Astronomy Reports, Vol. 44, No. 5, 2000, pp. 279–285. Translated from Astronomicheskiœ Zhurnal, Vol. 77, No. 5, 2000, pp. 323–330. Original Russian Text Copyright © 2000 by Kondrat’ev. Dynamics and Stability of Resonant Rings in Galaxies B. P. Kondrat’ev Udmurt State University, Krasnoarmeœskaya ul. 71, Izhevsk, 426034 Russia Received December 24, 1998 Abstract—We have developed a model for a spheroidal, ring-shaped galaxy. The stars move in a ring with an elliptical cross section at the 1 : 1 frequency resonance. The shape of the cross section of the equilibrium ring depends on the oblateness of the galaxy itself, so that the ellipse of the ring cross section is radially extended when the oblateness of the galaxy is small. If the oblateness of galaxy exceeds some critical value, the ellipse cross section is extended along the Ox3 axis. The shape of the ring cross section is circular for a galaxy with critical eccentricity. The stability of the ring over a wide range of perturbations is studied. A fundamental bicu- bic dispersion equation for the frequencies of small oscillations of a perturbed ring is derived. Application of the model to the ring galaxy NGC 7020 shows that its ring cross section should be approximately circular. Anal- ysis of the dispersion equation demonstrates that stellar orbits in the arm are unstable (but the instability incre- ment is small). We conclude that stars in the ring of this galaxy should drift from the 1 : 1 resonance, and the ring itself should evolve. © 2000 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION appearance of various commensurability ratios. Recall that our solar system is literally saturated with reso- The rings (or partial ring structures) existing in nances between the frequencies of the spin–orbital many galaxies have long attracted the attention of theo- motions of planets and satellites [7]. rists. The abundance of such stellar systems was empha- sized, for example, in [1], which notes about 200 flat gal- Galaxies are the oldest stellar systems and have axies with ring structures. The most prominent example undergone long periods of evolution. Therefore, we can of a galaxy possessing a distinct nucleus–ring combina- assume that resonances could be formed in galaxies in tion (outwardly similar to Saturn and its rings) is much the same way as in the solar system. Taking such NGC 7702. According to [2], this galaxy has not one resonances into account when developing galactic mod- but two well-separated rings. The well-known exam- els can radically change the dynamical properties of the ples [3–5] indicate that ring and spiral-arm structures models.1 Therefore, when constructing models for ring have similar compositions and stellar dynamics. Inves- galaxies, we should first and foremost investigate the tigations of such structures are necessary if we wish to dynamical consequences of resonances between the elucidate the important role of secular evolution pro- frequencies of small oscillations of stars in a plane of cesses in spiral and SO galaxies. the ring cross section. The existence of ring galaxies is a serious theoreti- The present paper is concerned with a stellar cal problem. A quite simple stellar dynamical model for dynamical model for a ring galaxy with a local 1 : 1 res- a galactic ring was developed and its stability studied in onance of the frequencies of stellar motions in a circu- [6]. That paper was primarily concerned with general lar arm with an elliptical cross section. The equilibrium collective effects in the dynamics of prolate structures, model itself is developed in Sections 2–5, and its stabil- whereas the dynamics of individual stars within such ity is studied in Sections 6–10. We apply this model to structures were not studied. However, the existence of the ring galaxy NGC 7020, whose structure was studied prolate structures and circular rings is possible only if in detail in [9]. there are certain definite relations between the parame- ters of stellar orbits and the corresponding first integrals of motion. Precisely the character of individual orbits is 2. EQUATIONS OF STELLAR MOTION the decisive factor determining the particular form of IN AN ELLIPTICAL ARM the components of the velocity dispersion tensor, and Let us consider an axially symmetric galaxy pos- pressure anisotropy in the “stellar gas,” which is char- ρ acterized by these components, will considerably affect sessing a uniform circular arm with density and ellip- the stability of rings in galaxies. 