The Magnitude scale
Relative brightness on a backwards (!) log scale. Dates to Hipparchus.
E.g., apparent relative luminosities of stars a & b are given by,
la
= 100(m b −m a )/ 5 = 10 0.4(m b −m a ) = 2.512 ×10 m b −m a ,
lb
Or, mb - ma = 2.5 log10(la/lb).
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A difference of 5 mags. = factor of 100 in luminosity.
mV(sun) ≈ -26.7.
Absolute Magnitude and Distance Modulus
To factor out the distance effect in apparent magnitudes (m) we define the
absolute magnitude M as - the magnitude a star would have if we put it at a
standard distance of 10 pc.
In terms of apparent luminosity l, and absolute luminosity L, we have,
2


L
d
=
 .
l 10 pc 
since luminosity falls off as the inverse square of the distance.
Then in magnitudes
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m - M = 2.5 log(L/l) = 5 log(d/10 pc) = 5 log(d) - 5.
or,
M = m + 5 - 5 log(d(pc)).
MV (sun) = 4.8.
The quantity m-M (generally > 0) is called the distance modulus.
E.g.,
m - M = 0 -->
d = 10 pc.
5
100 pc.
10
1000 pc.
Thus, if we can determine L or M independently, and measure m, we get the
distance. In practice we also have to worry about the effects of dust absorption.