## 2. Tilting of disk galaxiesThe 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 and Cartesian coordinate systems with the disk and halo equatorial planes in the plane. The total gravitational potential is written as where is the potential due to the inner part of the disk (as discussed below), is that for the outer disk, and is that for the halo. The particle orbits in the equilibrium disk are approximately circular with angular rotation rate , where The vertical epicyclic frequency (squared), gives a measure of the restoring force in the direction. The surface mass density of the inner (optical) disk is assumed to be with and constants and the total disk mass. For , (Binney and Tremaine 1987; hereafter BT, p. 409). Note that . The outer, neutral hydrogen disk can be described approximately by a Fermi function for the surface density where is the radius inside of which half the neutral hydrogen mass is located, , and . This dependence provides a good fit to the data of Broeils and Rhee (1966; their Fig. 6a). The total neutral hydrogen mass is . The halo potential is taken to be 6 where const. is the circular velocity at large distances, const. is the core radius of the halo, and is the axial ratio of the equipotential surfaces with their ellipticity. An oblate (prolate) halo corresponds to (). We have 0 The halo is usually assumed to be oblate with
-0.2 (Nelson and Tremaine 1995). This gives
which is of interest in the following. For the
radii of the warps observed in galaxies (),
. For , the halo mass
within a radius Perturbations of the galaxy are assumed to consist of small angle
tilting () of the outer disk with azimuthal mode
number (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 ## 2.1. One tilted ringThe tilting of the ring is completely described by the tilt angle , which is the angle between the upward normal to the ring and the axis, and the azimuthal angle , which is the angle between the line-of-nodes (where the ring intersects the plane) and the axis. The geometry is shown in Fig. 3. The perturbation is assumed to be small in the respect that . The position vector of a point on the ring is (Cartesian components), where is the height of the ring above the plane. We have where is the angle of the tilt about the axis, and is the angle about the axis. We have and .
The angular momentum of the tilted ring is where , Here, is the unit
tangent vector to the ring (to first order in ),
From Eq. (9) we have The torque on the ring is or The ring is assumed to exert no torque on itself. Consequently, the force (per unit circumference) which contributes to in Eq. (12a) is due to the inner disk and the halo. Thus we have where is the vertical epicyclic frequency excluding the contribution due to the outer disk [see Eq. (3)]. Also, where which excludes the outer disk contribution to [see Eq. (2)]. Thus we find The equations of motion are 15f 0 after multiplying by two and letting denote the moment of inertia of the ring. The terms of the left-hand side are due to the Coriolis force. It is useful to introduce 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 where Note that we can have for a prolate halo.
Solutions of Eq. (17) can be taken as with
In all cases considered here, . For
, the plus sign corresponds to the A general solution of Eq. (17) is where and are the two roots given in Eq. (19), and and are complex constants. The four real quantities determine the initial values , , , and . If only the fast or slow mode is excited ( or ), then = const and const with . A simple way to excite only the slow or fast mode is to take , , , and . Fig. 4 illustrates the two limiting cases of and .
We can recast Eqs. (15) in a form equivalent to that given by HT by multiplying Eq. (15a) by and Eq. (15b) by and subtracting the two. Noting that and , we obtain Note that is non-zero only for For a toroidal ring of minor radius , this
difference is of the order of . In the absence
of a halo, the right hand side of the equation for which is the same as the HT expression The term comes from the integration of over while the term is from the integration over . Multiplying Eq. (15a) by and (15b) by and adding the two gives , where the energy of the tilted ring is 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 (), we have , so that both terms in are non-negative which implies stability of the ring tilting. In the case of a prolate halo with 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 per unit circumference, with but . This drag could result from dynamical friction of the ring with the dark halo matter (Nelson and Tremaine 1995). The full calculation of 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 at the points (Weinberg 1985) gives where is the Coulomb logarithm,
is the circular velocity, and
accounts for the density profile of the halo,
with the halo core radius. Later, in Sect. 3,
where we represent the outer disk as The drag torque due to is and . Solution of Eq. (17) including gives Thus, both fast and slow precession modes with
are stable in the presence of friction,
whereas the slow precession mode with is
unstable . For the slow precession mode, the
kinetic energy term in
is smaller than the magnitude of the potential energy term
for . In the presence
of the friction force, . Thus, for the slow
precession mode, is negative if
, and the tilt angle grows. The instability is
therefore a A second quadratic integral of the motion of Eqs. (15), analogous to angular momentum (see following), can be obtained by multiplying Eq. (15a) by and Eq. (15b) by and adding. This gives , where 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 complex, . With the above-mentioned friction force included, we have , which is compatible with Eq. (23) which shows instabililty for and gives and . The second integral (24) can be understood by noting that Eqs. (15) follow from the Lagrangian A canonical transformation to the variables and gives Thus the canonical momentum = const because . On the other hand, depends on time. The Hamiltonian is and = const in that has no explicit time dependence. The dependent terms of
can evidently be viewed as an effective
potential for the motion. In general (for
), nutates between a
minimum and a maximum value. The angular frequency of the nutation is
so that the A counter-rotating ring with behaves in the same way as a co-rotating ring. For , 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 observationsWarps 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 plane at an angle to the 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 The spectrum of the ring is . The ring motion enters through the term involving . For a ring in the slow precession mode with , the ring motion has only a small affect on . The situation is very different for a ring in the fast precession mode where if . In this case the sign of the term in proportional to is reversed which corresponds to a change of the line-of-nodes angle by . Observations are interpreted assuming negligible ring motion, , 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 ringsConsider now the case of two tilted plane circular rings of mass and angular rotation rate at radii with . We have where and are the tilt angles and 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 where the is the gravitational potential due to ring 2, and the 1-subscript on the parenthesis indicates evaluation at . Note that where 30o 0 There is also the vertical force on ring 1, 31 where is the vertical force on ring 1 due to ring 2. This integral can be simplified by noting that 0 The sin term in Eq. (31c) does not contribute to the integral (31b). Thus we have Combining terms in Eqs. (12b) and (12c) gives where measures the strength of the gravitational interaction between the two rings, and is defined in Eq. (18). The two terms involving the integral cancel. Note that is symmetric between the two rings and that it is non-negative because [see Eq. (45) and Fig. 5].
