MIGRATION OF STAR CLUSTERS AND NUCLEAR RINGS

For a sample of 22 nuclear rings, Mazzuca et al. (2008) found that the rings are in the same plane as the disk of their host galaxy and that they are intrinsically circular. Typically, the radius of the ring is around ∼0.5 kpc, but it can be a factor of 3 smaller or larger. With a circular velocity at the ring radius of around ∼150 km s⁻¹, the (spherically) enclosed mass is about M_enc ∼ 2.5 × 10⁹ M_☉.

The gaseous mass in the ring may be estimated from high spatial resolution CO measurements, using the "standard" conversion to H₂ mass and adding 36% for He. In this way, we obtain for NGC 5457, NGC 3351, NGC 6951, NGC 3504 (Kenney et al. 1992, 1993) and NGC 4314 (Benedict et al. 1996) on average a mass inside the ring of M_d ∼ 5 × 10⁷ M_☉. Although the ring mass may be as much as a factor of 10 smaller or 2 larger, it correlates with the ring radius, so that the ring-to-enclosed mass ratio q_d ≡ M_d/M_enc remains within a narrow range of q_d = 0.01 − 0.03. While this is the molecular gas inside the ring, part of the gas is being ionized due to photons from newly formed massive stars or due to shocks from stellar winds or supernovae explosions, and at the inner edge of the ring possibly even due to an active galactic nucleus (AGN). However, the ionized gas mass estimated from the Hα (plus [N II]) emission is generally two orders of magnitude lower than the molecular gas mass (e.g., Planesas et al. 1997).

The nuclear rings are easily observable due to the emission from the massive stars inside the young star clusters in the ring (which in the UV even dominates the emission from a possible AGN). The combination of high spatial resolution (HST) images and multiwavelength coverage, makes it possible to isolate the individual clusters and after correction for (dust) extinction to estimate their mass and (relative) age. Below we use such a study by Benedict et al. (2002) to investigate in more detail the nuclear ring in NGC 4314. Overall, the cluster masses follow a power-law distribution with index −2, similar to young star clusters in merging galaxies such as the Antennae (Zhang & Fall 1999), but typically extending to a smaller upper limit in mass (few times 10⁵ M_☉), i.e., fewer of the so-called "super star clusters" (but see Maoz et al. 2001).

On the other hand, these young star clusters are most likely just the latest star formation in an ongoing succession of bursts (e.g., Allard et al. 2006). Most of the lower-mass clusters dissolve, either early on (≲50 Myr) due to violent relaxation after expelling the residual gas ("infant mortality") depending on the star formation efficiency (e.g., Parmentier et al. 2008), or later on due to internal relaxation ("evaporation") and external tidal effects (e.g., McLaughlin & Fall 2008). The resulting evolution toward a bell-shaped distribution implies a relative increase in more massive star clusters over time. Unfortunately, once the most massive stars have died, the luminosity of star clusters very rapidly fades, so that it becomes more difficult to distinguish them from the old stellar population of their host galaxy. Nevertheless, based on velocity dispersion measurements from stellar absorption lines, Hägele et al. (2007) estimate dynamical masses up to almost ∼10⁷ M_☉ for individual star clusters in the nuclear ring of NGC 3351. Henceforth, the typical ratio of cluster-to-ring mass ratios q' ≡ M_s/M_d depends on the (average) age of the star clusters. For the young, luminous star clusters we adopt the range q' = 0.001 − 0.01, whereas for the old(er), faint(er) star cluster q' = 0.01 − 0.1 is probably more appropriate. Below we investigate the two cases q' = 0.1 and q' = 0.01, corresponding to M_s ∼ 5 × 10⁶ M_☉ and M_s ∼ 5 × 10⁵ M_☉ in case of the above average gas mass in nuclear rings of M_d ∼ 5 × 10⁷ M_☉. The effect of a less massive star cluster with q' < 0.01 is similar to that of q' = 0.01.

The time for a star cluster to cross the nuclear ring is given by t_cross = 2π/Ω, where $\Omega = \sqrt{GM_{\rm enc}/r^3}$ is the orbital frequency and G is Newton's constant of gravity. Given a typical circular velocity of ∼150 km s⁻¹ at the observed mean nuclear ring radius of 0.5 kpc, the crossing time is around t_cross ∼ 20 Myr. A star cluster is subjected to dynamical friction with the bulge stars on a (minimum) timescale⁵ t_df = M_enc/(ΩM_sln Λ), with ln Λ the Coulomb logarithm. Using t_cross = t_df q'q_d 2πln Λ and substituting q_d ≃ 0.02 from above, we find that the dynamical friction time in units of the crossing time is approximately given by t_df/t_cross ≃ 1/q'. This means hundreds to tens of orbits for star clusters with masses that are 1% (q' = 0.01) to 10% (q' = 0.1) of the nuclear ring mass, or, with t_cross = 20 Myr, a dynamical friction time t_df from about 2 Gyr to 200 Myr for a nonrotating bulge, which we will assume for the remainder of this work.

