stars in an encounter. The velocity of fall from infinity alone is \J2 times
that appropriate to a circular orbit, so that when two stars meet, the value of
v must be greater than that assumed above by a factor of at least \/2.
As a consequence the value of R necessary for break-up is less than 84 per
cent, of that given by equation (376-2). But this still does not differ very
greatly from the simple equilibrium value given by formula (375'1), at any
rate so long as the masses M and M' are comparable.
Equation (3751) would however suggest that the limiting distance for
tidal break-up could be made very large by taking M' very large compared
with M. This is true on an equilibrium theory of the tides but equation
(376*2) shews it is not so when dynamical factors are taken into account.
When M' is very large the two stars shoot past one another with such
a high relative velocity that the more intense tidal forces have very little
time in which to operate ; the shortness of the time of action neutralises the
intensity of the forces.
377 . When there is great central condensation of mass the problem
assumes a different form. Let us consider the extreme case of a body whose
model (§ 229), so that its gravitational potential is always 7 Mfr.
If this mass is under the influence of its own gravitation, and also of a
tidal field of force of potential V T , the total gravitation potential il is
given by
il = ^ + V T (377-1).
r >
Let the tidal force originate in a second star S' which may be treated as
a point of mass M' at a distance R. If r' denotes distance from this point,
the whole potential of S' is 7 M'/r', but only a part of this produces tidal
forces. Part goes in producing the acceleration 7 M'/R 2 of S, which may be
supposed to originate from a field of force of potential yxM'/R 2 . Subtracting
this, the effective tide generating potential is
and the total potential il is
Following an equilibrium theory of the tides, the boundary of the surface
of S must be one of the system of equipotentials XI = constant. The tidal
disruption of S will commence when, if ever, there is no closed equipotential
capable of containing the whole volume of S.