189, i9o] Spherical Mass of Gas 191 



or to (1 /?)/(! + ft) times the increase in H, the total heat-content of the 

 gas. The total heat generated by contraction falls into two parts the first 

 part is radiated away ; the second is stored up in the gas and goes to increase 

 the total heat-content H. We have seen that these two parts are in the ratio 



(l-/3)to(l+/3). Hence 



Of the total heat generated by contraction, a fraction |(1 ft) is radiated 

 away, while a fraction J (1 + ft) is stored up in the gas. 



To form an idea of the way in which the mass gets hotter we suppose that 

 the contraction is a uniform one, so that after an interval of time each length 

 in the mass is reduced by the same fraction 6. The potential energy W, 

 which was initially Xmw'/r, becomes changed to 2mm'/r0, so that the con- 

 traction increases W to I/O times its initial value. Since 2T + WO t both 

 before and after the contraction, it follows that T must also have increased 

 to I/O times its initial value. The total heat-content H or T(l + ft) must 

 have similarly changed. Thus 



If a mass of gas contracts 'uniformly, its density being so low that the ideal 

 gas laws may be assumed to hold, then its heat-content varies inversely as its 

 linear dimensions*. 



SPHERICAL MASS OF GAS 



190. To consider secular changes more in detail we shall suppose the 

 mass of gas to have assumed a spherical form, its boundary being a sphere of 

 radius a. 



Let T, p, p denote the temperature, pressure and density at a distance r 

 from the centre, and let the density be everywhere so small that these may 

 be supposed connected by the ordinary gas equation 



.............................. (507), 



ra 



where R is the universal gas-constant, and m is the mass of a molecule of the 

 gas. When the matter consists of a mixture of different types of molecules, 

 ions, atoms, electrons, etc. m may be supposed defined by equation (507). 



rr 

 Let M r stand for 4?r I pr z dr, the mass inside a sphere of radius r\ then 



Jo 



the condition for mechanical equilibrium is 



Instead of using r as a coordinate, we may more conveniently use q, 

 defined by r = aq, so that q increases from zero at the centre to unity at the 



* This result is obtained much in the form in which I have given it by Poincare (Leqons sur 

 les Hypotheses Cosmogoniques, p. 95 (footnote) and p. 227) ; it may be obtained more rapidly 

 from a consideration of physical dimensions, H necessarily varying as yM 2 jr. 