We illustrate our basic model in a cartoon in Figure 1. Gas flowing in a bar potential along the X₁ orbits transitions to X₂ orbits making up the nuclear ring. The X₂ orbits are approximately circular and so we will approximate them as circles. Stars are formed in star clusters from the gas in the nuclear ring. We assume that the star clusters are formed preferentially near the outer edge. These newly formed and forming star clusters are subjected to dynamical friction with the background bulge stars⁶ and tidal interactions with the ring. However, the initial tidal interactions are strong compared with dynamical friction and thus these star clusters will initially move away from the ring.

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Indeed, in observations of nuclear rings young star clusters often appear to be at larger radii than the gas ring from which they are presumably formed. The color composite images of NGC 1512 (Maoz et al. 2001), NGC 4314 (Benedict et al. 2002), and NGC 7742 (Hubble Heritage⁷), show young star clusters as bright spots on the outside of gas traced by the dust. As pointed out by the referee, from these images it is also clear that, even though the nuclear rings are as we assume approximately circular with star clusters at the outer edge, the observations can be quite difficult and complex. First, when resolved in the optical, the rings can look like tightly wound spirals. This might be just appearance because the bar dust lanes connecting with the ring are also partly obscuring the ring in optical broadband images, whereas in, e.g., narrowband imaging around Hα the ring of young star clusters seem to be complete (see also Maoz et al. 2001). Nevertheless, our basic picture described above for a circular ring should still provide a fair description in the case of a tightly wound spiral. Second, very obscured star clusters could remain hidden below the gas ring even in the near-infrared images taken, whereas even longer-wavelength observations (yet) lack the high spatial resolution required to resolve the star clusters. Even so, the clearing of gas and dust from the newly formed star cluster is very fast and efficient, as even the youngest visible star clusters are only mildly reddened (e.g., Maoz et al. 2001). Finally, some (sectors of) star clusters seem to appear more on the inside of the nuclear ring, possibly along an inward extension of the gas inspiraling onto the ring along the large-scale bar. As we discuss briefly in Section 5.3, star clusters may form within the ring, in particular at the contact points where the gas flows into the ring. Moreover, the interaction of nonaxisymmetric structures inside the ring (such as nuclear bars and spirals) with the ISM may also induce the formation of star clusters. However, even though such additional mechanisms might be operating and the observations should be interpreted with care, they in general support our basic picture of a circular nuclear ring with young stars migrating outward.

Since the basic physics mirrors the "gas shepherding" scenario studied by C08, we adopt the same notation. In this case the satellite is a star cluster and the disk is a ring, but since only the (local) geometry at the edge matters the physics remains the same. There are however two important differences with respect to C08. First, the perturbing star cluster's initial radial position begins very close to the ring as opposed to a satellite reaching the disk from very far away in C08. Due to ring tidal interactions, the star cluster and ring will separate to establish an initial separation, Δr, which we show below is of the order of the "Hill radius" of the ring. We aim to study the manner by which this initial separation is achieved. Second, gas continues to flow in from the X₁ orbits so that at the transition between the X₁ to X₂ orbits, gas is continuously being added to the system. Therefore, a source of mass exists in this system, which changes the dynamics.

For now we ignore the source of mass to concentrate just on the tidal interaction between the ring and star cluster and the dynamical friction experienced by the star cluster. We begin by showing that this combined action of dynamical friction and ring tides results in a length scale in the problem, which we call the Hill radius of the ring. The torque due to dynamical friction on the star cluster is (Chandrasekhar 1943; Binney & Tremaine 1987)

$$\begin{equation} T_{\rm df}\sim \frac{M_s}{t_{\rm df}} \, \Omega _sr_s^2 \sim \frac{G M_s^2}{r_s}, \end{equation} \tag{ 1 }$$

where M_s is the mass of the star cluster, r_s is its distance from the center of the galaxy, Ω_s is its orbital frequency, and, as before, t_df is the timescale for dynamical friction. The torque due to the excitation of spiral density waves in a disk is (Goldreich & Tremaine 1980; Lin & Papaloizou 1986; Ward & Hourigan 1989; Artymowicz 1993; Ward 1997)

$$\begin{equation} T_{\rm disk}\sim \frac{GM_s^2}{r_s} \frac{M_d}{M_{\rm enc,s}} \left(\frac{r_s}{\Delta r}\right)^3, \end{equation} \tag{ 2 }$$

where M_d is the mass of the ring with radius r_d. The radial separation between the star cluster and the ring Δr = r_s − r_d is assumed to be small compared to the distance to the center, i.e., Δr ≪ r_d < r_s. While dynamical friction torques down the star cluster, the ring torques up the star cluster. Balancing these two torques, i.e., T_df ∼ T_disk, gives

$$\begin{equation} \Delta r\sim r_s\left(\frac{M_d}{M_{\rm enc,s}}\right)^{1/3} \equiv r_{H,d}, \end{equation} \tag{ 3 }$$

where the latter defines the Hill radius of the ring.

