781 



will cool down. Usually the loss of heal by radiation is given as 

 the principal canse of the cooling of a star; but the cooling from 

 adiabatic expansion is of much more importance. In a first approxi- 

 mation, therefore, radiation may be neglected. The force of gravity 

 is also left out of account, which to some extent diminishes the 

 force of expansion; this may be done the more legitimately as it is 

 to a greater or less extent compensated by the radiation-pressure. 

 We assume that all changes take place homocentrically. 



A volume-element at a distance )\ from the centre is found by 

 expansion after a time / at a distance r = r„ -\- h. We must then 

 have the relation 



d'A __ I dp 

 dt^ Q dr' 



A volume-element )\'Ub\dw shifts to the distance r and becomes 

 r' dr dw. By this the density changes according to: 



r' — 

 dr. 



As the change takes place adiabatically, p{f'^^ remaijis constant, 

 or 9 = Cons.t. X p'/?» therefore: 



d^^_pjh ^^_^^^ djy ^ _ 7 /p^\ dfh^ 

 dt' Q^ dr 2 V 9o / <^'' 



The index specifies the conditions at the time 0, which thus 

 remain a function of r. We shall indicate by the index 00 the 

 condition for ^ = and r ^ 0, at the centre therefore; putting 



Py'' Poo PO QOO rf 



= y, — = ((, = p; 



^PJ Qoo Poo Qc 



we find 



. d^L _ 7 p„ dy 

 ~dF ~~ ^Q^,dr' 



Then « is a constant for this gas ball, of the dimension UT ': 

 it is according to p = r^ 7/7' (// = gas-constant) proportional to the 

 temperature at the centre, and has the physical meaning of the 

 square of the speed of |)rüpagation of isolhermic disturbances of 

 equilibrium at the centre, /i is a number without dimensions, which 

 at the centre is = J , a function of r, which gives the course of a 

 from the centre outwards in the initial condition. The equation of 

 motion now becomes : 



