Kinematics of stellar associations: the epicycle approximation and the convergent point method

Employing analytical and numerical tools, we expound the Galactic epicycle theory to describe the kinematic evolution of stellar, gravitationally unbound associations in the vicinity of the Sun. We estimate the limits of applicability of the analytical epicycle approximation and harmonic vertical motion by numerical integration of test orbits with an axisymmetric potential. We consider mainly early times when the associations are not too stretched out by differential Galactic rotation. If the member stars are ejected from a compact parental molecular cloud simultaneously, but with different velocities of ejection, the association remains as an entity for a long time. However, at a given time, individual velocities may vary widely among the members. The systematic differences of individual velocities does not allow one to apply the classical convergent point method even to young associations only several million years old. We demonstrate, however, that certain coordinate- and age-dependent corrections can be applied to the proper motions as observed in order to ‘rectify’ them and retrieve a common convergent point. This method can be used for determining the Oort's constants A and B, and for estimating the vertical frequency in the local part of the Galaxy and the expansion ages of nearby associations. In combination with the global convergence mapping technique, it can be used to search for yet-undiscovered nearby associations using the Hipparcos and Tycho-2 catalogues.