190 The Evolution of Gaseous Masses [CH. vin 



and the value of E, the total energy, is 



E = H+ W=(I + /3)T+ W (505). 



From equation (504), a condition to be satisfied by a mass of gas in a 

 steady state is 2 T -f W = 0, or, in virtue of the equation just obtained, 



r (-!) = #....." (506). 



The special case of /3 = 1 or 7 = f demands attention. For a mass of gas 

 for which 7 = ^ in a steady state it appears that E = independently of the 

 radius of the mass. A small radial expansion of the mass can accordingly 

 take place, the mass passing from one configuration of equilibrium to an 

 adjacent configuration of equilibrium, without change of energy. Thus in 

 any configuration of equilibrium the frequency of one radial vibration is zero. 



It follows that on any linear series of configurations of equilibrium along 

 which 7 varies, there will be a change from stability to instability at the 

 value 7 = f, instability setting in through a radial vibration. Gases for 

 which 7 = oo are readily found to be stable, whence it appears that masses of 

 gas are radially stable when 7 > f and are radially unstable when 7 < f*. 



In illustration may be mentioned the period of radial vibration for a mass 

 of gas found by Ritter, subject to certain simplifying assumptions, to bef 



where p is the mean density in gravitational units. 



As a particular case of our result, it appears that a mass of gas for which 

 7 < f cannot rest in a state of stable equilibrium except when in a state of 

 infinite rarity. This result has been obtained only on the supposition that 

 the ideal gas laws are obeyed throughout. There will be other states of 

 equilibrium in which the density is so great that the ideal gas laws do not 

 hold. A mass of gas for which 7 < f and the total energy E is negative must 

 necessarily fall into one of these latter states of equilibrium. 



For a mass of gas for which 7> f, equation (506) requires that E shall 

 be negative ; in a steady state the energy is less than that in a state of 

 diffusion at infinity. As such a mass loses energy by radiation, there will be 

 a slow secular decrease of E, and therefore a secular increase of T. Thus the 

 mass will contract as it gets older and will get hotter at the same time, this 

 process of course continuing until the ideal gas laws are no longer obeyed. 



The energy lost by radiation during contraction is equal to the decrease 

 in E. This, from equation (506), is equal to (1 - ft) times the increase in T, 



* Eraden has obtained this result by a slightly different method (Gaskiigeln, 1907, 

 Chapter vm). 



f Emden, Gaskiigeln, p. 481. 



