## 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
$m_2 \gt m_1 \rightarrow b_1 \gt b_2$
.