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Abstract. Ring structures are a highly distinctive feature of many spiral galaxies. We analyze the current state of the results of both the observational and theoretical problems of ring galaxies. Also, we purpose the non-stationary theory of ring galaxies using Nuritdinov's non-linear disk model
Introduction. The intriguing behavior and theory of the ringed and ring-like galaxies wins over to one's side of every theorist on astrophysics. These objects display a wide variety of ring morphologies, e.g., uniformly bright almost symmetrical rings around the nucleus (Hoag's objects), elliptical rings (e.g. ESO 54-IG11), knotty ring-like structures without a nucleus (ESO 303-IG11), off center nuclei (ESO 46-IG10), half or full ring structures with nucleus (AM 0058-311, AM 0628-632), warped rings (ESO 316-IG32).Therefore we have a number of scientific questions on some of the interesting aspects of such galaxies.
The outline of this paper is as follows. First, we will discuss the different classification systems introduced in order to fully classify the ring-structure in galaxies. Next, we will discuss the origin theories and formation of rings in galaxies in which one emphasize on the collision-less gravitational system with a direct impact on the internal dynamics and evolution of disk galaxies, while others consider ring formation via mild tidal encounters as opposed to the more violent encounters of polar and Collisional ring galaxies. Then, we also suggest a new theory in which we can explain different observed properties of ring galaxies. Besides, we will raise some questions regarding the evolution & classification of ring galaxies
-Is the defined classification system enough to describe ring galaxies or is it high time we describe a new classification system in order to fully describe ring morphology?
-Can we construct a united class of ring galaxies?
-Does the morphology of ring galaxies also depend on some parameters which are not discussed yet?
- Is there any relation between collision-less and collisional theories of ring formation in galaxies?
- How often do the visible rings of stars have a gas ring stage in their period of evolution?
-At which conditions, can non-stationary stellar system can form ring subsystems?
Classification of ring galaxies
The (r) and (s) varieties are the well-established aspects of galaxy morphology that both the Hubble-Sandage and de Vaucouleurs types recognize as a part of normal galaxy morphology. However, most observers choose samples for follow-up work on the basis of published catalog types.
It is important to note that independent classifiers do not always agree on ring classifications. For example, some galaxies classified as (r)-variety by de Vaucouleurs  are classified as (s)-variety by Sandage and Tammann[2,3]. In the latter sources also, outer pseudo-rings are generally not recognized. The disagreement results because the spiral structure in many galaxies is very ring-like. In the Carnegie Atlas, strong "near rings'' such as those in NGC 3081, 1433, 6782, and 7217 are noted but not formally recognized in the classification because the rings are made of tightly wound spiral structure. In the case of NGC 6782, authors state that the 'ring' is two inner spiral arms that start in the nucleus (hence the subtype (s) rather than (r)) and nearly overlap after each has unwrapped by about half a turn. Apparently, this effect is caused by a combination of the leading dust lanes in the bar and the tight spiral pattern that defines the inner ring in this case. In the case of NGC 3081, where the "near ring'' has a very high contrast in the disk, authors of  state that the galaxy has one of the most complex morphologies of the RSA galaxies. There are rings within rings at the edges of disks within disks, as intricate a structure as in nesting, concentric "Russian dolls". None of these rings, however, is recognized in the Carnegie classification of SBa(s).The de Vaucouleurs classification of the galaxy is instead defined as (R)SAB(r)0/a.
Kostiuk  compiled a list of 143 galaxies with "outer ring-shaped structure'' as seen on the prints of the first Palomar Sky Survey. This list includes mostly true rings (not pseudo rings), and she identified three types of rings: RS, rings which are some type of spiral surrounding a bright main body (examples: NGC 2859, 3945, and 4736); RH, rings with a weak halo beyond the ring (examples: NGC 4340, 4371, and 4608); and R, lacking a central nucleus (now known as the conventional ``ring'' galaxies). This study does not distinguish inner from outer rings or early from late-type galaxies, but the sample emphasizes early types by default.
