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.
