Lectures on Stellar Statistics | Page 4

increases in arithmetical progression (1, 2,
3, 4, ...) simultaneously with an increase of the intensity in geometrical
progression (I, I^2, I^3, I^4, ...). 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 the ratio 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[3]), but considering that the Pole-star ([alpha] 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 [1911]), 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--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.
It follows from the definition of the scale-ratio that two stars for which
the light intensities are in the ratio 100:1 differ by exactly 5 magnitudes.
A star of the 6th magnitude is 100 times fainter than a star of the first
magnitude, a star of the 11th magnitude 10000 times, of the 16th
magnitude a million times, and a star of the 21st magnitude 100 million
times fainter than a star of the first magnitude. The star magnitudes are
now, with a certain reservation for systematic errors, determined with
an accuracy of 0m.1, and closer. Evidently, however, there will
correspond to an error of 0.1 in the magnitude a considerable
uncertainty in the light ratios, when these differ considerably from each
other.
Sun -26m.60 Full moon -11m.77 Venus - 4m.28 Jupiter - 2m.35 Mars -
1m.79 Mercury - 0m.90 Saturn + 0m.88 Uranus + 5m.86 Neptune +
7m.66
A consequence of the definition of m is that we also have to do with
negative magnitudes (as well as with negative logarithms). Thus, for
example, for Sirius m = -1.58. The magnitudes of the greater planets, as
well as those of the moon and the sun, are also negative, as will be seen
from the adjoining table, where the values are taken from "Die

Photometrie der Gestirne" by G. MÜLLER.
The apparent magnitude of the sun is given by ZÖLLNER (1864). The
other values are all found in Potsdam, and allude generally to the
maximum value of the apparent magnitude of the moon and the planets.
The brightest star is Sirius, which has the magnitude m = -1.58. The
magnitude of the faintest visible star evidently depends on the
penetrating power of the instrument used. The telescope of WILLIAM
HERSCHEL, used by him and his son in their star-gauges and other
stellar researches, allowed of the discerning of stars down to the 14th
magnitude. The large instruments of our time hardly reach much farther,
for visual observations. When, however, photographic plates are used,
it is easily possible to get impressions of