One-dimensional equilibrium structures of infinitely thin polytropic sheets

Details One-dimensional equilibrium structures of infinitely thin polytropic sheets One-dimensional equilibrium structures of infinitely thin polytropic sheets

Apr 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994a%26a...284.1007s&link_type=abstract

Astronomy and Astrophysics (ISSN 0004-6361), vol. 284, no. 3, p. 1007-1014

Statistics

Computation

1

Equilibrium Flow, Interstellar Matter, Kinematics, Perturbation, Polytropic Processes, Stability, Cartesian Coordinates, Chebyshev Approximation, Computation, Gravitational Fields, Numerical Analysis, One Dimensional Flow, Poisson Equation

Scientific paper

Equilibrium structures of infinitely thin sheets with a two-dimensional polytropic pressure are discussed. The structure of the sheets depends on one Cartesian coordinate. Elementary potential-surface density pairs are presented. Chebyschev polynomials are fundamental solutions of the Poission equation. The structure of the sheets is related to the structure of polytropic cylinders. All the characteristics of the solutions of Emden's equation of the polytropic cylinder occur in the behavior of the sheets. We present closed solutions for the case of a polytropic law with an index m = 1/2 and for the isothermal limit. As in the case of polytropic cylinders, for sheets with a negative polytropic index there are solutions with finite density at the origin which oscillate about self-similar solutions. We give the asymptotic form of these oscillations. Calculating the structure of a cylinder from that of a sheet by an approximation method, we get information on the range of application of this method. The sheets are limiting cases of rigidly rotating non-axisymmetric disks. The transition of a polytropic elliptic disk with an index m = 1/2 into the corresponding sheet is studied. The stability of the sheets against one-dimensional polytropic perturbations is discussed. The eigenfunctions of the m = 1/2-sheet are given analytically. The stability analysis of self-similar polytropic sheets is performed by use of marginal perturbations. The stability is compared with the stability of the corresponding cylinders against axisymmetric perturbations.

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