I'm not posting this in Q&A because I know the answer. My real question is, how come so many texts and internet sites indicate that high luminosity is caused by higher core temperature (in concert with nuclear burning rates), when in fact the causality proceeds in exactly the opposite direction? Also, it's fun to see how many "authoritative" websites are even confused about whether high-mass main-sequence stars have low or high core pressure/densities! (Check it out, I've never seen such widespread hooey from mainstream sources).

What? I thought they were heavy because they emitted a lot of light!

Which sites?

Almost all of them, believe it or not. Simply google the question: "Why do high-mass stars have such high luminosity?" and prepare to watch the fur fly. It's a major "oops" that I hope this thread can begin to stamp out.

Hmm, for some reason, the last time I did that query on Google I immediately got these four hits from "authoritative" sites:
http://imagine.gsfc.nasa.gov/docs/sc..._l2/stars.html
http://outreach.atnf.csiro.au/educat...nsequence.html
http://cassfos02.ucsd.edu/public/tutorial/StevII.html
http://www.astro.psu.edu/users/steinn/Astro1/lec21.html

yet now when I do that same query, I get irrelevant hits, and I'm quite confused by that! In any event, check out the above sites.
They all mistake the direction of the causality, as I mentioned in the OP, and worse, most of them assert that high-mass main-sequence stars are either high pressure, high gravity, or high density stars-- all of which is totally untrue! I had no idea that high mass stars were such a muddle on the internet, and even the textbooks get the causality wrong. Internet researchers-- beware!

So you are saying that the higher core temperature is caused by the high luminosity? If so, then what do you think causes the high luminosity?

Ah but that would be giving it away. I'll tell you, but I want you to see the links first, because it's amazing how such authoritative links can get it so fouled up. But I will give you a hint to keep your interest-- Eddington understoond the luminosity of massive stars, before nuclear reactions were even known to exist. Yes?

Wasn't that Henry Norris Russell, in his 1919 On the Sources of Stellar Energy? He essentialy derives, in a very non-mathematical way, why an the energy source in stars must have a strong temperature dependence to yield long-lasting stable structures. He is clear about this being a way to fix the energy generation to match the star's luminosity, and sneaks right up to comparison among stars of different masses. All this without any clear physical idea of where the energy arises. I should be so lucid on matters I don't understand.

The above quote is taken from the CSIRO site that Ken linked to, it clearly states that mass causes the higher temperature, which causes the higher luminosity.

Yes, and the CSIRO website is totally wrong. ngc3314 has it right-- you can derive the luminosity knowing nothing about central temperature, or radius. All you need is the mass, and knowledge of how opacity works, the virial theorem, and a statement that the star has some stable energy source. Gravity would work fine, for a Kelvin-Helmholtz time anyway! All nuclear fusion sets is the lifetime, and central T. Period. Not luminosity! CSIRO could not possibly be more blatantly incorrect on this, as is every other website I've seen. Too bad Russell, or ngc3314, didn't create those websites!

Okay, second draft of the post, thus a shorter draft:

I do not see where CSIRO actually screws up. Nowhere do they say that nuclear fusion sets the luminosity. Indeed, they explicitly state what you say, that from mass you get luminosity: I see comments on gravity exerting pressure on the core which means a high core pressure. I see them explicitely mention how gravity (a.k.a. mass) drives the fusion rate, and this comment leads into their discussion of stellar lifetimes:

So I don't see how CSIRO disagrees with your post. And note, that the gravity drives the core temperature, which sets the surface tempurature, explaining the mass luminosity relationship I quoted at the beginning of the post. They don't say the increased fusion rate creates a greater luminosity (though it does sustain the greater luminosity, which is your point).

Fasten your seat belt. I want to note that their mistakes are not unusual, they are absolutely the norm. But they are whoppers.

They certainly do state that nuclear fusion sets the luminosity. Here is a quote:
Faster fusion means higher luminosity, ergo they make the claim. Don't be confused by mass-luminosity relations, every Tom, Dick, and Harry points out that L correlates with M. I'm not talking about correlations, I'm talking about the explanations offered. These are what are totally bogus, here and elsewhere.

Precisely, they make exactly this claim:
Er... unless there's another parameter involved...
For fun, I'll leave it up to the others here to point out the staggering error in this claim, as applied to main-sequence stars. Why should I have all the fun?

