Relation between Lane-Emden solutions and radial solutions to the elliptic Heavenly equation on a disk

摘要

We provide a transformation between a type of solution to a Lane-Emden equation of second kind and a solution of the elliptic Heavenly equation on a disk. By doing so, we show that any solution of this Lane-Emden equation of second kind corresponds to an infinite family of solutions to the Heavenly equation. This Lane-Emden equation is naturally formulated as a boundary value problem, which makes it somewhat distinct from the initial value problem versions in the literature. We obtain simple analytical solutions of this Lane-Emden equation and associated boundary value problem, and then we use these analytical solutions to construct a family of solutions for the elliptic Heavenly equation. The obtained solutions are radial solutions to the Heavenly equation; that is, they exhibit radial symmetry. In effect, we obtain a relation between radially-symmetric self-dual gravitational instantons and the Lane-Emden approximation to the structure of a neutron star. In other words, the radially symmetric neutron star under the Lane-Emden model can be seen as a special type of gravitational instanton. (C) 2014 Elsevier B.V. All rights reserved.