Flux-Apparent Magnitude Relationship

Fechner's Law:  The intensity of a sensation increases as the logarithm of the stimulus.

    Because of Fechner's Law the magnitude difference of two stars (sensation) is proportional to the logarithm of the flux ratio (stimulus), i.e.,

m₁ - m₂ = k log (F₁/F₂).

Preserving Hipparchus' magnitude scale to the degree possible, since a magnitude difference of 5 corresponds to a flux ratio of ~100, with the lesser magnitude representing the greater flux,

5 = m₁ - m₂ = k log (F₁/F₂) = k log (1/100) = k ´ (-2), so

k = -2.5 and m₁ - m₂ = -2.5 log (F₁/F₂), or m = constant -2.5 log F.

We note that this implies that

F₁/F₂ = 10^-0.4(m1-m2) = 100^0.2(m2-m1) = 2.511886431^m2-m1,

^{that is,}

F₁/F₂ @ 2.512^m2-m1.

Also note that, since m = constant -2.5 log F, Δm = -2.5 log (e) ΔF/F , or

Δm = -1.086 ΔF/F.

For small Δm, an n% variation in F produces an » (0.01 ´ n) variation in m.

(Fechner's Law is also manifest in the logarithmic decibel scale used to express sound intensities.)