  The vertical distribution of stars in the solar neighbourhood is not in
equilibrium but contains a wave signature in both density and velocity space
originating from a perturbation. With the discovery of the phase-space spiral
in Gaia data release 2, determining the origin of this perturbation has become
even more urgent. We develop and test a fast method for calculating the
perturbation from a passing satellite on the vertical component of a part of a
disc galaxy. This fast method allows us to test a large variety of possible
perturbations to the vertical disc very quickly. We apply our method to the
range of possible perturbations to the solar neighbourhood stemming from the
recent passage of the Sagittarius dwarf galaxy (Sgr), varying its mass, mass
profile, and present-day position within their observational uncertainties, and
its orbit within different realistic models for the Milky Way's gravitational
potential. We find that we are unable to reproduce the observed asymmetry in
the vertical number counts and its concomitant breathing mode in velocity space
for any plausible combination of Sgr and Milky-Way properties. In all cases,
either the amplitude or the perturbation wavelength of the number-count
asymmetry and of the oscillations in the mean vertical velocity produced by the
passage of Sgr is in large disagreement with the observations from Gaia DR2. We
conclude that Sgr cannot have caused the observed oscillations in the vertical
disc or the Gaia phase-space spiral.
