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Astron. Astrophys. 338, 819-839 (1998)

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2. Tilting of disk galaxies

The equilibrium, unwarped galaxy is assumed to be axisymmetric and to consist of a thin disk of stars and gas and a slightly oblate or prolate halo of dark matter. We use an inertial cylindrical [FORMULA] and Cartesian [FORMULA] coordinate systems with the disk and halo equatorial planes in the [FORMULA] plane. The total gravitational potential is written as

[EQUATION]

where [FORMULA] is the potential due to the inner part of the disk (as discussed below), [FORMULA] is that for the outer disk, and [FORMULA] is that for the halo. The particle orbits in the equilibrium disk are approximately circular with angular rotation rate [FORMULA], where

[EQUATION]

The vertical epicyclic frequency (squared),

[EQUATION]

gives a measure of the restoring force in the [FORMULA]direction.

The surface mass density of the inner (optical) disk is assumed to be [FORMULA] with [FORMULA] and [FORMULA] constants and [FORMULA] the total disk mass. For [FORMULA],

[EQUATION]

(Binney and Tremaine 1987; hereafter BT, p. 409). Note that [FORMULA].

The outer, neutral hydrogen disk can be described approximately by a Fermi function for the surface density

[EQUATION]

where [FORMULA] is the radius inside of which half the neutral hydrogen mass is located, [FORMULA], and [FORMULA]. This dependence provides a good fit to the data of Broeils and Rhee (1966; their Fig. 6a). The total neutral hydrogen mass is [FORMULA].

The halo potential is taken to be 6

[EQUATION]

where [FORMULA] const. is the circular velocity at large distances, [FORMULA] const. is the core radius of the halo, and [FORMULA] is the axial ratio of the equipotential surfaces with [FORMULA] their ellipticity. An oblate (prolate) halo corresponds to [FORMULA] ([FORMULA]). We have

[EQUATION]

0 The halo is usually assumed to be oblate with [FORMULA]-0.2 (Nelson and Tremaine 1995). This gives [FORMULA] which is of interest in the following. For the radii of the warps observed in galaxies ([FORMULA]), [FORMULA]. For [FORMULA], the halo mass within a radius r is [FORMULA]

Perturbations of the galaxy are assumed to consist of small angle tilting ([FORMULA]) of the outer disk with azimuthal mode number [FORMULA] (or -1). The halo and the inner disk are assumed to be unaffected by the perturbation. We can describe the outer disk by a number N of tilted plane circular rings. This description is general for small tilting angles where the linearized equations are applicable. We do not need to assume that the disk is `razor thin' but rather that the disk half-thickness [FORMULA] is small, [FORMULA], and the radial wavelength of the warp is long compared with [FORMULA] (Papaloizou and Lin 1995; Masset and Tagger 1996). The vertical displacements of the disk [FORMULA] are in general much larger than [FORMULA].

2.1. One tilted ring

The tilting of the ring is completely described by the tilt angle [FORMULA], which is the angle between the upward normal to the ring and the [FORMULA]axis, and the azimuthal angle [FORMULA], which is the angle between the line-of-nodes (where the ring intersects the [FORMULA] plane) and the [FORMULA]axis. The geometry is shown in Fig. 3. The perturbation is assumed to be small in the respect that [FORMULA]. The position vector of a point on the ring is

[EQUATION]

(Cartesian components), where [FORMULA] is the height of the ring above the [FORMULA] plane. We have

[EQUATION]

where [FORMULA] is the angle of the tilt about the [FORMULA]axis, and [FORMULA] is the angle about the [FORMULA]axis. We have [FORMULA] and [FORMULA].

[FIGURE] Fig. 3. Geometry of one tilted ring of a disk galaxy. For the case shown, the line-of-nodes is along the [FORMULA]axis, that is, [FORMULA]. Also, [FORMULA], [FORMULA], and [FORMULA] denotes the normal to the ring plane.

