When Hipparchus first invented his magnitude scale, he intended each grade of magnitude to be about twice the brightness of the following grade. In other words, a first magnitude star was twice as bright as a second magnitude star. A star with apparent magnitude +3 was 8 (2x2x2) times brighter than a star with apparent magnitude +6.

In 1856, an astronomer named Sir Norman Robert Pogson formalized the system by defining a typical first magnitude star as a star that is 100 times as bright as a typical sixth magnitude star. In other words, it would take 100 stars of magnitude +6 to provide as much light energy as we receive from a single star of magnitude +1. So in the modern system, a magnitude difference of 1 corresponds to a factor of 2.512 in brightness, because

2.512 x 2.512 x 2.512 x 2.512 x 2.512 = (2.512)^{5} = 100

A fourth magnitude star is 2.512 times as bright as a fifth magnitude star, and a second magnitude star is (2.512)^{4} = 39.82 times brighter than a sixth magnitude star.

The following table shows how the difference in apparent magnitude between two stars (m_{2} - m_{1}) corresponds to the ratio of their apparent brightnesses (b_{1}/b_{2})

Apparent magnitude difference (m_{2} - m_{1}) |
Ratio of apparent brightness (b_{1}/b_{2}) |

1 |
2.512 |

2 |
(2.512)^{2} = 6.31 |

3 |
(2.512)^{3} = 15.85 |

4 |
(2.512)^{4} = 39.82 |

5 |
(2.512)^{5} = 100 |

10 |
(2.512)^{10} = 10^{4} |

20 |
(2.512)^{20} = 10^{8} |

This relationship can also be shown by the equation:

(m_{2} - m_{1}) = 2.5log_{10}(b_{1}/b_{2})

### Some examples to try:

1. Put these galaxies in order of magnitude from brightest to faintest:

NGC 4085: m = 12.94

M101: m = 8.30

M87: m = 9.60

IC1410: m = 15.94

NGC 5248: m = 10.97

2. How much brighter is a magnitude +2 star than a magnitude +4 star?

3. A variable star periodically triples its light output. By how much does the apparent magnitude change?

#### Answers:

1. M101, M87, NGC 5248, NGC 4085, IC1410

2. 6.31 times brighter

3. (m_{2} - m_{1}) = 2.5log_{10}(3) ; (m_{2} - m_{1}) = -1.19, so the star's brightness varies by 1.19 magnitudes