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Is there an exact relationship between color indices and temperature? I have seen some numerical formulas relating B - V color index to temperature, of the form $B - V = a_1 + \frac{a_2}{T} $ .

I am wondering is there any way to derive these formulas or are they completely empirical?

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They are mostly empirical. Found by measuring the $B-V$ for stars of known $T_{\rm eff}$ (which are in turn measured by knowing the luminosity and radius of a star, and this is only known for a small number of stars). The relationships also depend on stellar surface gravity and composition.

An alternative approach is to derive "synthetic" relationships by folding spectra calculated from model atmospheres through the appropriate filter responses.

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