The angular momentum of the tilted ring is

[EQUATION]

where [FORMULA], M is the ring mass, and [FORMULA] is its angular rotation rate which is unaffected by the perturbation to first order in [FORMULA]. We have

[EQUATION]

Here, [FORMULA] [FORMULA] is the unit tangent vector to the ring (to first order in [FORMULA]), s is the distance along the ring circumference measured from, say, the ascending node where [FORMULA], and

[EQUATION]

From Eq. (9) we have

[EQUATION]

The torque on the ring is

[EQUATION]

or

[EQUATION]

The ring is assumed to exert no torque on itself. Consequently, the force [FORMULA] (per unit circumference) which contributes to [FORMULA] in Eq. (12a) is due to the inner disk and the halo. Thus we have

[EQUATION]

where [FORMULA] is the vertical epicyclic frequency excluding the contribution due to the outer disk [see Eq. (3)]. Also,

[EQUATION]

where [FORMULA] which excludes the outer disk contribution to [FORMULA] [see Eq. (2)]. Thus we find

[EQUATION]

[EQUATION]

The equations of motion [FORMULA] are 15f

[EQUATION]

0 after multiplying by two and letting [FORMULA] denote the moment of inertia of the ring. The terms of the left-hand side [FORMULA] are due to the Coriolis force.

It is useful to introduce

[EQUATION]

This representation is well-known from treatments of spin precession in quantum mechanics. It was used previously for example by Hatchett, Begelman, and Sarazin (1981) in a treatment of twisted accretion disks around Kerr black holes. We can then combine Eqs. (15a) and (15b) to obtain

[EQUATION]

where

[EQUATION]

Note that we can have [FORMULA] for a prolate halo. Solutions of Eq. (17) can be taken as [FORMULA] with C a complex constant. From Eq. (17) we have the dispersion relation [FORMULA] which gives

[EQUATION]

In all cases considered here, [FORMULA]. For [FORMULA], the plus sign corresponds to the fast mode , [FORMULA], with prograde precession [ [FORMULA] increasing ], and the minus sign to the slow mode , [FORMULA], with retrograde precession [ [FORMULA] decreasing ] (HT). For [FORMULA] and [FORMULA], the slow mode has [FORMULA]. These two modes are the analogues of the normal modes of vibration of a non-rotating mechanical system. That there are two modes rather than one for each ring is due to the Doppler splitting from the rotation ([FORMULA]).

A general solution of Eq. (17) is

[EQUATION]

where [FORMULA] and [FORMULA] are the two roots given in Eq. (19), and [FORMULA] and [FORMULA] are complex constants. The four real quantities determine the initial values [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. If only the fast or slow mode is excited ([FORMULA] or [FORMULA]), then [FORMULA] = const and [FORMULA] const[FORMULA] with [FORMULA]. A simple way to excite only the slow or fast mode is to take [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. Fig. 4 illustrates the two limiting cases of [FORMULA] and [FORMULA].

[FIGURE] Fig. 4. Illustrative single ring orbits for a case shown in a where the slow mode is dominant ([FORMULA]) and in b where the fast mode is dominant ([FORMULA]). For both plots, i corresponds to the starting point, and f to the end point at [FORMULA], and [FORMULA].

We can recast Eqs. (15) in a form equivalent to that given by HT by multiplying Eq. (15a) by [FORMULA] and Eq. (15b) by [FORMULA] and subtracting the two. Noting that [FORMULA] and [FORMULA], we obtain

[EQUATION]

Note that [FORMULA] is non-zero only for r equal to the ring radius. From Eq. (2) we have

[EQUATION]

For a toroidal ring of minor radius [FORMULA], this difference is of the order of [FORMULA]. In the absence of a halo, the right hand side of the equation for h is

[EQUATION]

which is the same as the HT expression

[EQUATION]

The [FORMULA] term comes from the integration of over [FORMULA] while the [FORMULA] term is from the integration over [FORMULA].

Multiplying Eq. (15a) by [FORMULA] and (15b) by [FORMULA] and adding the two gives [FORMULA], where the energy of the tilted ring is

[EQUATION]

where a dot denotes a time derivative. This expression agrees with the result of HT (their Appendix B). In the absence of a halo or with an oblate halo ([FORMULA]), we have [FORMULA], so that both terms in [FORMULA] are non-negative which implies stability of the ring tilting. In the case of a prolate halo with [FORMULA] instability is possible.

