3. N tilted corotating rings
The generalization of Eq. (40) gives the Lagrangian for an N ring system as
Because the are non-negative, all of the terms in are non-negative if which is the case for oblate halo. Thus the ring system with is stable in the presence of dissipative forces. The total canonical angular momentum is
In the absence of dissipation and are constants of the motion.
As in the case of one or two rings, it is useful to let
The equations of motion can then be written as
with , where the generalize of Eq. (34).
As discussed in Sect. 2.1, a Newtonian drag force on the ring motion due to dynamical friction with the halo matter can be taken into account by including the term
on the right hand side of Eq. (55). The influence of the relative friction discussed in Sect. 2.3 can be accounted for by including the terms
on the right hand side of Eq. (55). Here, and with and . With the dissipation terms (56) and (57) included one can readily show that (see also Sect. 4). We have not found an analogous result for in the presence of dissipation.
We have developed and tested codes to solve Eqs. (55) (including the terms (56) and (57)) for and 49 rings. For the results presented here, the rings are taken to be uniformly spaced in r with ring one at and ring N at so that , with in units of kpc. The ring masses and moments of inertia are calculated with the sum of the surface density of the inner exponential disk and that of the neutral hydrogen given in Sect. 2. The are calculated using Eqs. (4), (6), and (18). The coupling coefficients are evaluated using Eqs. (34) and (30b) and stored. Eqs. (55) are solved as four first order equations for and . At the same time, an additional equation,
is also solved to give , which is the line-of-nodes angle relative to the axis.
Comparisons of the temporal responses of 25 and 49 ring sytems for different initial conditions show generally good agreement of the warps for the time intervals studied. This indicates that rings gives a valid representation of a continuous disk. However, in contrast with a continuous disk (HT), an ring disk has discrete modes with separate frequencies - N `low-frequency' modes and N `high-frequency' modes. Fig. 15 shows the power spectrum of for a 49 ring disk-halo system which is initially given a random perturbation with all radial wavelengths excited and no damping. The frequencies are normalized by the angular velocity of the inner ring . Note that Gyr for our reference values. The frequency differences between adjacent modes is . For treatment of initial value problems of disk warping for time intervals , frequency differences less than are irrelevant. It is sufficient to have which corresponds to rings. This suggests that a larger number of rings may be needed for our cases with Gyr (Figs. 18 and 19). This will be investigated in a future work.
The lowest frequency mode of the power spectrum, in Fig. 15 (see also Sect. 2.3 and Sect. 2.5), remains discrete (that is, isolated) as because our disk model has a sharp outer edge so that is integrable (HT). Power spectra obtained for cases with show that increases with much more slowly than the higher frequency modes which is consistent with the behavior observed in Figs. 6 and 12. For the conditions of Fig. 15, the magnitude is considerably smaller than that predicted for the limit of strong self-gravity between the rings where [equation(47)]. In any case, the frequency is so small (the corresponding period is Gyr) that it is irrelevant to the initial value problems considered below.
In the following four subsections we consider different possible origins of warps in an otherwise flat galaxy. Sect. 3.1 discusses warp excitation by a passing compact satellite; Sect. 3.2 treats warp excitation by a compact sinking satellite; Sect. 3.3 treats the case of a tilted halo potential. Sect. 3.4 considers the tilt evolution for the case where the initial plane of the disk material is tilted.
3.1. Warp excitation by a passing satellite
Consider the excitation of a warp in a galaxy due to the passage of a satellite of mass much less than the galaxy's mass. The satellite's orbit can easily be calculated exactly for , neglecting the back reaction of the perturbed galaxy on the satellite. However, we assume that even at closest approach the satellite is far from the center of the galaxy and therefore calculate the orbit in the halo potential, Eq. (6a), with the ellipticity of the halo neglected for simplicity (). At the closest approach at , the satellite's speed is , and the angle between the satellite and the plane is . Also, at closest approach, the satellite is taken to be at and . We consider both prograde and retrograde satellite passages.