1 See, for example, [8], which describes a self-consistent phase Therefore, when considering the orbits of individual model for a spheroidal galaxy with an additional resonant integral of the stellar motion. Torus-shaped vortex motions of the cen- stars in the arms, we should first and foremost examine troids arise in this model, in agreement with both observations possible resonances. This is especially important since and numerical experiments in the framework of the N-body the evolution of dynamical systems often leads to the problem. 1063-7729/00/4405-0279 $20.00 © 2000 MAIK “Nauka/Interperiodica” 280 KONDRAT’EV tical cross section. The arm is characterized by a cross motion along the corresponding axis is described by the section with semi-axes a1 and a3: equation of a one-dimensional harmonic oscillator 2 ξ2 x˙˙ = –2(α + A )x , (9) x3 ≥ 3 3 3 3 ----2 + ----2 = 1, a1 a3, (1) a3 a1 and obviously does not depend on motions along the R ξ and θ directions. (where = R – R0), and is filled with stellar orbits. Since the cross section and curvature of the generatrices of Let us simplify the problem and assume that the dis- the arm are small, we can take its inner potential at the tribution of mass in the galaxy and ring possesses cir- ξ point ( , x3) in the form cular symmetry; i.e., ϕ α ξ2 α 2 ∂ϕ p = const – 1 – 3 x3, (2) -∂---θ-- = 0. (10) which is typical for a uniform two-dimensional ellipti- cal cylinder with straight generatrices. The constant Next, let us consider a coordinate system rotating coefficients in expression (2) are equal to Ω with angular velocity c: a a α π ρ 3 α π ρ 1 θ( ) θ˜ ( ) Ω 1 = 2 G ----------------; 3 = 2 G ----------------. (3) t = t + ct. (11) a1 + a3 a1 + a3 A star in the arm is also affected by the gravitational Then, the first two equations of (8) take the form field of the galaxy itself. Let us expand the galactic field ∂ϕ ˙ 2 potential in a Taylor series in powers of ξ and x in the R˙˙ = ------ + R(θ˜ + Ω ) , 3 ∂R c neighborhood of the arm. Then, with accuracy up to (12) quadratic terms, we obtain d ( 2θ˜˙ ) Ω ˙ ---- R = –2R c R. ϕ Ω2 ξ ξ2 2 dt g = const – c R0 – A1 – A3 x3. (4) Here, Consequently, we obtain in a linear approximation with ˜˙ ∂ϕ respect to ξ and θ Ω2( ) 1 g c R0 = –----------- (5) R ∂R R = R 0 ˙ξ˙ (Ω2 α )ξ Ω θ˜˙ = c – 2 1 – 2A1 + 2R0 c (13) is the square of the angular velocity in a reference orbit with radius R0 from the galaxy center and the coeffi- and cients of the potential are equal to ˙˙ R θ˜ = –2Ω ξ˙ . (14) ∂2ϕ ∂2ϕ 0 c 1 g 1 g A1 = –-- ---------- , A3 = –-- ---------- . (6) η θ˜ 2 ∂ 2 2 ∂ 2 By substituting = R0 into the last two equations R R = R0 x3 R = R0 x3 = 0 x3 = 0 and using (9), the complete system of equations for the The total potential inside the arm is the sum of the stellar motion can be reduced to the form contributions: ˙ξ˙ = (Ω2 – 2α – 2A )ξ + 2Ω η˙ , ϕ ϕ ϕ c 1 1 c = P + g (7) η˙˙ = –2Ω ξ˙ , (15) Ω2 ξ (α )ξ2 (α ) 2 c = const – c R0 – 1 + A1 – 3 + A3 x3. (α ) x˙˙3 = –2 3 + A3 x3. The initial equations for the motion of a star inside θ the arm in cylindrical coordinates (R, , x3) are well- After the first integration (to within a constant, which is known: not important in our consideration), the middle equa- tion of system (15) gives ∂ϕ ˙˙ θ˙ 2 R = ------ + R , η Ω ξ ∂R ˙ = –2 c . (16) d 2 ∂ϕ ----(R θ˙ ) = ------, (8) Substituting (16) into the right-hand side of the first dt ∂θ equation of (15), we finally obtain the equations of a ∂ϕ two-dimensional harmonic oscillator, x˙˙3 = ∂------- . x3 ˙ξ˙ ω2ξ = – 1 , When solving this system of equations, we first note (17) ϕ x˙˙ = –ω2 x , the quadratic dependence (7) of on x3. As a result, 3 3 3 ASTRONOMY REPORTS Vol. 44 No. 5 2000 DYNAMICS AND STABILITY OF RESONANT RINGS IN GALAXIES 281 where we have introduced the parameters 4. SLOW EVOLUTION OF A RING GALAXY ω2 Ω2 (α ) In the case of slow evolution, the adiabatic invari- 1 = 3 c + 2 1 + A1 , (18) ants must be conserved. In accordance with (17), we ω2 (α ) ξ 3 = 2 3 + A3 . have a harmonic oscillator with respect to and x3. Consequently, as is well known [11], the products of α α In particular, when the arm is absent (i.e., 1 = 3 = 0), the squares of the amplitudes and the frequency are the first relation of (18) can be written in the equivalent conserved: form 2 2 2 2 a 3Ω + 2(α + A ) = a 2(α + A ). (26) 3 ∂ϕ ∂ ϕ 1 c 1 1 3 3 3 -----------g + ----------g < 0, (19) R ∂R ∂R2 In addition, the mass per unit length of the arm should be conserved: leading to the well-known criterion for the stability of ρ circular orbits (see, for example, [10]).