S.N.Nuritdinov & M.Usarova  formed the first ring galaxies catalogue of 350 objects in which they define a preliminary sequence of eight types on the popularity degree .These types include barred Spiral galaxies in which the ring length coincides with the bar one; spiral galaxies in which the arms are gone out of almost one point of the core and joined together in space; purely ringed galaxies without spiral arms with a ring and central core; the same ringed galaxies but core not central; barred Spiral galaxies with one ring without spiral arms; multi branch spiral galaxies with one ring and a central core; ringed galaxy with a half- bar and almost central core and a spiral galaxies with tightly twisted single-arm which looks like a ringed galaxy. It is also worth mentioning in  that this sequence strongly depends on the choice of samples.
Vorontsov-Velyaminov  also classifies ring phenomena in galaxies in a manner very different from the Hubble-Sandage and de Vaucouleurs systems. In his descriptive classification, there is no recognition of subclasses of rings, such as inner, outer, or nuclear types. Instead of it, any galaxy with a prominent ring or pseudo ring, whether luminous or dark (as in the absorption rings of S03 galaxies) is called a ring galaxy and classified as N;R in the Morphological Catalog of Galaxies . Vorontsov-Velyaminov suggested that the ring galaxies form a sequence parallel to and independent of spiral galaxies, and that there is a continuous transition of forms from pure amorphous disk galaxies to ring galaxies. These are not allowed to have a bar or spiral arms. He suggests a smooth transition also between ring galaxies with a nucleus to those without one. Thus, there is no distinction between what the others would call ringed galaxies and what are commonly referred to as collisional ring galaxies , unless perhaps there is a bar. Polar ring galaxies are also referred to as ring galaxies by Vorontsov-Velyaminov, who describes the edge-on cases as "lemons pierced by a needle". However polar ring has a different nature and we will not consider these ones. Vorontsov-Velyaminov also brought attention to ``double-staged" spirals, where an inner set of spiral arms is largely independent of an outer set of arms. The inner arms may be patchier than the outer ones and merit a later classification stage than the outer arms.In many of the galaxies where this ambiguity occurs, have rings or pseudo rings such as NGC 3504, NGC 1808, and NGC 1433. In the Hubble Atlas and , the inner set of arms determines the Hubble type.
The classification of non-interacting ringed galaxies has had one major revision early.Buta and Crocker discussed the morphology of outer rings and pseudo rings in barred galaxies which includes subcategories that strongly resemble the gaseous rings and pseudo rings that developed near the outer Lindblad resonance (OLR) in the test-particle simulations of Schwarz  Many barred galaxies display a pseudo ring of type R1' defined by a 180° winding of the spiral arms with respect to the ends of a bar or oval. Often, the pseudo ring displays a blunt oval shape with ``dimples'' at the points where the arms return to the ends of the bar. ESO 287-56, is an unusually strong example of this dimpling.
A substantial number of barred galaxies display a second type of feature known as an R2' pseudo ring.In this type, the pseudo ring is defined by a 270° winding of the outer arms with respect to the bar ends, so that in two opposing quadrants the arm pattern is doubled. The pseudo ring closes roughly along the line perpendicular to the bar in the galaxy plane, and dimpling is not generally seen. Detached true outer rings do not readily fit into these subcategories because the classification depends on the character of spiral arms. However, some true rings have a dimpled shape similar to R1' rings and are referred to as R1-type rings. The outer ring of NGC 1326 (see Buta & Crocker 1991 ) is such an example.
It is also noted in  a possible correlation between the presence of an R1 or R1' outer ring/pseudo ring morphology and the existence of a nuclear star formation. In an imaging survey of 29 early-type spirals, 9 out of 11 examples containing this type of outer feature included nuclear star formation in the form of blue nuclear rings, partial blue rings, or blue nuclei. In contrast, 8 out of 11 cases including a strong R2' outer feature showed no evidence for nuclear star formation. The R1' and R2' morphologies were predicted by Schwarz  as the kinds of patterns that would be expected near the OLR in a barred galaxy. Each pattern is linked mainly to one of two major families of periodic orbits expected near the OLR
The first attempt to classify collisional ring galaxies was by Theys and Spiegel, which mainly an observational, paper on ring galaxies They compiled a list of likely collisional rings based on miscellaneous early accounts of rings and developed a simple morphological classification scheme. Theys and Spiegel defined three basic ring types. The first, in some sense the "pure" ring, was the RE galaxy, defined as a galaxy which have crisp, elliptical rings with photo-graphically empty interiors (without a central nucleus), with archetypes such as Arp 146, Arp 147 and VII ZW 466; they also defined RN galaxies as galaxies containing an off-centered nucleus archetype II HZ 4; and the RK galaxies as ring-like galaxies with a prominent knot embedded within a markedly asymmetric light distribution characterized by being I Zw 45 and Zw 28 .
