3 Effective potential and Jacobi integral

In a rigid potential $\Phi(\vec{x})$ rotating at a constant frequency $\Omega_{\rm P}$ about the z-axis, the Hamiltonian of a test particle expressed in the rotating frame writes:

$$H(\vec{x},\dot{\vec{x}})=\frac{1}{2}\dot{\vec{x}}^2+\Phi_{\rm eff}(\vec{x}),$$ (1)

where $\Phi_{\rm eff}(\vec{x})=\Phi(\vec{x})-\frac{1}{2}\Omega_{\rm P}^2 (x^2+y^2)$ is the effective potential. If $\Phi(\vec{x})$ is non-axisymmetric and $\Omega_{\rm P}\neq 0$, the energy E and the z-component of the angular momentum L_z are not conserved individually, and the only known classical integral of motion generally is the value of the Hamiltonian $H=E-\Omega_{\rm P}L_z$, known as the Jacobi integral. Since $\dot{\vec{x}}^2$must be positive, this integral restricts the motion of a particle to the space region where $\Phi_{\rm eff}(\vec{x})<H$.

Figure 2: a) Effective potential in the x-y plane of a barred disc model corresponding to Eq. (5) with F=0.10. The bar is along the y-axis and the spacing between the contours increases by a factor 1.2 towards lower $\Phi _{\rm eff}$. The crosses indicate the Lagrangian points L1/2 (on the y-axis) and L4/5 (on the x-axis). b) Effective potential along the y-axis (thick line) and the three main classes of orbits related to the conservation of the Jacobi integral. The shaded area below the curve is forbidden for orbits with H<H12.

In a rotating barred potential, the contours of effective potential in the plane of symmetry z=0 look like a volcano with a sinusoidal crest, the extrema of which defining the locations of the Lagrangian points L_1/2 and L_4/5, corresponding respectively to the saddle points and maxima of $\Phi _{\rm eff}$ on the major and minor axis of the bar (Fig. 2a). Two critical values of the Hamiltonian are associated with stars corotating at these points, namely $H_{12}\equiv \Phi_{\rm eff}(L_{1/2})$ and $H_{45}\equiv \Phi_{\rm eff}(L_{4/5})$. The first of them can be used to classify stellar orbits into three dynamical categories (Sparke & Sellwood 1987; Pfenniger & Friedli 1991): the bar orbits and disc orbits with H<H₁₂, which cannot cross the H₁₂ contour and are therefore confined inside and outside corotation respectively, and the hot orbits with $H\geq H_{12}$, which are susceptible to cross the corotation barrier and explore all space except a small region around L_4/5 if H<H₄₅ (Fig. 2b). Stars with H₁₂<H<H₄₅ cannot cross the corotation radius at all azimuth and may therefore more likely be locked during several orbital periods on either side of corotation.

In the Solar neighbourhood, located confidently beyond corotation, only stars from the disc and hot populations are observed. Since these stars share about the same $\Phi _{\rm eff}$ if not too far from the Galactic plane, their H-values depend mainly on the velocities and thus one expects that the two populations occupy different regions in local velocity space. If v, u and w are measured with respect to the Galactic centre, Eq. (1) transforms into:

$$(v-R_{\circ}\Omega_{\rm P})^2+u^2+w^2=2(H-\Phi_{\rm eff}^{\circ}),$$ (2)

where $R_{\circ}$ is the galactocentric distance of the Sun and $\Phi_{\rm eff}^{\circ}$ the local effective potential. Thus the contours of constant Hamiltonian in velocity space are spheres centred on $(v,u,w)=(R_{\circ}\Omega_{\rm P},0,0)$ and of radius $\sqrt{2(H-\Phi_{\rm eff}^{\circ})}$ increasing with H. Stars on disc and hot orbits are respectively those inside and outside the H₁₂ sphere. If the vertical dimension is neglected, these spheres become circles with the same properties. In the axisymmetric limit and for a flat rotation curve of circular velocity $v_{\circ }$, the radius $c_{\hbox{\tiny CR}}$ of the H₁₂=H₄₅ contour and the low azimuthal velocity $\Delta v$ relative to $v_{\circ }$ at which this contour crosses the u=0 axis are then given by:

  

$$c_{\hbox{\tiny CR}}^2 \equiv 2[H_{12}-\Phi_{\rm eff}(R_{\circ})] = v_{\circ}^2\left[\ln{\left(... ...irc}} \right)^2}+\left(\frac{R_{\circ}}{R_{\hbox{\tiny CR}}}\right)^2-1\right],$$ (3)

$$\Delta v v_{\circ}\left(\frac{R_{\circ}}{R_{\hbox{\tiny CR}}}-1\right) -c_{\hbox{\tiny CR}},$$ (4)

where $R_{\hbox{\tiny CR}}=v_{\circ}/\Omega_{\rm P}$ is the corotation radius (see Fig. 3). For $R_{\hbox{\tiny CR}}/R_{\circ}=4.5/8$ and $v_{\circ }=200$ kms^-1, one gets $-\Delta v=0.227v_{\circ}\approx 45$ kms^-1, which coincides with the mean heliocentric asymmetric drift of the Hercules stream. Note however that this simple approximation is not truly a lower limit to the asymmetric drift of stars on hot orbits for several reasons: a non-zero w velocity component defines two circles on the H₁₂sphere with a reduced projected radius on the u-v plane (small effect, of order 1 kms^-1 for |w|=20 kms^-1), the presence of a bar lowers the effective potential at L_1/2 (larger effect, of order 5-10 kms^-1), and finally $\vert\Delta v\vert$ is smaller if $u\neq 0$. Hence most stars in the Hercules stream are likely to fall outside the H₁₂sphere and therefore may belong to the hot population.

Figure 3: Contours of constant Hamiltonian in the u-v plane of a realistic 2D barred model (Eq. (5) with $R_{\hbox{\tiny CR}}/R_{\circ}=4.5/8$, $v_{\circ }=200$ kms-1 and F=0.20). The velocities are relative to an inertial frame. The inner and outer solid circles give the H12 and H45 contours respectively, and the dotted circle is the axisymmetric limit of these contours.

From Eqs. (3) and (4), it also follows that the radius $c_{\hbox{\tiny CR}}$ and the velocity separation $\vert\Delta v\vert$ increase for larger galactocentric distances relative to $R_{\hbox{\tiny CR}}$. In particular, whatever the strength of the bar, one can always increase the fraction of the Hercules stream falling in the hot orbit region by reducing the value of $R_{\circ}/R_{\hbox{\tiny CR}}$.