397 
369, 37o] Laplace's Nebular Hypothesis 
been pretty much in its present state*. Its mass must have been about 
equal to the total present mass of the solar system, 2 x 10 33 grammes, and its 
angular momentum about equal to the present total angular momentum of 
the solar system. This can be calculated with fair accuracy, since it is con 
tributed mainly by the orbital momenta of Jupiter and Saturn, and is found 
to be 33 x 10 50 in C.G.s. units. Comparing equations (370T) and (370*2) we 
find that 
- s = 3-7 xlO 8 (370-3). 
r 0 
Putting r 0 equal to the sun’s present radius, 6"95 x 10 ]0 cms., we obtain 
Z.2 
—, = 0-072. 
r o 
The value of Jc' 2 /r<? for a homogeneous mass having the shape of a Roche’s 
critical lens-shaped figure is found to be 0"523. If a fraction x of such a mass 
is concentrated at its centre while the remaining fraction (1 — x) is uniformly 
spread through a critical lens-shaped figure, the value of k 2 /r 0 2 is given by 
k 2 
— = 0*523 (1 — x). 
'0 
This becomes equal to 0*072 when x = 0*863. Thus the present total 
angular momentum of the solar system would have 'sufficed to break up the 
sun if 86*3 per cent, of its mass had been concentrated at its centre while the 
remaining 137 per cent, had been uniformly distributed throughout its volume. 
This shews that if the sun ever broke up under rotation, there must have 
been great central condensation in the distribution of its mass. This necessity 
for great central condensation was apparent to Laplace, and has been 
recognised by most modern cosmogonistsf, although a few, starting from the 
supposition that the sun must have been of uniform density, have come, 
naturally enough, to the conclusion that it cannot have broken up by rotation. 
Since the sun’s density must decrease continuously as we pass from its 
centre outwards, the numerical result we have just obtained shews that, if 
ever the sun broke up rotationally, the density at its edge must have been less 
than 13 per cent, of its mean density. 
The theorem of Poincaré given in § 240 has, however, shewn that if a 
rotating mass sheds a ring of matter this will scatter into space under the 
disruptive effects of its own rotation, unless its mean density is more than 
0’36 times that of the main mass. Thus the matter ejected by the sun could 
only form planets if it immediately condensed to about three times its original 
density. 
* Jeffreys has always maintained this. When I first propounded the Tidal Theory I suggested, 
on grounds of probability, that the solar system had probably been formed when the sun had a 
very large diameter. Our knowledge of the ages of the earth and sun now makes this conjecture 
untenable. 
f Poincaré, Leçons sur les Hypotheses Cosmogoniques, p. 18.