Magnitude referred originally to the intensity of the visual sensation produced by a star. Ancient astronomers described the 20 brightest stars as of the first magnitude. Hipparchus in Greece extended the scale by dividing the stars into five magnitudes. Magnitudes were estimates without quantitative definition. The human eye in combination with the human brain perceives only relative differences. The eye-brain system responds not in a linear way. Steinhel and later Pogson (1856) assumed a logarithmic scale. Consider a star as bright as a candle at 10 kilometres. Let us define this brightness as 1 TKC (ten-km candle). The eye-brain system will sense the difference between a 1 TKC and a 2 TKC star as equal to the difference of a 10 TKC and a 20 TKC star.
One had to define the factor (or scale) that has to be multiplied with the ratio of two brightnesses to give a one magnitude step. This factor seemed to be about 2.5, and Pogson selected (arbitrarily!) the value 2.512. If F is the luminous flux coming from a star, the magnitude M is calculated by Pogson's classic expression:
M = -2.5 log F + C.
This is called the Pogson formula; the coefficient (-2.5) is called the Pogson scale; C is an arbitrary constant, generally called the zeropoint. A 1 TKC star (see previous paragraph) corresponds to a visual magnitude of 3.5. The difference in magnitude between two stars is expressed in function of the luminous fluxes F1 and F2 as follows:
M1 - M2 = -2.5 log (F1/F2).
Inversely, we can calculate a flux ratio from the known magnitude of two stars by:
F1/F2 = 10**((M2 - M1)/2.5) = 2.512**(M2-M1),
where ** means 'to the power'. The value 2.512 shows up again: this is not incidently! That is the value of the factor selected by Pogson. It will be clear now why precisely that value was selected: the reason is computational (not physical !).
The following table gives the relation between magnitude difference and flux ratio. From that table we see that a star being 100 times brighter than another, will have a difference in magnitude of 5.
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magnitude flux
difference ratio
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1 2.51
2 6.31
3 15.8
4 39.8
5 100.0
6 251.3
7 631.5
. .
. .
. .
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The text in this page is largely inspired by chapters in the two following splendid books:
Liller, W. (1992). The Cambridge guide to astronomical discovery. Cambridge University Press, Cambridge (USA), 257 p.
Sterken, Chr. and Manfroid, J. (1992). Astronomical photometry -- A guide. Kluwer Academic Publishers, Dordrecht, 272 p.
URL of this page is: [http://www.vub.ac.be/STER/www.astro/magnitud.htm]
Last modified: 1995 March 31. This page is ©1995.