Helios said:
Helios said:
Jenab6 said:
Ken G said:
That's a good question. In the vicinity of the Sun, you have a steeper dependence of L on M, yet convection is not yet dominant, so it's probably more about the complicated opacity in a cooler star. If you go to the lower masses, say <0.5 solar, you're right that convection does become dominant, and there is a general theory about how such stars behave, called the "Hayashi track." Adding the condition that nuclear fusion, rather than gravitational contraction as in protostars, is providing the luminosity, constrains the behavior on the Hayashi track. There might be a simple way to get luminosity under those conditions, I'd have to bone up on that. The problem is that convection is treated with "mixing length theory", which is an approximate approach that is pretty unwieldy and is used more for lack of an alternative than because of its dependability. So the short answer is, "yes you can estimate the luminosity of a convective main-sequence star of given M, but it's not easy, not reliable, and I don't recall how to do it."twofish-quant said:
Yes, it is due to the extreme T sensitivity of the fusion process, especially the CNO cycle in massive stars. Tiny increases in T produce huge increases in fusion rate, so this acts like a thermostat on T because it is stabilized by gas pressure (T rises cause adiabatic expansion which drops T back down).
Exactly. All you need fusion for is to be a thermostat, when coupled with the stabilizing effects of gas pressure. If you have an observational means of inferring the core T, you don't need fusion at all, which is how Eddington was able to model stars without knowing anything about fusion, he just needed on ad hoc element-- a characteristic value for the core T that had to be observed not explained.
over what that meant. I have encountered L \approx M3 for high mass stars before, but seeing your explanation along side it helped it gel in my head.
turboguppy said:
twofish-quant said:
Ha! Pretty familiar with the 'fudge factor' as I've also got to do atmospheric stuff like greenhouse effect and such. I'll keep digging on this, but it seemed to me that once you knew the mass and the expected luminosity for that mass, you should have a good idea if the star has to shed mass because of too much radiative pressure. Without modeling the star completely it seems I should be able to approximate or at least fudge a mass-loss rate based on the expected luminosity. Does that even make sense?Yes, that is for the most massive stars, that have a dominant contribution from radiation pressure rather than the gas pressure that dominates most stars. When radiation pressure dominates, it changes the way R depends on M. The radiative diffusion arguments, at roughly constant internal T, tells us that main-sequence luminosity scales roughly like R4/M. When gas pressure dominates, we find (from the virial theorem) that roughly R ~ M, so you get L ~ M3. But when M is really high and radiation pressure dominates, you reach the "Eddington limit", and internal force balance requires L ~ M, so this means the R ~ M we had for gas pressure gets replaced by R ~ M1/2, just because of the Eddington limit.turboguppy said:
Yes, there is still a lot that has not been nailed down. Even simple things like metal abundances keep changing with new data, and the ability to model a star from its spectrum and get agreement with binary data has only met with partial success.
I don't know-- it's hard to know how accurate the relationships are when we don't have access to the "raw numbers", only what we can infer indirectly. Even binaries are only a slam dunk when they are eclipsing, and even then, do we really know the metallicity of the interior? I think you should be happy with "close enough" rather than "correct."
There's a surface metallicity, and a core metallicity. In low-mass stars, the surface metallicity is mixed throughout the envelope by convection, possibly even into the core for very low-mass stars. In high-mass stars, the convection is only in the core, so the core metallicity is mixed around, but differs greatly from the envelope (and fusion can occur in shells in the envelope, so you get an "onion skin" model of varying metallicity). This develops with age. So I fear the answer is, there isn't any way to treat age and metallicity that is general across the main sequence, you'd need to treat each mass range differently, or else do the full evolutionary calculation, or access tables of those who have done them. Maybe they provide fits to their tables.
Yes, there is "CAK theory", which predicts roughly that the mass-loss rate scales like luminosity to the 1.6 power, at least in any star where this would matter. The constant in the relationship is not known, because mass-loss rates are hard to measure reliably. The constant appears to work out such that a star with mass of perhaps 50 solar masses is a star with a mass-loss rate timescale equal to what its main-sequence lifetime would have been without that mass loss (so the mass loss significantly shortens the main sequence lifetime).
turboguppy said:
