  We have derived exact solutions of the isothermal Lane--Emden equation with
and without rotation in a cylindrical geometry. The corresponding hydrostatic
equilibria are most relevant to the dynamics of the protosolar nebula before
and during the stages of planet and satellite formation. The nonrotating
solution for the mass density is analytic, nonsingular, monotonically
decreasing with radius, and it satisfies easily the usual physical boundary
conditions at the center. When differential rotation is added to the
Lane--Emden equation, an entire class of exact solutions for the mass density
appears. We have determined all of these solutions analytically as well. Within
this class, solutions that are power laws or combinations of power laws are not
capable of satisfying the associated boundary--value problem, but they are
nonetheless of profound importance because they constitute "baselines" to which
the actual solutions approach when the central boundary conditions are imposed.
Numerical integrations that enforce such physical boundary conditions show that
the actual radial equilibrium density profiles emerge from the center close to
the nonrotating solution, but once they cross below the corresponding
baselines, they cease to be monotonic. The actual solutions are forced to
oscillate permanently about the baseline solutions without ever settling onto
them because the central boundary conditions strictly prohibit such settling,
even in the asymptotic regime of large radii. Based on our results, we expect
that quasistatically--evolving protoplanetary disks should develop oscillatory
density profiles in their midplanes during their isothermal phase. The peaks in
these profiles correspond to local potential minima and their locations are
ideal sites for the formation of protoplanets ...
