Wrong Explanations for the Luminosity of Main-Sequence Stars

[Ken G]

Implying a constant heat flow

L ~ c l d/dr(4 pi r2 T4)

Where l~1/(rho sigma) is a mean free path measure

Yes, this is correct.

So the scaling relation simplification is

L ~ L / ( rho sigma T )

Rho sigma R ~ constant

I wasn't sure what you were getting at here. We both get L ~ d/dtau(r² T⁴), where dtau = sigma*rho*dr is the optical depth across dr, and r² T⁴ is like the radiative energy per shell of unit radial length. This is like setting dtau=1 and using the shells you were talking about above. Where I would go with that is just insert the characteristic proportionality tau ~ M/R² for constant sigma (for simplicity, or as a benchmark for comparison), yielding L ~ (RT)⁴/M. So that's all we can get from radiative diffusion, and if we take M as given, then the L depends only on RT if sigma is constant. Then the VT shows that RT is also constrained by M, since RT ~ M. Thus L depends only on M. You can use different opacity laws, which will introduce some independent dependence on R and T, which will in turn bring in the fusion physics (and the power n) because T will now matter, but if constant-sigma dominates, that won't occur, and if n is large (as for CNO cycle), then the T physics is highly thermostatic and it also won't matter much how the fusion changes as M changes, i.e., the explicit value of n won't matter if n is large. But you are right that for M ~ 0.5 solar, we have p-p fusion with n = 4, and it only goes up to about n = 5 for solar M, so those values of n are small enough that some fusion physics will end up showing up in L, as will some Kramers opacity effects. You only get the super-simple result L ~ M³ if you ignore those details, but then we are also ignoring convection and radiative forces, so we are already going to be stuck with some pretty rough results.

You are also looking at the scaling laws for R(M), so they will start with R ~ M/T and have to use some fusion physics to get T, but for higher mass stars, again T will be fairly thermostatic, around 20 MK to 30 MK, so you won't do too badly with R ~ M. For lower M stars, the lower n of p-p fusion makes R(M) drop less rapidly as M drops, and convection also becomes more of an issue, so again it's a matter of what subjective tradeoffs you want to make in your idealizations.

I wouldn't try to get surface T until the end-- once I have a handle on both L and R, I would then say the surface T is ~ L^1/4/R^1/2. The surface T rarely has much input into either L or R, so that last formula is not very useful as an input to either the L or the R determination. However, you have seen some nice scaling relations about the ratio of surface to core T.

 

[TEFLing]

Yes, this is correct.
I wasn't sure what you were getting at here. We both get L ~ d/dtau(r² T⁴), where dtau = sigma*rho*dr is the optical depth across dr, and r² T⁴ is like the radiative energy per shell of unit radial length. This is like setting dtau=1 and using the shells you were talking about above. Where I would go with that is just insert the characteristic proportionality tau ~ M/R² for constant sigma (for simplicity, or as a benchmark for comparison), yielding L ~ (RT)⁴/M. So that's all we can get from radiative diffusion, and if we take M as given, then the L depends only on RT if sigma is constant. Then the VT shows that RT is also constrained by M, since RT ~ M. Thus L depends only on M. You can use different opacity laws, which will introduce some independent dependence on R and T, which will in turn bring in the fusion physics (and the power n) because T will now matter, but if constant-sigma dominates, that won't occur, and if n is large (as for CNO cycle), then the T physics is highly thermostatic and it also won't matter much how the fusion changes as M changes, i.e., the explicit value of n won't matter if n is large. But you are right that for M ~ 0.5 solar, we have p-p fusion with n = 4, and it only goes up to about n = 5 for solar M, so those values of n are small enough that some fusion physics will end up showing up in L, as will some Kramers opacity effects. You only get the super-simple result L ~ M³ if you ignore those details, but then we are also ignoring convection and radiative forces, so we are already going to be stuck with some pretty rough results.

You are also looking at the scaling laws for R(M), so they will start with R ~ M/T and have to use some fusion physics to get T, but for higher mass stars, again T will be fairly thermostatic, around 20 MK to 30 MK, so you won't do too badly with R ~ M. For lower M stars, the lower n of p-p fusion makes R(M) drop less rapidly as M drops, and convection also becomes more of an issue, so again it's a matter of what subjective tradeoffs you want to make in your idealizations.

I wouldn't try to get surface T until the end-- once I have a handle on both L and R, I would then say the surface T is ~ L^1/4/R^1/2. The surface T rarely has much input into either L or R, so that last formula is not very useful as an input to either the L or the R determination. However, you have seen some nice scaling relations about the ratio of surface to core T.

Conceptually I think you have an equation like

L ~ d/dtau ( r2 T4) ~ ( Lcore - Lsurfsce)/( M/R2)

I think you can then use scaling relations like

Lcore ~ M rho T^n

And ignore Lsurface on the RHS since it's smaller than Lcore

Seemingly you're saying that the L seen at the surface is the original core luminosity stepped down like a voltage transformer , by the number of MFP i.e. optical depth

Assuming constant opacity, and using n=4,18 and RT ~ M, sqrt(M)...

You seem to get

L ~ M3 for low mass stars
L ~ M2 for intermediate mass stars
L ~ M for high mass stars

 

[Ken G]

I get L ~ M³ for all stars, independent of n, if I assume gas pressure dominates over radiation pressure, ideal gas T, ignore convection, and assume fixed opacity. I don't even need to say if fusion is even happening to get this, indeed look at the tracks of pre-main-sequence stars in an H-R diagram to see just how unnecessary it is to say if fusion is happening to get L.

If we instead use radiation pressure, instead of gas pressure, we get the Eddington limit value for L, which obeys L ~ M for constant opacity and no convection, and doesn't need to know anything about T or n. You'd actually get a lot of convection, but it might only carry some of the L, so the L ~ M is probably still not too bad.