Bingo, we are in complete agreement. And seriously, some emails to site administrators would be a good start. The textbooks are often not quite as egregious in their inaccuracies on this, but they still foment misconceptions that fester, just as a few insights have the opposite effect. Thanks for helping get this word out here in the forum.

So the luminosity of my 1kg physics book is proportional to it's mass{Edit: cubed, sorry]??

I think it only works if your physics book is on the main sequence.

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Interestingly, the calculation does not require that the star be on the main sequence. What it requires is that it obey the virial theorem, and have roughly the opacity of a hydrogen plasma. (The main problem with the textbook is that it does not obey the virial theorem-- by which I mean it is not an ideal gas held together by gravity, unless you are using a very nonstandard text). Incidentally, the rough scaling is with mass cubed, not mass.

What about red giaints? How do they fit into this thread?

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One of the oddities in fusion research, is muon catalyzed fusion. http://www.triumf.ca/welcome/text.only/h_fusion.html

Some astronomers believe ignition of protostars may occur this way, and at room temperature. Ciao. Pete

Good question, the proportionality to mass cubed assumed two things, at least one of which breaks down for red giants:
1) the dominant opacity does not depend sensitively on the stellar structure (this is true for free electrons, for example, but not for the bound-free opacity in cooler environments)
2) the star can be described adequately by characteristic numbers like radius and core temperature, without detailed reference to the internal structure.
Red giants have such bloated cool envelopes that both of these assumptions might break down, I don't know much about their interiors. But one thing is clear-- their luminosity is huge, so you have to include more physics than is in this thread.

I note that another case where the L ~ M^3 scaling breaks down is when the star gets so bright that it is nearly blowing itself away with its own luminosity. Then you reach the so-called "Eddington limit" where L~M, but this is mostly only thought to be important for the first generation of metal-free stars (which might have had an increased tendency to be really big and bright). The best region to apply this thread is for main-sequence stars between a few and 50 solar masses, although it also explains in the roughest terms why high-mass stars evolve off the main sequence at fairly constant luminosity.

The link didn't appear to work. In any event, it is certainly clear that protostars eventually reach the main sequence and fuse hydrogen in the conventional way.

http://www.triumf.ca/welcome/h-fusion.html is the link... my understanding was that D-D fusion happens in a limited way even in brown dwarfs. Muons are pretty short-lived beasts so without an artifical boost I don't see how they would influence the internal workings of protostars.

You beat me to it... and thanks for the answers also Ken, altough I'm not sure that the quesiton is truly answered yet. Red giants have low surface temperature but enormous volume, so high luminosity... I'm not sure about their temperature profile, but the fusion process is no longer central so anything could happen.

Yes, there is a weird tendency for complex fusion processes to have layers in the star that alternately expand and contract as you cross a zone of fusion. This means that the oversimplified use of the virial theorem in terms of "characteristic" numbers must be expanded to include more details of the internal structure to really get it right. Stars with fusion only in their cores are much easier to get away with those kinds of simplifications. The best description of a red giant I've heard is that the core contracts and heats the envelope, which then expands in response to this internal heating, like a two-point virial theorem instead of one-point. But you'd have to understand the interiors models pretty well to judge if this really captures the essence or not, I'm just not sure.

One probably can compare two post-MS of different masses at similar stages of evolution. i.e., there is likely to be a "helium core burning main sequence" that has a mass-luminosity relation. As a general rule of thumb for post-MS stars we know that the more massive the star the more luminous it is. Why? Well, as with MS stars the more massive stars cannot also be denser and be supported by pressure that depends on the thermal energy content. So all else equal (and that's not always easy to establish for post-MS stars), the more massive stars will be larger in size and thus for similar surface temperatures easily much more luminous.

Also, we do observe the flatter L-M relation on the upper MS due to the importance of radiation pressure in these stars. And since we observe them, their metal abundances aren't that different from solar. This is manifested in the approach to an asymptotic maximum surface temperature of roughly 50,000K for the most massive MS stars.

But yes, rapid mass loss in very massive stars due to radiation pressure (continuous and especially line transitions for solar like heavy element abundances) is a complicating factor in the structure and evolution of these stars.

Hi, Question

What is the proportionality of luminosity to mass cubed?

Skim back over the thread evanoconnor, you'll find your answers there.

Sorry I wasn't clear.

what are the constant that make the relationship if L proportional to M^3 (i.e. L = K*M^3) what is K, maybe this is in the thread but I don't think so.

In fact, more massive stars must have a higher central pressure. If they did not, then the high temperature would cause the core to expand & cool, shutting off the nuclear reactions, which are very sensitive to temperature. The high pressure causes the high temperature, in the same way that high pressure warms the santa ana winds of southern California, by compressional heating.

