4. Continuum limit
In the continuum limit, the Hunter and Toomre (1969) equation for the vertical displacement of the disk can be obtained from Eq. (55) by recalling that . We omit for the moment the damping terms (56) and (57). Multiplying Eq. (55) by and taking the imaginary part gives
where the left hand side of this equation follows from the same steps as for Eq. (21), where , and where with taken as a constant for simplicity. The continuum limit is and .
From the definitions (14) and (18), we have on the right hand side of Eq. (64), . From the decomposition of the potential (Sect. 1), it is clear that is the contribution to the disk rotation from the outer disk; that is,
where , and where the integration is over the outer part of the disk, , with chosen to be significantly larger than the scale-length of the inner disk (see Sect. 2). Note that the summations in Eq. (64) are also over the outer part of the disk. Thus, one term which contributes to the right hand side of Eq. (64) is
where , and the factor in the denominator to the power is the same as in Eq. (65). Using the definition of in Eq. (34), the first sum on the right hand side of Eq. (64) can be converted to an integral in the continuum limit,
Similarly, the continuum limit of the second sum in Eq. (64) gives
in view of Eq. (31).
Combining Eqs. (64)-(66) and dropping the j subscripts gives
where . Note that includes the vertical restoring force of the inner disk and the halo. Thus, Eq. (67) is the same as Hunter and Toomre's (1969) equation for h, in that they did not separate out the inner disk and did not include a halo potential.
The Newtonian drag term (56) due to dynamical friction gives a contribution
on the right hand side of Eq. (67). The damping term (57) due to relative friction between the rings gives a contribution
on the right hand side of Eq. (67).
Multiplying Eq. (67) by , integrating over the outer disk, and following the steps of HT (their Appendix B) gives, in the absence of dissipation, the constant of the motion
where is given by Eq. (30b). Substituting from Eq. (8), one readily finds that is identical to of Sect. 3. Note that is non-negative and therefore the disk is stable if .
Including the dissipation terms (68) and (69) gives
The relative friction is of course zero if is a function only of time which corresponds to a rigid tilt. Relative friction in warped accretion disks has been discussed earlier by Pringle (1992). However, we have not found a simple correspondence between his and the present work.
The second constant of the motion , found in Sect. 2.3 and Sect. 3 [Eq. (53c)], can of course be expressed in terms of in the continuum limit. Instead, we obtain this new constant directly from the HT equation for h by multiplying Eq. (67) by and integrating over the outer disk. This gives
and in the continuum limit, in the absence of dissipation. An anonymous referee has pointed out that our is the same as the constant found by Nelson and Tremaine (1995). With dissipation included, we have not found a simple expression for .
© European Southern Observatory (ESO) 1998
Online publication: September 17, 1998