Continuous Stellar Radiation 1
Stellar Magnitude

by George Hrabovsky

Brightness of Starlight

All stars in the sky seem to differ in brightness.  Brightness is in fact the energy flux (energy per unit area per unit time) received from the star.  Brightness can be measured with modern instruments.  We speak of the brightness of stars in terms of magnitudes, with the lower the number indicating a brighter star.
	It turns out that each jump in magnitude represents a fixed flux ratio, and not a flux difference.  It is because of this that we use a logarithmic scale for magnitude.
	There was a need to make the scale continuous so that useful calculations could be made on stars that could be seen only through telescopes might be included.  It is here that we develop our first true definition,

A difference of five magnitudes represents a factor of 100 in energy flux.

This is very important in our development of the magnitude scale.  Let us assume that two stars have observed energy fluxes f1 and f2 with corresponding magnitudes m1 and m2, we can rewrite our definition as,

f1/f2 = 100^((m2 - m1)/5)		(1)

This gaurantees that every time we increase m2 - m1 by 5, f1/f2 decreases by a factor of 100. It is important to remember that an increase in energy flux decreases the magnitude!
	Equation (1) gives flux ratios as powers of 100, but it is often more convenient to use powers of ten.  We convert (1) to this by writing 10^2 instead of 100 and substituting this into (1),

f1/f2 = 10^((m2 - m1)/2.5)		(2)

Sometimes we will want to determine the magnitude difference of a given brightness ratio.  Simply take the base 10 logarithm of both sides of (2),

Log[10, f1/f2] = (m2 - m1)/2.5

Or,

m2 - m1 = 2.5 Log[10, f1/f2]		(3)

For visual observation, the human eye can detect 1-6 magnitudes, this corresponds to a flux ratio of,

f1/f2 = 10^((6 - 1)/2.5)

The largest of ground-based telescopes increse our range from 6 to 26 magnitudes, or a flux ratio of,

f1/f2 = 10^((26 - 6)/2.5)

So, we now can see that the magnitude difference corresponding to a factor of ten change in energy flux is,

m2 - m1 = 2.5 Log[10, 10]

The Sun has an apparent magnitude of -26.8, if we want to know how much brighter than any other star this is,

f1/f2 = 10^((1 + 26.8)/2.5)

thus the Sun is nearly a trillion times brighter than any other star in the sky!