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We present a numerical study of the properties of the stellar velocity distribution in stellar discs which have developed a saturated, two‐armed spiral structure. We follow the growth of the spiral structure deeply into the non‐linear regime by solving the Boltzmann moment equations up to second order. By adopting the thin‐disc approximation, we restrict our study of the stellar velocity distribution to the plane of the stellar disc. We find that the outer (convex) edges of stellar spiral arms are characterized by peculiar properties of the stellar velocity ellipsoids, which make them distinct from most other galactic regions. In particular, the ratio  σ 1  : σ 2   of the smallest versus largest principal axes of the stellar velocity ellipsoid can become abnormally small (as compared to the rest of the disc) near the outer edges of spiral arms. Moreover, the epicycle approximation fails to reproduce the ratio  σ φφ  : σ  rr    of the tangential versus radial velocity dispersions in these regions. These peculiar properties of the stellar velocity distribution are caused by large‐scale non‐circular motions of stars, which in turn are triggered by the non‐axisymmetric gravitational field of stellar spiral arms.  The magnitude of the vertex deviation appears to correlate globally with the amplitude of the spiral stellar density perturbations. However, locally, there is no simple correlation between the vertex deviation and the density perturbations. In particular, the local vertex deviation does not correlate with the local gravitational potential and shows only a weak correlation with the local gravitational potential gradient. We find that the local vertex deviation correlates best with the spatial gradients of mean stellar velocities, in particular with the radial gradient of the mean radial velocity.

Shape and orientation of stellar velocity ellipsoids in spiral galaxies