Theys and Spiegel were aware of the sometimes difficult task of distinguishing between rings which may have been formed by collisions and the "resonance" ringed or pseudo-ringed galaxies first discussed by De Vaucouleurs and recently catalogued by Buta The RN class could potentially contain rings formed by processes other than that of galaxy collisions, although most of the resonance rings appear to contain symmetrical nested rings, whereas the RN rings are notable because the inner ring or nucleus is offset from the geometrical center of the ring. In the presentation of their small ring sample, Theys and Spiegel  discussed a number of possible interpretations for ring galaxies. They noticed that with the exception of II Zw28 (a small RK ring), all of the other ring galaxies had small companions within a few ring radii and that many of them lay within a few degrees of the projected minor axis of the ring. They were concerned that the degree of flattening of the rings did not appear to be consistent with the position of the companion galaxy in the sky. The drawback of their classification is that it does not contain a number of observed large-scale structures in ring galaxies.
By careful analysis of southern ESO and SERC J Schmidt plates, Few and Madore separated 69 ring galaxies into two main classes, the O-type and P-type galaxies-type rings, which have a smooth structure and a centrally located nucleus, without companions; whereas, the P-type rings, which have a crisp, knotty structure and often an off-center nucleus (and that happen to have a significant number of companions lying within a distance of about two ring diameters). Both classes contained a fraction of barred galaxies.The O-type galaxies did not show any enhancement of possible companions over that expected by chance. They concluded that the P-type galaxies were good candidates for collisional galaxies and that the majority of the O-type galaxies were not. The O-type galaxies were found to be similar to the (R)S galaxies of de Vaucouleurs.
On origin theories of ring galaxies
There are two major divisions of the theory of origin of ringed galaxies. The first one is collision-less theory which associates problems of rings in galaxies with its internal dynamics such as with spiral arms. It is supposed that ring formation in galaxies are not a result of gravitational interaction between two galaxies. Other schools consider it as colliding galaxies, as a result of the interaction of galaxies.
It is interesting to note that earlier publications (Randers, Icke , Lesch ) proposed a theory of ring formation based on viscous torques. In principle, the viscous torques act in regions of differential rotation, and accumulates matter where the angular velocity becomes constant, i.e., rings are expected at the turnover point of the rotation curves. Renders applied this theory to the stellar population, where he claimed that collisions and scatterings were equivalent to a viscosity effect. The problem of diffusion and heating of the stellar component was debated much later (see &references there in)
Danby  discussed the existence of ringlike structures in barred galaxies, associated with the equipotential surfaces in the rotating frame of the bar; this might be the closest approach to the modern theory of ring formation, as that of resonant accumulation of gas in a non-axisymmetric potential. The considerable success of the density wave theory of Lin & Shu  led to the abandonment of the non-gravitational approaches, and the growing importance of N-body simulations focused attention on bars, which were the only robust density waves found numerically in a self-consistent stellar disk (e.g Miller; Hohl). With this advancement, bars were no longer considered as solid body entities in rotation around the center, but as waves interacting with the rest of the disk and possibly triggering spiral structure. Simulations of gas flows in barred galaxies were initiated by Sanders & Huntley , who showed convincingly that a long-lived spiral structure could be generated in the gas component by the stellar bar, owing to the properties of periodic orbits in a non-axisymmetric potential. No resonant rings were formed in these simulations, which could be as result of an effect of the gas viscosity. The first work where resonance rings were formed, and explained through gravitational torques on the gas, was done by Schwarz , who modeled the gas component by ballistic particles undergoing collisions. This representation, which corresponds much better to the cloudy and clumpy interstellar medium, minimizes the artificial viscosity, and strongly helps the formation of highly contrasted rings.
