MOTION OF THE GAS SPHERE 337 



tional to the translational momentum acquired by buoyancy during the 

 first expansion, and the numerical factor of proportionality was ob- 

 tained by empirically fitting values obtained by numerical calculations. 

 Shiffman and Friedman have done essentially the same thing, using a 

 momentum function s in reduced units. They have also added the 

 refinement of using a relation of the form A6* = Kis + ^25^ instead of 

 A6* = KiS, where s is the value at the minimum. In order to express 

 such a formula in usable form, it is necessary to obtain an expression for 

 s in terms of initial conditions, and Shiffman and Friedman's derivation 

 (102) is outlined here, as it is needed also for calculations of pressure in 

 section 9.3. 



The translational momentum s at the minimum is assumed to be 

 twice the value Sm at the first maximum, in good agreement with numeri- 

 cal calculations. In the absence of surfaces, the momentum s is given 

 by (a*^/3) (dh*/dt*), but in their presence, the more general equation 

 (8.53) must be used, which is 



dt^ dt^y ^^^^^'^ dt* ^"^ ^'dt* 



26*2 Ida \dt* ) da \dt* ) V dty da\ dt*) J 



+ \ «*' 



If, as in the period calculation, the vertical velocity is neglected during 

 the first expansion, we obtain for Sm 



Sm — 



o o 



1 r''"*a=^^/Vl - a*3 - A;*a*-=^/4 dfo 



•^ a* 



Vl+/, da 



da' 



a*9/2\/l + /, 



+ -J^-^^= ^ ''' da* 



the second step following on eliminating t* by Eq. (8.58). In addition 

 to the value of v 1 + /^ already used in the period calculation, the value 

 of dfo/ da is n eeded. This is adequately approximated by using 

 dfo/da/Vl -j-/o = 1 — a*/46o* in the first integral. A numerical 