Just looking at the average properties of the star, like its average density or temperature, won't do. Even solar type stars have most of their mass concentrated in the core. That effect is enhanced in more massive main sequence stars, and goes to extremes in red giant & supergiant stars, or AGB stars, where the core is extremely dense & massive, and the vast bulk of the star's volume is very tenuous by comparison. Since the gravitational acceleration is inversely proportional to the radius squared, a high mass concentrated in a small volume will produce a large surface gravity. That's just what happens in the massive core, which will be under extreme pressure due to its own large mass & small size. That pressure is what allows the core of a massive star to maintain temperatures in excess of 1,000,000,000 Kelvins during the late stages of nuclear burning.

These massive cores commonly "breathe", expanding & contracting, alternately turning the nuclear reactions off & on. This kind of thermal pulse instability is responsible for several types of variable star (cepheids are of this type, I think). The pulses come faster, later in the stars life. The conspicuous shells of the Egg nebula are the result of thermal pulsations, about 100 years apart.

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I'm sure if you look over the thread and test the suggested simple calculations, you will find the error in your reasoning. Or if you don't feel like actually following the thread, then just pick up a book on stellar structure. You'll be quite surprised to find how low the core pressure of massive main-sequence stars are.

I have often been surprised by how often the cause-and-effect in the ideal gas law gets massacred by perfectly intelligent people. I would argue that it is quite rare for the ideal gas law to set either the pressure or the temperature-- it is usually a law about density, given the pressure and temperature, as usually holds in our atmosphere (and by the way, are you sure that Santa Anna winds involve "compressional heating"? That would seem to be quite unlikely in subsonic flows). And that's how the ideal gas law works in massive stellar cores.

This must be your day for going out on a limb! Don't worry, we've all been guilty of it. Cepheids pulsate for reasons that have to do with the opacity in their outer layers, nothing at all to do with nuclear burning. They are a fine example of the way opacity controls luminosity, but in a transient way in this case.

Tim -

The pressure required by hydrostatic equilibrium is a quantity that looks like this:

P_req = alpha * GM^2/R^4, where M is the star's mass, R its full radius, and alpha is a dimensionless 1/R^4 weighted integral over the mass distribution:

alpha = (1/4pi) * integral{m(x) * dm(x)/x^4},

where x is the dimensionless radial dimension, r/R, and m(x) is the dimensionless mass m(r)/M. If this integral describing alpha is evaluated between x = 0, 1 (star's center and surface), then this is the required *central* pressure. For MS stars alpha is ~10 and it doesn't vary too much. It's larger for less massive MS stars and smaller for more massive ones, or at least that's the general trend.

Now think of the MS scaling relations with mass. A MS star's radius scales something like R~M^2/3 on the upper end of the MS and something like R~M on the lower MS and R~M^0.6 over most places in between. What this means is that, as stated above, less massive MS stars are as a rule MORE dense than high mass MS stars, with density scaling like M^-0.8. This is required by hydrostatic equilibrium for stars whose pressures depend on the temperature (gas, radiation). Why?

Look again at the expression for the required pressure - it requires that Pressure scales like (density^4/3 * M^2/3). Since the available gas pressure scales linearly in density (and T), and radiation pressure is independent of density (scaling like T^4), a more massive star MUST be less dense to be in hydrostatic equilibrium. This makes sense, since density contributes to gravity, not just gas pressure. It is also for this reason that the more massive MS stars MUST have higher T throughout m(r)/M. Of course, this analysis assumes that the numerical constants (such as alpha) in front of the physical variables (e.g., M^2/R^4) are similar for MS stars of different mass, i.e., homology holds. MS stars are homologous to a reasonable approximation (or they woudn't form a sequence), and the minor deviations from homology don't change the the global effect that the required pressures inside more massive MS stars are *generally* smaller than those for less massive ones:

P_req ~ density^4/3 * M^2/3 ~ (M^-0.8)^4/3 * M^2/3 ~ M^-0.4 .

The same result it obtained, by looking at the required T inside the star:
T ~ mu * M/R, where mu is the mean mass per particle exerting gas pressure (for stars in which normal gas pressure predominates).
If R scales like M^0.6, then T scales like M^0.4, and so the available gas pressure then scales like density*T ~ M^-0.8 * M^0.4 ~ M^-0.4.

Of course, you can't compare the cores of supergiant stars with those of MS stars - homology definitely does not hold in this comparison. Yes, the pressures within the cores of evolved stars are greater than they were as MS stars.

As Ken G. said, find some results from the full numerical models of ZAMS stars.
There will be exceptions to the above "rules of thumb" for MS stars, but the big ideas will hold.

(some editing for clarification of a few points)

Actually, the "foehn", chinook and, I believe, the Santa Anna winds are indeed due to compressional heating - air mass moving over mountain range is driven downward on the leaward side of the range. You don't need to shock a gas to heat it via compression.