Majorana transformation of the Thomas–Fermi equation demystified

Abstract

The Majorana transformation makes it possible to reduce the Thomas–Fermi equation to a first-order differential equation. This reduction is possible due to the special scaling property of the Thomas–Fermi equation under homology transformations. Such reductions are well known in the context of stellar astrophysics, where the use of homology-invariant variables has long proved useful. We use homology-invariant variables in the context of the Thomas–Fermi equation to demystify the origin of the otherwise mysterious Majorana transformation.