In the presence of a satellite of finite mass , the description of the galaxy disk in terms of tilted circular rings breaks down. The centers of the rings are shifted from the origin, and the rings become non-circular. For this reason we consider the response of the galaxy to a `symmetrized' perturbation of a satellite obtained by replacing the actual satellite by two satellites, each of equal mass , with one satellite in the orbit described above and the other in the orbit . With this prescription we then calculate the torque of the satellites on each ring as a function of time,
where , with given by Eq. (8a), is the position vector to a point on the jth ring, and runs over the different rings. Here, is the gravitational potential of the satellite and a is its characteristic radius. To obtain the warp response of the galaxy we evaluate Eqs. (59) numerically for and include the complex torques on the right hand sides of Eqs. (55).
Fig. 16a shows the Briggs plot of the warp response of the galaxy resulting from the retrograde passage of a satellite of mass . Note that the dependence of at the times shown is of the form of a leading spiral wave and that it is qualitatively similar to the leading spiral waves observed in many warped galaxies (Briggs 1990). Note also that the form of the spiral wave is preserved for a long time. However, the amplitude of the warp is significantly smaller than the warps observed in many cases, and this agrees with the conclusions of HT. Nevertheless, it is of interest to understand the behavior.
Fig. 16b shows the warp response for the retrograde case with and without gravitational interactions () between the rings. Clearly these interactions give a strong phase-locking of the inner rings of the disk (roughly, to or 10 to 18 kpc) which have the same as a function of time. The phase-locking of two and three rings was disscussed in Sect. 2.2 and Sect. 2.4.
The results shown in Fig. 16 are obtained from a code which solves Eqs. (55) including the torques of Eq. (59) with rings equally spaced between kpc and kpc. Thus the separation between rings is kpc. The torques are evaluated numerically at each time step. The Newtonian drag is neglected, in Eq. (56). A small relative friction is included, in Eq. (57). [For a disk half-thickness pc and sound speed km/s, this corresponds to a value of Shakura's (1973) viscosity parameter . This friction acts to smooth out sharp `corners' which exist in the curve . Owing to this , decreases by 16% between and 3 Gyr.] The mass of the satellite is , its core radius is kpc, and at closest approach it is at a distance kpc where it has a speed of 400 km/s and is located at an angle above the plane. The mass of the inner disk in Eq. (4) is , and the disk radial scale is kpc. The halo potential is given by Eq. (6a) with ellipticity , core radius kpc, and circular velocity km/s so that for . The mass of halo matter inside of kpc is . The neutral hydrogen has a total mass of and is distributed according to Eq. (5) with kpc.
Fig. 17a shows the Briggs plot for the warp resulting from the prograde passage of the satellite. For early times ( Gyr) the form of is a trailing spiral wave which is distinctly different from that for the retrograde passage at a similar time. The amplitude of the warp is noticably larger than for the retrograde case but still smaller than many observed warps. For later times ( Gyr) the trailing wave `unwraps,' evolving into a leading spiral wave of form similar to that for the retrograde passage for radial distances out to about 35 kpc. For larger r, the curve has a `spur' where increase with r while remains roughly constant (see Sect. 3.2). This unwrapping appears to be due to the phase-mixing of the fast precession modes which propagate over the radial extent of the disk in times less than 4 Gyr.