They disagree with me by having two totally wrong physical points:
1) they think more massive stars have higher central pressure
2) they think this causes a higher central temperature (huh?), which in turn causes the high luminosity via nuclear burning rates. In fact, point (1) is false and point (2) reverses the correct cause and effect.

They say precisely this, in my quote above. Yet,
you understand my point, fusion sustains the luminosity without being the explanation for it, so you have not fallen for their hooey. In truth, a snapshot of the star explains its luminosity without even knowing what the energy source is, but details of the energy source determine how long it can be sustained. I could not find a single website that made this simple point!

Okay...care to actually explain what the actual operating core pressures and densities for high mass stars are?

Cause I'm pretty sure the massive stars have a higher core temp (thus allowing efficient CNO cycles) and relatively sure they also have a higher pressure in the core. IIRC All main sequence stars have relatively similar surface pressures as the collisional broadening of spectral lines is used to determine which luminosity calss they belong to...which means they use surface pressure to define this.

Now, I think you're reading what you want into the CSIRO statements. They say more mass means higher core temp, faster fusion rates. They do Not say the higher fusion rates feed the luminosity. Read the section again. You're drawing a conclusion from the statements that they don't.

I.e. they do not draw the line connecting fusion rates and luminosity.

I do agree that they should far more clear on this point.

Anyway, I think people have had enough time to think about it (and some of the responses have taken the bait) so...out with it!

Sorry, I was hoping someone else would chime in with that, to give this thread a more group-based flavor! But there aren't enough folks following it at present. Pity, there's a lot of stellar physics to be learned. But here's your answer: the most important thing to know about high-mass main-sequence stars is that they are low-density, low-pressure, low-gravity objects. That's why they supernova-- they have a lot more gravitational collapsing to do yet!

Yup, but that's an insignificant detail for determining their luminosity
No, see above.

Of course higher fusion rates must lead to higher luminosity. I cite energy conservation. The issue is, which is the dog and which is the tail. The first argument given should be for why the luminosity is high, and then you can invert their reasoning to infer why the core temperature is (a little) higher. Basically, this was also the historical process, but that's been forgotten somehow!

All right, your curiosity is worthy of an explanation. This is why higher mass stars have higher luminosity, it doesn't even matter what their core temperature is:

If you take a snapshot of the star, its luminosity can be inferred by looking at the density of radiant energy inside the star (its temperature), and the rate that volumes of that radiant energy are "dumped" into space (via diffusion). The latter is determined by the surface area times the average photon diffusion speed, which is c/tau, where tau is the optical depth of the star. Since tau is given by the opacity times the density times the radius (characteristic numbers throughout), that is proportional to opacity (lets assume this is constant, as for the free electrons in higher mass stars) times M/R^2. The radiant energy is proportional to T^4, so L is now proportional to T^4 R^4 / M. But the virial theorem says that the average kinetic energy of a particle, set by T, is proportional to the average gravitational energy, set by M/R. This says T*R may be replaced by M in my earlier result-- giving L proportional to M^3 with no reference to R or T! That's all you need, no CNO chain, nothing.

Physically, this says that if a higher mass star has a similar core T, it must be far less compressed than a lower mass star, so the photons diffuse out more easily and L is higher. Or, if it is as compressed as the lower mass star, the central T is much higher so there are more photons to escape. Main-sequence stars correspond to the former case, but it's just a subclass. All you really need is stability in your energy generation mechanism. That's it!

Let me further add that all of the websites I quoted above either say that high-mass main-sequence stars are high density, high pressure, or high gravity. All of these are patently false, as any introductory stellar structure textbook will tell you. How did the situation get this out of hand? I can only guess that the mistake got made somewhere authoritative, then repeated, then gained further authority. It's called an urban legend folks, and it's not just the woo-woos that fall for it! Beware relying too much on the internet for factual information. So who's going to write to the site administrators? (I will if nobody else wants to.) Matthew, you brought the site up...

Or expanding Ken G.'s explanation a bit more generally:

a) take kappa to be the opacity in cm^2/g (a larger value means that light interacts more strongly with matter), which Ken G. took to be a fixed value - nothing wrong with that, but let's put it in explicitly. It generally decreases inward through the star.

b) set mu to be the mean mass per particle (depends on the star's composition), then from the virial theorem (or equivalently, hydrostatic equilibrium) the temperature inside the star scales as
T ~ (mu*beta) * M/R , where beta = the ratio of gas/total pressure. beta is smaller for higher temperatures, for which radiation pressure ~ T^4 becomes increasingly important.

c) and set eta = (L(r)/M(r)) / (L*/M*), where L(r) and M(r) are the run of luminosity (due to fusion or lacking that: graviational contraction) and enclosed mass as a function of radius r through the star, while L* and M* are the star's surface luminosity and its full mass. So eta = 1 at the surface and generally increases inward.

