Title: A Comprehensive Perturbative Formalism for Phase Mixing in Perturbed Disks. II. Phase Spirals in an Inhomogeneous Disk Galaxy with a Non-responsive Dark Matter Halo

Abstract: We develop a linear perturbative formalism to compute the response of an inhomogeneous stellar disk embedded in a non-responsive dark matter halo to perturbations like bars, spiral arms and satellite galaxy encounters. Without self-gravity to reinforce it, the response of a Fourier mode phase mixes away due to an intrinsic spread in the vertical ($\Omega_z$), radial ($\Omega_r$) and azimuthal ($\Omega_\phi$) frequencies, giving rise to local phase-space spirals. Collisional diffusion due to scattering of stars by structures like giant molecular clouds causes super-exponential damping of the phase-spiral amplitude. The $z-v_z$ phase-spiral is 1-armed (2-armed) for vertically anti-symmetric (symmetric) bending (breathing) modes. Only transient perturbations with timescales ($\tau_{\mathrm{P}}$) comparable to the vertical oscillation period ($\tau_z \sim 1/\Omega_z$) trigger $z-v_z$ phase-spirals. Each $(n,l,m)$ mode of the response to impulsive ($\tau_{\mathrm{P}}<\tau=1/(n\Omega_z+l\Omega_r+m\Omega_\phi)$) perturbations is power law ($\sim \tau_{\mathrm{P}}/\tau$) suppressed, but that to adiabatic ($\tau_{\mathrm{P}}>\tau$) perturbations is exponentially weak ($\sim \exp{\left[-\left(\tau_{\mathrm{P}}/\tau\right)^\alpha\right]}$) except resonant ($\tau\to \infty$) modes. Slower ($\tau_{\mathrm{P}}>\tau_z$) perturbations, e.g., distant encounters with satellite galaxies, induce stronger bending modes. If the Gaia phase-spiral was triggered by a satellite, Sagittarius is the leading contender as it dominates the Solar neighborhood response of the Milky Way disk to satellite encounters. However, survival against collisional damping necessitates that the impact occurred within $\sim 0.6-0.7$ Gyr ago. We discuss how the detailed galactic potential dictates the phase-spiral shape: phase mixing occurs slower and phase-spirals are less wound in the outer disk and in presence of an ambient halo.