Buta also suggest that since the rings are relatively narrow features, they are always associated with gas, and are usually the site of enhanced star formation. This suggests that gas is at the basis of ring formation. The high velocity dispersion of the old stellar component will prevent it from forming very sharp structures. Even after young stars have formed out of the gas, they will acquire velocity dispersion in a time-scale of 109 yrs, and will no longer follow thin ring morphology. Gas is therefore necessary also for the persistence of a ring structure. He also emphasized that rings are preferentially found in barred galaxies. They have a preferential orientation with respect to the bar, either parallel or perpendicular, and their sizes are also related to that of the bar. The inner ring is always encircling the bar, forming the famous -shape, which shows that the bar must be an essential element in ring formation. Bars can be considered as long-lived density waves, and long-lasting modes. When a density wave of constant pattern speed is present in a galaxy, resonant phenomena with this forced pattern speed has time to develop and will be conspicuous. The idea of rings formed at resonances with the bar is therefore entirely natural.
A detailed color decomposition of an inner ring  has shown that the ring is the superposition of two gaussians: a narrow blue one, containing the more recent stars, and a wider red one. The latter has about twice the width of the former. The mean central positions of the two gaussians coincide which means that the position of the inner ring has not significantly evolved from the time of its formation. The underlying pattern speed must therefore be nearly constant over the ring formation time.The breakthrough idea of collisional ring galaxies came from two sources, namely the work of Lynds and Toomre , and Theys and Spiegel . In both cases ring galaxies were hypothesized to be the result of a head-on collision between a compact companion galaxy and a larger disk system. The resulting gravitational perturbation was shown to generate rings in the disk of the larger system. A common feature of the collisional ring formation picture is the generation of radially expanding density waves resulting from the crowding of stars in the disk. Although the most coherent effects are likely to be found in the dynamically cool disk stars and gas, the central perturbation will also have substantial consequences on the halo stars and any dark matter present.
Lynds and Toomre  first presented the conceptual model that forms the basis of our understanding of the ring galaxy phenomenon. In the simplest form of this model, a small companion galaxy is assumed to pass down the symmetry axis of the larger primary galaxy, and move through the disk center. Prior to the collision the stars in the primary disk are assumed to be in circular orbits. At the time of impact the disk stars feel a strong pull toward the center as a result of the companion's gravity. In the simplest case, we also assume that the collision is so rapid that the disk stars do not have time to adjust to the sudden inward impulse. Specifically, in the so-called impulse approximation (IA), the stellar positions are assumed to be the same immediately before and after the collision, but after the collision each star has acquired an inward radial velocity 
Freeman and de Vaucouleurs  also proposed their model, which was a collisional one, involving the gas dynamical interaction of a cool intergalactic hydrogen cloud (IGC) with a pre-existing galaxy containing an external HI ring. This collision was expected to strip away the HI ring from the underlying galaxy, leaving behind the bulge which would lie on the minor axis of the HI ring. Subsequent star formation in the HI was believed to provide the characteristic ring galaxy appearance. De Vaucouleurs  also suggested a similar mechanism for the formation of the ring in NGC 985 (a Seyfert 1 galaxy). Although many ring galaxy candidates were mentioned by Freeman and De Vaucouleurs, two particular galaxies were discussed in detail. They were Arp 144 and NGC 2444 / 5 = Arp 143. The case of Arp 144 was particularly compelling, since the morphology of the ring seemed to suggest that the intruding cloud was caught in the act of peeling the ring away from the disk of the target galaxy.
New non-stationary theory of ringed galaxies
One of the urgent problems of up-to-day astrophysics, the galactic and extra galactic astronomy is to study the early non stationary evolution stage and the development processes of large scale structure  of galaxies and other self gravitating systems among which we are particularly interested in ring-like structures  .This problem calls for complex numerical calculations with modern computers or constructing nonlinear, non-stationary models as well as studying the problem of their stability; The second method seems more preferable, as at the numerical experiments it is difficult to reveal some non linear effects and to take notice of gravitational instabilities origin at a proper time.
