## 5. ConclusionsThis paper develops a representation for the antisymmetric small
amplitude warp dynamics of a self-gravitating disk in terms of
We first consider in detail the tilting dynamics of one ring in the potential of the halo and inner disk. The equations of motion are shown to have a particularly simple form when written in terms of the complex tilt angle , where is the actual tilt angle, and is the angle of the line-of-nodes relative to the inertial axis. The single ring has a slow and a fast precession mode which are analogous to the normal modes of vibration of a non-rotating mechancial system. For the slow (fast) mode rotates in the clockwise (counter clockwise) direction in the complex plane. The dynamical equation for of course has an energy constant of the motion, . Additionally, we show that there is a second constant of the motion. By writing out the Lagrangian for one ring, , this constant of the motion is recognized as the canonical angular momentum for the coordinate, const. We examine the influence of the drag on the motion of one ring through the halo matter due to dynamical friction. The full calculation of this drag (Nelson and Tremaine 1995) is beyond the scope of this work. Instead, we include a linear drag torque in the equation of motion for which we estimate following the approach of Weinberg (1985). Inclusion of this term shows that dissipation can destabilize the ring tilting in a prolate halo potential. The instability is of the negative energy type; that is, it occurs only for . We next consider the particularly interesting case of two tilted, gravitationally interacting rings of different radii in the potential of the halo and inner disk. We derive dynamical equations for and . This case shows the phenomenon of phase-locking of the line-of-nodes of the two rings. The gravitational interaction between the two rings is appropriately measured by the coupling strength of Eq. (48). As increases the ring motion changes from that of independent precession for to phase-locked precession for where the lines-of-nodes coincide approximately, . This phase-locking, which is a new finding of the present work, persists for the case of rings. We examine the influence of dissipative forces on the tilting motion of two rings. In addition to dynamical friction, which is included as linear drag torques on each ring, we include the `relative' friction which can result from the differential vertical motions of two gaseous rings due to turbulent viscosity. The corresponding drag torques on each ring have a definite form owing to the requirement that the total angular momentum of the two rings be conserved in the absence of external torques. The turbulent viscosity is assumed to be described by the model of Shakura (1973) and Shakura and Sunyaev (1973). We find that the relative friction, as well as the dynamical friction, can destabilize the motion of two rings in a prolate halo potential. We study the case of two tilted rings which rotate in opposite directions. This situation, which is pertinent to observed counter-rotating galaxies (see, for example, Jore, Broeils, and Haynes 1996), is found to be unstable in a prolate halo potential for certain conditions in the absence of dissipation. That is, a dynamical instability may occur. For the general case of tilted,
gravitationally interacting rings of radii in
the potential of the halo and inner disk we obtain equations of motion
for for . The
Lagrangian has the form , and this gives, in
the absence of dissipative torques, both an energy constant of the
motion and a total canonical angular momentum constant of the motion,
const. We comment on the numerical solutions
for for different values of In the continuum limit , the ring model is shown to give the Hunter and Toomre (1969) dynamical equation for the vertical displacement of the disk . Dissipative torques due to dynamical friction and/or relative friction (for gaseous rings) are shown to cause the energy of the warp to decrease with time. The behavior of the total canonical angular momentum is indeterminate in the presence of dissipation. We have numerically solved the dynamical equations for for for four different types of initial conditions which may give rise to observed warps in galaxies: (1) warp excitation by a passing satellite with relatively large impact parameter; (2) excitation by a slowly sinking compact minor satellite; (3) warp evolution in a tilted halo potential (Dekel and Shlosman 1983; and Toomre 1983); and (4) warp evolution resulting from an initially tilted disk plane due to the tidal breakup of a gas cloud. The nature of the disk response at different times is most clearly shown in the polar plots of versus which we refer to as Briggs plots. This is one of the forms used by Briggs (1990) to describe the warp geometry as deduced by fitting a kinematic ring model (Rogstad et al. 1974) to HI observations. We find strong evidence for phase-locking of the line-of-nodes ( const) in the inner regions of the disks, and we show that this is due to the self-gravity of the disk. The fact that the phase-locking occurs in the inner part of the disk indicates that it is independent of the distant outer boundary of the disk. The phase-locking is a new finding of the present work, and it explains rule of behavior No. 2 of Briggs (1990). For case (1) we find that the polar plots have the shape of a leading spiral wave qualitatively of the form observed, but the warp amplitude is smaller than observed in many galaxies. The small warp amplitudes found here are in agreement with the conclusions of HT. The leading spiral in the curves in the outer part of the disk (where the self-gravity between the rings is small) results from the dominance of the slow precession mode which gives very roughly , where is the single ring slow precession frequency. The fast precession modes disappear in a few Gyrs due to phase-mixing. For case (2) of a sinking satellite, we find that the
curves have a large noise component and are
unlike the observed curves in the absence of relative friction between
the rings. This is due to the violent excitation from the satellite
passing through the disk. However, with relative friction included,
the curves are smooth leading spiral waves
qualitatively similar in shape to the observed curves. The warp
amplitude, although significantly larger than for case (1), is still
smaller than that for many observed warps. We find that the inner part
of the disk where is small has an approximately
straight line-of-nodes ( independent of For case (3) of an oblate halo potential rigidly tilted with respect to the inner disk of the galaxy, there is a unique time independent Laplacian warp (BT, p. 413) which has a straight line-of-nodes ( const). However, it is unlikely that a galactic HI disk is `set up' with the Laplacian tilt. Deviations from evolve to give a line-of-nodes which is not straight. Over the age of the galaxy, the deviations from may damp out due to dynamical friction and/or relative friction between the rings. This case is possibly relevant to the warp of NGC 3718 which shows an approximately straight line-of-nodes out to (Briggs 1990). Case (4) of an initially tilted disk plane with a straight
line-of-nodes gives: (1) polar plots for
-6 Gyr with the leading spiral shape
qualitatively of the form observed; (2) straight line-of-nodes
( const) in the inner part of the disk as
observed as a result of phase-locking due to self-gravity; and
(3) a warp amplitude linearly dependent on the initial tilt of
the disk plane. Further, the line-of-nodes angle
in the outer part of the disk is found to have
an inverse power law dependence on © European Southern Observatory (ESO) 1998 Online publication: September 17, 1998 |