All responses calculated in this paper only account for the direct response to a perturbing potential. In general, though, the response also has an indirect component that arises from the fact that neighboring regions in the disk interact with each other gravitationally. This self-gravity of the response, which we have ignored, triggers long-lived normal-mode oscillations of the slab that are not accounted for in our treatment. Several simulation-based studies have argued that including self-gravity is important for a realistic treatment of phase spirals (e.g., Darling & Widrow 2019a; Bennett & Bovy 2021). Using the Kalnajs matrix method (Kalnajs 1977; Binney & Tremaine 2008), we have made some initial attempts to include the self-gravity of the response in our perturbative analysis, along the lines of Weinberg (1991). Our preliminary analysis shows that the self-gravitating response is a linear superposition of two terms: (i) a continuum of modes given in Equation (12), dressed by self-gravity, that undergo phase mixing and give rise to the phase spiral; and (ii) a discrete set of modes called point modes or normal modes (see Mathur 1990; Weinberg 1991) that follow a dispersion relation. The continuum response can be amplified by self-gravity when the continuum frequencies, nΩ
z
 + kv

x
, are close to the point-mode frequencies, ν. Depending on the value of k, the normal modes can be either stable or unstable. Araki (1985) finds that in an isothermal slab the bending normal mode undergoes fire hose instability below a certain critical wavelength if σ

z
/σ ≲ 0.3, while the breathing normal mode becomes unstable above the Jeans scale. In the regime of stability, the normal modes are undamped oscillation modes in absence of lateral streaming (Mathur 1990) but are Landau damped otherwise (Weinberg 1991). For an isothermal slab with typical MW-like parameter values, the point modes are strongly damped since their damping timescale (inverse of the imaginary part of ν) is of order their oscillation period (inverse of the real part of ν), which turns out to be of order the vertical dynamical time, h

z
/σ

z
. Moreover, the normal-mode oscillations are coherent oscillations of the entire system, independent of the vertical actions of the stars, and are decoupled from the phase spiral in linear theory since the full response is a linear superposition of the two. Based on the above arguments, we conclude that self-gravity has little impact on the evolution of phase spirals in the isothermal slab, at least in the linear regime. We emphasize that Darling & Widrow (2019a), who found their phase spirals to be significantly affected by the inclusion of self-gravity, assumed a perturber-induced velocity impulse with magnitude comparable to the local velocity dispersion in the solar neighborhood; hence, their results are likely to have been impacted by nonlinear effects. Moreover, the self-gravitating response of an inhomogeneous disk embedded in a DM halo, as in the simulations of Darling & Widrow (2019a), can be substantially different from that of the isothermal slab. We intend to include a formal treatment of self-gravity along the lines of Weinberg (1991) in future work.
Citation: Mathur 1990
Citation function: Uses; Uses
Citation: Mathur 1990
Citation function: Uses