In order to understand the stability of the ring tilting, it is useful to consider the influence of a small Newtonian drag or friction on the ring motion, a force [FORMULA] per unit circumference, with [FORMULA] but [FORMULA]. This drag could result from dynamical friction of the ring with the dark halo matter (Nelson and Tremaine 1995).

The full calculation of [FORMULA] due to dynamical friction on a precessing ring is complicated (Nelson and Tremaine 1995), but an estimate based on treating the ring as two point masses [FORMULA] at the points [FORMULA] (Weinberg 1985) gives

[EQUATION]

where [FORMULA] is the Coulomb logarithm, [FORMULA] is the circular velocity, and [FORMULA] accounts for the density profile of the halo, with [FORMULA] the halo core radius. Later, in Sect. 3, where we represent the outer disk as N tilted interacting rings, the pertinent estimate for [FORMULA] for a single ring is this formula with M the total mass of the outer disk rather than the mass of a single ring (Weinberg 1985).

The drag torque due to [FORMULA] is [FORMULA] [FORMULA] and [FORMULA]. Solution of Eq. (17) including [FORMULA] gives

[EQUATION]

Thus, both fast and slow precession modes with [FORMULA] are stable in the presence of friction, whereas the slow precession mode with [FORMULA] is unstable [FORMULA]. For the slow precession mode, the kinetic energy term [FORMULA] in [FORMULA] is smaller than the magnitude of the potential energy term [FORMULA] for [FORMULA]. In the presence of the friction force, [FORMULA]. Thus, for the slow precession mode, [FORMULA] is negative if [FORMULA], and the tilt angle grows. The instability is therefore a negative energy dissipative instability. An analogous instability was predicted and observed for the precession of laboratory collisionless relativistic electron rings (Furth 1965; Beal et al. 1969). The instability also occurs in the presence of a non-Newtonian (nonlinear) friction force on the ring.

A second quadratic integral of the motion of Eqs. (15), analogous to angular momentum (see following), can be obtained by multiplying Eq. (15a) by [FORMULA] and Eq. (15b) by [FORMULA] and adding. This gives [FORMULA], where

[EQUATION]

An analogous constant of the motion exists for cases of two or more rings (see Sects. 2.3, 3, and 4). This constant of the motion is a new result of the present work. For the case of instability, with [FORMULA] complex, [FORMULA]. With the above-mentioned friction force included, we have [FORMULA], which is compatible with Eq. (23) which shows instabililty for [FORMULA] and gives [FORMULA] and [FORMULA].

The second integral (24) can be understood by noting that Eqs. (15) follow from the Lagrangian

[EQUATION]

A canonical transformation to the variables [FORMULA] and [FORMULA] gives

[EQUATION]

Thus the canonical momentum [FORMULA] [FORMULA] [FORMULA] = const [FORMULA] because [FORMULA]. On the other hand, [FORMULA] depends on time. The Hamiltonian [FORMULA] is

[EQUATION]

and [FORMULA] = const in that [FORMULA] has no explicit time dependence.

The [FORMULA]dependent terms of [FORMULA] can evidently be viewed as an effective potential for the [FORMULA] motion. In general (for [FORMULA]), [FORMULA] nutates between a minimum and a maximum value. The angular frequency of the nutation is [FORMULA] so that the nutation period is [FORMULA], where [FORMULA] is the period of the orbit. This can also be seen by noting that with [FORMULA] given in general by Eq. (20), we have [FORMULA] and [FORMULA]. For [FORMULA], the period of the slow mode [FORMULA], while the fast mode period [FORMULA] is slightly longer than the nutation period [FORMULA]. The nutation of the ring is clearly evident in Fig. 4. For [FORMULA] (Fig. 4a), [FORMULA] nutates many ([FORMULA]) times in the period [FORMULA] of motion of [FORMULA] about the origin. On the other hand, for [FORMULA] (Fig. 4b), [FORMULA] has an elliptical path with the azimuth of say the maximum of [FORMULA] precessing slowly in the clockwise direction with a period [FORMULA].

