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Ken G

Gold Member

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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 ∝ M

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 M

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.

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 ∝ M

^{3}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 M

^{3}, 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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