DIMENSIONS

OF

THE FIXED STARS,

WITH ESPECIAL REFERENCE TO

BINARIES AND VARIABLES OF

THE ALGOL TYPE.

BY

EDWARD C. PICKERING.

REPRINTED FROM THE PROCEEDINGS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES, VOL. XVI.

CAMBRIDGE:

JOHN WILSON AND SON.

University Press.

DIMENSIONS OF THE FIXED STARS, WITH ESPECIAL REFERENCE TO BINARIES AND VARIABLES OF THE ALGOL TYPE.

BY EDWARD C. PICKERING.

Presented May 25, 1880.

SINCE direct measurements cannot at present be made of the disks of the fixed stars, any information with regard to their dimensions derived from the amount and character of their light will have a value. This is the course ordinarily taken in the case of a satellite or small planet, and appears to deserve a more extended trial beyond the limits of the solar system than it has yet received. The principal objection to this method is the uncertainty in the numerical value of the intrinsic brightness or other constants involved, which cannot at present be measured with accuracy. The exact formula will therefore be first given, and hypotheses then introduced regarding the values of these constants.

Let B, b the diameters of the Sun and of any given star, as seen from the Earth, expressed in seconds of arc.

Let l

=

the intrinsic brightness of the star, that of the sun being taken as unity; in other words, let denote the ratio borne by the quantity of light emitted by the star to that emitted by the Sun from

the same

superficial area.

VOL. XVI. (N. S. v111.)

1

Let S, s the light of the Sun and of the star expressed in stellar magnitudes by means of the scale of Pogson, in which a difference of one magnitude corresponds to the logarithmic ratio, 0.4. This ratio, expressed in numbers, is approximately 2.512.

Let p the parallax of the star in seconds of arc.

The observed light of the star will be to that of the Sun as 762 is to B2; the difference in their stellar magnitudes, or

Hence,

log blog B+ 0.2 S— 0.2 s — 0.5 log l.

The radius of the Sun equals 16' 2", and accordingly B = 1924". The value of S is more uncertain. Various determinations of the ratio of the light of the Sun to that of Sirius have been made by different observers. In 1698, Huyghens found the value 756,000,000 by reducing the light of the Sun by a minute hole.* Wollaston, in 1829, compared the image of the Sun and of a lamp reflected in a silvered bulb of glass, and deduced the ratio 20,000,000,000.† Steinheil, in 1836, using the Moon as an intermediate standard of comparison, gave the value 3,840,000,000. In 1861, Bond determined the relative light of the Sun and Moon by comparing their reflections in a glass globe with that of a Bengola light. Combining his measures with the comparisons of the Moon and Sirius by Herschel and Seidel, he deduced the value 5,970,500,000.§. In 1863, Clark found that, if the Sun was removed to 1,200,000 times its present distance and Sirius to 20 times its distance, they would appear equally bright, and equal to a sixth-magnitude star. Their ratio, consequently, equals 3,600,000,000. Reducing these measures to magnitudes, we obtain the values, Huyghens, 22.20; Wollaston, 25.75; Steinheil, 23.96; Bond, 24.44; and Clark, 23.89. The mean of all of these is 24.05, with an average deviation of 0.84. The last three agree well, and give 24.10, with an average deviation of 0.23. Probably 24.0 is not far from the truth, and may be assumed to represent this ratio as closely as it is at present known. If we adopt -1.5 for the magnitude of Sirius, from the measures of Herschel and Seidel, we obtain for the stellar magnitude of the Sun -25.5.

* Cosmotheoros, La Haye, 1698.

† Phil. Trans., cxix. 28.

Elemente der Helligkeits-Messungen, Munich, p. 24.

§ Mem. Amer. Acad., viii. N. s., p. 298.

Amer. Jour. Sci., xxxvi. 76.

Substituting in the formula for log b giyen above, B 1924" = 3.2845.100 -0.2 s 0.5 log l

and S=25.5, we obtain log b:

= 8.184 0.2 s

--

0.5 log l. This formula is exact, and would give

the true diameter of any star if I was known.

An approximate value of 7 might be determined by the following method. Suppose that an electric current be passed through a platinum-iridium wire heating it to incandescence, and that the brightness of a short portion of it be compared with an artificial star when the current is varied by a known amount. As the current increases, the color of the light changes, the amount of the blue light increasing more rapidly than that of the red. The ratio of the two may be determined by inserting a double-image prism in the collimator of a spectroscope and viewing the wire through it. The two images may be made to overlap by any desired amount by varying the distance of the doubleimage prism from the slit of the collimator. The blue rays may thus be combined with the red, yellow, or green, as desired. The relative brightness of the two images may be varied by a Nicol placed in the eyepiece and turned through a known angle. We may thus combine any portion of the spectrum with any other part in such a proportion as to produce a tint to which the eye is especially sensitive. From the readings of the Nicol when different currents are passed through the wire, we may determine the varying proportion of any two rays, as the red and blue, when the wire is emitting a given amount of light. Observing in the same way the spectra of the Sun and star, and applying to them the law deduced from the observations of the wire, we obtain an approximate value of the comparative light emitted by equal areas of the two bodies. This will not be exact, since the effect of absorption is not allowed for, a difference of temperature being assumed to be the only cause of the observed difference in color. Probably the error will not be large, except perhaps in the case of the red stars. Until these measurements are made, we can do no better than to assume that = 1, or that the emissive power is the same for the Sun and star. As a large portion of the stars have nearly the same color as the Sun, and a similar spectrum, this assumption will probably not be far from the truth. The term equivalent diameters may be conveniently applied to the quantities thus computed. They may be defined as the diameters the Sun would have if removed successively to such distances that it would equal in light stars of the given magnitudes. The expressions, equivalent densities and equivalent masses, will be used in the same manner to denote the densities or masses of bodies in their other properties resembling the Sun.