Swing amplification and the Gaia phase spirals

We explore the interplay between in-plane and vertical dynamics in stellar discs within the framework of the shearing box approximation. Julian and Toomre used the shearing sheet to show that leading density waves are amplified as they swing into trailing ones. We extend their formalism into the dimension perpendicular to the disc to obtain explicit solutions for the disc's response to an impulsive, external excitation. An excitation that is symmetric about the mid-plane produces a density/breathing wave as well as two-armed phase spirals in the vertical phase space plane. On the other hand, an excitation that is antisymmetric about the mid-plane leads to a bending wave and single-armed phase spirals. In either case, self-gravity plays a crucial role in driving the evolution of the disturbance and determining the amplitude and pitch angle of the ensuing spirals. We also show that when the disc is excited by a co-rotating cloud, it develops stationary phase spirals in the wake of the cloud. The results call into question simple kinematic arguments that have been used to determine the age of the phase spirals seen in the Gaia survey.