Resumen
A new family of three-dimensional Newtonian models for galaxies is constructed. The models describe a thin disk and a matter halo, whose gravitational potentials satisfies the equation (2) presented in Gonzalez & Pimentel (2016, Phys. Rev. D, 93, 044034), and therefore, they satisfy the energy conditions for a gravitational system. The expressions for the potential of the disk and the halo are obtained by applying the "displace, cut, and reflect" method to the solution of the Laplace equation in cylindrical coordinates. Analytical expressions that describe the rotation curves and the mass distributions in the disk and in the halo are computed for the first three models of the family of solutions. It is shown that the mass densities of the disks and the haloes present a maximum at the center of the system and go to zero at infinity. Finally, for some values of the free parameters, the obtained rotation curves present a flat region for larger values of the radial coordinate. The model was obtained considering the total gravitational potential, which satisfies the equation (2) imposed by Gonzalez & Pimentel (2016). The potential generated by the spheroidal halo of matter is constructed considering a multipolar expansion, expressed in cylindrical coordinates. As this solution of the Laplace equation a substitution is done so that the Laplacian is nonzero, the z coordinate (Kuzmin, 1956, AZh, 33; Toomre, 1963, ApJ, 138, 385). So the new potential satisfies Poisson equation and represents the distribution of three-dimensional material. From the gravitational potential analytical expressions were derived for the surface density of the disk, halo density of matter and from the rotation was derived. We found that the surface densities of the disks present a maximum at the center, vanishing at infinity; and the halo density is maximum at the disk surface, also vanishing at infinity. For some values of the parameters, the derived rotation curves present a flat region for larger values of the radial coordinate.
