?) and you cannot simply change them at will because a star is a giant self-correcting feedback system.
3. Hotter objects will have a higher luminosity, all else being equal. Conversely, cooler objects have lower luminosity.
 
  • #28
Drakkith said:
I understand your position, but I simply don't agree. Without knowing that the surface emits radiation because it is hot, you will not understand why the star has the luminosity that it has.
We don't need to enter a different world where radiation is caused by something other than the temperature of its source, I'm answering a very simple question:
How do you derive from first principles the luminosity of a star?
The source I just cited in my last post answers that in exactly the way I have been talking about (the Wiki kind of did, but only if you dig under the completely unnecessary reference to surface T, and it's obviously unnecessary because the second source didn't use it). All you need to assume is radiative energy balance, and understand diffusion of light, and understand the connection between radiative energy density and temperature (that's really the Stefan-Boltzmann law, but not at the surface, you don't need it there at all).

I can prove all this quite easily. Imagine surrounding the Sun with a spherical half-silvered mirror. It allows half the light through, and reflects the other half. What happens to the luminosity observed at Earth? Nothing! (Or at least, very little.) What happens to the temperature of the gas at the surface of the Sun? A great deal, it would get significantly hotter. Do you agree? I assume so, and this should show you that the detailed physics of how light last leaves the surface of the Sun plays no essential role in the luminosity of the Sun.

Sorry, by "stove" I meant the heating element on top. AKA the burners. Imagine one of the old style heating elements where the dial determines the current through it and there is no temperature control. In that case the internal generation of energy through ohmic heating is constant, and the temperature of the element is determined by the amount of heat radiated away from the element (Again, assuming the element is in a vacuum). And before criticizing the analogy, remember that it's just an analogy.
I wouldn't criticize that analogy, it's perfectly fine-- and demonstrates my point to a tee. You have an element and you push current through it, which dissipates heat at a given rate. That's exactly like the light diffusing up through the interior of the Sun! That will set the luminosity of your burner, and will also set the temperature of the burner. It's exactly what I'm saying.

And this is wrong. The luminosity MUST change because the surface is heated by the light.
What I mean by the net luminosity is the radiative flux integrated over the surface of the star. That will not change if you shine light from the outside, absolutely not. But shining light on the outside will change the surface T, which again should be showing you that the surface T is not the way to understand the integrated flux over the surface of a star, because you have to know the physical cause of that integrated flux before you will ever know the surface T. In particular, you will have to know the rate that light diffuses up through the star, and in a close binary, you will need to know that for both stars, and the very last thing you will ever be able to calculate is their surface T. If you think that's not true, show me how to calculate the surface T, from first principles, any other way. I've already showed you the way to do it, in that second link.
Your entire explanation after this requires knowing the Stefan-Boltzmann law in order to find the amount of radiation emitted inside the star. It's right there in BOTH of of your links.
One certainly needs to know how temperature of a radiation field determines its energy density. Technically, that is not the Stefan-Boltzmann law, because that gives the energy flux at a surface, but I grant you the physics is essentially the same and could be used to derive the energy density. My point is not what we want to call this law, it is that one does not need the Stefan-Boltzmann law to apply at the surface, as I showed in the situation where the surface is a half-silvered mirror. My point is that you never need to know squat about the surface temperature, because it is set by the luminosity and the physical details (like if there is a binary companion, and/or a half-silvered mirror). You do need all the laws of physics, however, so you need to know that you have light diffusing through a gas that reaches thermal equilibrium with that light over the bulk of the star. That's where you need Stefan-Boltzmann, but not for the flux version at the surface, you need it for the energy density version in the interior. The latter determines the luminosity of that big leaky bucket of light that is a star. What's more, no one can have any insight whatsoever into why the luminosity of a star is what it is until they understand this crucial fact.
So let's see... They determine the energy flux (aka the luminosity as you've said) from the core to the surface by first finding the temperature of the core, and by saying that this energy is radiated outwards. The surface temperature is approximated as zero in order to simplify things to get an approximation of the gradient of the photon energy density.
And that really should have showed you that the actual value of surface temperature cannot possibly be important to the luminosity of a star, if they can infer the luminosity by approximating the surface temperature by zero! I just can't see how anyone can maintain that I need to first know the surface temperature, before I can know the luminosity, given that fact, but I can certainly see how someone (like me) can claim that I can first know the luminosity (to a reasonable approximation), before I know anything about the surface T, beyond that it is much smaller than the core T.
I'm sorry, Ken, but, as you can see, this entire derivation is utterly dependent on temperature.
Goodness gracious, I never said the luminosity of a star could be understood without understanding the connection between the energy density of a radiation field and the temperature of an ideal gas! Just look at how many times I referred to the importance of that relation in my summary of how you can know the luminosity of a main-sequence star. I said you don't need to know the surface temperature, which is just clearly true.

