Absolute Magnitude
The absolute magnitude of a star is that magnitude that it would appear to have as viewed from a standard distance, d0. This standard distance is chosen to be 10 pc. From this definition, if a star is actually at a distance of 10 pc, the absolute and apparent magnitudes will be the same.
For two identical stars, one at a distance d and the other at the standard distance d0. Where m is the apparent magnitude of the star at distance d, and M that of the star at distance d0. (Of course, M will be the absolute magnitude for both stars, because the two stars are identical.).
\[m_1-m_2=2.5 \log(b_2/b_1)\]
\[m-M=2.5 \log(b_2/b_1)\]
The energy flux falls off inversely as the square of the distance:
\[\frac {b_2}{b_1}=(\frac {d_1}{d_2})^2\]
in this case: \(d_1=d\) and \(d_2=d_0=10\ pc\) so:
\[m-M=2.5 \log(d/10\ pc)^2\]
\[m-M=5 \log(d/10\ pc)\]
\[m=M+5 \log(d/10\ pc)\]
This aquation helps in determaining the absolute magnitude or the distance, because m which is the apparent magnitude which is always measurable.
The quantity 5 log10(d/10 pc), which is equal to (m − M), is called the distance modulus of the star.
Determining the absolute magnitude, is really only determining it over some wavelength range, corresponding to the wavelength range of the observations, which differs it from the the bolometric magnitude.