12 Conclusion

The Galactic bar induces a characteristic splitting of the disc phase space into regular and chaotic orbit regions, with the latter regions owing only to the non-axisymmetric part of the potential in the limit of no vertical motion. In this paper, we have isolated these two kind of regions, as well as the quasi-periodic orbit sub-regions inside the regular regions associated with the stable x₁(1) and x₁(2) periodic orbits respectively, within the same analytical 2D rotating barred potential with flat azimuthally averaged rotation curve as in D2000. We then have run test particle simulations in this potential and a more realistic self-consistent 3D N-body simulation to find out how the disc distribution function outside the bar region relates to such a phase space splitting and in particular how chaos may explain features in the Solar neighbourhood stellar kinematics like the Hercules stream.

Beside the bar strength, the regular versus chaotic splitting of phase space, investigated via the largest Liapunov exponent, is mainly determined by the value of the Hamiltonian H (or Jacobi's integral) and by the bar related resonances. In two dimensions and at fixed space position, the constant-Hcontours in the galactocentric u-v velocity plane are circles centred on $(v,u)=(R_{\circ}\Omega_{\rm P},0)$ and of radius $\sqrt{2(H-\Phi_{\rm eff}^{\circ})}$, where $R_{\circ}$ is the galactocentric distance, $\Omega_{\rm P}$ the rotation frequency of the bar and $\Phi_{\rm eff}^{\circ}$ the local effective potential. The fraction of chaotic orbits increases with H and there is a sharp average transition from regular to chaotic behaviour in the u-v plane when crossing the contour corresponding to the effective potential at the saddle Lagrangian points, $H_{12}\equiv \Phi_{\rm eff}(L_{1/2})$, which generally intersects the v-axis at lower velocity than the circular orbit in the axisymmetric part of the potential. At H<H₁₂, the orbits are rather regular, while at H>H₁₂, which defines the category of hot orbits susceptible to cross the corotation radius, they are rather chaotic.

The resonances, on the other hand, generate an alternation of regular and chaotic orbit arcs in the velocity plane which, contrary to the low-v part of the H-contours, are opened towards lower angular momentum. At bar inclination angles $\varphi=0$ and $\varphi=90^{\circ}$, the maxima or minima of these stochasticity waves are in phase with the resonance curves derived from the axisymmetric limit and the arcs are symmetric in u, reflecting the four-fold symmetry of the potential. At intermediate angles, these extrema become offset with respect to the resonance curves and the u-symmetry breaks. In particular, at and $\varphi \sim 30^{\circ}$, a prominent regular region of eccentric quasi-x₁(1) orbits extends well within the hot orbit region at , while the u>0 counterpart of the OLR curve is surrounded by a wide chaotic region consistent with the location of the Hercules stream.

For a moderate bar strength (F=0.10), the low-eccentricity and non-resonant quasi-x₁(2) orbit regions exist only for and over an angle range around $\varphi=90^{\circ}$ increasing towards smaller R. There is no such region near the default position considered in D2000, i.e. $R/R_{\hbox{\tiny OLR}}\approx 1.1$ and $\varphi =25^{\circ }$, compromising the quasi-x₁(2) orbit interpretation given by Dehnen for the Hercules-like mode occuring in his simulations at the most realistic positions of the Sun relative to the bar.

The test particle simulations, started from axisymmetric initial conditions and progressively exposed to the full rotating barred potential, reveal a decoupled evolution of the disc distribution function within the regular and chaotic phase space regions. In the regular regions, the phase space density after phase mixing is roughly the same as the initial one, whereas in the chaotic regions, the particles quickly evolve towards a uniform population of the easily available phase space volume via chaotic mixing, resulting in a substantial density re-adjustment. Because the space region within corotation represents a large initial reservoir of hot chaotic orbit particles which are spread throughout the disc by this process, yielding a net outward migration of such particles, the chaotic regions in the u-v plane outside corotation become more heavily crowded than the regular regions at . In particular, the wide and predominantly u>0 chaotic region mentioned above for realistic space positions of the Sun appears as an overdensity in the u-v distribution, providing a coherent interpretation of the Hercules stream and explaining the u>0 property of this stream. According to this interpretation, the Hercules stream involves stars on essentially chaotic orbits which are forced to avoid the regular x₁(1)region at negative u.

The time averaged disc u-v velocity distributions inferred from the N-body simulation are remarkably similar to those of the test particle simulations, despite the action of the transient spiral arms which allows at least some of the particles to diffuse from the regular to the chaotic regions and vice versa. At low eccentricity, the orbits are less sensitive to the inner features of the potential and the azimuthal properties of the velocity distributions essentially align with the average local phase shift of the potential well relative to the bar major axis induced by the spiral arms.

The velocity distributions may be very time dependent if for instance the bar has formed recently, because of the phase mixing occuring in the disc during at least $\sim$10 bar rotations after the growth of the bar according to the test particle simulations. The u-v distributions in the N-body simulation at fixed space position relative to the bar also display a strong temporal behaviour (see the mpeg movies at http://www.mso.anu.edu.au/~fux/streams.html), as expected from the presence of the transient spiral waves. However, since the simulation has been run for only about 1.25 Gyr after the formation of the bar, phase mixing is still operating in the disc component, rendering difficult to disentangle from it the individual effects of such waves. The N-body simulation also gives some insight on the consequences of evolving bar parameters: the slowly decreasing pattern speed of the bar mainly introduces a delayed response of the disc distribution function to the outward moving resonances, so that the velocity distributions at a given time rather reflect a higher value of $R/R_{\hbox{\tiny OLR}}$ than the true instantaneous one when compared with the constant $\Omega_{\rm P}$ test particle simulations. For completeness, one should mention that other perturbations than the bar and the spiral arms may provide alternative explanations of the local stellar streams, like for example the interactions of the Milky Way with its satellite companions.

Finally, the process of chaotic mixing, combined with the possible stellar exchanges between the regular and chaotic phase space regions resulting from the diffusion of stars by transient spiral arms or molecular clouds, may provide an new and efficient way of heating galactic discs which remains to be explored.

Acknowledgements

I would like to thank K.C. Freeman for a careful reading of the manuscript and A. Kalnajs for many enriching interactions. I am also thankful to Walter Dehnen for having partly inspired the present investigation and for several enlightening discussions, and to the University of Geneva where the N-body simulation presented in Sect. 10 has been performed. This work was mainly supported by the Swiss National Science Foundation.