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Pitch Angle Restrictions in Late Type Spiral Galaxies Based on Chaotic and Ordered Orbital Behavior

We built models for low bulge mass spiral galaxies (late type as defined by the Hubble classification) using a 3-D self-gravitating model for spiral arms, and analyzed the orbital dynamics as a function of pitch angle, going from 10$°$ to 60$°$. Testing undirectly orbital self-consistency, we search for the main periodic orbits and studied the density response. For pitch angles up to approximately $\sim 20°$, the response supports closely the potential permitting readily the presence of long lasting spiral structures. The density response tends to "avoid" larger pitch angles in the potential, by keeping smaller pitch angles in the corresponding response. Spiral arms with pitch angles larger than $\sim 20°$, would not be long-lasting structures but rather transient. On the other hand, from an extensive orbital study in phase space, we also find that for late type galaxies with pitch angles larger than $\sim 50°$, chaos becomes pervasive destroying the ordered phase space surrounding the main stable periodic and quasi-periodic orbits and even destroying them. This result is in good agreement with observations of late type galaxies, where the maximum observed pitch angle is $\sim 50°$.