The Newtonian limit of a boson star is analyzed as a function of the number of its component particles (N). For moderate N, the sphere is in the collisionless regime and its radius ( R ) satisfies the law R a N -' m -3, m being the elementary boson mass. For increasing N, this dependence stops when the interparticle repulsion takes over in the equation of state and the cluster turns into an n -1 polytrope: Thus R becomes constant, independent of N. The physical scales involved and some phenomenological implications are briefly discussed.Although the actually observed cold astrophysical bodies are fermion-composed entities, relativistic boson clusters, usually referred to as boson stars,' have recently aroused considerable interest. This attention is due basically to theoretical grounds because a good number of integer-spin elementary particles are predicted by modern grand unification t h e~r i e s .~ The existence of these elementary bosons, their clustering, and their hypothetical role in cosmology and astrophysics, for instance, with regard to the dark-matter problem,3 is obscure. Other, even more exotic, boson entities have also been i n t r o d~c e d .~The structure of boson stars derives from the Lagrangian 4 is a complex field, and the inclusion of the third term, in general, permits the description of interactions between the particles. Unless stated explicitly, natural units, h = c g l , are used throughout the paper. The equilibrium configurations of these objects were analyzed by Colpi, Shapiro, and Wasserman, ' who continued the original work of Ruffini and ~o n a z z o l a ,~ while their stability was studied by Gleiser and Watkins.In Ref. 5, the Newtonian limit of a boson star is studied when h -0, i.e., in the collisionless regime. This situation, which will be denoted by F, is also discussed in Ref. 7. In Ref. 8, without departing from any fundamental Lagrangian, we studied the equilibrium structure of a Newtonian cluster such that its equation of state is that of a dilute Bose gas. There, bosons repel each other, and this is effectively parametrized through a scattering length a. The self-gravitating body emerging from this equation of state is an n -1 polytrope; this case will be denoted by the subscript I.Our intention in this paper is twofold: First, we want to show how, for fixed microscopic parameters m and a, a Newtonian boson cluster evolves from the F regime and to the I one; and second, to establish the connection of I with the Newtonian limit derived from the Lagrangian of Eq. (11, when h#O. Thus, this discussion completes a general overview of the nonrelativistic limit of a boson star. Final-ly, definite values of m and a will be inserted-specifically those presumed for the axion-to obtain a better idea of the physical scales in which these objects should reside.In Newtonian gravity a structure in equilibrium is described by and by This second equation is the condition of statistical equilibr i~m ,~ with p standing for the chemical potential. 4 is the Newtonian potential and n represen...