A counter-rotating ring with [FORMULA] behaves in the same way as a co-rotating ring. For [FORMULA], the ring slow mode precession is retrograde relative to the ring particle motion but of course prograde relative to our coordinate system.

2.2. Interpretation of ring observations

Warps of spiral galaxies are deduced from measurements of 21 cm line neutral hydrogen emission. Here, we consider the spectral signature of a single ring in either the slow precession mode or the fast precession mode. The observer is considered to be in the [FORMULA] plane at an angle [FORMULA] to the [FORMULA]axis. The velocity of the ring matter is given by Eq. (10). Thus, the velocity of the HI in the direction of the observer is

[EQUATION]

The spectrum of the ring is [FORMULA] [FORMULA]. The ring motion [FORMULA] enters through the term involving [FORMULA]. For a ring in the slow precession mode with [FORMULA], the ring motion has only a small affect on [FORMULA]. The situation is very different for a ring in the fast precession mode where [FORMULA] if [FORMULA]. In this case the sign of the term in [FORMULA] proportional to [FORMULA] is reversed which corresponds to a change of the line-of-nodes angle [FORMULA] by [FORMULA]. Observations are interpreted assuming negligible ring motion, [FORMULA], and this is wrong if the fast precession mode(s) is excited. The fast precession modes of a system with many rings may disappear from view over a long enough time due to phase-mixing (see Sects. 3.1 and 3.2).

2.3. Two tilted rings

Consider now the case of two tilted plane circular rings of mass [FORMULA] and angular rotation rate [FORMULA] at radii [FORMULA] with [FORMULA]. We have

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] are the tilt angles and [FORMULA] the moments of inertia for the two rings.

To obtain the torque on, say, ring 1, we exclude as before the force due to this ring on itself. Thus the horizontal force on ring 1 is

[EQUATION]

where the [FORMULA] is the gravitational potential due to ring 2, and the 1-subscript on the parenthesis indicates evaluation at [FORMULA]. Note that

[EQUATION]

where 30o

[EQUATION]

0 There is also the vertical force on ring 1, 31

[EQUATION]

where

[EQUATION]

is the vertical force on ring 1 due to ring 2. This integral can be simplified by noting that

[EQUATION]

0 The sin[FORMULA] term in Eq. (31c) does not contribute to the integral (31b). Thus we have

[EQUATION]

Combining terms in Eqs. (12b) and (12c) gives

[EQUATION]

[EQUATION]

where

[EQUATION]

measures the strength of the gravitational interaction between the two rings, and [FORMULA] is defined in Eq. (18). The two terms involving the integral [FORMULA] cancel. Note that [FORMULA] is symmetric between the two rings and that it is non-negative because [FORMULA] [see Eq. (45) and Fig. 5].

[FIGURE] Fig. 5. Dependence of the dimensionless integral [FORMULA] defined in Eq. (45) on [FORMULA]. The dashed lines correspond to approximations discussed in the text.

The torque on ring 2 is found in the same way to be

[EQUATION]

[EQUATION]

The equations of motion are

[EQUATION]

For the limit where the only torques are those due to the two rings, [FORMULA] and [FORMULA], we find

[EQUATION]

which is a necessary result.

We can follow the steps leading to Eq. (22) to obtain [FORMULA] for the energy [FORMULA] of the two tilted rings, where

[EQUATION]

where [FORMULA] and [FORMULA] are the moments of inertia of the two rings. This energy is non-negative and the rings are stable if both [FORMULA] and [FORMULA] are non-negative. Also, in this case the rings are stable in the presence of dissipative forces such as the friction force of Sect. 2.1. Thus a necessary condition for instability with or without dissipation is that [FORMULA] and/or [FORMULA] be negative.

A second quadratic integral of the motion can be obtained by following the steps leading to Eq. (24). We find [FORMULA], where

[EQUATION]

For the case of instability, [FORMULA], [FORMULA], we have [FORMULA]. Thus, a further necessary condition for instability in the absence of dissipation (in addition to [FORMULA] and/or [FORMULA] being negative) is that [FORMULA] and [FORMULA] have different signs; that is, [FORMULA] for [FORMULA]. For the slow precession modes with [FORMULA], this corresponds to counter-rotating rings, [FORMULA].

