Exact Axisymmetric Solutions of the 2-D Lane-Emden Equations with Rotation

TL;DR

This paper derives exact axisymmetric solutions to 2-D Lane-Emden equations with rotation, revealing intrinsic oscillatory behaviors and density profiles relevant to spiral galaxies and protoplanetary disks.

Contribution

It provides the first exact solutions to the 2-D Lane-Emden equations with rotation, describing intrinsic oscillatory behaviors and specific density profiles.

Findings

Solutions oscillate around intrinsic solutions.

Density profiles follow power-law and Bessel function forms.

Results applicable to spiral galaxies and protoplanetary disks.

Abstract

We have derived exact axisymmetric solutions of the two-dimensional Lane-Emden equations with rotation. These solutions are intrinsically favored by the differential equations regardless of any adopted boundary conditions and the physical solutions of the Cauchy problem are bound to oscillate about and remain close to these intrinsic solutions. The isothermal solutions are described by power-law density profiles in the radial direction, whereas the polytropic solutions are described by radial density profiles that are powers of the zeroth-order Bessel function of the first kind. Both families of solutions decay exponentially in the vertical direction and both result in increasing or nearly flat radial rotation curves. The results are applicable to gaseous spiral-galaxy disks that exhibit flat rotation curves and to the early stages of protoplanetary disk formation before the central star is formed.