::Interactive Astronomy::

Magnitude Scale

The magnitude scale is a logarithmic system that divides the stars’ brightness into classes, where the brightest stars have negative magnitudes, 6th magnitude stars were the dimmest, barely visible to the naked eye. How bright a star appears to us in the sky is its apparent magnitude. The stars of the 1st magnitude are brighter than their 2nd magnitude counterparts by 2.5 times, and so on. However, the magnitude scale is unreliable when comparing the brightness of stars as stars may vary widely in brightness. Hence, there was the birth of Absolute Magnitude.

Absolute magnitude is the appearance of a star at 10 parsecs (1 parsec is 3.26 light years) away. This is a good way of comparing the absolute brightness of a star to that of others. Our Sun appears to have an apparent magnitude of -26.7, but at 10 parsecs away, it is of an absolute magnitude of 4.8; while the star Deneb (Cygnus) appears to be of magnitude 1.3 to us on Earth, at 10 parsecs away it is of an absolute magnitude -8.7.

Calculating Stellar Magnitude using Logarithms

A star’s apparent magnitude is commonly expressed by astronomers as m and a star’s luminosity in terms of absolute magnitude as M.

If I have 2 stars, with apparent magnitudes m1 and m2, and apparent bright nesses b 1 and b 2 respectively, the ratio of their apparent bright nesses

will correspond to a difference in their apparent magnitudes (m 2-m 1)

Each step in magnitude corresponds to a factor of 2.512 in brightness. Therefore, a -1 magnitude star is 2.512 times brighter than a 0 magnitude star and (2.512) 2 times the brightness of a 1 magnitude star.

An equation is commonly used to relate the difference in brightness between two stars’ apparent magnitudes to the ratio of their bright nesses:

Worked Examples:

1. A certain star triples its light output periodically. By how much does its apparent magnitude change?

Since the brightness varies by a factor of 3, the ratio of its maximum brightness = 3

Substitute this into the formula

The change in apparent magnitude:

= 2.5 log 2 (Use a calculator)

= 0.75

2. (Difficult question!) Two stars orbiting each other have apparent magnitudes of 1 and 3 respectively. Assuming that neither eclipses the other and there is no obstruction to block its light, find the combined magnitude of these two stars.

On the first look this question looks simple enough. One might be tempted to exclaim, “This is obvious! The combined magnitude is 1 + 3 = 4!” No, that is a big mistake many of us make.

We will draw a table, taking a star of magnitude 6 as a starting point with all the other brightness of stars building upon it:

Apparent Magnitude of star	Apparent brightness of star
1	(2.512) X 5 y (correspond to 5 steps up in magnitude) =100y
2	(2.512) X 4 y (correspond to 4 steps up in magnitude) =39.81y
3	(2.512) X 3 y (correspond to 3 steps up in magnitude) =15.85y
4	(2.512) X 2 y (correspond to 2 steps up in magnitude) =6.31y
5	2.512 X y (correspond to 1 step up in magnitude) =2.512y
6	Let brightness be y

For this question, we will modify the formula a little.

Let m x be the combined magnitude of the 1 st and 3 rd magnitude stars and b x is thecombined brightness of the 1st and 3rd magnitude stars.

m 1 is the magnitude of the 1 st magnitude star, b 1 is the brightness of the 1 st magnitude star

Therefore, the combined magnitude of a 1 st and 3 rd magnitude star is 0.84