The Lagrangian for the two ring system is

[EQUATION]

The corresponding Hamiltonian [FORMULA] is equal in value to [FORMULA] of Eq. (38). The second constant of the motion, [FORMULA], where [FORMULA]. Note that [FORMULA] and that [FORMULA] so that [FORMULA].

As in the case of a single ring, it is useful to introduce

[EQUATION]

[EQUATION]

The equations of motion (36) can then be written as

[EQUATION]

[EQUATION]

after dividing through by [FORMULA].

For [FORMULA] and [FORMULA], we get the dispersion relation

[EQUATION]

where

[EQUATION]

with [FORMULA]. Eq. (43) can readily be solved for the four roots. Two of these roots are high-frequency, fast precession modes with [FORMULA], assuming [FORMULA] and [FORMULA]. The other two roots are the slow precession modes with much smaller frequencies, typically, [FORMULA].

In order to get simple analytic results, we first consider the low-frequency modes [FORMULA], assuming [FORMULA] and [FORMULA]. We can then neglect the second time derivatives in Eqs. (42) and the [FORMULA] terms in Eq. (43), and this leads to a quadratic dispersion equation for [FORMULA]. The roots of this equation are

[EQUATION]

Here, [FORMULA] for [FORMULA], are the slow precession frequencies of the two rings in the absence of gravitational interaction between them ([FORMULA]); [FORMULA], and [FORMULA], where [FORMULA] is a frequency which measures the strength of the gravitational interaction between the two rings, and where [FORMULA] are the angular momenta of the two rings. We have

[EQUATION]

where

[EQUATION]

with [FORMULA] For [FORMULA], [FORMULA]. For [FORMULA], [FORMULA]. We assume that [FORMULA], where [FORMULA] is the disk thickness. The dependence of [FORMULA] on [FORMULA] is shown in Fig. 5.

A simple limit of Eq. (44) is that where [FORMULA]. The two roots are then [FORMULA] and [FORMULA]. The zero frequency mode corresponds to both rings tilted by the same angle, [FORMULA], which is the rigid tilt mode (HT). The other mode has [FORMULA]; that is, the rings are tilted in opposite directions for [FORMULA].

We first consider co-rotating rings where the four [FORMULA] roots of Eq. (43) are real in the absence of dissipation. Note that the low frequency roots of Eq. (44) are real for [FORMULA]. A general solution for the motion of the two rings is then

[EQUATION]

where [FORMULA] are the four mentioned frequencies, and [FORMULA] are 8 complex constants. However, only 8 real quantities are needed to specify general initial conditions because the [FORMULA] are related to the [FORMULA] by Eqs. (42). If only a single mode of the system is excited, then [FORMULA] and [FORMULA]. Eqs. (42) then imply that [FORMULA] is real. Because [FORMULA], we conclude that [FORMULA] is real so that [FORMULA] is either 0 or [FORMULA]. That is, the rings are either tilted in the same direction and precess together or they are tilted in opposite directions and also precess together. From Eqs. (42), [FORMULA] [FORMULA].

Fig. 6a shows the [FORMULA] dependence of the frequencies of the two slow-precession modes obtained from Eqs. (43) and (44) for the case where both rings are co-rotating and the halo is oblate . The `slower' of the two modes has [FORMULA], while the other has [FORMULA] as shown in Fig. 6b. The larger gravitational torque in the second case accounts for the faster retrograde precession. As [FORMULA] increases, the frequency of the slower mode approaches

[EQUATION]

and [FORMULA] for [FORMULA], where the [FORMULA] are defined in Eq. (45). This result can be derived from Eqs. (43) or (44). For the case of N rings and sufficiently large [FORMULA], we find in general that the lowest frequency approaches the angular momentum weighted average of the single ring precession frequencies [FORMULA] [Eq. (45a)]. In contrast with the dependence of [FORMULA], [FORMULA] decreases monotonically as [FORMULA] increases.

