# Ratios of magnitudes instead of differences

The color index in the photometric system is defined by the differences in magnitudes between two wavelength filters, which gives the ratio of intensities. For example, 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.

But what about other operations? For instance, would there be any significance in taking, say, the ratio $B/V$ or $B\times V$?

Consider for example these bright stars (V and B-V values from Yale Bright Star Catalog). Near V=0, B/V is ill-behaved, and B*V is near 0.

Name       B     V     B-V    B/V   B*V
Arcturus  1.19 -0.04  +1.23 -29.7  -0.05
Vega      0.03  0.03   0.00   1.00  0.00
Capella   0.88  0.08  +0.80  11.0   0.07
Rigel     0.09  0.12  -0.03   0.75  0.01


Now imagine that each of those stars were 10 times as far away. Neglecting interstellar extinction, this would decrease the apparent brightness by a factor of 100 and increase the apparent magnitude by 5.0. As B and V increase together, B/V approaches 1 and B*V approaches V^2, telling us nothing meaningful about a given star.

Name       B     V     B-V   B/V    B*V
Arcturus  6.19  4.96  +1.23  1.25  30.7
Vega      5.03  5.03   0.00  1.00  25.3
Capella   5.88  5.08  +0.80  1.16  29.9
Rigel     5.09  5.12  -0.03  0.99  26.1


B-V, on the other hand, has stood the test of time as a distance-independent metric of a star's color and surface temperature.

The definition of a magnitude is something like $$B = -2.5\log_{10} f_B + Z_B,$$ where $f_B$ is a physical flux in whatever unit system you are using and $Z_B$ is a zeropoint for the magnitude system and the minus sign is there to ensure small magnitudes are brighter.

A colour index is therefore something like $$B-V = -2.5\log_{10} f_B + 2.5\log_{10}f_V + Z_B - Z_V = -2.5\log_{10} \frac{f_B}{f_V} + Z_{BV}$$

$$BV = (-2.5\log_{10} f_B + Z_B)(-2.5\log_{10} f_V + Z_V)$$ $$BV= -2.5Z_v\log_{10} f_B -2.5Z_B\log_{10} f_V +Z_{B}Z_V -2.5 \log_{10} f_B^{-2.5\log_{10}f_V}$$
This numerology has no physical significance at all and, because the fluxes themselves are distance-dependent, then like $B$ and $V$ individually, $BV$ and $B/V$ would be distance-dependent and therefore have no relationship to anything physical that is intrinsic to the star.