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

• A map between the elliptic Heavenly and Lane–Emden equations is provided.
• Simple analytical solutions are constructed through this transform.
• Quasi-stationary symmetric Heavenly solutions can be viewed as Lane–Emden solutions.

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.