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Partial answer so far This is a really interesting question! I've just started looking at Inferring the interplanetary dust properties from remote observations and simulations and surprisingly the power law fit to observed solar system dust temperature between 0.5 and 1.5 AU as shown in Eq. 2 varies as $d^{-1/3}$ instead of $d^{-1/2}$ in your equation. It ...

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Okey in agree with @fasterthanlight a star can be treated as a black body. The derivation of the black body law is through thinking about a gas of photons wich obbeys the Bose-Einstein statistics distribution function. This function dependes on the temperature and frecuency, so that is why the radiation intensity varies in function of the two variables ...

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The core of the star is the seat of nuclear fusion, yes, but by the time this energy reaches the surface (a few hundred thousand years at least in the case of our Sun), it has time to dissipate (from a [comparatively] small core to a huge outer surface). What’s left at the photosphere (the apparent surface of the star) is not nuclear fusion anymore, but ...

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