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13.3. Rotating frame
It should be stressed that an orbit can appear open in one frame of
reference and closed in another. In fact, suppose we move to a rotating
frame for which the polar coordinates are (r,
), with
=
-
p; here
p is the
angular velocity of the rotating frame. Then
orbits are described by the new Hamiltonian ("Jacobi integral")
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(30) |
with
p = J, so that H = E - J
p. In the
rotating frame, the important ratio
2
/
becomes
2(
-
p) /
, which then may be
rational or not depending on our choice of
p. In the
dynamics of galaxies there are sometimes
physical reasons that identify a specific value of the angular velocity
of the rotating frame. The three important possible conditions of
2(
-
p) /
= -1, 0, +1 are often
called condition of Inner
Lindblad Resonance, Corotation, and Outer Lindblad Resonance,
respectively (see
Fig. 13.4). Thus, in the rotating frame, at
the Lindblad resonances orbits appear
closed into ellipses centered at r = 0. This feature, and the
fact that
-
/ 2 can be
approximately constant on a wide radial range, led B. Lindblad
to conjecture that two-armed spiral structure could persist as a kinematical
wave in a differentially rotating disk (see
Fig. 13.5).
The shear flow pattern associated with the differential rotation in an axisymmetric disk, with the flow "reversal" at the corotation circle is somewhat reminiscent of certain magnetic surface configurations noted in magnetically confined toroidal plasmas, where a suitable projection of the magnetic field changes sign on a "neutral" surface.