An axially-symmetric Newtonian Boson Star

A new solution to the coupled gravitational and scalar field equations for a condensed boson field is found in Newtonian approximation. The solution is axially symmetric, but not spherically symmetric. For N particles the mass of the object is given by M = Nm − 0.02298NG N m, to be compared with M = Nm− 0.05426NG N m for the spherically symmetric case. Recent developments in particle physics and cosmology suggest that evolving scalar fields may have played an important role in the evolution of the early universe, for instance in primordial phase transitions, and that they may make up the missing dark matter. Models for galaxy formation using cold dark matter and the inflationary scenario suggest that the ratio of baryonic (luminous) matter to dark matter can be of the order of 10 %. These facts naturally raise the question whether cold gravitational equilibrium configurations of massive scalar fields Bose stars may exist and whether such configurations are dynamically stable. For bosonic fields interacting only via gravity spherically symmetric equilibrium solutions were studied by Kaup [1] as well as Ruffini and Bonazzola [2] by solving the coupled Einstein-Klein-Gordon equations. They analyzed only the zero-node solutions, corresponding to the lowest energy state. The results of ref.[1,2] have been confirmed and extended later on. For reviews we refer to ref.[3,4]. Recently the suggestion has been made, that the halo of galaxies is itself a condensed bosonic object[5,6]. This model was studied in the Newtonian approximation, where reasonable agreement with experimental rotation curves was found. All studies sofar have been restricted to non-rotating objects, i.e. spherically symmetric solutions. In this letter we make a first approach in studying axially symmetric solutions. For simplicity we restrict ourselves here to the Newtonian approximation. The Newtonian treatment of self-gravitating bosons of mass m interacting only gravitationally has been studied in ref.[2,7]. For condensed bosons at T=0 the equations reduce to two coupled equations for the gravitational potential and the Schrödinger field. The gravitational potential V satisfies the Poisson equation ∆V = 4πGNρ (1) where the mass density is given by