  Recent studies have shown that the passage of a massive satellite through the
disk of a spiral galaxy can induce vertical wobbles in the disk and produce
features such as in-plane rings and phase-space spirals. Here we analyze a
high-resolution N-body simulation of a live stellar disk perturbed by the
recent passage of a massive dwarf galaxy that induces such perturbations. We
study the implications of the phase-space structures for the estimate of the
matter density through traditional Jeans modelling. The dwarf satellite excites
rapid time-variations in the potential, leading to a significant bias of the
local matter surface density determined through such a method. In particular,
while the Jeans modelling gives reasonable estimates in the most overdense
regions of the disk, we show that it tends to overestimate the dynamical
surface density in underdense regions. In these regions, the phase-space spiral
is indeed more marked in the surface of section of height vs. vertical
velocity. This prediction can be verified with future Gaia data releases. Our
finding is highly relevant for future attempts at determining the dynamical
surface density of the outer Milky Way disk as a function of radius. The outer
disk of the Milky Way is indeed heavily perturbed, and \textit{Gaia} DR2 data
have clearly shown that such phase-space perturbations are even present
locally. While our results show that traditional Jeans modelling should give
reliable results in overdense regions of the disk, the important biases in
underdense regions call for the development of non-equilibrium methods to
estimate the dynamical matter density locally and in the outer disk.
