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It must also have a continuous spectrum which is dependent only on temperatue, and is determined by Planck's law?

It must, but the "continuous" part is mis-leading.

Planck's perfect black body is - like everything in physics - just a model, and contains assumptions and approximations.

The assumptions are that if your body has a length L, then the wavelengths lambda all have the property L = n * lambda, where n is an integer, (and that all absorption and transmission happens between two sharp energy levels; not 100% about this one).

The simplification is that in order to calculate the spectral power distribution, one replaces a sum by an integral, and thus looses the quantization condition on the way.

So in the model of the Planck black body radiator the continuous spectrum is just an artifact of a mathematical approximation, not a physical property.

Note that there are other mechanism that take care of smoothing the spectrum. For example all particles in the sun move (brownian motion), so the radiation has a red- or blue shift. The second mechanism is that all processes - including those of photon emissions - take finite time, so that have (by virtue of the Heisenberg uncertainty principle) a non-sharp energy distribution (called "intrinsic" line width)

My point is though, shouldn't it? If the main mode of EM emission from a black body is by atomic excitation/de-excitation, shouldn't black body theory take account of this, or use it as its starting point?

I'm not too firm with the astronomy of our sun, so I could be wrong, but... I think the "black body" approximation holds true for the bulk of the sun, but not for the out regions (Corona, whatever), and these outer regions could be enough to filter all the spectral lines that you see when looking at the sun.

You're trying to mix two different models (atomic spectra vs. black body, a thermodynamic model), so you have to be very careful which part of which model still holds true in the mixed case.

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