Apparent Magnitude

Flux Ratio:

Let b1 and b2 be the observed brightnesses of two stars, and let m1 and m2 be the corresponding magnitudes. The statement that a five-magnitude difference gives a flux ratio of 100 corresponds to:

\[b_1/b_2=100^{(m_2-m_1)/5}\]

increasing the brightness decreases the magnitude.

\[b_1/b_2=10^{(m_2-m_1)/2.5}\]

\[m_1-m_2=2.5 \log(b_2/b_1)\]

Objects brighter than magnitude 1 can have magnitude 0 or even negative magnitudes.

Changing of Apparent Magnitude:

Changing the distance of a star changes its apparent magnitude, but it does not change any of its colors. Because colors are defined to be differences in magnitudes, each is changed by the distance modulus:

\[m_B=M_B+5 \log(d/10\ pc)\]

\[m_V=M_V+5 \log(d/10\ pc)\]

So:

\[m_B-m_V=M_B-M_V\]

Therefore, the distance modulus never appears in the colors.