An infinite family of axially symmetric thin discs of finite radius is presented. The family of discs is obtained by means of a method developed by Hunter and contains, as its first member, the Kalnajs disc. The surface densities of the discs present a maximum at the centre of the disc and then decrease smoothly to zero at the edge, in such a way that the mass distribution of the higher members of the family is more concentrated at the centre. The first member of the family has a circular velocity proportional to the radius, thus representing a uniformly rotating disc. On the other hand, the circular velocities of the other members of the family increase from a value of zero at the centre of the discs to a maximum and then decrease smoothly to a finite value at the edge of the discs, in such a way that, for the higher members of the family, the maximum value of the circular velocity is attained nearest the centre of the discs.

A family of analytical potential-density pairs for flat galaxies with spheroidal halos is presented. The potential are obtained by means of the sum of two independent terms: a potential associated with a thin disc and a potential associated with a spheroidal halo, which are expressed as appropriated superpositions of products of Legendre functions, in such a way that the model implies a linear relationship between the masses of the thin disc and the spheroidal halo. By taking a particular case for the halo potential, we found that the circular velocity obtained can be adjusted very accurately to the observed rotation curves of some specific galaxies, so that the models are stable against radial and vertical perturbations. Two particular models for the galaxies NGC4389 and UGC6969 are obtained by adjusting the circular velocity with data of the observed rotation curve of some galaxies of the Ursa Mayor Cluster, as reported in Verheijen and Sancisi (2001). The values of the halo mass and the disc mass for these two galaxies are computed obtaining a very narrow interval of values for these quantities. Furthermore, the values of obtained masses are in perfect agreement with the expected order of magnitude and with the relative order of magnitude between the halo mass and the disc mass. © 2016. Acad. Colomb. Cienc. Ex. Fis. Nat.Key words: Potential Theory; Disk Galaxies; Celestial Mechanics; Galactic Mass. Potenciales analíticos para galaxias planas con halos esferoidales ResumenSe presenta una familia de pares analíticos potencial-densidad para galaxias planas con halos esferoidales. Los potenciales son obtenidos por medio de la suma de dos términos independientes: un potencial asociado al disco delgado y un potencial asociado al halo esferoidal, los cuales son expresados apropiadamente como la superposición de productos de funciones de Legendre, de tal manera que el modelo implica una relación lineal entre las masas del disco delgado y el halo esferoidal. Tomando un caso particular para el potencial del halo, encontramos que la velocidad circular obtenida puede ser ajustada muy precisamente con la curva de rotación de algunas galaxias específicas, de tal manera que los modelos son estables contra perturbaciones radiales y verticales. Dos modelos particulares para las galaxias NGC4389 y UGC6969 son obtenidos ajustando la velocidad circular del modelo con datos de la curva de rotación observada de algunas galaxias del Cluster de la Osa Mayor, reportados en Verheijen and Sancisi (2001). Los valores de la masa del halo y la masa del disco para estas dos galaxias son calculados obteniendo un intervalo muy estrecho de valores para dichas cantidades. Además, los valores de masa aquí obtenidos están en perfecto acuerdo con el orden de magnitud esperado y con el orden de magnitud relativo entre la masa del halo y la masa del disco.

Regular and chaotic test particle motion in axially symmetric gravitational fields due to bodies with quadrupolar and octupolar deformation are studied using Poincaré surfaces of section. We find that inclusion of the octupolar term induces a distortion in the KAM curves corresponding to regular trajectories, as well as an increasing in chaoticity. The fact the switching on of the octupolar moment it increases chaoticity and leads to the apparition of spindle torus, can be seen even in the case corresponding to oblate deformation, which commonly presents regular motion. Thus, the results here obtained are the generalization, for the case of Newtonian gravity, of those previously obtained by Heiss, W. D., Nazmitdinov R. G. & Radu, S. (1994) and Li, J. (1998) for harmonic oscillators.Key words: Stellar dynamics; galaxies: kinematics and dynamics; chaotic dynamics.Movimiento caótico y regular alrededor de objetos con deformación cuadrupolar y octupolar. ResumenMovimiento regular y caótico de partículas de prueba en campos gravitacionales axialmente simétricos debido a cuerpos con deformación cuadrupolar y octupolar es estudiado usando superficies de sección de Poincaré. Encontramos que la inclusión del término octupolar induce una distorsión en las curvas KAM correspondiente a trayectorias regulares, así como un aumento en la caoticidad. El hecho de que la activación del momento octupolar incremente la caoticidad y lleve a la aparición de regiones inestables, puede ser visto aún en el caso correspondiente a deformación oblata, la cual comúnmente presenta movimiento regular. Así, los resultados aquí obtenidos son la generalización, para el caso de gravedad Newtoniana, de los obtenidos previamente por