Equilibrium Halo Solutions of the Gross-Pitaevskii-Poisson System: The Role of the Particle Number
We investigate stationary halo-like solutions of the Gross-Pitaevskii-Poisson (GPP) system, which describes self-gravitating Bose-Einstein condensates with repulsive self-interactions, as a dark matter model. The boson mass $m_ϕ$, scattering length $a_s$, and total particle number $N$ are kept explicit, with $N$ treated as an independent macroscopic control parameter. Solving the stationary GPP equations over a broad parameter space, we identify ground-state, excited-state, and unbound solution branches according to their binding properties and nodal structure. The ground-state branch occupies a well-defined region of the $(m_ϕ,N)$ plane whose location depends strongly on the self-interaction strength, whereas the excited-state and unbound regions are largely insensitive to the initial ansatz. From the converged solutions, we derive empirical scaling relations connecting the characteristic halo radius $R_{99}$ to $m_ϕ$, $a_s$, and $N$. In the weakly interacting regime, the results reproduce the standard Schrodinger-Poisson mass-radius relation, while finite self-interactions reveal an intermediate regime in which gravity, quantum pressure, and repulsive interactions jointly determine the equilibrium structure. As an astrophysical application, we show that ground-state solutions can reproduce representative dwarf-galaxy rotation curves using only the solitonic component. We also examine the implications of current Lyman-$α$ forest constraints and find that, although increasing $a_s$ shifts equilibrium solutions toward larger boson masses compatible with existing bounds, the resulting configurations do not reproduce the observed dwarf-galaxy kinematics. These results provide a systematic characterization of stationary GPP halos and establish a direct connection between microscopic particle properties and observable galactic quantities.