272 The Origin and Evolution of the Solar System [CH. xn 



probably large, but whatever reasonable value is assigned to n, the value 

 of 0'101/\/tt or A^/V 2 comes out very small. Thus there must have been very 

 extreme central condensation in the primitive mass. 



General dynamical theory has shewn that there are two, and only two, 

 distinct types of rotational break-up. The fissional break-up happens in a 

 mass in which great variations of density do not occur, while the equatorial 

 break-up happens in masses with considerable central condensation. We 

 have seen that, if our system broke up by rotation, there must have been 

 very extreme central condensation, so that we may be confident that the 

 break-up, if ever it occurred, must have been by equatorial ejection*. 



The values of rf/ZTryp for equatorial ejection range from about 0'31 to 

 O36075. With central condensation as extreme as that we are now con- 

 sidering o> 2 /27ry/5 must be very nearly equal to the latter value. Within an 

 error of about one per cent, we may suppose it to be 0*36. 



287. We have now determined three numerical data, 



M = 3'3 x 10 50 , M = 2 x 10 33 , tf/Ziryp = 0'36, 



all probably accurate to within about one per cent. Equation (590) now 

 determines the further value 



-=3'7xl0 8 cms. 



r 3 



This gives the following values for & 2 /r 2 , these still being exact to about 

 one per cent. : 



r = Radius of present sun & 2 /r 2 = 072, 



r = orbit of earth Ic 2 /r * = 0'005, 



r = Neptune & 2 /r 2 = 0"00090. 



Exact analysis has not so far sustained the objection of Babinet ( 14) to 

 Laplace's Theory. The smallness of the present angular momentum does 

 not shew that the system cannot have broken up rotationally ; it merely 

 shews that the value of & 2 /r 2 must have been very small if the system ever 

 did break up rotationally. This necessity for extreme central condensation 

 was, however, apparent to Laplace, and has been fully recognised by subse- 

 quent cosmogonistsf. It should, however, be added that See and also 

 Moulton and Chamberlin, starting apparently from the tacit assumption that 



* For an adiabatic mass of gas, the transition occurs when 7 = 2-2 (about), 7 here denoting 

 (momentarily) the ratio of the two specific heats. The value of fc 2 /r 2 for a spherical mass in 

 which 7 = 1-66 is 0'22, for one in which 7 = 2 is 0'26, and for one in which 7 =00 is 0'40. Thus it 

 appeais that for a mass in which 7 = 2-2 the value of /c 2 /r 2 will be about 0-29. The flattening 

 produced by rotation naturally increases fc a /r 2 , so that there is a wide margin of safety in sup- 

 posing that & 2 /ro 2 = 0-101n~* corresponds to equatorial break-up. 



t Cf. Poincare^ Lemons sur les Hypotheses Coanwgoniques, p. 18. 