[FIGURE] Fig. 6. The top panel a shows the dependences of the two slow precession frequencies of two co-rotating rings in an oblate halo on the strength of the gravitational interaction measured by [FORMULA] obtained the full Eq. (43) and the approximate Eq. (44). The frequencies are measured in units of [FORMULA]. For this figure, [FORMULA], [FORMULA], and [FORMULA]. [Note that the two fast precession modes, [FORMULA] and [FORMULA], have frequencies [FORMULA] for [FORMULA], and [FORMULA] for [FORMULA]. The [FORMULA] mode has [FORMULA] and thus involves mainly motion of ring 2, whereas the [FORMULA] mode has [FORMULA] and involves motion mainly of ring 1.] - The bottom panel b shows the dependences of the amplitude ratios [FORMULA] for the two slow precession modes. For small values of [FORMULA], the [FORMULA] mode involves motions mainly of ring 2, whereas the [FORMULA] mode involves motion mainly of ring 1.

From the approximate Eq. (44) for the slow precession modes, we can distinguish weak and strong -coupling limits of two co-rotating rings. The coupling strength is measured by the dimensionless parameter

[EQUATION]

which is symmetrical in the ring indices (unlike [FORMULA]). For weak coupling , [FORMULA], the two rings are affected little by their gravitational interaction, and the two slow mode frequencies are [FORMULA] with [FORMULA] and [FORMULA] with [FORMULA]. In the opposite limit of strong coupling , [FORMULA], the gravitational interaction of the two rings is important, and [FORMULA] is given approximately by Eq. (47) with [FORMULA], and [FORMULA] with [FORMULA].

For nearby rings, [FORMULA], of equal mass [FORMULA] (with [FORMULA] the surface mass density of the disk at r), we have [FORMULA], [FORMULA], and thus

[EQUATION]

Evidently, there is strong (weak) coupling for [FORMULA] ([FORMULA]), where [FORMULA] corresponds to [FORMULA] so that

[EQUATION]

Assuming a flat rotation curve ([FORMULA] const.) and [FORMULA] const., we obtain

[EQUATION]

Rings closer together than [FORMULA] will be strongly coupled.

Fig. 7 shows sample orbits of [FORMULA] and [FORMULA] for two co-rotating rings in an oblate halo for initial conditions [FORMULA] and [FORMULA]. The top panel shows a case of weak coupling (small [FORMULA]) where the line-of-nodes [FORMULA] of the two rings regress essentially independently with the result that [FORMULA] increases linearly with time. The bottom panel is for a case of strong coupling (large [FORMULA]) where the line-of-nodes are phase-locked in the sense that the time average of [FORMULA] is zero. Fig. 8 shows the azimuthal angle difference of the line-of-nodes for the two cases of Fig. 7. In the case of weak coupling, a low frequency initial perturbation excites both low frequency modes which have similar frequencies ([FORMULA]), and this gives the unlocked behavior. For strong coupling, the initial perturbation excites mainly the [FORMULA] mode because [FORMULA], and this gives the locked behavior. For the parameters of Figs. 4 and 5, note that [FORMULA].

[FIGURE] Fig. 7. Illustrative orbits [FORMULA] for two co-rotating rings in an oblate halo for weak (a , top panel) and strong (b , bottom panel) coupling for the same parameters as Fig. 6. The initial conditions are [FORMULA] and [FORMULA]. The points labeled by [FORMULA]s all correspond to the final time [FORMULA] (which is arbitrary). The constants of the motion [FORMULA] and [FORMULA] are constant in the numerical integrations with fractional errors [FORMULA]. For the top panel, [FORMULA] ([FORMULA]), and the rings precess almost independently. For the bottom panel, [FORMULA] ([FORMULA]), and the rings precess together with their lines-of-nodes phase-locked as shown in Fig. 8. The dividing value separating independent and phase-locked precession is [FORMULA] or [FORMULA]. Of course, this value depends on the inital value [FORMULA] for [FORMULA].

[FIGURE] Fig. 8. Azimuthal angle difference of the line-of-nodes of two rings [FORMULA] for cases a and b of Fig. 7. At the final time [FORMULA], for the phase-locked case [FORMULA], whereas for the unlocked case [FORMULA].

