Polar view of Milky Way galaxy showing L4, L5 Lagrangian points and the solar neighborhood.
As I noted from an earlier (May 15, 2011) post, analysis of the motions in the spiral galaxies like the Milky Way is not easy or straightforward. As an example, assume the (polar) coordinates for a galactic rotating frame are given as (r, φ) with:
dφ/dt = dΘ/dt - W p
H = ½(pr2 + pφ2/r2) + V(r) - pφ Wp
where the pr, pφ are the particle momenta referred to the associated coordinates, and V(r) is the gravitational potential. The point is that H can change depending on the coordinates, and what is presented for the previous frame as H = E - J Wp (with simplification, pφ = J) may well be different for another frame.
Such considerations enter into investigating the cause of the motion of the so-called "Hercules stream". This is a group of stars (about 10 percent in the solar neighborhood) which exhibits an oddly distinctive collective motion. It includes stars that are moving away from the galaxy center while actually falling behind the galaxy's general rotation.
What this means is that the group's average orbital angular speed is less than the angular speed of the galaxy (e.g. W < W p )
According to standard theories of stellar evolution, new stars typically condense with other new stars out of the same dense, cool region of dust and gas. Hence, since stars are born more or less at the same time they "inherit" the motion at that position of the galaxy coincident with their birth place. The problem is that the Hercules stream is different with spectroscopic observations revealing stars of widely varying ages. How can this be? Whatever the basis of the streaming motion it operates across vast scales of time and space.
One suspect is the dense central bar of stars (see highlighted in graphic) from which two of the galaxy's spiral arms sprout. The solar neighborhood lies beyond the orbit of the ends of the bar, but the Sun (and stars of the Hercules stream) are close enough to feel the bar's gravitational influence. As the stars orbit the galaxy they oscillate toward and away fro the galactic center.
The number c of epicycle oscillations per orbit about the galactic center is given by the ratio of the star’s epicycle frequency (ko) to its orbital angular speed, W.
Or: c = ko / W
If it happens that the star's radial oscillation resonates with the bar's rotation the star can end up with an extra 'kick' away from the galactic center. The resonance, the outer Lindblad resonance, has been invoked to explain the Hercules stream but more recent observations have identified flaws. In fact one major flaw in that since the bar rotates more slowly than originally thought the outer Lindblad resonance is too distant to be the stream's primary mover.
Recognition of this disparity has been incorporated in a new dynamical model proposed by Angeles Perez - Villegas and colleagues at the Max Planck Institute Their model has succeeded in not only re-creating the positions and velocities of the Hercules stream but also revealed the cause of the unusual motion.
It turns out the stream's stars are orbiting two local maxima, identified as the L4 and L5 Lagrange points. (See diagram at top). The Lagrange points mark positions where the combined gravitational pull of the two large masses provide precisely the centripetal force required to orbit with them. In this case, the masses are the galaxy central bar, and the Hercules stream. The key point to note is that the L4 and L5 points are in the bar's effective gravitational potential, V(r) = - GMm/r.
Note more specifically that the L4 and L5 points are situated on the bar's perpendicular bisector close to the distance where stars co-rotate with the bar. In the Perez -Villegas et al model the majority of stars in the Hercules stream originate from the inner part of the galaxy where stars tend to be older than the Sun. Then their looping orbits, i.e. around L4 and L5, take them all the way into the solar neighborhood but not much beyond it.