The torque on ring 2 is found in the same way to be The equations of motion are For the limit where the only torques are those due to the two rings, and , we find which is a necessary result. We can follow the steps leading to Eq. (22) to obtain for the energy of the two tilted rings, where where and are the moments of inertia of the two rings. This energy is non-negative and the rings are stable if both and 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 and/or be negative. A second quadratic integral of the motion can be obtained by following the steps leading to Eq. (24). We find , where For the case of instability, , , we have . Thus, a further necessary condition for instability in the absence of dissipation (in addition to and/or being negative) is that and have different signs; that is, for . For the slow precession modes with , this corresponds to counter-rotating rings, . The Lagrangian for the two ring system is The corresponding Hamiltonian is equal in value to of Eq. (38). The second constant of the motion, , where . Note that and that so that . As in the case of a single ring, it is useful to introduce The equations of motion (36) can then be written as after dividing through by . For and , we get the dispersion relation where with . Eq. (43) can readily be solved for
the four roots. Two of these roots are high-frequency, In order to get simple analytic results, we first consider the low-frequency modes , assuming and . We can then neglect the second time derivatives in Eqs. (42) and the terms in Eq. (43), and this leads to a quadratic dispersion equation for . The roots of this equation are Here, for , are the slow precession frequencies of the two rings in the absence of gravitational interaction between them (); , and , where is a frequency which measures the strength of the gravitational interaction between the two rings, and where are the angular momenta of the two rings. We have where with For , . For , . We assume that , where is the disk thickness. The dependence of on is shown in Fig. 5. A simple limit of Eq. (44) is that where . The two roots are then and . The zero frequency mode corresponds to both rings tilted by the same angle, , which is the rigid tilt mode (HT). The other mode has ; that is, the rings are tilted in opposite directions for . We first consider co-rotating rings where the four roots of Eq. (43) are real in the absence of dissipation. Note that the low frequency roots of Eq. (44) are real for . A general solution for the motion of the two rings is then where are the four mentioned frequencies,
and are 8 complex constants. However, only 8
real quantities are needed to specify general initial conditions
because the are related to the
by Eqs. (42). If only a Fig. 6a shows the 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
and for , where the
are defined in Eq. (45). This result can be
derived from Eqs. (43) or (44). For the case of
From the approximate Eq. (44) for the slow precession modes, we can
distinguish which is symmetrical in the ring indices (unlike
). For For nearby rings, , of equal mass
(with the surface mass
density of the disk at Evidently, there is strong (weak) coupling for (), where corresponds to so that Assuming a flat rotation curve ( const.) and const., we obtain Rings closer together than will be strongly coupled. Fig. 7 shows sample orbits of and
for two co-rotating rings in an oblate halo
for initial conditions and
. The top panel shows a case of weak coupling
(small ) where the line-of-nodes
of the two rings regress essentially
independently with the result that increases
linearly with time. The bottom panel is for a case of strong coupling
(large ) where the line-of-nodes are
Fig. 9a shows the 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
As in the case of a single ring in a prolate halo potential [see Eq. (23)], small friction torques, with friction coefficients and , 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, , are unstable in the presence of dissipation. That is, the mode in Fig. 9 is unstable (stable) for 0.02, whereas the mode is unstable for all . The two high frequency modes are prograde but are damped by friction for all . As in the case of an oblate halo (Fig. 8), we observe phase-locking for sufficiently large values of .
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 on the right hand sides of the two Eqs. (42) respectively, where is the friction coefficient (with units of angular frequency). For a gas with kinematic shear viscosity (with units of ) the terms (51) arise from the momentum flux density or stress due to the different vertical velocities of the two rings as a function of . This gives , where . The relevant viscosity is that due to turbulence in the gas owing to the smallness of the microscopic viscosity. An estimate of can be made following the proposal of Shakura (1973) and Shakura and Sunyaev (1973) that , where is the sound speed, is the half-thickness of the disk, and is a dimensionless constant thought to be in the range to 1. In Sects. 3 and 4 we discuss further the frictional torques between adjacent rings. ## 2.4. Two tilted, counter-rotating ringsHere, 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 in agreement with our discussion of Eq. (39). Thus, consider ring 2 to be counter-rotating ( and ), while ring 1 is co-rotating. The region of instability is bounded by the two curves 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
and/or negative. For
fixed values of , , and
, instability occurs in general
() for larger than a
critical value but smaller than a second larger critical value. The
maximum growth rate is for (so that
), and it is . This is a
Fig. 11a shows the dependence of the
frequencies of the two slow-precession modes obtained from Eq. (43)
for the case where ring is counter-rotating
() and ring 1 is co-rotating
() and the halo is
## 2.5. Three tilted ringsExtension 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 with
const. For increasing ,
the slowest mode ()
© European Southern Observatory (ESO) 1998 Online publication: September 17, 1998 |