Nuritdinov  constructed the non stationary phase pulsating models, which are adequate to investigate the instabilities of early evolution stages of galaxies and their superclusters. In particular, he obtained the models that generalize the Einstein and Camm equilibrium models for spherical self gravitating systems in non-stationary case.
Phase density of spherical self gravitating system the Einstein-Nuritdinov model is 
Phase density of spherical self gravitating system in generalized Camm's model is 
Where variables are
Here is the matter density in physical space and it not depend on r ; and are the symbols of Dirac and Heviside function; are azimuthal and meridian components of velocity ; is the expansion factor of sphere; the variable is connected with t by correlation ; is the frequency of star revolution in the orbit of the stationary system; is the pulsation amplitude, where is the virial ration with , T and U being the system kinetic and potential energies, respectively. is the angular velocity of the system rotation.Real evolution of collapsing galaxies is described by function, which should satisfy to the system of Jeans Poisson equations. The above mentioned models are applicable to early evolution stages of elliptical E and lense like Spiral galaxies. For super galaxy clusters which are Zeldovich pancakes, construction of disk like non-linear, non-equilibrium models are required. In this case, the isotropic model for phase density 
Deduction of NADE's of isotropic model of ring and ring like mode
Let us consider the concrete oscillation modes (4,0) and (4,2) , the instability of which could lead to the formation of the ring and ring like structure on the background of non-stationary model (1).Here, we impose a small disturbance onto the nonlinear non-stationary model (1) and deduce the equation for the particle drift in the in the perturbed state
where the operator , Î´Ð¤ is a perturbation of the potential. As long as the divergence of the particle in the perturbed state at the current moment of time depends on condition of the field in the previous moments of time and our aim is to search for instability, one could figure, that at , . In the current moment of time at every point, there are particles with the different velocities, therefore for the calculation of the density perturbation or deformation of the perturbed system border it is worthwhile to turn to the centroid deviation , averaging (2) upon the velocity space. Hence, we find
where S(,1) in an analogue of the Green's function. Taking into consideration the nature of the investigated model by using the analogy with the theory of stationary model stability we describe it as
where is a function of time sought for, m is the azimuthal wave number.
The formula for the disturbed density
From which, we get Non-stationary Dispersion Equation for ring mode (4,0) 
(2T/|U|)0 (2T/|U|)0 (2T/|U|)0
Also, non-stationary Dispersion Equation for ring like structure mode (4,2) is
. The integral equations (6) and (7), we have found that it is inconvenient for the analytical studies and numerical calculations needed to calculate values of the pulsation amplitude for mode (1), we therefore introduce the notation
The particular case corresponds to asymmetric perturbation being supposed on an equilibrium mode without pulsation, which satisfies the Kalnaj's results .
The following criterion for the formation of SB-galaxies with the ring-like structure on the early non-stationary stage of their evolution is found: the beginning total kinetic energy of the self-gravity disc should be not more than 5.2 percent of initial potential energy. It is interesting to know whether the similar criteria for mode (4,2) has more than 5.2 percent .Herewith, the nature of instability is connected with the mechanism of radial motion instability, which has non-periodic character for the ring structure; the result and character of instability do not depend on the values of Î©.
Given above we have implemented a comparative analysis of both the observational progress and theoretical advances and have come to conclude that it is a time to implement a new statistical measures in order to fully describe ring morphology by which we can construct a general united class for ringed galaxies. This new statistics must take into account all parameters concerning rings in galaxies so that we can completely understand the morphology; we have also found some evidence that there is a relation between collision-less and collisional theories of ring formation in galaxies and most of the visible rings of stars have a gas-ring stage of evolution which we will discuss in the next paper .
I am grateful to the honorable Professor Dr.S.Nuritdinov for suggesting me to use the earlier discovered models and methods for my use and to Professor M.Khalid for his helpful suggestions in numerical computing.
This research work is supported by the grant of Higher Education Commission, Pakistan.