Spatial eigenvalue problems for stars in hydrostatic equilibrium: Generalized Lane–Emden equations as boundary value problems

Abstract

We derive a generic spatial eigenvalue problem governing stars in hydrostatic equilibrium. Our approach generalizes the various Lane-Emden equations finding use over the past century, allowing for more general equations of state (EoS) while ensuring a stellar structure with finite size (without the need for artificial truncation of the radius). We show that the resulting stellar structure is encoded into two quantities: the eigenvalue, which determines the total size or mass of the star, and the density distribution, which encodes the internal structure. While our formalism recovers known results for polytrope and white dwarf EoS, we also study additional EoS, such as those incorporating excluded volumes or those calibrated through viral expansions. We obtain numerical values for the stellar structure under a variety of frameworks, comparing and contrasting stellar structure under different EoS. Interestingly, we show how different EoS can be calibrated to give solutions with the same stellar structure, highlighting the arbitrariness of a particular EoS for replicating observations. This leads us to comment on general properties EoS should obey to describe physically realistic stars. We also consider hydrostatic gas clouds immersed in larger regions having non-zero ambient density. We compare three analytical methods for finding solutions of these eigenvalue problems, including Taylor series solutions, the variational approximation, and the non-perturbative delta-expansion method. Although each method has benefits and drawbacks, we show that the delta-expansion method provides the most accuracy in replicating stellar structure.