  We present a new analytic study of the equilibrium and stability properties
of close binary systems containing polytropic components. Our method is based
on the use of ellipsoidal trial functions in an energy variational principle.
We consider both synchronized and nonsynchronized systems, constructing the
compressible generalizations of the classical Darwin and Darwin-Riemann
configurations. Our method can be applied to a wide variety of binary models
where the stellar masses, radii, spins, entropies, and polytropic indices are
all allowed to vary over wide ranges and independently for each component. We
find that both secular and dynamical instabilities can develop before a Roche
limit or contact is reached along a sequence of models with decreasing binary
separation. High incompressibility always makes a given binary system more
susceptible to these instabilities, but the dependence on the mass ratio is
more complicated. As simple applications, we construct models of double
degenerate systems and of low-mass main-sequence-star binaries. We also discuss
the orbital evolution of close binary systems under the combined influence of
fluid viscosity and secular angular momentum losses from processes like
gravitational radiation. We show that the existence of global fluid
instabilities can have a profound effect on the terminal evolution of
coalescing binaries. The validity of our analytic solutions is examined by
means of detailed comparisons with the results of recent numerical fluid
calculations in three dimensions.
