Introduction

In the conventional understanding of galaxies embedded in a kinematically hot spheroidal or triaxial halos of dark matter, the frequently observed pronounced outer warps of neutral hydrogen, interstellar dust and, to a lesser noticeable extent, optical disks set a challenge for explaining their presumably long-lived existence, and particularly, their frequent quasi-straight line of nodes (LON), or pair of quasi-straight LON (Briggs [1990]). At least half the spirals possess a detectable warped HI and stellar disk(e.g., Briggs [1990], and more should remain undetected due to projection confusion.

Hypotheses like cosmic infall (Jang & Binney [1999]), gravitational interactions (Hernquist [1991]; Weinberg [1995], [1998]), normal modes (Sparke [1984]; Sparke & Casertano [1988]), misaligned dark halos (Kuijken [1991]; Dubinski & Kuijken [1995]; Debattista & Sellwood [1999]) or magnetic fields (Battaner et al. [1990]) have been proposed, without clearly satisfying all the constraints provided by the observations. See Binney ([1992]) or Kuijken ([2000]) for reviews.

But, as argued by Arnaboldi et al. ([1997]), Becquaert & Combes ([1998]) and Reshetnikov & Combes ([1998]), also in the context of polar rings, the evidence for non-self-gravitating HI disks is actually weak. Instead, the opposite assumption of coexistence of warps containing most of the dynamical mass is not in contradiction with the available observations. Such an assumption of thick and flaring disks of visible and dark matter in the form of cold gas approximately proportional to neutral hydrogen has been proposed by Pfenniger et al. ([1994]) to account for many more known facts about spirals.

For the warp problem, the self-gravitating disk assumption is attractive for several reasons. HI observations reveal that most of the galaxies possess an inner flat disk, while the warp develops beyond a specific radius (Briggs [1990]; Burton [1992]). Such a comportment, which is the signature of a radical change of the dynamic along the radius, can hardly be explained by a hot halo which would impose a uniform dynamic over a larger scale. In contrast, a strong rotational support in the dark matter component offers a natural explanation for the straight lines of nodes. They would result as a natural consequence of the weak diffusive properties of angular momentum in a self-gravitating disk: if, following for example an accretion event, angular momentum is deposited in a self-gravitating disk, bending its outer parts, the angular momentum will slowly diffuse across the whole disk in a monotonous way, because angular momentum is a quasi conserved quantity. As its diffusion is slow, the line of nodes must be quasi-straight and persist for many rotational periods. Finally, self-gravitating optical disks are in agreement with the often stated maximum disk interpretation of observations of the Milky Way and other spirals (see for example Gerhard [2000]).

In numerous models of warped galaxies, and also polar ring models, stars are assumed to move along circular rings which are tilted as a function of radius. Because stars are driven by periodic orbits, which are the backbone of galaxies, such models implicitly presuppose the existence of stable circular tilted periodic orbits. Yet, to our knowledge they have never been verified, and do not appear obvious in non-spherical geometries. It is far from obvious that a perturbation in the form of a warp will conserve the well known circular orbits of a flat disk. In order to verify the latter assumptions, it is useful to find the exact periodic orbits existing in a warped disk.

Moreover, understanding the stable periodic orbits allows us to grasp the basic properties of the other quasi-periodic orbits in an efficient way, particularly when no analytical tools exist. Such an approach has been very useful in the context of barred galaxies for which the complexity of motion is substantial (e.g., Contopoulos & Papayannopoulos [1980]; Pfenniger [1984]).

For studying the principal periodic orbits in a galactic potential, even rough representations of the potential are sufficient to yield the basic properties of the main periodic orbits, as that shown in numerous situations. In this work, the disk autogravitation will play a dominant role. Therefore, the following results will be applicable as long as the local density is dominated by the disk.

The alternative case would be the existence of a hot dark matter halo. In this case, one must admit a center dominated by the halo density, because the density of a warm gravitating system increases at the center. The following implications should be explored:

1)

A hot halo is aligned with the central stellar disk. The outer gaseous disk acts as a tracer of the main periodic orbits in the halo. The only reason for the disk to warp is then to follow the orbits generated by a vertical 1:1 resonance. But in spheroidal halos, stable warped orbits do not exist due to the invariance in azimuth of the potential. To avoid this problem, one has to take in account both the mass of the halo and the self-gravity of the disk (Sparke [1984]; Sparke & Casertano [1988]). In triaxial halos more periodic 1:1 orbit families exist. 1:1 resonances are strong only in rapidly rotating triaxial potentials (e.g. Mulder [1983]), and associated orbits would be compatible with warps only outside the corotation radius: the dynamics would resemble the one of a rescaled triaxial bar (Pfenniger [1984]). But such triaxial halos appear difficult to reconcile with the axisymmetrization that follows the inclusion of even a weak fraction of dissipative matter (Dubinski [1994]).

2)

The gaseous disk is aligned with the outer part of a hot halo, and the optical disk is tilted. This case appears dynamically highly unstable because the optical disks appear maximum (cf. Gerhard 2000), i.e., self-gravitating to a large extent.

3)

The halo follows the warp of the disk. Such a halo can survive if it possesses enough angular momentum and therefore should be flattened. This case becomes very similar to the case of the massive disk, and the comportment of the periodic orbits is not changed dramatically.

In this paper we present succinctly a study of periodic orbits in a simple model of warped disk galaxies, in which the stability and consistency with the density distribution is also determined. The influence of a global rotation of the warp pattern is examined too. A subsequent study will investigate the orbits in self-consistent N-body models of warps.