Galactic Spiral Arms: Structure and Dynamics Given by an Equation of Motion

Using an equation of motion for a self-gravitating filament, we show how galactic spiral arms might be created and sustained. We find that the combination of differential rotation of the galactic disk and the self-gravity of the arm (as given by the equation) leads to a rotating spiral structure. Moreover, using this analysis, we then find a second differential equation that explicitly relates this spiral structure to the rotation curve of the galaxy -- it connects several factors, including spiral shape and pattern speed. We also describe a simple many-body numerical experiment that supports our approach. The findings are with consistent with observational evidence concerning arm structure and rotation curves, including leading arm structures.