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I am reading my textbooks and a few "Astro101" websites, but I cannot find a final answer to this:

In the $B-V$ colour index that is plotted in an H-R diagram, are $B$ and $V$ the (observed) apparent magnitudes, or should they be corrected for extinction first, i.e. with known $A_V$ and $A_B$ values?

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Unless you know all the stars in the diagram suffer the same extinction, then you should de-redden them individually.

If you do know they have the same extinction (e.g. they are all nearby stars and it is zero; or they are all stars well above the Galactic plane in a similar direction; or stars in a cluster at the same distance) then you can plot a diagram using the observed magnitudes. In the latter scenario of a cluster, this diagram can be used to estimate the reddening of the cluster.

If you want to compare with models, to estimate temperatures or masses, then you must de-redden (or redden the model!)

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If you can correct them because you know A_V and A_B (which can be hard), it's probably better, although I don't think there is a general rule here.

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