11 Models versus observations

Before concluding, we now present a selection of test particle and N-body velocity distributions yielding a good match to the observed one, confront the quasi-x₁(2) orbit and chaotic orbit interpretations of the Hercules stream, paying also attention to the case of the Hyades stream, and discuss how the models could be further improved.

Beside the parameters in the initial conditions of the simulations, the free model parameters are $R_{\circ}/R_{\hbox{\tiny OLR}}$, $\varphi$, the velocity scale specified by $v_{\circ }$ (defined as the local circular velocity in the axisymmetric part $\Phi _{\circ }$ of the potential for the N-body simulation), which should lie between 180 and 230 kms^-1 (e.g. Sackett 1997), and the velocity of the Sun $(v_{\rm s},u_{\rm s})$ relative to the circular orbit in $\Phi _{\circ }$. A commonly adopted velocity reference in the Solar neighbourhood is the LSR, defined as the velocity of the most nearly circular closed orbit passing through the present location of the Sun according to Binney & Merrifield (1998). This definition, which is merely an attempt to generalise the circular LSR orbit of the axisymmetric case to non-axisymmetric potentials, is not always well adapted. The most reasonable LSR orbit candidates near the OLR of a barred potential indeed are the stable low-eccentricity x₁(1) and x₁(2) orbits, but some space positions near the OLR circle are visited by neither of these orbits in our models (see for example $R_{\circ}/R_{\hbox{\tiny OLR}}=1.0$ and $\varphi =30^{\circ}$ in Fig. 12). However, for , there always exists a prominent peak of low-eccentricity quasi-x₁(1) orbits in the model velocity distributions, which, as pointed out in Sect. 7, not necessarily coincides with the trace of the non-resonant x₁(1) orbit when there is one. The maximum of this peak will therefore be taken as the model "LSR'' and will be preferably associated to the Coma Berenices stream, which is the local maximum in the observed velocity distribution that lies closest to the heliocentric velocity of the LSR $(v,u)_{\rm LSR}\approx (-5,10)$ kms^-1 derived from the Hipparcos data (Dehnen & Binney 1998).

For , the quasi-x₁(1) peak is always close to the circular orbit of the axisymmetrised potential, except near the OLR radius and $\varphi=90^{\circ}$ where it reaches a maximum positive v-offset of $\sim 0.05v_{\circ}$ for all explored bar strengths. Under the above circumstances and for realistic space positions, the azimuthal velocity of the Sun should exceed the circular velocity by 5-10 kms^-1.

Figure 20: Selection of scaled velocity distributions from the test particle (frame  a) to d)) and N-body (frame  e)) simulations, with the various parameters indicated on the top of the frames and the velocity origin at the adopted Solar motion. The velocity window and the velocity contours are the same as in Fig. 1. The filled circles represent the mean velocities of the streams listed in Table 1, excluding the Arcturus stream. All distributions from the test particle simulations are time averages over $55\leq t/t_{\rm b}\leq 65$. The H-contours and the resonance curves are as in Fig. 7 and are not plotted for the N-body model because of the time delay problem discussed in Sect. 10.

The selected model velocity distributions are displayed in Fig. 20. The distributions are derived according to the same procedures as described in Sects. 7 and 10. Frame (a) shows a case where the Hercules-like stream is induced exclusively by chaotic orbits and peaks inside the H₁₂ contour, illustrating the fact that chaotic modes not necessarily only occur in the hot orbit region. Here the Hyades stream coincides with a chaotic overdensity associated with a narrow and low-H chaotic breach roughly along the OLR curve, i.e. an interpretation similar to the one proposed in Sect. 7 for the u<0 extension of the LSR mode. Frame (c) gives a case where the Hercules-like stream now falls entirely in the hot orbit region and where the Hyades stream has the same chaotic origin as in frame (a).

Frame (e), derived from the N-body simulation and presenting a larger velocity dispersion, provides a remarkable example of a Hercules-like stream sustained exclusively by quasi-x₁(2) orbits. The test particle simulations develop quasi-x₁(2) modes which cannot be as easily matched to the Hercules stream in our scaling procedure, generally falling right between this stream and the Hyades stream. This can be explained by the different local slope of the circular velocity $v_{\rm c}$ in the N-body and the test particle models. As explained by D2000 in terms of orbital frequencies, the separation between the quasi-x₁(1) and the quasi-x₁(2) modes increases with ${\rm d}v_{\rm c}/{\rm d}R(R_{\circ})$. Since the average N-body model has a slightly raising rotation curve near the OLR radius (Fig. 16d), its circular velocity gradient is larger than for the flat rotation curve test particle simulations and thus the quasi-x₁(2) mode is found at higher asymmetric drift relative to the quasi-x₁(1) mode. However, the fact that observations support a rather gently declining rotation curve at $R_{\circ}$ and that a large inclination angle of the bar is needed ( for $R_{\circ}>R_{\hbox{\tiny OLR}}$) are arguments against the quasi-x₁(2) interpretation of the Hercules stream. On the other hand, the displacement of the quasi-x₁(2) peak towards the H₁₂contour with increasing integration time mentioned in Sect. 7 for F=0.10 and $R_{\circ}\approx R_{\hbox{\tiny OLR}}$ may reinforce this interpretation.

Frame (d) is an example with two distinct low angular momentum peaks, the most negative v one being related to chaotic hot orbits and the other one to quasi-x₁(2) orbits. It would be interesting to check whether a sufficiently negative ${\rm d}v_{\rm c}/{\rm d}R(R_{\circ})$ is able to shift the quasi-x₁(2) mode more towards the true location of the Hyades stream and thus yield a model velocity distribution with a better overall match to the observed one. Note that the chaotic orbit mode will not necessarily be shifted as the quasi-x₁(2) mode, because its location in the u-v plane does not actually depend on the local slope of the circular velocity, but rather on the difference of $\Phi _{\rm eff}$ between the current space position and the Lagrangian point L_1/2, which determines the u-vlocation of the H₁₂ contour. Finally, frame (b) displays a surprising case where the velocity distribution in the quasi-x₁(2) region of the u-v plane (see Fig. 9) seems to have split into two peaks coinciding well with the locations of the Hercules and Hyades streams, i.e. both these streams have a quasi-x₁(2) origin. However, this is likely to be a transitory situation resulting from the unachieved phase mixing near $R_{\circ}=R_{\hbox{\tiny OLR}}$ (see Sect. 7).

These examples illustrate the variety of possible interpretations for the Hercules and Hyades streams, and it is very hard at this stage to decide with certainty which one is the most appropriate. The splitting of the LSR mode into the Pleiades, Coma Berenices, Sirius and other streams is probably not related to the bar itself and has a more local origin, like for instance the effect of time dependent spiral arms.