4. Pattern velocity and Lindblad resonances
The non-axisymmetric structure in the disk has an angular pattern speed given by the angular velocity . Three radii in each galaxy are of specific interest because the pattern speed resonates with the eigenfrequencies of the unperturbed galaxy. The first order epicyclic theory (e.g. Binney and Tremaine 1987) superimposes harmonic oscillations on the rotation in both the radial and tangential directions with a characteristic frequency called the epicyclic frequency . Thus, in the proper rotating frame, the particle will move in a retrograde sense around a small ellipse, called an epicycle, with axial ratio . The resulting motion in the inertial frame is a rosette orbit, generally not closed. To the first approximation, the motion of a gas cloud or a star is thus governed by two frequencies. In a frame corotating with a spiral these are and . When they are comparable, i.e. when the relative frequency is equal to a rational number, we have a resonance. The three main resonances in a galaxy (with a two-arm spiral pattern ) are
- the inner Lindblad resonance (ILR) where and
- the corotation resonance (CR) where and
- the outer Lindblad resonance (OLR) where and
At the Lindblad resonances, the disk and the pattern (bar, spiral arms) intensively exchange energy, mass and angular momentum. The radial velocity, for example, changes its sign at CR. Inside CR, the material flows inward and outside CR it moves outward. Due to the energy and angular momentum transfer, the orbits of the particles change by 90 degrees at every resonance. As we shall see, this simple picture is complicated by the fact that a bar as well as a spiral system has its own resonances. Due to the interaction of the particles with the non-axisymmetric structure, the bar and spiral arms are slowed down, thereby moving the resonance radii outward. Larger portions of the disk are then involved in the radial inflow of the material. Since the resulting disk velocity field, which we use as an input parameter for our magnetic field calculation, reflects this dynamical behaviour, we analyze the pattern velocity and the radii of the Lindblad resonances and corotation in our simulated galaxies.
Based on a Fourier analysis of the density distribution or of the gravitational potential, which allows us to decompose the perturbation present in the galaxy (Junqueira & Combes 1996), we are able to calculate the pattern velocity of our simulation at different time steps. By plotting the pattern velocity together with the curves we determine the radii of the resonances in the disk.
The uncertainty of the pattern velocity amounts to about 2 km/s/kpc. For the Lindblad radii the error increases from the inner to the outer part of the disk.
© European Southern Observatory (ESO) 1998
Online publication: April 15, 1998