# Relationship between Color Index and Temperature

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?

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.