Stellar Orbital Studies in Normal Spiral Galaxies I: Restrictions to the Pitch Angle

We built a family of non-axisymmetric potential models for normal non-barred or weakly-barred spiral galaxies as defined in the simplest classification of galaxies: the Hubble sequence. For this purpose a three-dimensional self-gravitating model for spiral arms PERLAS is superimposed to the galactic axisymmetric potentials. We analyze the stellar dynamics varying only the pitch angle of the spiral arms, from 4$°$ to 40$°$, for an Sa galaxy, from 8$°$ to 45$°$, for an Sb galaxy, and from 10$°$ to 60$°$, for an Sc galaxy. Self-consistency is indirectly tested through periodic orbital analysis, and through density response studies for each morphological type. Based on ordered behavior, periodic orbits studies show that for pitch angles up to approximately $15°$, $18°$, and $20°$ for Sa, Sb and Sc galaxies, respectively, the density response supports the spiral arms potential, a requisite for the existence of a long-lasting large-scale spiral structure. Beyond those limits, the density response tends to "avoid" the potential imposed by mantaining lower pitch angles in the density response; in that case the spiral arms may be explained as transient features rather than long-lasting large-scale structures. In a second limit, from a phase space orbital study based on chaotic behavior, we found that for pitch angles larger than $\sim30°$, $\sim40°$ and $\sim50°$ for Sa, Sb, and Sc galaxies, respectively, chaotic orbits dominate all phase space prograde region that surrounds the periodic orbits sculpting the spiral arms and even destroying them. This result seems to be in good agreement with observations of pitch angles in typical isolated normal spiral galaxies.