Annu. Rev. Astron. Astrophys. 1985. 23: 147-168 Copyright © 1985 by Annual Reviews. All rights reserved

4.2. Theoretical Considerations

The formation of two disks at nearly right angles seems unlikely to happen in a single collapse event, thus the atypical disk (i.e. the material in the polar orbits) must be ascribed to a second event occurring much later, possibly quite recent. Two candidates for second events have been considered (116). The first of these is accretion of a companion galaxy. Such a hypothesis is necessary for fairly isolated galaxies. Presumably the dwarf companion would start out in a near-polar orbit and be captured as a result of dynamical friction. In (116) it is assumed that the stellar component will disperse into the halo around the main disk, whereas the gas and dust will settle into a new disk with an angular momentum vector roughly perpendicular to the original orbital plane. Thus polar rings would consist of (a) the gas and dust of the captured companion and (b) stars formed since the capture, in agreement with what is observed in several cases.

The other candidate for the second event is mass transfer during encounters. Such a process is frequently noted in numerical simulations (e.g. 140) and is suggested by pictures in Arp's Atlas of Peculiar Galaxies (2), e.g. Arp 87 and Arp 238-240. It has recently been suggested that polar rings are in the process of forming around M82 (116) and NGC 1023 (98).

Are polar rings long lived or short lived? Before answering this question, we must look at the possible steady-state configurations for gas and dust in various potentials and geometries. Two methods have been used so far for this. One is the straightforward calculation of families of periodic nonintersecting orbits in a suitable potential (34, 54, 55, 71, 80). The other is the preferred plane' method, in which a perturbation analysis provides the location of the stable preferred orbital planes of gas (stable equilibrium configurations) in a near-spherical potential. Orbits not in the preferred planes will process about them, with those at smaller radii precessiog more rapidly, and an initially planar disk will be twisted (58). This will lead to cloud-cloud collisions, so that the gas dissipates and eventually settles into one of the preferred planes.

The preferred planes are calculated by finding extrema in the relevant Hamiltonian (35, 38, 126-129, 136). This method is in principle valid only for near-circular orbits in near-spherical potentials, but comparison (129) of its results with those of direct orbit calculations (55) shows that it may be further extended. Although a number of the results that we use have been or can be derived from periodic orbit calculations, we adopt the preferred-plane method here as more suited to our particular needs. Most conveniently, it yields analytical results and thus permits an estimate of the fraction of all possible orbit orientations leading to gas settling in a given preferred plane ("capture area"). The settling time itself depends on the amount of differential precession and dissipation.

A static or slowly tumbling oblate (prolate) figure has a preferred plane perpendicular to the short (long) axis. In the static or near-static triaxial surface figure, there are two preferred planes, one perpendicular to the long axis and the other perpendicular to the short one, while the plane perpendicular to the intermediate axis is not a preferred plane (cf. 54). A small deviation from axial symmetry is sufficient to have a sizable capture area for stable polar orbits (126). Thus one would expect more polar rings to be found around SBO galaxies than SAO galaxies.

For a tumbling triaxial figure, several preferred planes exist (38, 127, 129, 136). Their number and orientations are a function of the orientation of the tumble axis, the tumbling speed, and the radius. In the outermost region there are only two preferred planes, one parallel and one antiparallel to the tumble axis of the galaxy. The innermost and intermediate regions have a more complicated behavior, and one can find preferred planes that are initially parallel but that further out bend gradually toward a configuration perpendicular to the tumble axis. These planes are called anomalous, since they correspond to the anomalous orbits (55). Thus, continuous warped surfaces as found, for example, in some dust lanes in elliptical galaxies can be understood (cf. 142).

How do these results apply to polar rings? The underlying galaxy has been shown in the three cases studied to be an S0. Unless we are prepared to consider such extreme cases as a prolate halo or a halo tumbling along an axis in or close to the plane of the disk, we are left with only two alternatives. The first is that the galaxy potential is oblate and the polar rings are unstable. In this case, either the time it takes them to settle to the preferred planes is very long, of the order of or longer than a Hubble time, or the number of polar rings formed is large enough to account for the few percent observed despite their short settling times. The second alternative is that the galaxy is somewhat triaxial and nontumbling. This explanation is very attractive, since exact axisymmetry might well be a theoretician's dream, and since the majority of observed galaxies show sizable nonaxisymmetric deviations in their main plane (albeit mainly in their inner parts). Yet it should not be forgotten that polar rings are rare objects, while captures of dwarfs, of intergalactic gas, or of gaseous fragments of other galaxies are believed to be quite common. On the other hand, polar rings can be destroyed by violent tides from nearby passages of other galaxies. An estimate of all these factors would set a limit on the allowable triaxiality of the galaxy potential.

Varying estimates of the settling time for gas initially in a plane at a given angle to a preferred plane have been reported (49, 125, 128). These differences, which can be as much as an order of magnitude, are due to the different rotation curves and different modeling of the interstellar medium used. What matters here is not the settling time itself, but rather the "coherence time" beyond which the initially planar disks are caused by differential precession to deviate from a plane by an amount larger than what is permissible by observations. This is shorter than the settling time, and for thick disks or annuli, as in A0136-0801, it is quite short (of the order of or less than a few Gyr) unless the galaxy is nearly spherical. Thus if galaxies are oblate with a considerable eccentricity, then the capture of gas into polar orbits must be a common occurrence.

If the main body of the galaxy is oblate and the polar ring is transient, then the inner radii of the rings should show signs of the instability first, and one would expect the formation of an inner edge with an odd warped shape. On the other hand, if galaxies are triaxial, then the observed well-defined inner edges can be explained, for example, by the interaction of the gas in the ring with that in the disk.

Cases with two rings at an angle to each other, such as ESO 474-G26, can be understood with the help of the anomalous orbits and planes (55, 127). NGC 3718 (106) provides a spectacular application of such orbits to a complicated gas distribution.