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Suppose you have a star with radius $R_s$ which radiates with some power $P$, suppose we then envelop the star with a thin gas shell (and assume the gas behaves as a blackbody) with a radius $R_{gc}$.What is the new power radiated by the combined system $P'$?

So clearly this problem requires the use of the Stefan-Boltzmann equation $P= \sigma A T^4$, what's tripping me out is that the gas cloud will radiate energy back to the star presumably raising the temperature (bad assumption maybe?) of the star which increases the temperature of the gas cloud and so on....

Any advice on how to move forward?

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    $\begingroup$ What happens to the gas if the power radiated from the combined system is less than $P$? If it's more than $P$? $\endgroup$ – tfb Oct 17 '17 at 19:17
  • $\begingroup$ I think your instinct that there will be a back-reaction on the star from the gas-cloud is a good one, and it could indeed be complicated. But try to think about the long-term equilibrium behavior of the combined system. $\endgroup$ – kleingordon Oct 17 '17 at 19:21
  • $\begingroup$ I don't think this is very different than just adding a little more hydrogen to the star. An equilibrium will quickly be hit at a slightly different temperature. $\endgroup$ – kbelder Oct 17 '17 at 19:25
  • $\begingroup$ Remember that a radiating body cools off when it radiates. So there can't be any runaway reaction where the star heats the cloud and the cloud heats the star and so on. $\endgroup$ – probably_someone Oct 17 '17 at 20:16
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Assuming you only ultimately care about calculating the total radiated power, and don't specifically care about various temperatures, I would recommend using conservation of energy to approach this problem.

Think about this from a system-level perspective. Before adding the gas, the star has some distributed volumetric heat generation rate that, when integrated over the whole star, results in a total power output P. Where does that power go? Ultimately at steady state conditions, all that power can only escape out the surface of the star, hence it radiates out power P. Yes, some power goes back into the star and then back out etc., but we're thinking steady state conditions.

Now you add a shell of gas. The volumetric heat generation of the star has not changed. Where must that energy ultimately end up going at steady state conditions?

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With the s-b law it´s easy. The star emits according to temperature, with a source power density according to the inverse square law. You add the gas-shell and let it come to a steady state of emission and convection, Q+W. The temperature difference of the star and the cloud determines the potential for heat flow according t the s-b law for heat transfer. So, in any case, it will be a transfer of energy into the colder gas, and heat transfer from the star. This is the only exchange there is. One might argue that energy flows back and forth in ways not included in the s-b net transfer, but that is irrelevant for heat flow. The net transfer is what we can calculate. There is no hidden heat.

So with an added gas or atmosphere, the same amount of energy is used to heat a bigger volume simultaneously. The star emits $T^4$, and also transfer $T1^4-T2^4$.

Adding mass to a constant limited heat flow means less energy per molecule. Therefore you will not have any addition of energy from the radiating layer of gas, only a diffuse reduction.

$T^4$ is a measure of energy density. When using the s-b heat transfer equation you balance the densities of different temperature regions. It´s just like how you mix different concentrations of a solution. They don´t add to each other, what you get is dilution.

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