334
The Great Nebulae
[ch. XIII
as to how far the observed figures of the regular nebulae can be explained as
the figures assumed by masses rotating under their own gravitation. The
theoretical material necessary for such an inquiry has already been assembled
in Chapters vm and ix.
In Chapter ix we examined the configurations assumed under rotation by
a mass of matter of a type which we described as the extended Roche’s model,
this consisting of a central nucleus of homogeneous incompressible matter
surrounded by a light atmosphere of negligible density. The possible con
figurations for a given mass rotating with a given uniform angular velocity are
of the general type of those shewn in fig. 54. Here the shaded part represents
the central mass, which assumes the shape of the Maclaurin spheroid appro
priate to the given rotation, while the outer curves are the external closed
equipotentials of this mass in rotation.
The rotating mass may have any one of
these external equipotentials as its boun
dary, and selects that particular one whose
volume is just adequate to contain its own
atmosphere. If even the outermost of the
closed equipotentials is inadequate to con
tain the whole atmosphere, the mass fills
the outermost lenticular shaped equipotential, and the remainder spills over
into the equatorial plane.
We notice at once that this series of equipotentials have very much the
shape of the “ elliptic ” nebulae which occupy the lower half of the Y-diagram.
A mass rotating with given angular velocity assumes these various forms
according to the extent of the atmosphere which surrounds it. We add a bit
more atmosphere to an E 3 figure and it becomes E 4 ; subtract a bit and it
becomes E 2. But the sequence of figures possible for a given velocity of
rotation is limited at both ends ; it is limited at one end by the bare Maclaurin
spheroid and at the other by the last closed (sharp-edged) equipotential of the
Maclaurin spheroid.
The limits in both directions can be extended by varying the angular
velocity of rotation ; an increase of speed changes an A 1 3 mass into E 4, while
a decrease of speed changes an E‘S mass into E 2. With zero rotation every
mass, no matter how great or how small its atmosphere, becomes E 0, so that
this fixes the limit in one direction. The limit in the other direction is that of
the sharp-edged equipotential surrounding the critical Maclaurin spheroid
which is just about to elongate itself into a Jacobian ellipsoid. This is not,
however, of shape E 7 but of shape E 5 3.
The same general sequence of configurations is exhibited by almost any
model in which the mass is well concentrated towards the centre, as for
instance the adiabatic model discussed in § 235. The limits here are again
E0 at one end, and at the other end the sharp-edged equipotential of the