146 ANNUAL, REPORT SMITHSONIAN INSTITUTION, 1923 



star contracts, its density must increase; and the pressure will in- 

 crease too, for the various parts of the mass are nearer one an- 

 other, and attract one another more strongly. When the star has 

 shrunk to half its original diameter, the mean density will be eight 

 times as great. 



If the star, after contraction, continues to be " built on the same 

 model," so to speak — that is, if the law according to which the 

 density increases proportionally toward the center remains the same, 

 except for the altered scale of miles provided by the shortened 

 radius, the density at any point, after contraction, will also be eight 

 times the original density at the corresponding point (distant from 

 the center by the same fraction of the radius) . 



How will the pressures at the two points compare? The portion 

 of the star nearer the center than the point under consideration is 

 compressed by the weight of the overlying portions. After the 

 contraction, every part of these is twice as near the center as be- 

 fore, and will, therefore, be attracted four times more strongly. 

 The whole compressive force will, therefore, be four times as great 

 as at first; but the area over which this force is distributed will 

 have shrunk to one-fourth of its former amount. Hence the pres- 

 sure per unit of area will increase sixteenfold, as against an eight- 

 fold increase of density. Applying the familiar laws of gases, we 

 find that the temperature of the gas, after contraction, must be 

 twice its original value in order that equilibrium shall still exist 

 when the star has shrunk to half its former size. More generally, 

 during the whole process of contraction, the temperatures at cor- 

 responding points will be inversely proportional to the star's 

 radius — so long, indeed, as the star continues to be built on the same 

 model, and the simple gas laws hold good. This proportion was 

 first proved by Lane of Washington, in 1870, and is known as 

 Lane's law. 



It appears at first sight paradoxical that a star may grow hotter 

 by losing heat ; but the difficulty disappears when it is realized that 

 the heat produced by the contraction exceeds the amount which is 

 required to raise the temperature of the mass to the extent de- 

 manded by Lane's law. The remainder is available for radiation, 

 and it is only as it is gradually lost into space that the process of 

 contraction can take place. The manner in which the surface tem- 

 perature of a star, which determines its color and spectral type, 

 will vary as it contracts is somewhat different. As has already been 

 shown, the light from the far interior of a star stands no chance 

 of getting out to the surface, but practically all of it will be scattered 

 away by the gases through which it passes, and remain inside the 

 star. Light can only reach us directly from a relatively shallow 

 layer close to the surface, and it is a certain sort of average of the 