The different behavior of the prograde and retrograde cases for early times ( Gyr) can be understood by considering the response of a single ring with or without gravitational interactions between rings. Fig. 17b shows the orbits of ring No. 25 (radius kpc) including the gravitational interactions for prograde and retrograde encounters. Consider the torque on the ring which enters on the right hand side of Eq. (55). The tilting of the ring during the encounter is relatively small and can be neglected for this discussion. Consequently, is an even function of t if the pericenter is taken as . Further, is the same for prograde and retrograde encounters. For the considered geometry of the satellite orbit, , and the magnitude of decreases monotonically with increasing . On the other hand, the torque , although smaller in magnitude, is an odd function of time, and it has opposite signs for prograde and retrograde encounters. Very roughly for the prograde case whereas for the retrograde case. As a function of time, traces out a closed clockwise path in the complex plane for the prograde case, whereas this path is counterclockwise for the retrograde case. For the prograde case, the path of is in the same clockwise sense as the path of the orbit for the fast precession mode (see Sect. 2.1), whereas for the retrograde case the path of is in the same direction of motion as for the slow precession mode. For this reason a prograde encounter couples strongly to the fast precession mode, whereas a retrograde encounter couples mainly to the slow precession mode.
3.2. Warp excitation by a sinking satellite
Here, we discuss the behavior of warps excited by a slowly sinking compact minor satellite. The satellite is assumed to be minor in the respect that its mass is much smaller than that of the galaxy plus halo matter within say 30 kpc. The sinking of a more massive satellite is likely to cause substantial thickening of the disk (Walker, Mihos, and Hernquist 1996) which is not observed. Initially, the satellite is assumed to be in an approximately circular bound orbit in the halo gravitational potential at a large radius in the plane at an ange above the plane. The satellite slowly sinks owing to dynamical friction with the halo matter [Eq. (6a) with ] which in the simplest description (BT, p. 428) causes the specific angular momentum of the satellite to decrease as
where is the Coulomb logarithm. The time for the satellite to sink from to the galaxy's center is
where we have taken . The disk response is found by integrating Eqs. (55) including the numerically evaluated torques (59) for at each time step for a `symmetrized' sinking analogous to the approach discussed in Sect. 3.1.
Fig. 18 shows the Briggs plots of the warp response of the galaxy at different times after the sinking () of a satellite of mass in retrograde and prograde orbits. A number of points are observed:
(1) The warp amplitudes are larger than for the passing satellite (of mass ) discussed in Sect. 3.1, but the amplitudes are still smaller than some observed warps (Briggs 1990).
(2) Coiling in the curves at early times ( Gyr) tends to disappear at later times ( Gyr). We believe but have not proven that this is due to the phase-mixing of the fast precession modes which propagate over the radial extent of the disk in times Gyr (see also Sect. 3.1).
(3) For both the retrograde and prograde cases exhibits a leading spiral wave for Gyr qualitatively of the form observed for M 83 (see Fig. 2c). However, the tilt amplitude is significantly smaller than that for M 83.
The dissipative torques due to dynamical friction [Eq. (56)] and that due to relative friction [Eq. (57)] have different consequences for the warp evolution. For a dynamical friction coefficient , the warp amplitudes for the cases of Figs. 18a and 18b are reduced by factors while the line-of-nodes angles are roughly the same. On the other hand, for relative friction coefficients , short radial wavelengths are damped out without appreciably affecting the overall shape of the curves.
3.3. Tilted halo potential
Dekel and Shlosman (1983) and Toomre (1983) proposed that observed warps of galaxies may result from the fact that the dark matter halo is oblate and is rigidly tilted with respect to the inner disk. To consider this possibility we generalize Eq. (55) to the case of N tilted rings in a halo potential which is rigidly tilted by an angle, say, (with ) with respect to the axis of the inner disk. We find
where and . The damping terms are still given by Eqs. (56) and (57).
For a special disk tilt, , the right-hand-side of Eq. (61) is zero so that is time-independent. This corresponds to the Laplacian surface of the disk (BT, p. 413). It is given by
where is the Kronecker delta and the prime on the summation means that the diagonal terms are omitted. The damping terms (56) and (57) are of course zero for the Laplacian tilt. If the self-gravity of the rings () is negligible, . Eq. (62) can always be inverted to give if . This is because the ring system is stable for (see discussion following Eq. (53b)) which implies .
