In several papers I see something equivalent to the following expression for the entropy of radiation given by an astronomical object such as the Sun (assuming the object can be approximated as a black body):
$$
s = \frac{4}{3}\frac{u}{T},
$$
where $u$ is the total flux of radiation energy through a spherical shell surrounding the object, $s$ is the entropy flux through the same imaginary surface, and $T$ is the black-body temperature of the object (and hence also of the radiation). It is also generally stated that no entropy is produced during the process of emitting radiation.

In a much smaller number of papers I see a formula that corresponds to just $s=u/T$, which is what I would expect. So the short version of my question is, which of these is correct for an astronomical body? But please read the rest, so you can understand why I'm so confused about it.

I understand the derivation of the 4/3 formula by considering a photon gas enclosed in a piston (see below), but in the context of a continuously emitting body like the Sun it doesn't seem to make sense. The issue is that if, over a given time period, the body loses an amount of heat $Q$ then this must balance the increase in energy of the radiation field $U$, i.e. $U=Q$. The body loses entropy at a rate $Q/T$, and if radiation is really a reversible process then this should equal the gain in entropy of the radiation, which should therefore be $U/T$. But according to the above formula it's actually $4U/3T$, meaning that the total entropy increases by $Q/3T$.

The 4/3 formula above was derived by Planck (in his book "the Theory of Heat Radiation", of which I've read the relevant chapter), who considered a photon gas in a sealed cylinder of finite volume. At one end of the cylinder is a black body and at the other is a piston. The radiation comes into equilibrium with the black body and exerts a pressure on the piston. If one reversibly (i.e. slowly) allows the piston to move then this causes some heat to be lost from the black body. In this case it turns out that $U=3Q/4$, with the discrepancy being due to the fact that the radiation field loses energy when it does work on the piston. The entropy balance then requires that the entropy of the photon gas increases by $4U/3T$, as above.

The point is, I can't see how the radiation being emitted by the Sun can be seen as doing work on anything. At first I thought it might be doing work on the outgoing radiation field. So let's draw an imaginary shell around the Sun again, but this time let the shell be expanding at the speed of light. Perhaps the radiation inside the shell is doing work on the radiation outside it, and that's what's "pushing" it away from the Sun? But it seems to me that for anything inside the shell to have an effect on anything outside it, some kind of influence would have to travel faster than light, so I don't think that can be right.

In any case it's well known that for a normal gas (made of matter), expanding against a piston is quite different from just expanding into a vacuum. In the former case the temperature and internal energy decrease, because the molecules lose energy in pushing the piston, whereas in the latter case they both remain constant. I haven't found any source that addresses the question of why this would be different for a photon gas.

So it seems like the emission of radiation from a body like the Sun into space is quite different from the emission of radiation into a sealed piston, and I'm puzzled as to how the same formula can apply. Below I've listed some possible resolutions to this which seem plausible to me. I haven't been able to find any sources that directly address this issue, or state any of these positions.

1. The emission of radiation into space is an irreversible process after all. $U=Q$, and the total entropy increases by $Q/3T$. (But then, what happens when radiation is absorbed by a body at a similar temperature? Surely the entropy doesn't decrease.)
2. There is some weird sense in which the outgoing radiation can be thought of as doing work on something, so $U\ne Q$. (If so, why does nobody ever explain this subtle and important point?)
3. $Q=U$, but the entropy of radiation emitted into space is actually different from the entropy of a photon gas in a sealed cylinder, such that its entropy is given by $U/T$, not $4U/3T$. (This one actually seems the most reasonable to me. The radiation in the closed cylinder has rays travelling in all directions, whereas the radiation coming off the Sun only has rays travelling in directions away from its surface, so it seems reasonable that they would have different entropies. This would mean that the 4/3 formula doesn't apply to the radiation emitted by astronomical bodies after all - but if that's the case then it's an extremely widespread mistake.)

Any insights into which, if any, of these is correct would be greatly appreciated - as would any references to sources that directly address the relationship between Planck's photon-gas piston and the emission of radiation into empty space.