Ken G
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Yes, this is the virial theorem, but there are a few important caveats. First of all, the virial theorem is a kind of average statement, so is really only useful when the whole star can be treated as "all one thing," where the temperature is characterized by the temperature over most of the interior mass, and the radius characterizes that mass. So it's best for pre-main-sequence and main-sequence stars, failing badly for giants and supergiants which have decoupled outer radii. Secondly, we must be very clear that the temperature we mean is the interior temperature, not the surface temperature you find in an H-R diagram. You may well know this, but this confusion comes up in a lot of places where people try to marry the Stefan-Boltzmann law, applying only to surface temperature, to the interior temperature. The surface is like the clothes worn by the star, much more than it is like the star itself, but since we only see the surface this can cause confusion.JohnnyGui said:- Temperature in a star is inversely proportional to ##r##
This is the Stefan-Boltzmann law, but one must be wary of the cause and effect. In protostars that are fully convective and have surface temperatures controlled to be about 3000-4000 K or so, this law is quite useful for understanding the rate that energy is transported through the star. In effect, the luminosity is controlled outside-in, because the convective interior will pony up whatever heat flux the surface says it needs to (via the relation you mention). However, when stars are not fully convective, or when they have fully convective envelopes controlled by tiny interior fusion engines (like red giants), the cause and effect reverses, and the luminosity is handed to the surface. In that case, it is not that the luminosity is proportional to ##1/r^2##, it is that the radius is proportional to the inverse root of the luminosity.- Light (photon) loss per unit time has several relationships with ##r##:
1. It's proportional to the area, thus proportional to ##r^2##
When radiative diffusion controls the luminosity (pre-main-sequence and main-sequence, and also giants and supergiants to some degree), what you mention is one of the factors. But not the only one-- diffusion is a random walk, so optical depth enters as well, not just distance to cross.2. It's also inversely proportional to the time it takes that a photon travels from within the star to the surface, thus it's inversely proportional to ##r##
Best not to think in terms of energy per photon, but rather energy per unit volume. That scales like ##T^4## (that's the other half of the Stefan-Boltzmann law), not T.3. It's proportional to the temperature (energy per photon-wise) and since temperature is inversely proportional to ##r##, that means it's again inversely proportional to ##r##
If you are testing dynamical stability (the usual meaning of a "kick," you would kick it on adiabatic timescales, i.e., timescales very short compared to the energy transport processes that set the luminosity. So for dynamical timescales, use adiabatic expansion, and ignore all energy release and transport. If you want to know how the luminosity evolves as the stellar radius (gradually) changes, that's when the above considerations about the leaky bucket of light come into play.4. This all means that the net change in light loss per unit time when you expand a main-sequence star is 0, before a star contracts again after the expansion "kick". This conclusion is without taking the fusion rate into account that is also affected during expansion.
The simplest way to treat fusion is to pretend the exponent of T is very high, and just say T makes minor insignificant adjustments until the fusion rate matches the pre-determined luminosity. For p-p fusion, the exponent is a little low (about 4, as you say), so that's not a terrific approximation, but it's something. For all other fusion (including CNO cycle hydrogen fusion), it's a darn good approximation. So if you are making this approximation, you don't care about the value of the exponent, the fusion just turns on at some T and self-regulates. However, in red giants, where the fusion T cannot self-regulate, there you do need the full exponent, you need to explicitly model the T dependence of the fusion because T is preset to be quite high.- Fusion rate is proportional to the temperature ##T## but to different extents depending on what is being fused. Fusion rate of hydrogen is proportional to ##T^4##, fusion rate of helium is proportional to ##T^{40}##. Thus it's proportional to ##r## and influences the light loss per unit time in different amounts depending on what is being fused.
Yes, for p-p hydrogen fusion, say like in the Sun. In fact, this is not a bad approximation for what would actually happen if you added mass to the Sun-- you wouldn't need to keep the interior temperature the same, the thermostatic effects of fusion would do that for you. The Sun would expand a little, and its luminosity would go up a little because it is now a bigger leakier bucket of light. Fusion would simply increase its own rate to match the light leaking out, and it would do that with very little change in temperature, expressly because it is so steeply dependent on T. But this story would work even better if the fusion rate was even more sensitive to T, say for the CNO cycle fusion in somewhat more massive stars than the Sun. (Ironically, many seemingly authoritative sources get this reasoning backward, and claim that the temperature sensitivity of fusion is why the luminosity is higher for higher mass, on grounds that adding mass will increase the temperature which will increase the fusion rate which will increase the luminosity. They are saying that the sensitivity of the fusion rate to T is why it rules the star's luminosity, when the opposite is true-- it is why the fusion rate is the slave of the luminosity. The situation is similar to having a thermostat in your house, and throwing open the windows in winter-- opening the windows is what causes the heat to escape, not the presence of a furnace, but the extreme sensitivity of a thermostat is what causes the furnace burn rate to be enslaved to how wide you open the windows.)One other question @Ken G ; you said fusion rate is proportional to mass to the 3rd or 4th power. Is this apart from the temperature being higher or lower with mass? So if I add more mass to a star while keeping the temperature constant per unit mass, fusion rate would still go up?
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