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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 $B\times V$?

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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.

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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 $$ B-V = -2.5\log_{10} f_B + 2.5\log_{10}f_V + Z_B - Z_V = -2.5\log_{10} \frac{f_B}{f_V} + Z_{BV}$$

Now, let's think about what you are asking.

$$BV = (-2.5\log_{10} f_B + Z_B)(-2.5\log_{10} f_V + Z_V)$$ $$BV= -2.5Z_v\log_{10} f_B -2.5Z_B\log_{10} f_V +Z_{B}Z_V -2.5 \log_{10} f_B^{-2.5\log_{10}f_V}$$

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.

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The magnitude is logarithmic, so taking the difference between two values is effectively the same as taking the ratio between the two unlogged values.

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  • $\begingroup$ Which is pretty much what I said. My question remains: is there any physical meaning in the ratio or product of the magnitudes? $\endgroup$
    – Jim421616
    Commented Aug 16, 2018 at 23:34
  • $\begingroup$ There is no such physical significance. @Jim421616 $\endgroup$ Commented Aug 17, 2018 at 7:03

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