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# 2.6 The Magnitude System

Astronomers generally use linear brightness units to talk about the light intensity of objects in the sky. However, they also continue to measure brightness using the 2000-year-old system of magnitudes. The magnitude system uses a non-linear scaling with brightness ratio, and it probably has its origin in the fact that the eye is a non-linear detector of light. People at the time of the great astronomer Hipparcos found they could categorize the visible stars in the night sky into roughly 5 or 6 "bins" or brightness ranges, which map to the different numbers for magnitude. If b1 is the apparent brightness of star 1 and b2 is the apparent brightness of star 2, the magnitude difference is given by:

A portrait of Hipparchus of Nicaea from "The School of Athens" by Raphael. Click here for original source URL.

m1 - m2 = 2.5 log (b2 / b1)

We can also define the relationship in terms of the brightness ratio:

b2/b1 = 10x, where x = 0.4 (m1-m2)

The?apparent magnitude of large asteroid 65 Cybele (apmag 11.6) and stars HD 217121 (apmag 8.7) and HD 216932 (apmag 9.1). The two bright stars would be near the limit of typical 50mm binoculars. Also visible towards the upper right are the galaxies PGC194570 and PGC1016451. Click here for original source URL.

In addition to being on a logarithmic scale, the magnitude system is an inverse scale — larger magnitudes correspond to fainter objects. The brightest stars in the sky are around magnitude zero (m = 0), and some are even negative (negative magnitude does not imply negative brightness). The faintest are around magnitude six (m = 6). With a small telescope from a dark site, you could see magnitude twelve (m = 12). The faintest objects visible to the Hubble Space Telescope are around magnitude thirty (m = 30). The list below gives the relationship between these two measures of brightness (HST is the Hubble Space Telescope). After the description of the objects being compared, the first number is the magnitude difference, m1-m2 and the second is the brightness ratio, b2/b1

• Two identical stars at the same distance: 0, 1

• Minimum difference easily discernable by eye: 1, 2.5

• Venus at its brightest to Mars at its brightest: 2, 6.3

• Venus (at its brightest) to brightest star (Sirius): 3, 15

• Limit for eye to limit for binoculars: 4, 40

• Brightest to faintest star visible by eye: 5, 100

• Full Moon to Mars at its brightest: 10, 104

• Brightest star to Pluto: 15, 106

• Limit for binoculars to limit for HST: 20, 108

• Sun to the brightest star: 25, 1010

• Brightest to faintest star visible with HST: 30, 1012

The magnitude system is clumsy and non-intuitive. One difference of one magnitude corresponds to a brightness ratio of a factor of slightly more than 2.5. It is perhaps convenient to remember that a difference of five magnitudes is a brightness ratio of exactly a factor of 100. Magnitudes represent the most persistent intrusions of non-metric units into the world of astronomy. You do not need to be burdened by history — for most applications, linear brightness units can be used.