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

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4. Continuum limit

In the continuum limit, the Hunter and Toomre (1969) equation for the vertical displacement of the disk [FORMULA] can be obtained from Eq. (55) by recalling that [FORMULA] [FORMULA]. We omit for the moment the damping terms (56) and (57). Multiplying Eq. (55) by [FORMULA] and taking the imaginary part gives

[EQUATION]

where the left hand side of this equation follows from the same steps as for Eq. (21), where [FORMULA], and where [FORMULA] with [FORMULA] taken as a constant for simplicity. The continuum limit is [FORMULA] and [FORMULA].

From the definitions (14) and (18), we have on the right hand side of Eq. (64), [FORMULA]. From the decomposition of the potential (Sect. 1), it is clear that [FORMULA] is the contribution to the disk rotation from the outer disk; that is,

[EQUATION]

where [FORMULA], and where the integration is over the outer part of the disk, [FORMULA], with [FORMULA] chosen to be significantly larger than the scale-length [FORMULA] 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

[EQUATION]

where [FORMULA], and the factor in the denominator to the [FORMULA] power is the same as in Eq. (65). Using the definition of [FORMULA] in Eq. (34), the first sum on the right hand side of Eq. (64) can be converted to an integral in the continuum limit,

[EQUATION]

Similarly, the continuum limit of the second sum in Eq. (64) gives

[EQUATION]

in view of Eq. (31).

Combining Eqs. (64)-(66) and dropping the j subscripts gives

[EQUATION]

[EQUATION]

where [FORMULA]. Note that [FORMULA] 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

[EQUATION]

on the right hand side of Eq. (67). The damping term (57) due to relative friction between the rings gives a contribution

[EQUATION]

on the right hand side of Eq. (67).

Multiplying Eq. (67) by [FORMULA], 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

[EQUATION]

where [FORMULA] is given by Eq. (30b). Substituting [FORMULA] from Eq. (8), one readily finds that [FORMULA] is identical to [FORMULA] of Sect. 3. Note that [FORMULA] is non-negative and therefore the disk is stable if [FORMULA].

Including the dissipation terms (68) and (69) gives

[EQUATION]

The relative friction is of course zero if [FORMULA] 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 [FORMULA], found in Sect. 2.3 and Sect. 3 [Eq. (53c)], can of course be expressed in terms of [FORMULA] in the continuum limit. Instead, we obtain this new constant directly from the HT equation for h by multiplying Eq. (67) by [FORMULA] and integrating over the outer disk. This gives

[EQUATION]

and [FORMULA] in the continuum limit, in the absence of dissipation. An anonymous referee has pointed out that our [FORMULA] is the same as the constant [FORMULA] found by Nelson and Tremaine (1995). With dissipation included, we have not found a simple expression for [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: September 17, 1998
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