198 The Evolution of Gaseous Masses [CH. vm 



Consider a configuration in which equation (525), the condition for homo- 

 logous contraction, is satisfied everywhere except in the neighbourhood of 

 some point P. Let there be an excess of heat to the right of P and a deficiency 

 of equal amount to the left of P. Then the temperature gradient from right 

 to left at P will be in excess of that determined by equation (525), so that 

 the flow of heat from right to left will be greater than that in the permanent 

 homologous series. This flow may be regarded as made up of two parts : first 

 a flow of amount given by equation (525), and, second, a flow in the neigh- 

 bourhood of the point P, this latter flow being necessarily from right to left. 

 The first flow results in a homologous contraction of the whole mass; the 

 second flow reduces the excess of heat to the right of P and reduces also the 

 deficiency to the left. Thus the final state of the mass is nearer to the per- 

 manent series than was the original state. 



By an obvious extension of this argument it can be seen, although not by 

 strict mathematical proof, that any configuration not on the permanent homo- 

 logous series always moves towards that series as contraction proceeds. Thus 

 a mass of gas which has been contracting for a sufficient length of time may be 

 assumed to be on the permanent homologous series. 



Stellar Radiation 



197. We have considered the mechanism by which heat is brought to the 

 surface of a star, but have not yet considered the mechanism by which it is 

 radiated away. 



To an approximation which will prove to be good enough for our present 

 purpose, the radiation from a gaseous mass may be thought of as the free 

 radiation into space from a definite " photosphere," this being roughly identical 

 with the deepest layer of gas to which we can see from outside*. 



As a mass of gas contracts, the depth of the photosphere below the surface 

 will naturally diminish. On account of the increase of density produced by 

 the lateral contraction of the surface layers, the depth of the photosphere 

 must decrease more rapidly than the radius a. Thus when a mass of gas 

 moves through a series of homologous configurations, the various photo- 

 spheres will not form homologous points on this series. 



The position of the photosphere may be supposed to be determined by the 

 condition that the mass per unit area between it and the surface of the star 

 is always the same quantity p. Thus if g is the value of gravity at the sur- 

 face, the pressure at the photosphere will be pg, and the temperature will be 

 given by 



W .............................. (529), 



* For a more exact treatment, see Monthly Notices R.A.S. 78 (1918), p. 28. 



