Your eye sees color, but a telescope measures it with filters. The standard Johnson filters pass bands of light centered near 440 nm (B, blue) and 550 nm (V, visual, yellow-green). Measure a star's magnitude through each, and their difference is its color index $B - V$. It builds directly on magnitude arithmetic.
Why it works
A difference of two magnitudes is a ratio of two fluxes: $$B - V = -2.5\log_{10}\frac{F_B}{F_V} + \text{const},$$ with the constant chosen so that Vega has $B - V \approx 0$. A hotter blackbody pushes its peak toward the blue, so it sends relatively more light through the B filter than through the V filter. A larger $F_B/F_V$ means a smaller (even negative) $B - V$. Distance dims both bands by the same factor, so it cancels in the ratio: color is a thermometer that works at any distance (as long as no dust is in the way).
Key idea
$B - V = m_B - m_V$. Smaller $B - V$ means a hotter, bluer star; larger $B - V$ means a cooler, redder one.
| Star | Type | $T_{\text{eff}}$ | $B - V$ |
|---|---|---|---|
| Spica | B1 | about 22,000 K | $-0.23$ |
| Vega | A0 | about 9,600 K | $0.00$ |
| Sun | G2 | 5,772 K | $+0.65$ |
| Arcturus | K1.5 | about 4,300 K | $+1.23$ |
| Betelgeuse | M1–M2 | about 3,600 K | $+1.85$ |
Worked examples
Worked example: Reading a color index
A star has $B = 6.20$ and $V = 5.40$. Find $B - V$ and compare the star with the Sun ($B - V = 0.65$).
- $B - V = 6.20 - 5.40 = 0.80$.
- It is larger than the Sun's 0.65.
Answer: $B - V = 0.80$: the star is a little cooler and redder than the Sun.
Worked example: From a flux ratio to a color
Relative to Vega, a star's ratio of blue to visual flux, $F_B/F_V$, is $0.40$ times Vega's. What is its $B - V$?
- $B - V = -2.5\log_{10}(0.40)$.
- $\log_{10}0.40 = -0.398$, so $B - V = +0.99$.
Answer: $B - V \approx 0.99$, an orange K-type star.
Worked example: Bright is not the same as hot
Star X has $V = 2.0$ and $B - V = 1.5$. Star Y has $V = 8.0$ and $B - V = -0.1$. Which is hotter?
- Brightness ($V$) depends on luminosity and distance, not only on temperature, so ignore it here.
- Star Y has the smaller color index.
Answer: Star Y is hotter (a blue star), even though star X looks 250 times brighter.
Common mistake
Thinking a larger $B - V$ means a bluer star. Remember that magnitudes grow as light gets fainter. A large $B - V$ means $m_B$ is large: the star is faint in blue light compared with yellow-green light, so it is red and cool. Quick check: Betelgeuse, a red supergiant, has $B - V = +1.85$; Spica, a blue-white star, has $-0.23$.
Dust also reddens starlight (see extinction and reddening), so astronomers check colors against spectral lines, which dust does not change.