Besides, the whole issue with surface temperature is a sidelight, the real point is that you don't need to know anything about fusion! You don't seem to be objecting to that part, although that was entirely the focus of the objections I got before. To get back on track, let's see if you agree with this statement:

You will never get anywhere trying to figure out the rate that a main-sequence star generates and radiates heat by first calculating the fusion rate and then equating that to the luminosity, but you will get to an excellent understanding of both rates if you start with the diffusion of light and find the rate the star loses heat, and say that must control the fusion rate.

1. The energy radiated from the star is equal to the energy generated internally by the star. (Ignoring external energy sources like light)
2. The mass of the star is by far the biggest factor in determining its radius, temperature, and luminosity. By that I mean that the mass is the one variable we can freely change in our examples without having to worry about magic pixie dust or something. The other three are intimately related (what's the correct term for this?) and you cannot simply change them at will because a star is a giant self-correcting feedback system.
3. Hotter objects will have a higher luminosity, all else being equal. Conversely, cooler objects have lower luminosity.
I do agree with all those statements. What I want to know is how to understand the luminosity of these stars from first principles. I know how to do that, and I do not need to know that the star is fusing anything, and I do not need to know that its surface obeys the Stefan-Boltzmann law, I only need to know the physics of the diffusion of light inside that leaky bucket-- and this should be quite clear from that link I gave.
 
  • #29
Here's what I need you to do, Ken. Tell me exactly what you're saying using a short, bullet-like format like I did at the end of my last post. I think I understand what you're saying, but I'm having a difficult time keeping two pages worth of information in my head so I can't be certain. I don't need any explanations, just basic facts.
 
  • #30
OK, fair enough, we kind of got off on a tangent, in large part because I used that pretty awful Wiki instead of http://www.astro.virginia.edu/~jh8h/...er5/box5b.html
as the link to look at.

What is clear from that link (and consistent with the arguments I've been giving) is that you can know, to a good approximation, the luminosity of an internally radiative star that has a simple enough internal structure to be characterized by global variables like average T (not surface T!), R, and M, just from the physics of radiative diffusion. You therefore do not need to calculate the surface T (except to verify it is way less than the core T, which is kind of obvious), nor do you need to do any calculations that involve fusion. Hence you do not need to know anything about the processes that set the surface T (like if there's a half-silvered mirror surrounding the star, which changes surface T but not L), nor do you need to know anything about fusion (you don't even need to know the star is undergoing fusion, it never shows up in that link and Eddington didn't even know fusion existed when he first did this calculation). I regard everything I just said as demonstrable fact.

So we have that we can know L (fairly well) without knowing the surface T or anything about fusion. Can we say the same thing about either of the other two? Can we know the surface T without calculating the interior radiative diffusion physics? No. Can we know the fusion rate without calculating the radiative diffusion physics? No! So it's very clear that the radiative diffusion physics is what you need to know to get L, and it in turn sets both the fusion rate, and the surface T (although to get surface T you have to also know R, which you get from knowledge about the interior T at which fusion initiates. Interestingly, you don't need to know R to get L, which is why you don't need to know anything about fusion).
 
  • #31
Ken G said:
Apparently I am not making myself clear. I am not talking about how to measure the temperature, or the luminosity, of a star. I am talking about how to know why the luminosity is what it is (and then the surface temperature). I'm talking about how to calculate it if you only know the laws of physics, but have never seen a star in your life. Understanding the luminosity, not observing the luminosity. Like what the Wiki is trying to do, I don't see spectra there.

Understanding the luminosity means knowing by which physical mechanism the radiation is produced in the star's atmosphere (which is really all we can see of the star). Only if we have answered that question, does it make any sense to try to connect the luminosity to any other physical parameters of the star.
 
  • #32
Ken G said:
So we have that we can know L (fairly well) without knowing the surface T or anything about fusion.

See, this is where I have a problem. You can find L without finding surface T, but I just don't see the significance in that. In fact, your link even says that setting the temperature of the surface to 0k only gives them an approximation. You can get a "fairly well" answer, but the surface temperature still matters if you want the best understanding of what the luminosity of the star is.

Also, to respond to something from your earlier post:

Surely, we can agree that understanding in physics equates to derivations from first principles?

No, I don't quite agree with this.

And you didn't stick to bullet-like format. Shame on you! :tongue:
 
  • #33
I've been reading this post with some interest, and I have a few questions with regards to the spectrum analysis procedures. From the light waves various frequencies emitted we can infer much about the composition of the internal workings on the star in question. Different elements have different temperature absorption rates. The viscosity and turbulence also plays a factor. Shock waves also occur causing disruptions. Probably the best article I can think of to describe some of the processes is "physics of the intergalactic medium". Although this article is developed for plasma measurements I would think that much of the same metrics are involved just on a more dense scale with fusion reactions.

http://arxiv.org/abs/0711.3358

Wouldn't you also have to be concerned by the variations in temperature absorbtion, shock waves etc? To be honest I'm not sure how much of this article applies to stars itself, however it demonstrates some of my question in regards to analyzing the luminosity relations via processes within a star

line from the wiki article

"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"

So at best this method is an approximation. However I'm unclear if the method your proposing is a better or worse approximation. Seems to me you still need to understand the stars composition to get an accurate luminosity relation.
 
