## Apparent Magnitudes

(Y |
0 |
- |
f\ | |

dA |
-— |
dco | ||

VL |
w |

Fig. 4.5. In time di, the radiation fills a volume d V = c di dA, where d A is the surface element perpendicular to the propagation direction of the radiation

Fig. 4.5. In time di, the radiation fills a volume d V = c di dA, where d A is the surface element perpendicular to the propagation direction of the radiation

As early as the second century B.C., Hipparchos divided the visible stars into six classes according to their apparent brightness. The first class contained the brightest stars and the sixth the faintest ones still visible to the naked eye.

The response of the human eye to the brightness of light is not linear. If the flux densities of three stars are in the proportion 1:10:100, the brightness difference of the

first and second star seems to be equal to the difference of the second and third star. Equal brightness ratios correspond to equal apparent brightness differences: the human perception of brightness is logarithmic.

The rather vague classification of Hipparchos was replaced in 1856 by Norman R. Pogson. The new, more accurate classification followed the old one as closely as possible, resulting in another of those illogical definitions typical of astronomy. Since a star of the first class is about one hundred times brighter than a star of the sixth class, Pogson defined the ratio of the brightnesses of classes n and n +1 as ^100 = 2.512.

The brightness class or magnitude can be defined accurately in terms of the observed flux density F ([ F] = Wm—2). We decide that the magnitude 0 corresponds to some preselected flux density F0. All other magnitudes are then defined by the equation F

Note that the coefficient is exactly 2.5, not 2.512! Magnitudes are dimensionless quantities, but to remind us that a certain value is a magnitude, we can write it, for example, as 5 mag or 5m.

It is easy to see that (4.8) is equivalent to Pog-son's definition. If the magnitudes of two stars are m and m +1 and their flux densities Fm and Fm+1, respectively, we have m — (m + 1) = —2.5 lg ^ + 2.5 lg ^ F0 F0

whence

In the same way we can show that the magnitudes m1 and m 2 of two stars and the corresponding flux densities F1 and F2 are related by mi — m2 = —2.5lg F F2

Magnitudes extend both ways from the original six values. The magnitude of the brightest star, Sirius, is in factnegative — 1.5. The magnitude of the Sun is — 26.8 and that of a full moon — 12. 5. The magnitude of the faintest objects observed depends on the size of the tele-

scope, the sensitivity of the detector and the exposure time. The limit keeps being pushed towards fainter objects; currently the magnitudes of the faintest observed objects are over 30.

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