# Comparing the magnitudes of different objects

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