196-198] Spherical Mass of Gas 199 



the quantities with suffix 1 referring to the photosphere. The total emission 



of radiation is now 



(m \ T 3 

 9) ^4 (530). 

 KJ p l 



On the permanent homologous series, E must remain unaltered by con- 

 traction, so that m l T l z lp l must also remain unaltered by contraction. In a 

 homologous contraction T 3 /p remains unaltered for homologous points, and m 

 may be supposed to remain unaltered. In a mixture of gases, there must be 

 a.certain amount of rearrangement when a mass contracts, and the increase of 

 temperature must alter the degree of ionisation when any is present ; these 

 complications prevent a strictly homologous contraction occurring at all, but 

 we may, as an approximation, neglect them and suppose that the mass can 

 contract homologously, so that in remains always the same function of q. 

 Assuming this, mT 3 /p will be unaltered by contraction, whence it follows that 

 i^! TV/pi will remain unaltered for the photosphere if, and only if, mT 3 /p has 

 initially a uniform value throughout the range within which the photosphere 

 moves. 



This range may be regarded as infinitesimal in comparison with the radius 

 of the star, so that the condition just found may be put in the form of a 

 boundary condition, namely that at the boundary of the star 



(531). 



r p 



This boundary condition together with the differential equation (525) 

 suffice to determine uniquely the series of permanent homologous configu- 

 rations. 



Mechanical Stability 



198. We have not so far discussed the question of mechanical stability 

 of the permanent homologous series. 



From the fundamental equations p = kp? and p = (R/m) Tp, we readily 

 find that 



We are only considering masses for which 7 > f , so that _p( 3 ~ 4 /v) will in- 

 crease with p as we pass inwards. It follows that on the permanent series 

 km? will decrease as we pass inwards. 



Strictly speaking, a permanent homologous series only exists when m 

 remains constant throughout contraction i.e. when no ionisation or chemical 

 change occurs. In such a case m will increase as we pass inwards, so that k 

 must decrease as we pass inwards. Thus dk/dr will be positive everywhere, 

 which is the condition for mechanical stability without convection. 



