Cardinal kinematics: I. Rotation fields of the APOGEE Survey
  Correlations between stellar chemistry and kinematics have long been used to
gain insight into the evolution of the Milky Way Galaxy. Orbital angular
momentum is a key physical parameter and it is often estimated from
three-dimensional space motions. We here demonstrate the lower uncertainties
that can be achieved in the estimation of one component of velocity through
selection of stars in key directions and use of line-of-sight velocity alone
(i.e. without incorporation of proper motion data). In this first paper we
apply our technique to stars observed in the direction of Galactic rotation in
the APOGEE survey. We first derive the distribution of azimuthal velocities,
$v_\phi$, then from these and observed radial coordinates, estimate the stellar
guiding centre radii, $R_g$, within $6.9\leq R \leq 10$ kpc with uncertainties
smaller than (or of the order of) 1kpc. We show that there is no simple way to
select a clean stellar sample based on low errors on proper motions and
distances to obtain high-quality 3D velocities and hence one should pay
particular attention when trying to identify kinematically peculiar stars based
on velocities derived using the proper motions. Using our azimuthal velocity
estimations, we investigate the joint distribution of elemental abundances and
rotational kinematics free from the blurring effects of epicyclic motions, and
we derive the $\partial v_\phi / \partial R_g$ trends for the thin and thick
discs as a function of radius. Our analysis provides further evidence for
radial migration within the thin disc and hints against radial migration
playing a significant role in the evolution of the thick disc.
