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

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