## The Relationship Between Apparent Magnitudes and Apparent Brightnesses of Different Stars

The magnitude of a celestial object - star, planet, comet etc - is a measure of its brightness. Magnitudes are measured on a logarithmic scale that is used to determine levels of brightness between other stars. The ratio of apparent brightness between two stars is\[m_2-m_1=-2.5 log_{10} (\frac{b_2}{b_1})\]

.In this equation

\[m_1, \: m_2\]

are the apparent magnitudes of two stars and \[b_1, \: b_2\]

their apparent brightnesses (if the power output of a star at a distance \[d\]

is \[L\]

then \[b= \frac{L}{4 \pi d^2}\]

).Suppose then that the apparent brightnesses of two stars differs by a magnitude of 1, so that

\[m_2-m_1=1\]

, then\[1=-2.5 log_{10} (\frac{b_2}{b_1})\]

\[log_{10} (\frac{b_2}{b_1})= \frac{1}{-2.5}=- \frac{2}{5}\]

\[\frac{b_2}{b_1}=10^{- \frac{2}{5}} \simeq \frac{1}{2.512}\]

Hence

\[b_1=2.512b_2\]

.Note that

\[m_2 \gt m_1 \rightarrow b_1 \gt b_2\]

.