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

We have derived exact solutions of the isothermal Lane–Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are relevant to the dynamics of the solar 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, a new 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 the matching of the two types of solutions. This oscillatory behavior of the isothermal solutions to the Lane–Emden boundary–value problem is entirely generic and extends to polytropic models as well. Based on our results, we expect that quasistatically–evolving protoplanetary disks should develop oscillatory radial density profiles in their midplanes Math Methods, 54 Middlesex Tpke, Bedford, MA 01730. E-mail: [email protected] University of Massachusetts Lowell, Dept. of Mathematical Sciences, Olney Hall, Room 428, Lowell, MA 01854. E-mail: Dimitris [email protected] NASA/GSFC, Code 663, Greenbelt, MD 20771. E-mail: [email protected]