In this Appendix the epicyclic KIA model is used to further examine the effects of a second symmetric collision described in Section 4.3. Specifically, immediately after the second collision, Equation (4.5) can be replaced with
![]() | (A1.1) |
where t2 is the time of the second collision (with t = 0 at the first collision). The amplitude A2 is defined through the velocity equation at time t2,
![]() | (A1.2) |
where
v2
is the velocity impulse in the second collision, and the
second equality defines A2. Equations (A1.1) and
(A1.2) can be combined
to eliminate the phase
,
![]() | (A1.3) |
with r2 = r(t2), the radius of the given star at the time of the second impact. This can be simplified using the relations
![]() | (A1.4) |
Thus, we obtain
![]() | (A1.5) |
The orbit equation then becomes
![]() | (A1.6) |
In zones where the last term in the square brackets is small after the
second impact, i.e.,
sin(t2
+
)
0, Equation
(A1.6) is essentially
the same as Equation (4.5), but with a position-dependent amplitude.
In zones where the first term in the square brackets in Equation
(A1.6) is small, i.e.,
cos(t2
+
)
0, that
equation is similar to
Equation (4.5), but the perturbation term is off by a phase of
/ 2. As a result we expect a
change in the morphology and rate of
propagation at phases where there is a change of dominance from one
term to the other. The models confirm-that the rings do not propagate
as rapidly outward following the second impact as they did during the
first.
When
cos(t2)
> 1, all of the terms in Equation (A1.5) add,
and the wave amplitude will be relatively large (unless
cos(
t2
+
) is small). When
cos(
t2)
< 1 the last two terms on the rights-hand-side of
Equation (A1.5) will tend to cancel, resulting in weak waves in that
region. Indeed if the phase
t2 is
such that
sin(
t2)
0, then
A2
0, i.e., the second perturbation can essentially cancel the
first. Such trajectories are seen in the models. If the amplitude of
nearby trajectories is large enough they can cross the
A2
0
trajectory. The result is a ring that does not propagate outward, but
its width varies with time. Though it will be affected by stellar
diffusion, such a stationary ring could be long-lived. Because their
locations depend on the details of the collisions as well as the disk
structure, they will in general have no relation to classical Lindblad
resonances.