Structure, kinematics and time evolution of the Galactic Warp from Classical Cepheids

The warp is a well-known undulation of the Milky Way disc. Its structure has been widely studied, but only since Gaia DR2 has it been possible to reveal its kinematic signature beyond the solar neighbourhood. In this work we present an analysis of the warp traced by Classical Cepheids by means of a Fourier decomposition of their height ($Z$) and, for the first time, of their vertical velocity ($V_z$). We find a clear but complex signal that in both variables reveals an asymmetrical warp. In $Z$ we find the warp to be almost symmetric in amplitude at the disc's outskirts, with the two extremes never being diametrically opposed at any radius and the line of nodes presenting a twist in the direction of stellar rotation for $R>11$ kpc. For $V_z$, in addition to the usual $m=1$ mode, an $m=2$ mode is needed to represent the kinematic signal of the warp, reflecting its azimuthal asymmetry. The line of maximum vertical velocity is similarly twisted as the line of nodes and trails behind by $\approx 25^\circ$. We develop a new formalism to derive the pattern speed and change in amplitude with time $\dot{A}$ of each Fourier mode at each radius, via a joint analysis of the Fourier decomposition in $Z$ and $V_z$. By applying it to the Cepheids we find, for the $m=1$ mode, a constant pattern speed in the direction of stellar rotation of $9.2\pm3.1$ km/s/kpc, a negligible $\dot{A}$ up to $R\approx 14$ kpc and a slight increase at larger radii, in agreement with previous works.