# Distance to star using color index

I'm doing the following question:

A star is observed with UBV magnitudes $$m_u = 16.31$$, $$m_b = 14.52$$, $$m_v = 13.76$$. Spectral analysis gives $$M_{bol} = 7.31$$, $$BC = -1.02$$, $$(U − B)_0 = 1.222$$. Determine the distance to the star.

As $$m_u - m_b ≠ (U-B)_0$$, I gathered that the distance modulus can't be used directly, since extinction is present. From the bolometric correction (BC), we can determine that the absolute visual magnitude $$M_v = 8.33$$.

After that, I'm unsure of what to do; normally I think in these questions you can obtain the visual extinction $$A_v = 3.0E_{B-V}$$, where $$E_{B-V}$$ is the color excess, calculated by $$E_{B-V} = (m_b - m_v) - (B-V)_0$$. Then, the visual extinction can be added to the distance modulus equation to account for extinction in the visual band, and then we can easily obtain distance.

But, while we know $$m_b$$, $$m_v$$, and $$M_v$$, we don't really have a way of obtaining $$(B-V)_0$$ to my knowledge, unless there's some way we can relate it to $$(U-B)_0$$. Any help would be much appreciated!

There is a relationship between $$E(B-V)$$ and the reddening in any other colour. The exact value depends on the type of dust, the extinction value itself and the intrinsic spectrum of the star (as does the 3.0 coefficient mentioned in your question).
However, for the purposes of estimation, the canonical relationship is $$E(U-B) = 0.72 E(B-V)$$ (e.g. Pandey et al. 2003).
Thus you can calculate $$E(U-B)$$, then $$E(B-V)$$, then $$A_V$$, then $$V_0$$ and use this with $$M_V$$ to get the distance.