A general solution of Eq. (61) can evidently be written as , where obeys Eq. (55) which has no reference to the halo tilt. Over a sufficiently long period of time, damping given by Eq. (56) and/or (57) will give as discussed in the paragraph after next.
Fig. 19a shows the Laplacian tilt for a representative case with and without the self-gravity of the rings. Fig. 19b shows the Briggs plots at times and 8 Gyr for a disk started from a Laplacian tilt and from a deviation from a Laplacian tilt. The initial deviation is taken to be . This deviation vanishes at and it has a maximum at the outer edge of the disk. Dissipative torques are neglected. This deviation from gives rise to a leading spiral feature at the outer edge of the disk at Gyr. For a deviation with the factor replaced by -0.2 the line-of-nodes angle at the outer edge of the disk () at Gyr has a similar magnitude to that in Fig. 19b, but it is negative. The Briggs plot for the Laplacian tilt is in contrast with the leading spiral wave observed, for example, in M 83 (Figs. 1 and 2) (Briggs 1990). However, the Laplacian tilt may be relevant to cases such as NGC 3718 which shows a relatively straight line-of-nodes.
There is no evident reason for a disk to be initially `set up' in a Laplacian tilt. Fig. 19b shows that deviations from the Laplacian tilt evolve to give a line-of-nodes which is not straight. As mentioned, dissipative torques due to dynamical friction [Eq. (56)] and/or relative friction [equation(57)] act to damp out the deviation () from the Laplacian tilt over a period of time. For example, for in Eq. (56) and no relative friction, the maximum line-of-nodes deviation () at 8 Gyr is reduced by a factor from the case with no dissipation shown in Fig. 19b. On the other hand for and no dynamical friction, the maximum line-of-nodes deviation is reduced by a factor .
3.4. Initially tilted disk plane
Here we consider the possibility that at some intial time an outer gaseous disk is formed which is tilted with respect to the plane of the inner disk. This situation could arise by the capture, tidal-disruption, radial-spreading, and cooling of a low mass gas cloud by a disk galaxy. For a spherical cloud of mass and radius in an approximately circular orbit of radius r, tidal breakup occurs roughly for , where is the circular velocity in the flat rotation region of the galaxy [Eq. (6b)]. Treatment of the cloud disruption is beyond the scope of the present work. Instead, we study the initial value problem where at there is a smooth transition from an untilted inner disk to a tilted outer disk with a straight line-of-nodes. Specifically, we take
where is the initial tilt angle of the outer disk. Similar behavior is found for other smooth variations of . The smooth variation of avoids the excitation of short radial wavelength modes.
Fig. 20 shows the nature of the warp evolution resulting from the initial conditions (63). The spiral for Gyr is qualitatively similar to that observed for M 83 (see Fig. 2c). From results not shown here the similarity continues up to Gyr after which time and become large at the outer edge of the disk.
Fig. 21 shows a surface plot of the disk at Gyr. A leading spiral wave appears in the Briggs plot (panel c ), and its amplitude grows with time. Progressively, the interior region of the disk (say, kpc) flattens to the plane of the inner disk while the exterior region warp amplitude grows. The inner part of the disk maintains a straight line of nodes ( const), which is due to the self-gravity of the disk (see Sect. 2.2, Sect. 2.4, and Sect. 3.1). The dependences shown in Fig. 20 for -4 Gyr are qualitatively similar to those observed for M 83 (see Fig. 2). The outer part of the disk shows an inverse power law dependence of the line-of-nodes on r (panel b of Fig. 20) which is qualitatively similar to that for M 83 where .
The dissipative torques due to dynamical friction [Eq. (56)] and that due to relative friction [Eq. (57)] affect the results of Fig. 20 in different ways. For a dynamical friction coefficient , and no relative friction, the maximum warp amplitude at Gyr is reduced by about . On the other hand, for no dynamical friction, but a relative friction coefficient , the warp at Gyr is essentially unchanged.
© European Southern Observatory (ESO) 1998
Online publication: September 17, 1998