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We have derived exact solutions of the isothermal Lane--Emden equation withand without rotation in a cylindrical geometry. The corresponding hydrostaticequilibria are most relevant to the dynamics of the protosolar nebula beforeand during the stages of planet and satellite formation. The nonrotatingsolution for the mass density is analytic, nonsingular, monotonicallydecreasing with radius, and it satisfies easily the usual physical boundaryconditions at the center. When differential rotation is added to theLane--Emden equation, an entire class of exact solutions for the mass densityappears. We have determined all of these solutions analytically as well. Withinthis class, solutions that are power laws or combinations of power laws are notcapable of satisfying the associated boundary--value problem, but they arenonetheless of profound importance because they constitute "baselines" to whichthe actual solutions approach when the central boundary conditions are imposed.Numerical integrations that enforce such physical boundary conditions show thatthe actual radial equilibrium density profiles emerge from the center close tothe nonrotating solution, but once they cross below the correspondingbaselines, they cease to be monotonic. The actual solutions are forced tooscillate permanently about the baseline solutions without ever settling ontothem because the central boundary conditions strictly prohibit such settling,even in the asymptotic regime of large radii. Based on our results, we expectthat quasistatically--evolving protoplanetary disks should develop oscillatorydensity profiles in their midplanes during their isothermal phase. The peaks inthese profiles correspond to local potential minima and their locations areideal sites for the formation of protoplanets ...

Exact Solutions of the Isothermal Lane--Emden Equation with Rotation and Implications for the Formation of Planets and Satellites