4 Working potential

The analytical 2D barred potential adopted for the orbital structure analysis and the test particle simulations is the same as in D2000:

$$\Phi(R,\phi)=\Phi_{\circ}(R)+\Phi_{\rm b}(R,\phi),$$ (5)

with

 

$$\hspace*{1cm} \Phi_{\circ}(R) v_{\circ}^2\ln{R},$$ = (6)

$$\Phi_{\rm b}(R,\phi) \frac{1}{2}Fv_{\circ}^2\cos{(2\phi)}\left\{ \begin{array}{cl} 2-\... ...& R\leq a% \\ \left(\frac{R}{a}\right)^{-3} & R\geq a. \end{array}\right.\cdot$$ = (7)

This represents the sum of an axisymmetric potential $\Phi _{\circ }$ with constant circular velocity $v_{\circ }$ and a barred potential $\Phi_{\rm b}$falling off as a quadrupole at $R\geq a$. The inner and outer parts of the latter component are described by two distinct functions which connect together at R=a such as to ensure continuous potential and forces. The bar major axis is taken to coincide with the y-axis, contrary to the convention in D2000. The parameter F is the bar strength, defined as the maximum azimuthal force on the circle of radius a divided by the radial force of the axisymmetric part of the potential at the same radius (in absolute value). It is related to Dehnen's parameter $\alpha$ by $F=8.89\alpha$. The potential is rotating at a constant pattern speed $\Omega_{\rm P}$ such as to place corotation at $R_{\hbox{\tiny CR}}=1.25 a$, in agreement with numerical simulations and analyses of observations in early-type barred galaxies if a is associated with the bar semi-major axis (e.g. Elmegreen 1996). Unlike D2000, only flat rotation curve models will be examined. In this case, the OLR and corotation radii are related via $R_{\hbox{\tiny OLR}}=(1+1/\sqrt{2})R_{\hbox{\tiny CR}}$, and a value of $R_{\circ}/R_{\hbox{\tiny OLR}}=1.1$ corresponds to a corotation radius $R_{\hbox{\tiny CR}}=4.26$ kpc if $R_{\circ}=8$ kpc. Some considerations in the case of a non-constant rotation curve can be found in Sect. 11.

The next sections present a study of the periodic orbits outside corotation in the adopted rotating barred potential, identify the regular and chaotic regions in phase space associated with this potential, and discuss how stars may populate the available orbits. All orbits in these sections are integrated in double precision using an 8 order Runge-Kutta-Fehlberg algorithm (Fehlberg 1968). Two values of the bar strength will be considered, F=0.10 and F=0.20. The larger value corresponds to a rather strong bar (see Sect. 10 for a quantification with respect to real galaxies), but has the advantage to clearly point out the effect of chaos in the test particle simulations. Some of the key results for the intermediate case F=0.15 are presented in Fux (2001a). For comparison, Dehnen's simulations were done in the range F=0.062-0.116.