# Magnitude Arithmetic

Written by Glow

*Last Updated: January 18, 2011, 05:37 am (UTC)Originally created on January 18, 2011*

The brightness of astronomical objects are listed by magnitudes. In astronomy, each magnitude is 10

^{0.4}(about 2.512) times higher than the previous. This makes the combining of magnitudes a bit different from say that of earthquakes which follow and base 10 scale.

Magnitude arithmetic is rarely used in astronomy outside of binary stars and clusters. In most cases though, it has no real scientific application but there are rare exceptions.

Brightness can be defined as follows:

B = 10

^{-0.4M}

Where B is brightness in a linear scale and M is the brightness in the magnitude scale. The magnitude uses a base of 10

^{-0.4}rather than 10

^{0.4}since lower magnitudes are brighter than higher magnitudes. A magnitude of 0 equates to a linear brightness of 1 in this conversion.

Addition of magnitudes extends this concept of linear brightness:

10

^{-0.4M}= 10

^{-0.4Ma}+ 10

^{-0.4Mb}

This equation, in short, sets the linear brightness of magnitude M equal to the sum of the linear brightness of M

_{a}and M

_{b}. Finding the magnitude equivalent of the resulting linear brightness takes some rearranging of the equation:

M = log(10

^{-0.4Ma}+ 10

^{-0.4Mb})/ -0.4

For subtraction, change the addition sign to subtraction:

M = log(10

^{-0.4Ma}- 10

^{-0.4Mb})/ -0.4

Division is close to the same except it is not converted back into magnitude form since how many times one star is brighter is not a magnitude. Multiplication replaces 10

^{-0.4Ma}with a scalar value since the multiplication of two magnitudes is close to meaningless.