142 



NATURE 



[March 31, 192 1 



Stellar Magnitudes and their Determination. 

 By H. Spencer Jones, Chief Assistant, The Royal Observatory, Greenwich. 

 I. — Apparent Magnitudes : (a) Visual. 



THE magnitude of a star, as determined by 

 direct astronomical observation, is a measure 

 of its apparent brightness on a scale which has 

 been precisely defined only within recent years. 

 Hipparchus was, so far as is known, the first to 

 assign magnitudes to the stars, and his results 

 have been preserved for us by Ptolemy in the 

 Almagest. The classification of Hipparchus was 

 a crude one, the stars being divided into six 

 classes, all the brightest stars being assigned to 

 the I St magnitude, and all those only just visible 

 to the nakfed eye to the 6th. Ptolemy extended 

 the classification by recognising the gradation in 

 brightness between the stars in a given class, this 

 gradation being indicated by the words /xei^wv 

 and cAao-o-wv, used to denote that a star was 

 brighter or fainter than the average star of its 

 class. Ptolemy's estimations were adopted 

 almost universally until the time of Sir William 

 Herschel, who developed a plan for representing 

 various degrees of difference in brightness between 

 stars by the use of arbitrary symbols, and made 

 observations of the magnitudes of nearly three 

 thousand stars. It was not until Argelander 

 carried out the great project of the " Bonn Durch- 

 musterung " (1852 onwards) that magnitudes were 

 first estimated to tenths, and even in this great 

 work the scale adopted, though made to corre- 

 spond fairly closely with the then existing scales, 

 was an arbitrary, and not a uniform, one. 



Sir John Herschel was the first to attempt to 

 formulate a numerical relationship between the 

 apparent brightnesses of stars of successive mag- 

 nitudes, and he concluded that the best repre- 

 sentation was afforded by a relationship accord- 

 ing to which a decrease in light in geometrical 

 progression corresponds to an increase in magni- 

 tude in arithmetical progression. He also esti- 

 mated that the actual ratio of the light of a star 

 of the I St magnitude to one of the 6th is at 

 least 100: I. Herschel's conclusion is in accord- 

 ance with a psycho-physical law, enunciated by 

 Fechner, that, as a stimulus increases in geo- 

 metrical progression, the sensation produced by it 

 increases in arithmetical progression, the law 

 being departed from, however, in the case of very 

 intense or very weak stimuli. According to this 

 law, if !„ denotes the apparent brightness of a 

 star of magnitude m, then I^ : Ito+aw= >^'^'", 

 where fe is a constant, which is called the "light 

 ratio." 



Using this relationship, the value of fe (or log fe) 

 corresponding to various early series of magnitude 

 determinations, after standardisation by various 

 photometric devices, can be found. These show 

 a somewhat wide variation around a mean of about 

 040 for log fe. Thus a few values are : — 



Herschel ... 0407 Argelander ... 0-431 



Struve ... 0383 Groombridge ... 0-388 



NO. 2683, VOL. 107] 



The values are not, in general, constant within 

 any given series. Thus for the " Bonn Durch- 

 musterung " of Argelander we have : — 



For magnitudes 3 to 5 ... 0-29 



M ,, s to 6 ... 0-30 



„ ,, 6 to 7 ... 039 



,, ,, 7 to 8 ... 039 



,, ,, 8 to 9 ... 0-44 



It was, therefore, suggested by Pogson that 

 the value 040 for log fe should be definitely adopted 

 as a basis for accurate photometric determinations 

 of magnitude. This value is in sufficiently close 

 agreement with the values derived from the older 

 series of determinations to ensure that the magni- 

 tudes derived on this basis will not deviate greatly 

 from the older estimates. Owing to the conveni- 

 ence of this figure, all modern photometry has 

 been based on this convention, which assigns a 

 value to fe of 2-512... The convenience of the 

 figure is due to the facility with which it enables 

 estimates of brightness to be transformed into 

 magnitude differences (Aw =2-5log I„/I^+^^). 

 In the case of two stars one of which is 100 times 

 as bright as the other, we then have Am = 5 mag- 

 nitudes, exactly in accordance with Sir John 

 Herschel's estimate. 



Having adopted this convention, it becomes 

 necessary, before a magnitude can be assigned 

 to any star, to fix the zero from which the magni- 

 tudes are to be estimated, it being agreed that 

 the scale shall be continued in both directions » 

 stars brighter than a star of the ist magnitude 

 being assigned zero or negative magnitudes. The 

 use of the term " negative magnitude " may be 

 misleading to those who are not astronomers, but 

 the conception is a useful one if the scale of mag- 

 nitude is to be considered — as theoretically it must 

 be considered — capable of infinite extension at 

 each end. It has the further advantage of not 

 causing a break with the old-established conven- 

 tion that the brighter the star the smaller (alge- 

 braically) is the quantity denoting its magnitude. 

 It is convenient so to choose the zero that the 

 modern precise photometric magnitudes shall 

 agree as closely as possible with the older values, 

 which we have seen also corresponded closely with 

 a value of 04 for the logarithm of the light ratio. 

 In actual practice the zero has been fixed some- 

 what indirectly ; in the extensive visual photo- 

 metric work carried out at the Harvard Observa- 

 tory all the stars were compared with the Pole- 

 star, for which a provisional magnitude was 

 assumed. Thus differences of magnitude only 

 were determined. All the magnitudes were finally 

 increased by a quantity so chosen that the mean 

 of the magnitudes deduced for 100 circumpolar 

 stars, between the 2nd and 6th magnitudes agreed 

 with the corresponding mean of the values as- 

 signed in the "Bonn Durchmusterung. " In the 



