Eccentricity samples: implications on the potential and the velocity distribution

Planar and vertical epicycle frequencies and local angular velocity are related to the derivatives up to the second order of the local potential and can be used to test the shape of the potential from stellar disc samples. These samples show a more complex velocity distribution than halo stars and should provide a more realistic test. We assume an axisymmetric potential allowing a mixture of independent ellipsoidal velocity distributions, of separable or Staeckel form in cylindrical or spherical coordinates. We prove that values of local constants are not consistent with a potential separable in addition in cylindrical coordinates and with a spherically symmetric potential. The simplest potential that fits the local constants is used to show that the harmonical and non-harmonical terms of the potential are equally important. The same analysis is used to estimate the local constants. Two families of nested subsamples selected for decreasing planar and vertical eccentricities are used to borne out the relation between the mean squared planar and vertical eccentricities and the velocity dispersions of the subsamples. According to the first-order epicycle model, the radial and vertical velocity components provide accurate information on the planar and vertical epicycle frequencies. However, it is impossible to account for the asymmetric drift which introduces a systematic bias in estimation of the third constant. Under a more general model, when the asymmetric drift is taken into account, the rotation velocity dispersions together with their asymmetric drift provide the correct fit for the local angular velocity. The consistency of the results shows that this new method based on the distribution of eccentricities is worth using for kinematic stellar samples.