Animation redrawn from Sellwood & Carlberg (2021)
The top left panel of the animation, which loops back after a short
pause, shows the time evolution of the net overdensity of the disk
response to two rings at R=2.5R0
and R=4R0 each of three very light disturbance
masses that are driven at the local corotation speed. These rings of
particles, marked by large dots and each having a mass of
10-4R0V02/G or
almost 1000 times the mass of a background disk particle, were added
into a simulation of the stable half-mass Mestel disk in
which Q=1.5.
The disk response in the simulation was fitted by two steadily
rotating m=3 waves of fixed shape and amplitude that are
illustrated in the top panel. The fitted pattern speeds of each of
the responses are Ωp,1=0.404 and
Ωp,2=0.252, and the principal resonances are marked
by circles. As in the animation, distances are in units
of R0 and times are in units
of R0/V0.
The top right panel of the animation shows the time evolution of the
superposed fitted disturbances, each turning at the fitted rate, that
appear as a shearing transient pattern. From time to time, the
pattern breaks into two separate spiral segments that quickly rejoin
into a more open spiral that winds up again. The similarity of the
more intense parts of the disturbances in the top left and top right
panels is evident.
The bottom two panels illustrate the evolution of the logarithmic
spiral spectrum in the simulation on the left and the superposed two
waves on the right. The abscissa in these panels is
tan γ, where γ is the angle between the
radius vector and the tangent to the spiral; it is therefore the
complement to the pitch angle α. Here one can see a
repeating pattern of a peak moving to the right from leading to
trailing as the amplitude rises to a maximum near tan γ =
+2, followed by a decrease as the spiral begins to wind more
tightly.
This is a clear demonstration that the superposition of two steady
disturbances can create the appearance of swing amplification.
Reference: Sellwood, J. A. & Carlberg, R. G. 2021, MNRAS, 500, 5043