Galactic echoes

Abstract

Gaia has revealed a variety of substructures in the phase space of stars in the Solar neighborhood, including the vertical `Snail' in $(z,v_z)$ space. Such substructures are often interpreted as the incompletely phase-mixed response of the disc stars to a single perturbation, such as an impulsive encounter with a satellite galaxy. In this paper we consider the possibility that such structures contain manifestations of phase space echoes. First established in plasma physics in the 1960s, echoes arise when a collisionless system is perturbed twice: the macroscopic responses to both perturbations mix to small scales in phase space, whereupon they couple nonlinearly, producing a third macroscopic `echo' response without the need for a third perturbation. We derive the galactic analogue of the plasma echo theory using angle-action variables and apply it to a one-dimensional model of vertical motion in the Milky Way. We verify the predicted echo behavior using idealized test particle simulations, both with and without the inclusion of diffusion through orbital scattering off molecular clouds. While we conclude that the Gaia Snail itself is unlikely a (pure) echo effect, the basic physics we uncover is sufficiently generic that we expect phase-space echoes to be common in disc galaxies.

Figures (7)

• Figure 1: Phase-space evolution of an isothermal slab perturbed by two successive impulsive perturbations at $t_1=0.2 \,{\rm Gyr}$ (blue) and $t_2=0.8 \,{\rm Gyr}$ (green). Nonlinear coupling between the two perturbations results in a third phase spiral---the echo---which unwinds until $t=1.4 \,{\rm Gyr}$ (red) and then rewinds again.
• Figure 2: Time evolution of the mean height of the disc $\langle z \rangle$, its thickness $[\langle z^2 \rangle-\langle z \rangle^2]^{1/2}$, and the rms vertical coordinate $\langle z^2 \rangle$, all in the absence of diffusion. The mean height undergoes damped oscillation after the first uniform kick at $t_1$ due to phase mixing. The second antisymmetric kick at $t_2$ does not directly affect the mean height, although it distorts the pre-existing one-armed phase spiral, generating a renewed oscillation (an echo) near $t_{\rm echo}$. The echo signals in the other panels are weaker (note the shrunken vertical axis).
• Figure 3: Winding time (top) and amplitude (bottom) of the phase spirals as a function of time. Blue and green mark those of the phase spirals directly induced by the first and second kick, respectively, while red represents the echo. Results both with (solid) and without diffusion (dotted) are plotted. The shaded regions around each line represent the 1-$\sigma$ uncertainty of the fit, which are invisibly small in the collisionless case (dotted). Since the amplitudes decay super-exponentially in the presence of diffusion (black dashed curves), the echo becomes much larger than the original phase spirals at late times.
• Figure 4: Amplitude of the echo, $A_{\rm echo}$, in the absence of small-scale diffusion, as a function of the time interval between the two large-scale impulsive kicks, $t_2-t_1$, and the strength of the second kick, $\Delta V_2$. The white curves are the contours of $A_{\rm echo}$ obtained via bivariate spline interpolation, and the red dashed curves mark the boundary within which $A_{\rm echo}$ exceeds $A_1$. The echo amplifies with both parameters but eventually saturates and begins to decline.
• Figure 5: As in Fig. \ref{['fig:t2t1_dV2_Aecho_NoSSDiff']}, but with small-scale diffusion by, e.g., molecular cloud scattering.