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Is there a formula to convert the B-V index to U-B index or are they totally different and separate observations and can't be convert like you can with Fahrenheit->Celsius? (Updated)
I know F->C is not astronomy related directly but there is a formula to convert between the two measurements.
Both $B-V$ and $U-B$ are a reflection of how hot the surface of a star is. To first order (inhomogeneous surface temperatures do exist) this has a single value for a given stellar photosphere and so there ought to be a direct relationship between $B-V$ and $U-B$. Here it is, shown below as a graph appropriate for main sequence stars with a composition similar to the Sun. You can reproduce this yourself using data found (for example) in the Handbook of space Astronomy and Astrophysics (pp68-69) compiled by M.V. Zombeck.
You can see that the relationship is complicated; this is because stellar surfaces are not blackbodies and the absorption by various chemical species cuts out chunks of the spectrum, particularly in the blue and ultraviolet. However, given a $B-V$ one could look up the appropriate $U-B$. Note that this is not straightforward the other way around, where there may be two values of $B-V$ possible for a given $U-B$.
In practice things are much more complicated. The plot above assumes the stars are on the main sequence. Stellar atmospheres change a bit as stars evolve and get bigger, with lower surface gravity. Thus changes the relationship a little. The relationship also changes if the composition is not similar to the Sun, especially around the bump at $B-V \simeq 0.4$. Finally, this relationship is intrinsic, it assumes you are not viewing the star through extinction caused by dust. If that is not the case then the whole relationship slides redward in both colours (the vector direction of which is shown in the plot above for about 2 magnitudes of visible extinction) but also changes shape slightly, depending on how much extinction there is.
You can convert from B-V to U-B by assuming spectral energy distribution (SED).
For example, if you assume blackbody SED (BBSED). The BBSED is determined by only the temperature (+ scaling constant which is irrelevant here). Therefore, there is a mapping from B-V to the certain temperature. Once you get the temperature, you can find U-B by applying the temperature to the BBSED (again, there is a mapping).
Becareful in the situation where the object's SED is not BB (e.g., galaxy). Or, the object's SED in the range from U to V has issues such as strong line emission or absorption (especially 4000-A break). In these cases, you cannot assume BBSED.
People might provide you more informative idea how to deal with the problem if you specify what kind of object you are dealing with.
In photometry, we determine the color index (or colour index if you're from my part of the world) as the difference in magnitudes between two wavelength filters. The reason that we use the difference, is because the magnitude system is logarithmic, so using the difference tells us the ratio of the intensities of the two wavelength bands.
To determine the B-V index, you'd need the magnitudes in the B band (which is centered on $\lambda = 445$nm) and in the V band, centered on $\lambda = 551$nm.
To find the U-B index, you'd also need the U-magnitude, at $\lambda = 365$nm.
So, as @zephyr said, there's no way to convert between the two color indices.
