239-241] Summary 265
If inequality (240*3) is not satisfied, d 2 A/dt 2 must be positive, so that the
mass must continually increase its rate of expansion, or, if it is contracting,
the contraction will be checked and ultimately replaced by an expansion.
According to the ideas of Laplace and Roche the ring of matter which
was thrown off from the sun, and ultimately formed the planets, was rotating
at one time as a closed ring with approximately the same angular velocity
as the main mass of the sun. If p s was the mean density of the sun, <w 2 would
be given by
a ) 2 = 0'36075 x 27r r yp 8
whence, from inequality (2403)
p > 0-36075 p s (240-4).
This shews that unless the ring condensed at once so as to have a density
of at least a third of the mean density of the main mass, it could not rotate
steadily but would continually expand under the centrifugal forces arising
from its own rotation.
Summary.
241. This and the preceding chapter have been occupied with an investi
gation into the configurations assumed by masses rotating freely in space under
their own gravitational forces. Before leaving the theoretical discussion, and
turning our attention to the actual problems of astronomy, it may be profitable
to summarise the main theoretical results which have been obtained. Some of
these results have been quite general, but we have also investigated in detail
the behaviour of certain simplified model masses. These models have been four
in number :
(A) The incompressible model, consisting of a mass of homogeneous
incompressible matter of uniform density.
(B) Roche’s model, consisting of a point nucleus of very great density,
surrounded by an atmosphere of negligible density.
( C ) The generalised Roche’s model, consisting of a homogeneous incom
pressible mass of finite size and of finite density, surrounded by an atmosphere
of negligible density.
(D) The adiabatic model, consisting of a mass of gas in adiabatic
equilibrium, so that the pressure and density are connected at every point by
the relation p = Kp K , where K and k retain the same values throughout the
mass.
The two models A and B are limiting cases of the more general models C
and D. If s denote the ratio of the volume of the atmosphere to that of the
nucleus in the generalised Roche’s model G, then model '0 degenerates into
model A when s = 0, and degenerates into model B when s — oo . Similarly
the adiabatic model D degenerates into model A when 7 = oo and into model
B when 7 = 1 ^ (cf. § 228). The relation between the four models is represented
diagrammatically in fig. 44.
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