MOTION OF THE GAS SPHERE 273 



du 2pu 



dr r 



Integrating this equation, we have 



(8.2) u{r, = ^ 



Avhere the constant of integration Ui{t) is the velocity f or r = 1 and may 

 depend on time. The radial velocity in noncompressive flow thus falls 

 off as the inverse square of the distance from the origin, as is, of course, 

 evident from elementary principles. With this result the second of 

 Eqs. (8.1), becomes 



r^ dt 2 dr dr 



Integrating from the surface of the gas sphere, for which r = a, 

 Ua = da/dt = Ui/o?j P = Pa, to infinite distance where P = Po and 

 w = 0, gives 



M(-f)-l'-(l)" -"•"-'■•'=» 



Integrating with respect to time leads to the result 

 (8.4) I p^a^ (^\ + ^ Poa' ~ J ^" ''''^'' " ^' 



where C is a constant of integration. Except for a factor iir, the inte- 

 gral over a is easily seen to represent the work done by the pressure Pa 

 in expanding the sphere to its radius a{t), as the element of volume is 

 dV = ^iraHa, and the integral must therefore equal the decrease in 

 internal energy of the gas to E{a) from its initial value. Absorbing this 

 initial value into a new constant of integration Y gives after rearrange- 

 ment 



Written in this form, it is easily seen that the first time integral of the 

 equation of motion is merely the expression of conservation of energy, 

 as the first term is readily shown to be the kinetic energy of radial flow 