L ~ M^3 * (mu*beta)^4 / (kappa * eta)

Sir Arthur Eddington derived the "Standard Stellar Model" that had this form roughly a century ago, well before anyone understood the role of nuclear fusion. While kappa and eta are not constant within the star, their product roughly is.

Nevertheless, nuclear fusion shouldn't be viewed as an innocent bystander in determining the star's internal structure and surface properties. The pre-main sequence objects contracted (via gravity) *sufficiently* until density and temperature in their central regions resulted in the production of Joules/sec through fusion that offset the loss of Joules/sec at the surface due to the emitted light there.

The more massive stars thus arrive on the main sequence with lower densities and thus lower pressures, despite their higher temperatures, than their lower mass siblings.

Here are two more commonly made, erroneous statements in introductory level astronomy texts and websites:

1) stars shine because of the energy released from fusion.

Stars shine because they're hot, and they're especially good at it because they're opaque radiators. That they produce energy in their cores that offsets the energy lost at their surfaces due to their emitted light just allows stars to remain relatively stable longer in time.

2) the energy released from fusion keeps the star in pressure-gravity balance (or more technically termed 'hydrostatic equilibrium').

Again, fusion is irrelevant to the basic process of hydrostatic equilibrium. A star without fusion going on inside would simply contract on something like the Kelvin-Helmholtz time scale, due to the very slow "energy leak" (photons escaping from the surface), and remain in hydrostatic equilibrium every step of the way. In contracting it becomes hotter and denser and more luminous (if not contracting along the 'Hyashi track'). This equilibrium is established because the star's interior naturally generates pressure (gas, radiation, ...) and a gradient thereof. If hydrostatic equilibrium is not established in some layer within the star, the structure will adjust rapidly until it is (or rapidly pulsate in size trying to find an equilibrium - as in the Cepheid variables). A main sequence star steadily fusing hydrogen into helium at a rate sufficient to put the star in energy balance (energy in = energy out) no longer has a net energy leak, and thus will not contract.

Right on spaceman, your generalization to the eta*kappa product is a very useful addition to the general scheme. And I want to be clear that I'm not saying fusion doesn't affect the internal structure (it's crucial for central temperature, of course), I'm saying it does not affect the luminosity in any fundamental way. Any stable heat-producing mechanism works fine. The biggest irony is that the extreme temperature sensitivity of fusion, which is almost always cited as the cause of high luminosity, is exactly the reason that fusion becomes the slave of the luminosity. Since the temperature sensitivity implies that insignificant rearrangements of the internal structure are all you need to provide the necessary luminosity, fusion ends up being rather powerless to alter the internal stellar structure! What has much greater power is opacity, so most of the structure changes you see on the main sequence come when bound-free opacity (Kramers type) in lower mass stars gives way to free-electron opacity (which is constant) in higher mass stars. For the really high-mass stars, the argument breaks down because the virial theorem as stated in this thread does not include radiation pressure, but that only becomes important for the very highest mass stars (near the so-called "Eddington limit"). Eddington knew all this a century ago-- when did it get forgotten?

And I'll complete the lesson by saying that all fusion can do is slow the inevitable gravitational contraction (good for Earth, we needed longer than a Kelvin-Helmholtz time to develop civilizaton). But it can't stop the contraction because it runs out of fuel eventually. The only thing that can prevent the collapse to a black hole, ultimately, is degeneracy pressure (of either electrons, for low-mass stars, or neutrons, for high-mass stars). The key asset of degeneracy pressure is that it does not rely on heat content-- you can radiate away all the heat available and degeneracy pressure still works fine.

See, I agree completely with everything you've said.

Except that the CSIRO site does not say the higher fusion rates cause the higher luminosity. They state in two different comments that Luminosity is due to mass, and that fusion rates are moderated by mass. I.e. they are linked via mass (as you've even stated), but nowhere do I see them state that fusion rates lead to higher luminosity, which is the explicit complaint you have against them.

Anyway, as far as the physics you've presented: No complaint.

As far as websites needing to clean up the explaination a bit (i.e. to much repetition of the same ole' spiel)..I agree.