306-310] The Time-Scale 287 



and negative charges rushing together and annihilating one another. Taking 

 the intrinsic energy of an electron to be mC 2 , where C is the velocity of light, 

 the reduction of the sun's mass by only one per cent, would set free 1*8 x 10 5 ' 2 

 ergs of energy, or sufficient to furnish radiation at the present rate for 

 150,000 million years. 



309. Before either giving up the question as insoluble, or calling to our 

 aid stores of energy such as that just mentioned, it will be well to examine 

 how far the gravitational source of energy is really proved to be inadequate. 

 The situation is that the total energy of contraction of our sun provides for 

 radiation at the present rate i.e. as a star of absolute magnitude 5*0 for 

 only 18*3 million years. Let us consider the problem first in reference to the 

 universe as a whole, and afterwards with special reference to our sun. 



The energy set free by gravitational contraction varies as the square of 

 the mass of the contracting body. Taking the average star to be of mass 17 

 times that of our sun, the energy lost in contracting to the radius of our sun 

 i.e. to a density of 2'3 would be 2'9 times that generated by the contrac- 

 tion of our sun to its present size. Thus it would provide for radiation for 53 

 million years as a star of luminosity equal to our sun. 



Our sun, however, is somewhat exceptionally bright. To estimate how 

 far its brightness is above the average, we must limit ourselves to the nearest 

 stars, for distant faint stars escape observation altogether. Of the nineteen 

 stars of parallax greater than 0'2" tabulated by Eddington*, only nine have 

 luminosities greater than j 1 ^, that of the sun being taken as unity, while 

 nine have luminosities less than T ^, the nineteenth star being of estimated 

 luminosity equal to -fa. Moreover no fewer than seven have luminosities less 

 than T J-^ suggesting that we ought to add to the least luminous stars others 

 whose nearness has not been suspected on account of their faintness, such as 

 the near star recently discovered by Barnard. Taking the luminosity of the 

 average star to be ^, we find that contraction provides for radiation at this 

 rate for 530 million years, a period which agrees well enough with our other 

 estimates of the age of the universe. 



310. Thus as regards the universe as a whole, there is no difficult 

 problem associated with the time-scale : the problem only arises in con- 

 nection with special stars, and our sun happens to be one of these. Both 

 Russell's theory and the theoretical investigation of Chap. VIII suggest that 

 in the comparatively recent past our sun must have been radiating even 

 more energy than at present. On the other hand, when we pass to the 

 remote past, there is no justification for believing the rate of radiation to 

 have been so great. There must be a long period between the stage at 

 which a star is formed in the nebulous state and the period at which it 



* Stellar Movements, p. 41. 



