ASTRONOMY: SHAPLEY AND NICHOLSON 
417 
ON THE SPECTRAL LINES OF A PULSATING STAR 
By Harlow Shapley and Seth B. Nicholson 
Mount Wilson Observatory, Carnegie Institution or Washington 
Communicated by G. E. Hale, July 21, 1919 
The hypothesis that the periodic variations in hght and spectrum of 
a Cepheid variable are due to radial expansion and contraction of a single 
stellar body demands a behavior in the periodic oscillation of the spectral 
lines which is decidedly different from the behavior of the absorption 
lines of an ordinary spectroscopic binary star. In the latter case the 
line-shifts show, in accordance with the Doppler effect, the orbital 
motion of the star as a whole; in the former case the displacements of 
the spectral lines depend on the radial velocity of the part of the stellar 
disk from which the light emanates. The radial velocity due to a 
spherical pulsation diminishes from a maximum value at the center of 
the stellar disk to zero at the limb; the shifts, therefore, are not true 
bodily displacements of the normal spectral line, as in the case of orbital 
motion, but represent a broadening toward the violet while the star 
expands and toward the red while it contracts. Obviously a lack of 
symmetry as well as a widening is produced; but if the total displace- 
ment, due to radial movement of the absorbing layers, is of the same 
magnitude as the inherent width of the lines, it is clear that the asym- 
metrical broadening may not be readily noticed or measured. The 
object of this note is to examine in some detail the character of the 
spectral lines of a pulsating star. 
For a star of unit radius undergoing alternate expansion and con- 
traction symmetrically about the center, the radial velocity, in terms 
of its ma ximum value, for any point on the visible hemisphere is = 
"n/ 1 — r^, where r is the distance from the center of the apparent 
disk. Hence 
r = Vl - (1) 
The usual law of darkening, which has been found satisfactory in the 
case of the sun and eclipsing stars^ and has subsequently been derived 
by Jeans^ in his theory of stellar photospheres, may be written 
L I ^x^x Vl - r2 
Jo 
where J /Jo is the surface intensity at the distance r from the center 
in terms of the central intensity, and x is the so-called darkening 
