400
The Solar System [ch. xvi
from those postulated by the planetesimal hypothesis. This investigation
further suggested that mere tidal action of itself would suffice to explain the
origin of the solar system, without calling in the various complicated and
wholly hypothetical mechanisms of intermittent eruptions, of smaller erup
tions to form satellites, of planetesimals, and so on, and led me to put forward
the simple tidal theory which follows.
The Dynamical Tidal Theory.
374 . Just as with rotation, the effects produced in a star by tidal action
prove to be very different according as the star is of fairly uniform density or
is arranged so as to have high central condensation of mass.
Tidal Effects in an Incompressible Mass.
375 . Let us examine first the effects which would be produced in a liquid
star of homogeneous incompressible matter.
Following Roche, we examined in §206 the effect produced on a small
mass S of approaching too near to a big mass S' around which S was supposed
to revolve in orbital motion. If S and S' were of the same density we found
that S could not approach to within 245 radii of S' without being broken
up. At this distance the difference in the gravitational pull of S' on the
nearer and further halves of S became so great that the attraction of the two
halves of S for one another failed to keep S together as a single body. The
mass S accordingly broke in two, and as the broken halves found themselves
again torn to pieces in precisely the same way, the process of disruption
continued indefinitely.
A similar situation arises when the two masses only approach one another
for a short time and then recede again, as happens in an ordinary gravitational
encounter between two stars whose orbits happen to pass fairly close to one
another.
When a second star S' approaches the star S whose behaviour we are
considering, its first effect is to raise tides of the ordinary kind on S. First
suppose that the distance R between S' and S changes very slowly, so that an
equilibrium theory is adequate to follow the changes in the tides. In this case
the equations of equilibrium are precisely of the type already discussed in
§206; they are in fact equations (206-6) with co 2 put equal to zero, /4 still
denoting 7 M'/R 3 .
A discussion of the general type already given in § 206* shews that the
configuration of S remains stable until it has assumed the shape of a prolate
spheroid of eccentricity e = 0-8826, the value of R at this stage being
R = 2198 (¿gy r * (375-1),
* Fora full account of the whole investigation see the paper already mentioned (p. 391), or
Problems of Cosmogony and Stellar Dynamics, pp. 43 ff, 118 ff.