Disk heating by more than one spiral density wave

We consider a differentially rotating, 2D stellar disk perturbed by two steady state spiral density waves moving at different patterns speeds. Our investigation is based on direct numerical integration of initially circular test-particle orbits. We examine a range of spiral strengths and spiral speeds and show that stars in this time dependent gravitational field can be heated (their random motions increased).This is particularly noticeable in the simultaneous propagation of a 2-armed spiral density wave near the corotation resonance (CR), and a weak 4-armed one near the inner and outer 4:1 Lindblad resonances. In simulations with 2 spiral waves moving at different pattern speeds we find: (1) the variance of the radial velocity, sigma_R^2, exceeds the sum of the variances measured from simulations with each individual pattern; (2) sigma_R^2 can grow with time throughout the entire simulation; (3) sigma_R^2 is increased over a wider range of radii compared to that seen with one spiral pattern; (4) particles diffuse radially in real space whereas they don't when only one spiral density wave is present. Near the CR with the stronger, 2-armed pattern, test particles are observed to migrate radially. These effects take place at or near resonances of both spirals so we interpret them as the result of stochastic motions. This provides a possible new mechanism for increasing the stellar velocity dispersion in galactic disks. If multiple spiral patterns are present in the Galaxy we predict that there should be large variations in the stellar velocity dispersion as a function of radius.Comment: 20 pages, 13 figures. Submitted to MNRA

Harmonic Twistor Formalism and Transgression on Hyperk\"ahler manifolds

In this paper we continue our study of the fourth order transgression on hyper\"ahler manifolds introduced in the previous paper. We give a local construction for the fourth-order transgression of the Chern character form of an arbitrary vector bundle supplied with a self-dual connection on a four dimensional hyperk\"ahler manifold. The construction is based on the harmonic twistor formalism. Remarkably, the resulted expression for the fourth order transgression is given in terms of the determinant of the $\bar{\partial}$-operator defined on fibers of the twistor fibration.Comment: 10pp., Late