Emergence of the Gaia Phase Space Spirals from Bending Waves

We discuss the physical mechanism by which pure vertical bending waves in a stellar disc evolve to form phase space spirals similar to those discovered by Antoja et al. ( arXiv:1804.10196) in Gaia Data Release 2. These spirals were found by projecting Solar Neighbourhood stars onto the $z-v_z$ plane. Faint spirals appear in the number density of stars projected onto the $z-v_z$ plane, which can be explained by a simple model for phase wrapping. More prominent spirals are seen when bins across the $z-v_z$ plane are coloured by median $v_R$ or $v_\phi$. We use both toy model and fully self-consistent simulations to show that the spirals develop naturally from vertical bending oscillations of a stellar disc. The underlying physics follows from the observation that the vertical energy of a star (essentially, its "radius" in the $z-v_z$ plane) correlates with its angular momentum or, alternatively, guiding radius. Moreover, at fixed physical radius, the guiding radius determines the azimuthal velocity. Together, these properties imply the link between in-plane and vertical motion that lead directly to the Gaia spirals. We show that the cubic $R-z$ coupling term in the effective potential is crucial for understanding the morphology of the spirals. This suggests that phase space spirals might be a powerful probe of the Galactic potential. In addition, we argue that self-gravity is necessary to properly model the evolution of the bending waves and their attendant phase space spirals.