Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results

TL;DR

This paper derives an analytical mass-radius relation for Newtonian self-gravitating Bose-Einstein condensates with short-range interactions, relevant for astrophysical objects like boson stars and dark matter halos, and analyzes their stability and collapse conditions.

Contribution

It provides an approximate analytical solution for the mass-radius relation of BECs with short-range interactions, connecting previous non-interacting and Thomas-Fermi limits.

Findings

Repulsive interactions allow arbitrary mass with a minimum radius.

Attractive interactions limit the maximum mass before collapse.

Stable configurations exist only above a negative scattering length.

Abstract

We provide an approximate analytical expression of the mass-radius relation of a Newtonian self-gravitating Bose-Einstein condensate (BEC) with short-range interactions described by the Gross-Pitaevskii-Poisson system. These equations model astrophysical objects such as boson stars and, presumably, dark matter galactic halos. Our study connects the non-interacting case studied by Ruffini & Bonazzola (1969) to the Thomas-Fermi limit studied by B\"ohmer & Harko (2007). For repulsive short-range interactions (positive scattering lengths), there exists configurations of arbitrary mass but their radius is always larger than a minimum value. For attractive short-range interactions (negative scattering lengths), equilibrium configurations only exist below a maximum mass. Above that mass, the system is expected to collapse and form a black hole. We also study the radius versus scattering length relation for a given mass. We find that stable configurations only exist above a (negative) minimum scattering length. Our approximate analytical solution, based on a Gaussian ansatz, provides a very good agreement with the exact solution obtained by numerically solving a nonlinear differential equation representing hydrostatic equilibrium. Our treatment is, however, easier to handle and permits to study the stability problem, and derive an analytical expression of the pulsation period, by developing an analogy with a simple mechanical problem.