What is equation of straight line for a ideal blackbody on the color color diagram?

The color-color diagram is a plot between $U-B$ vs $B-V$ values.

For a ideal blackbody, what is the equation of this line?

In other words, what does the linear relation between $U-B$ and $B-V$ look like?

• I saw this in a book (carol ostlie) where a st line was given. Apr 10 '18 at 14:19

Use Planck's law: $$F(\lambda,T) = c \lambda^{-5} (e^{d/(\lambda T)} - 1)^{-1}$$.
Since magnitude is defined as, $M = -2.5 \log_{10}(F) + \textrm{constant}$, the color index $A-B$ is $$(A - B) = -2.5 \log_{10}(F(\lambda_A,T)/F(\lambda_B,T))$$.
To answer your question, use an appropriate effective wavelength of $U,B,V$. Link $U-B$ and $B-V$ via temperature, you should get $U-B = f(B-V)$.
• Thanks for the answer, this site has MathJaX enabled, so you can format formulas using $\TeX$ Apr 8 '18 at 22:41
• It depends on temperature evolution because $U-B = f(T)$ and $B-V = g(T)$. Nov 5 '18 at 11:21