8 Changing the initial conditions

How can the initial conditions be changed in order to increase the population of the hot chaotic orbits, and in particular the density in the Hercules-like chaotic overdensity? Figure 14 shows how the three parameters of the initial axisymmetric distribution function (Eq. (12)) individually affect the final velocity distribution for a realistic position of the Sun and F=0.10, using exceptionally the backward integration technique which, as mentioned in Sect. 7, is very convenient for this purpose. A long time integration is adopted to reduce the phase mixing problem and the u-vdistributions are smoothed the same way as the previous ones based on the direct integration method. A comparison of the default parameter velocity distribution with the corresponding distribution (at $\varphi =30^{\circ}$) in Fig. 12 indicates that both integration techniques give very similar results.

Figure 14: Velocity distribution in the u-v plane after 120 bar rotations as a function of initial conditions and using the backward integration technique. The space position is $R_{\circ}/R_{\hbox{\tiny OLR}}=1.1$ and $\varphi =25^{\circ }$ and the bar strength F=0.10. The left frame shows the result for the default values of the parameters, i.e. $\tilde{\sigma}_{\circ}/v_{\circ}=0.2$, $h_{\sigma }/R_{\circ }=1$ and $h_{\rm R}/R_{\circ }=0.33$, and the other frames the results when changing only one parameter at a time to the indicated value. The velocity contours are as in Fig. 1 and the circular arcs represent the H12 and H45contours.

Increasing the overall initial velocity dispersion (top frames in Fig. 14) yields a larger final velocity dispersion. Hence in this kind of simulations the particles remember the initial conditions and the action of the bar is unable to completely erase them. Also, the u-v density in the Hercules-like stream is enhanced relative to the density within the main velocity mode. This is because the larger velocity dispersion lowers the peak of the latter mode, and because a larger $\tilde{\sigma}_{\circ}$ increases the average Hamiltonian value of the particles and thus the fraction of hot chaotic particles. Reducing the initial velocity dispersion scale length keeping the same velocity dispersion at $R_{\circ}$ (middle frames) renders the inner regions hotter and hence mainly increases the density of the velocity distribution at low angular momentum. To increase the relative fraction of stars in the Hercules-like stream, the most efficient way seems to start with a smaller disc scale length (bottom frames). This represents a higher initial space density of the disc in the inner regions where the particles have larger H-values on the average and thus again a larger fraction of hot chaotic particles which will be spread over the whole disc by the barred potential.

By changing the initial conditions, it is therefore possible to achieve a more pronounced main chaotic overdensity than in the results based on the default parameters and also to match more precisely the observed velocity dispersions in the Solar neighbourhood.