Fig. 9a shows the [FORMULA] dependence of the frequencies of the two slow-precession modes obtained from Eq. (43) for the case where both rings are co-rotating and the halo is prolate . (We have adopted the convention of ordering the mode frequencies by their magnitudes for [FORMULA].) Fig. 9b shows that the [FORMULA] mode has [FORMULA]. As in the case of an oblate halo, for large [FORMULA] or [FORMULA], [FORMULA] approaches the limit given by Eq. (47) and [FORMULA].

As in the case of a single ring in a prolate halo potential [see Eq. (23)], small friction torques, with friction coefficients [FORMULA] and [FORMULA], on two co-rotating rings in a prolate halo lead to a negative energy dissipative instability. We find that the slow modes with prograde precession, [FORMULA], are unstable in the presence of dissipation. That is, the [FORMULA] mode in Fig. 9 is unstable (stable) for [FORMULA] [FORMULA] 0.02, whereas the [FORMULA] mode is unstable for all [FORMULA]. The two high frequency modes are prograde but are damped by friction for all [FORMULA]. As in the case of an oblate halo (Fig. 8), we observe phase-locking for sufficiently large values of [FORMULA].

[FIGURE] Fig. 9. The top panel a shows the dependences of the two slow precession frequencies of two co-rotating rings in an prolate halo on the strength of the gravitational interaction measured by [FORMULA] obtained Eq. (43). The frequencies are measured in units of [FORMULA]. For this figure, [FORMULA], [FORMULA], and [FORMULA]. - The bottom panel b shows the dependences of the amplitude ratios [FORMULA] for the two slow precession modes. For small values of [FORMULA], the [FORMULA] mode involves motions mainly of ring 2, whereas the [FORMULA] mode involves motion mainly of ring 1. For large [FORMULA] the [FORMULA] mode frequency approaches the limit given in Eq. (47) and [FORMULA].

Frictional torques between two co-rotating rings in a prolate halo also give rise to a negative energy dissipative instability. Such torques could arise from the viscous interaction between adjacent gaseous rings. These frictional torques do not change the total angular momentum of the two rings. Therefore, these frictional torques, if Newtonian (linear) in nature, are necessarily of the form that they contribute terms

[EQUATION]

on the right hand sides of the two Eqs. (42) respectively, where [FORMULA] is the friction coefficient (with units of angular frequency). For a gas with kinematic shear viscosity [FORMULA] (with units of [FORMULA]) the terms (51) arise from the momentum flux density or stress [FORMULA] due to the different vertical velocities of the two rings as a function of [FORMULA]. This gives [FORMULA], where [FORMULA]. The relevant viscosity is that due to turbulence in the gas [FORMULA] owing to the smallness of the microscopic viscosity. An estimate of [FORMULA] can be made following the proposal of Shakura (1973) and Shakura and Sunyaev (1973) that [FORMULA], where [FORMULA] is the sound speed, [FORMULA] is the half-thickness of the disk, and [FORMULA] is a dimensionless constant thought to be in the range [FORMULA] to 1. In Sects. 3 and 4 we discuss further the frictional torques between adjacent rings.

2.4. Two tilted, counter-rotating rings

Here, we consider the case of two tilted rings which are rotating in opposite directions. This situation is pertinent to observed counter-rotating galaxies (see, for example, Jore, Broeils, and Haynes 1996). From Eq. (44), instability is possible (in the absence of dissipation) only for [FORMULA] in agreement with our discussion of Eq. (39). Thus, consider ring 2 to be counter-rotating ([FORMULA] and [FORMULA]), while ring 1 is co-rotating. The region of instability is bounded by the two curves

[EQUATION]

on which the square root in Eq. (44) is zero. The curves are shown in Fig. 10. For parameters in the region between the two curves there is instability, which is seen to occur only for [FORMULA] and/or [FORMULA] negative. For fixed values of [FORMULA], [FORMULA], and [FORMULA], instability occurs in general ([FORMULA]) for [FORMULA] larger than a critical value but smaller than a second larger critical value. The maximum growth rate is for [FORMULA] (so that [FORMULA]), and it is [FORMULA]. This is a dynamical instability in that it does not depend on dissipation as in the case of a single ring with [FORMULA]. For conditions of maximum growth, [FORMULA], so that [FORMULA] and [FORMULA]. We can thus view the co-rotating ring as `torquing up' the more slowly precessing counter-rotating ring by the gravitational interaction or vice versa.

