# Stellar Statistics

## Stellar Statistics

This depends upon a general law in psycho-physics, known as Fechner's law, which says that changes of the apparent impression of light are proportional not to the changes of the intensity but to these changes divided by the primitive intensity. A similar law is valid for all sensations. A conversation is inaudible in the vicinity of a waterfall.[Pg 9] An increase of a load in the hand from nine to ten hectograms makes no great difference in the feeling, whereas an increase from one to two hectograms is easily appreciable. A match lighted in the day-time makes no increase in the illumination, and so on.

A mathematical analysis shows that from the law of Fechner it follows that the impression increases in arithmetical progression (1, 2, 3, 4, ...) simultaneously with an increase of the intensity in geometrical progression (I, I2, I3, I4, ...). It is with the sight the same as with the hearing. It is well known that the numbers of vibrations of the notes of a harmonic scale follow each other in a geometrical progression though, for the ear, the intervals between the notes are apprehended as equal. The magnitudes play the same rôle in relation to the quantities of light as do the logarithms to the corresponding numbers. If a star is considered to have a brightness intermediate between two other stars it is not the difference but theratio of the quantities of light that is equal in each case.

The branch of astronomy (or physics) which deals with intensities of radiation is called photometry. In order to determine a certain scale for the magnitudes we must choose, in a certain manner, the zero-point of the scale and the scale-ratio.

Both may be chosen arbitrarily. The zero-point is now almost unanimously chosen by astronomers in accordance with that used by the Harvard Observatory. No rigorous definition of the Harvard zero-point, as far as I can see, has yet been given (compare however H. A. 50), but considering that the Pole-star (α Ursæ Minoris) is used at Harvard as a fundamental star of comparison for the brighter stars, and that, according to the observations at Harvard and those of Hertzsprung (A. N. 4518 ), the light of the Pole-star is very nearly invariable, we may say that the zero-point of the photometric scale is chosen in such a manner that for the Pole-star m = 2.12. If the magnitudes are given in another scale than the Harvard-scale (H. S.), it is necessary to apply the zero-point correction. This amounts, for the Potsdam catalogue, to -0m.16.

It is further necessary to determine the scale-ratio. Our magnitudes for the stars emanate from Ptolemy. It was found that the scale-ratio[Pg 10]—giving the ratio of the light-intensities of two consecutive classes of magnitudes—according to the older values of the magnitudes, was approximately equal to 2½. When exact photometry began (with instruments for measuring the magnitudes) in the middle of last century, the scale-ratio was therefore put equal to 2.5. Later it was found more convenient to choose it equal to 2.512, the logarithm of which number has the value 0.4. The magnitudes being themselves logarithms of a kind, it is evidently more convenient to use a simple value of the logarithm of the ratio of intensity than to use this ratio itself. This scale-ratio is often called the Pogson-scale (used by Pogson in his “Catalogue of 53 known variable stars”, Astr. Obs. of the Radcliffe Observatory, 1856), and is now exclusively used. 