  • #34
Fantasist said:
Understanding the luminosity means knowing by which physical mechanism the radiation is produced in the star's atmosphere (which is really all we can see of the star). Only if we have answered that question, does it make any sense to try to connect the luminosity to any other physical parameters of the star.
The beauty of science is that it allows us to test the validity of statements like this. So let's say we have two stars that are exactly like the Sun, but one of them has at its surface a thin spherical half-silvered mirror that allows half the light through, and reflects the other half. So we must admit we have here two totally different physical mechanisms for emitting light from the surfaces of those two stars, and indeed their surface T will be quite different. Now the question: will their luminosity be different?

When you realize the correct answer is "no, not measurably so", you will be able to see that your assertion does not test out.
 
  • #35
Drakkith said:
See, this is where I have a problem. You can find L without finding surface T, but I just don't see the significance in that.
Well at least that's progress, you see the truth in what I'm saying. Whether it has significance is another matter-- I'd say it only has significance to all the posters who seem to think the statement is flat out incorrect!
In fact, your link even says that setting the temperature of the surface to 0k only gives them an approximation.
The approximate nature of that solution goes waaaay beyond that assumption!
You can get a "fairly well" answer, but the surface temperature still matters if you want the best understanding of what the luminosity of the star is.
So do a lot of other things that are equally unhelpful in obtaining understanding. Don't tell me you've never heard of a device called idealizaton?
No, I don't quite agree with this.
Well, if you don't agree that to "understand" we must derive from first principles, at least I'm sure we can agree that derivations from first principles is quite important in physics-- even (especially?) when idealizations are included!
 
<h2>1. What is luminosity?</h2><p>Luminosity refers to the total amount of energy that a star emits in all directions. It is often measured in units of watts or solar luminosities, which is the amount of energy emitted by our Sun.</p><h2>2. How is luminosity related to the brightness of a star?</h2><p>Luminosity and brightness are related, but they are not the same thing. Luminosity is a measure of the total energy emitted by a star, while brightness is a measure of how that energy appears to us on Earth. A star's brightness may be affected by factors such as distance and the amount of dust and gas between the star and Earth.</p><h2>3. What factors affect the luminosity of a star?</h2><p>The luminosity of a star is primarily determined by its mass and its surface temperature. Generally, the more massive and hotter a star is, the more luminous it will be. Other factors that can affect luminosity include the star's age, composition, and any ongoing nuclear reactions within the star.</p><h2>4. How do scientists measure the luminosity of a star?</h2><p>Scientists use a variety of methods to measure the luminosity of stars. One common method is to measure the star's apparent brightness and distance, and then use mathematical equations to calculate its luminosity. Another method is to analyze the star's spectrum and use models to determine its luminosity based on its temperature and composition.</p><h2>5. Why is understanding the luminosity of stars important?</h2><p>Understanding the luminosity of stars is crucial for many reasons. It helps us to classify and categorize stars, which allows us to better understand their evolution and behavior. Luminosity also plays a significant role in the study of exoplanets, as it can help us determine the habitability of a planet and its potential for hosting life. Additionally, knowing the luminosity of stars is essential for accurately measuring distances in the universe and for studying the history and structure of our galaxy.</p>

1. What is luminosity?

Luminosity refers to the total amount of energy that a star emits in all directions. It is often measured in units of watts or solar luminosities, which is the amount of energy emitted by our Sun.

2. How is luminosity related to the brightness of a star?

Luminosity and brightness are related, but they are not the same thing. Luminosity is a measure of the total energy emitted by a star, while brightness is a measure of how that energy appears to us on Earth. A star's brightness may be affected by factors such as distance and the amount of dust and gas between the star and Earth.

3. What factors affect the luminosity of a star?

The luminosity of a star is primarily determined by its mass and its surface temperature. Generally, the more massive and hotter a star is, the more luminous it will be. Other factors that can affect luminosity include the star's age, composition, and any ongoing nuclear reactions within the star.

4. How do scientists measure the luminosity of a star?

Scientists use a variety of methods to measure the luminosity of stars. One common method is to measure the star's apparent brightness and distance, and then use mathematical equations to calculate its luminosity. Another method is to analyze the star's spectrum and use models to determine its luminosity based on its temperature and composition.

5. Why is understanding the luminosity of stars important?

Understanding the luminosity of stars is crucial for many reasons. It helps us to classify and categorize stars, which allows us to better understand their evolution and behavior. Luminosity also plays a significant role in the study of exoplanets, as it can help us determine the habitability of a planet and its potential for hosting life. Additionally, knowing the luminosity of stars is essential for accurately measuring distances in the universe and for studying the history and structure of our galaxy.

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