[FIGURE] Fig. 10. Region of instability of two rings, one co-rotating ([FORMULA]) and the other counter-rotating ([FORMULA]), in a prolate halo potential as discussed below Eq. (51).

Fig. 11a shows the [FORMULA] dependence of the frequencies of the two slow-precession modes obtained from Eq. (43) for the case where ring [FORMULA] is counter-rotating ([FORMULA]) and ring 1 is co-rotating ([FORMULA]) and the halo is oblate . Fig. 11b shows that the mode [FORMULA] with prograde precession involves mainly motion of ring 2, whereas the [FORMULA] mode with retrograde precession involves mainly ring 1. For the conditions studied ([FORMULA]), we find no phase-locking.

[FIGURE] Fig. 11. The top panel a shows the dependences of the two slow precession frequencies of two counter rotating rings ([FORMULA] and [FORMULA]) in an oblate halo on the strength of the gravitational interaction measured by [FORMULA] obtained from Eq. (43). The frequencies are measured in units of [FORMULA]. For this figure, [FORMULA], [FORMULA], and [FORMULA]. - The bottom panel b shows the dependences of the amplitude ratios [FORMULA] for the two slow precession modes.

2.5. Three tilted rings

Extension of the results of Sect. 2.2 to the case of three tilted rings is straightforward. Fig. 12a shows the dependence of the three slow precession frequencies on the strength of the gravitational interaction measured by [FORMULA] with [FORMULA] const. For increasing [FORMULA], the slowest mode ([FORMULA]) slowly approaches the limiting frequency given by the generalization of Eq. (47) to three rings. The other two frequencies decrease monotonically with increasing [FORMULA]. Fig. 12b shows the geometrical nature of the three slow precession modes. Fig. 13 shows sample orbits [FORMULA] for three rings for cases of weak and strong coupling. Fig. 14 shows different cases of phase-locking for a three ring system.

[FIGURE] Fig. 12. The top panel a shows the dependence of the three slow precession frequencies of a three ring system on the strength of the gravitational interaction measured by [FORMULA]. The frequencies are measured in units of [FORMULA]. For this figure, [FORMULA] [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. For small values of [FORMULA], the [FORMULA] mode involves motion mainly of ring 3, the [FORMULA] mode motion of ring 2, and the [FORMULA] mode motion of ring 1. For larger values of [FORMULA] the motions of the different rings become coupled. For large [FORMULA] the slowest mode ([FORMULA]) approaches the limiting value given by the generalization of Eq. (47) to three rings. [The three fast precession modes have frequencies [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA].] - The bottom panel b shows the r-dependence of the vertical displacements h (along a line [FORMULA] to the line-of-nodes) for the three slow precession modes for [FORMULA]. The vertical scale is arbitrary except for the condition [FORMULA]. Smooth lines have been drawn through the calculated values indicated by circles.

[FIGURE] Fig. 13. Illustrative orbits [FORMULA] for three co-rotating rings in an oblate halo for weak (a , top panel) and strong (b , bottom panel) coupling for the same parameters as Fig. 12. The initial conditions are [FORMULA] and [FORMULA]. The points labeled by [FORMULA]s all correspond to the final time [FORMULA] (which is arbitrary). For the top panel, [FORMULA] and the rings precess almost independently. For the bottom panel, [FORMULA] and the rings precess together with their lines-of-nodes phase-locked as shown in Fig. 14.

[FIGURE] Fig. 14. Azimuthal angles [FORMULA] of the line-of-nodes for three rings for cases of: a  weak ([FORMULA]), b  intermediate ([FORMULA]), and c  strong coupling ([FORMULA]).

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Online publication: September 17, 1998
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