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

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1. Introduction

The large-scale integral-sign warps of the outer regions of spiral galaxies has been a long-standing theoretical puzzle. The strongest warps are observed in the HI disks when these extend well beyond the optical disks (Briggs 1990). The warps are antisymmetric and consequently can be well-fitted by the kinematic tilted-ring model of Rogstad, Lockhart, and Wright (1974). For a set of 12 galaxies with high-quality data on the extended HI disks, Briggs (1990) found empirically a set of `rules of behavior' of warps: (1) The warps develop with increasing radius r for r between [FORMULA] and [FORMULA] (the Holmberg radius, [FORMULA]), where the subscript denotes the surface brightness in B in mag/arcsec2; (2) The line-of-nodes of the warp is straight for [FORMULA]; and (3) For [FORMULA], the line-of-nodes forms an open leading spiral. An illustrative case is shown in Fig. 1 which gives a surface plot of the warp of the disk of the spiral galaxy M 83 obtained from the data of Briggs (1990). The rotation of M 83 is clockwise so that the warp is a leading spiral wave. Fig. 2 also shows the data of Briggs (1990) for M 83, in panel (a) the radial (r) dependence of the tilt angle [FORMULA], in (b) the r-dependence of the angle of the line-of-nodes [FORMULA], and in (c) the dependence of [FORMULA] on [FORMULA], which we refer to as a Briggs plot. A different behavior is observed for the galaxy NGC 3718 which shows an approximately straight line-of-nodes out to [FORMULA] and a large tilt angle ([FORMULA]) at this radius (Briggs 1990).

[FIGURE] Fig. 1. Surface plot of the warped disk of M 83 (NGC 5236) obtained from the data of Briggs (1990). The rotation of the galaxy is clockwise so that the warp of the disk is a leading spiral wave (Briggs 1990). The scales are in units of 10 kpc for a Holmberg radius of M 83 of [FORMULA] arcmin and a distance of M 83 of 5.9 Mpc which assumes a Hubble constant [FORMULA] km/s/Mpc (Briggs 1996, private communication). The gray scale is in 16 steps uniformly spaced between [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. Nature of the warp of the disk of M 83 from Briggs (1990). The top panel a shows the radial (r) dependence of the tilt angle [FORMULA]. The middle panel b shows the r-dependence of the angle of the line-of-nodes [FORMULA]. The bottom panel c shows the dependence of [FORMULA] on [FORMULA], which we refer to as a Briggs plot. The radii are in units of 10 kpc for the parameters mentioned in the caption of Fig. 1. In these units the Holmberg radius is [FORMULA]. In panel b , the dashed curve shows the least-square-fit of a power law dependence to the curve for [FORMULA] which gives [FORMULA] (with an R value of 0.993). Note that in the Briggs plot [FORMULA] has been taken to increase in the clockwise direction to account for the fact that the galaxy rotates in the clockwise direction. Except for our dicussion of counter rotating rings, we assume that the matter of the rings (representing the galaxy disk) rotates in the counter-clockwise direction. Note that the data points given here have only an approximate correspondence with those given by Briggs in that we have scanned and digitized the data from his figures.

A partial listing of the theoretical works on warps of disk galaxies include Kahn and Woltjer (1959), Hunter and Toomre (1969; hereafter HT), Toomre (1983), Sparke and Casertano (1988); Hunter (1994), Merritt and Sellwood (1994), and Nelson and Tremaine (1995). Since the pioneering analysis by Hunter and Toomre (1969) and the earlier related work by Lynden-Bell (1965), the theoretical studies have focused mainly on determining the eigenmodes and eigenfrequencies of warped self-gravitating disks. The studies used the HT dynamical equation for the vertical displacement of the disk surface [FORMULA], where [FORMULA] is the azimuthal angle.

In contrast with most of these earlier works, the present study investigates different initial value problems of a warped disk in an oblate (or prolate) halo potential, and it develops a different representation for the disk warping. Instead of describing the disk in terms of [FORMULA], we develop a representation in terms of N independently tilted, self-gravitating, concentric rings which is suggested by the kinematic ring model of Rogstad et al. (1974). Toomre (1983) and May and James (1984) mention a model of this kind, but they give no details. We show that the [FORMULA]ring representation is equivalent to the HT equation for [FORMULA] for [FORMULA]. However, new insight is given by the [FORMULA]ring model: The notion of the phase-locking of the line-of-nodes of nearby rings due to self-gravity is demonstrated. We propose that this explains Briggs's rule No. 2. Furthermore, treatment of initial value problems for the age of a galaxy can be done with sufficient accuracy with relatively small values of N ([FORMULA]). Thus our approach is analogous to that of Contopoulos and Grosbol (1986, 1988) where self-consistent galaxy models are constructed from a finite number of stellar orbits.

In Sect. 2 we first give basic parameters of the disk and the halo. In Sect. 2.1 we treat the case of one tilted ring and in Sect. 2.2 comment on the interpretation of observations of a ring. Sect. 2.3 considers the case of two tilted rings. Sect. 2.4 treats the case of two tilted, counter-rotating rings. Sect. 2.5 treats the case of three corotating rings. Sect. 3 treats the general case of N tilted corotating rings. In this section we consider different possible excitations or origins of warps - that due to a passing satellite (Sect. 3.1), a sinking satellite (Sect. 3.2), a tilted halo potential (Sect. 3.3), and an initially tilted outer disk plane (Sect. 3.4). Sect. 4 discusses the continuum limit where [FORMULA]. Sect. 5 gives conclusions of this work.

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Online publication: September 17, 1998