1 Introduction

The kinematics of disc stars in the Solar neighbourhood displays several long known properties, such as the increase of velocity dispersion with age, the tendency of young stars to appear in moving groups or streams, and the classical vertex deviation affecting stars with asymmetric drift down to $\sim$25 kms^-1 relative to the Sun and mainly owing to the Hyades and Sirius streams. Disc heating is traditionally attributed to the diffusion of stars by transient spiral arms or by massive compact objects like molecular clouds, the streams to dissolving ensembles of stars born at the same place, and the vertex deviation to local gravitational perturbations like spiral arms or local departures from a steady state.

Beside these properties, the local disc velocity distribution also betrays a broad stream of low angular momentum and mainly outward moving stars with a mean heliocentric asymmetric drift $s\approx 45$ kms^-1, i.e. typical of the thick disk (Gilmore et al. 1989), which hereafter will be referred to as the "Hercules'' stream, according to the comoving Eggen group $\zeta$ Herculis (Skuljan et al. 1999). The mean outward motion of stars with high asymmetric drift, also known as the "u-anomaly'' and seen up to over s=100 kms^-1 in metal rich samples (Raboud et al. 1998) and in Mira variables with period between 145 and 200 days (Feast & Whitelock 2000), is already apparent in early stellar kinematical samples (Eggen 1966; Woolley et al. 1970) and was recognised long ago by Mayor (1972), but the clearest evidence for the Hercules stream comes from the Hipparcos proper motions combined with (Fig. 1) or without (Dehnen 1998) available radial velocities. This stream is very likely to have a dynamical origin because its stars are older than $\sim$2 Gyr (Caloi et al. 1999) and present a wide range of metallicities (Raboud et al. 1998).

The existence of the Hercules stream is most probably related to the influence of the Galactic bar. It is now indeed widely accepted that the Milky Way is a barred galaxy, as are the majority of disc galaxies. Evidence for the bar comes from longitudinal asymmetry in the bulge surface photometry (e.g. Blitz & Spergel 1991; Binney et al. 1997), star counts (e.g. Nakada et al. 1991; Nikolaev & Weinberg 1997; Stanek et al. 1997), interpretation of the observed gas kinematics in the central few kpc (Binney et al. 1991; Englmaier & Gerhard 1999; Fux 1999; Weiner & Sellwood 1999), large microlensing optical depths towards the Galactic bulge (Paczynski et al. 1994; Kuijken 1997; Gyuk 1999; Alcock et al. 2000) and possibly inner stellar kinematics (Sevenster et al. 1999; see also Gerhard 1999 for a recent review). Although still not very well constrained, the most quoted values for the main bar parameters are an in-plane inclination angle with respect to the Galactic centre direction $\varphi\approx 15^{\circ}-45^{\circ}$, with the near side of the bar in the first Galactic quadrant, and a corotation radius $R_{\hbox{\tiny CR}}\approx 3.5-5$ kpc.

Barred N-body models of the Milky Way produce a mean outward motion of disc particles at realistic positions of the Sun relative to the bar (Fux et al. 1995; Raboud et al. 1998), but the precise bar induced dynamical process leading to the observed kinematical properties of the Hercules stream is still a matter of debate. Dehnen (1999b, 2000 - hereafter D2000) relates this stream and the main mode of high angular momentum stars in the observed velocity distribution to the coexistence near the outer Lindblad resonance (OLR) of two distinct types of periodic orbits replacing the circular orbit close to the OLR in a rotating barred potential, i.e. the same idea introduced by Kalnajs (1991) to explain the Hyades and Sirius streams. Linear theory indeed predicts that the orientation of orbits closing in the bar rotating frame changes across the main resonances associated with the bar (Binney & Tremaine 1987). In particular, periodic orbits outside and inside the OLR radius are respectively elongated along the major and minor axis of the bar, and both types of orbits, as well as the quasi-periodic orbits trapped around these orbits, can overlap in space near the OLR. According to D2000, the Hercules stream and the main velocity mode, respectively "OLR'' and "LSR'' mode in his terminology, result from the anti-bar and bar elongated orbits respectively, and the valley between the two modes from off-scattered stars on unstable OLR orbits. Raboud et al. (1998), on the other hand, suggest that the Hercules stream involves stars merely on chaotic orbits and susceptible to cross the corotation radius and wander throughout the Galaxy, but do not explicitly justify why such stars should move outwards on the average in the Solar neighbourhood. One motivation for this interpretation is that of order $10\%$ of the particles in N-body models of barred galaxies indeed follow such orbits (e.g. Pfenniger & Friedli 1991).

This paper investigates how the barred potential of the Milky Way divides the phase space of the stellar disc into regions of regular and chaotic motion and how this segregation may explain some properties of the observed local stellar kinematics and in particular help to clarify the real nature of the Hercules stream. The investigation is first performed in details using the same analytical two-dimensional rotating barred potential as in D2000 and then complemented with the results from a more realistic high-resolution three-dimensional N-body simulation.

The structure of the paper is as follow: Sect. 2 briefly presents the observed stellar velocity distribution in the Solar neighbourhood and some further informations about the Hercules stream. Section 3 recalls a dynamical classification of orbits in rotating barred potentials based on the Jacobi integral and determines the location in local velocity space of the class of orbits that may cross the corotation radius. Section 4 describes the analytical barred potential adopted in the 2D study and Sect. 5 the main periodic orbits supported by this potential outside corotation. Section 6 derives the associated regular and chaotic regions in velocity space as a function of space position relative to the bar. Section 7 presents the velocity distributions at the same space positions resulting from test particle simulations and examines the role of chaos in shaping these distributions. Section 8 shows how the derived velocity distributions depend on the initial conditions of the simulations and Sect. 9 how the particles initially on OLR orbits eventually contribute to these distributions. Section 10 gives the results inferred from the 3D N-body simulations. Section 11 makes a quantitative comparison of the model velocity distributions with the observed one and discusses the most likely origin of the Hercules stream. Finally, Sect. 12 sums up.

Figure 1: Heliocentric velocity distribution in the u-v plane of all the Hipparcos single stars with $\sigma (\pi )/\pi <0.1$, d<100 pc and radial velocities in the Hipparcos Input Catalogue (left) and of the sub-samples with B-V>0.6 (middle) and B-V<0.4 (right). For the sake of comparison, the contours are as in Dehnen (1998), containing 2, 6, 12, 21, 33, 50, 68, 80, 90, 95, 99 and 99.9 percent of all stars. The diagram for the full sample is exactly the same as in Fux (2000), except for a different labelling of the contours.