Given the observed range in ring-to-enclosed mass of q_d = 0.01 − 0.03 (Section 2.1), the Hill radius is around 20%–30% of the radius r_s of the star cluster. Since r_s ∼ r_d ≫ r_H,d, the observed range in nuclear ring radii of r_d = 0.2 − 1.7 kpc (e.g., Mazzuca et al. 2008) results in r_H,d = 50–500 pc. We expect the radial separation between the star cluster and nuclear ring to be of the order of this Hill radius. However, the exact amount and nature by which this separation will be achieved depends strongly on the strength of the tidal torque between the cluster and the ring, which we intend to quantify in this paper.

The second process which we wish to study is the effect of an additional source of mass on the dynamics of the cluster-ring system. The gas merges onto the nuclear ring at the two contact points where the X₁ orbits transition to X₂ orbits. We assume that the region where the gas is added to the nuclear ring is radially narrow compared to the radius of the nuclear ring. For a mass inflow rate of $\dot{M}$, the amount of angular momentum added to a system per unit time is

$$\begin{equation} \dot{L} = \dot{M} \, \Omega r^2. \end{equation} \tag{ 4 }$$

However, not all of this angular momentum can be immediately applied to affecting the qualitative dynamics of the system. By qualitative, we mean that the mass inflow rate, which acts as a source of angular momentum, changes the behavior of the cluster-ring system. For instance, the source of angular momentum from mass inflow balances the sink of angular momentum from dynamical friction on the star cluster. To estimate what mass inflow rate would be needed to effect such a change, we suppose this mass is added just inside of the satellite radial position, i.e., r ≃ r_s. The mass will then be pushed to the edge of the ring, r_d, due to tidal interactions. The change in radial position will be of the order of the Hill radius of the disk, r_H,d. Hence, for such a parcel of gas of mass M, the change in angular momentum moving from r_s to r_d is

$$\begin{equation} \Delta L \sim M \, \Omega r \, \Delta r\sim M \, \Omega _sr_s\, r_{H,d}. \end{equation} \tag{ 5 }$$

For a constant mass inflow rate $\dot{M}$, we expect the torque from this inflow of gas, acting on the star cluster, to be

$$\begin{equation} T_{\rm infl}= \frac{{\rm d}\Delta L}{{\rm d}t} \sim \dot{M} \, \Omega _sr_s\, r_{H,d}. \end{equation} \tag{ 6 }$$

Setting this inflow torque equal to the dynamical friction torque in Equation (1), we find the scale for the mass inflow rate to be

$$\begin{equation} \dot{M} \sim \frac{M_s}{t_{\rm df}} \left(\frac{M_{\rm enc,s}}{M_d}\right)^{1/3}, \end{equation} \tag{ 7 }$$

where we have used the definition of the Hill radius of the ring given in Equation (3). Had we instead (naively) set Equation (4) to be equal to Equation (1), we would have found $\dot{M} \sim M_s/t_{\rm df}$, which can be significantly smaller than the more physical scale in Equation (7).

As shown in Section 2.1 above, t_df ≃ t_cross/q', so that $\dot{M} \sim (M_d/t_{\rm cross}) \, q^{\prime 2} / q_d^{1/3}$. Substituting the average (gas) mass inside the ring of M_d ∼ 5 × 10⁷ M_☉, the crossing time of t_cross ∼ 20 Myr, and the observed range in ring-to-enclosed mass of q_d = 0.01 − 0.03, the typical scale for the mass inflow rate becomes $\dot{M} \sim 10 \, q^{\prime 2}$ M_☉ yr⁻¹. For star clusters with masses that are 1% (q' = 0.01) to 10% (q' = 0.1) of the nuclear ring mass, this implies a scale $\dot{M}$ from about 0.001 to 0.1 M_☉ yr⁻¹. To balance the combined dynamical friction from the many star clusters that are formed, the total mass inflow rate might easily have to be as high as ∼1 M_☉ yr⁻¹, which is the typical star formation rate in nuclear rings (e.g., Mazzuca et al. 2008). For mass inflow rates significantly below this scale, we expect the effect of this source of (gas) mass to be insignificant. For mass inflow rates at or above this scale, the effect of dynamical friction to push the cluster-ring system as a whole to smaller radii is halted.

We now present the governing equations for our model. We do not present a detailed and complete derivation because these equations have been already derived in much of the literature (e.g., Lin & Papaloizou 1979b, 1979a, 1986; Hourigan & Ward 1984; Ward & Hourigan 1989; Rafikov 2002), with two differences. First, we include the effect of dynamical friction of the background bulge stars, which is extensively discussed in C08 (to which we refer the interested reader). Second, we will include a mass source term for, which models the inflow of gas mass transitioning from the X₁ to X₂ orbits.

The evolution of a viscous gas ring under the influence of an external torque is given by equations for continuity and angular momentum. The equation of continuity is

$$\begin{equation} \frac{\partial \Sigma }{\partial t} + \frac{1}{r} \frac{\partial (r v_r \Sigma)}{\partial r} = S(r,t), \end{equation} \tag{ 8 }$$

where r is the radial coordinate, v_r is the radial component of the velocity, Σ is the surface mass density of the gas in the ring, and S(r, t) is a position- and time-dependent source term. The angular momentum equation is (Pringle 1981; Rafikov 2002)

$$\begin{eqnarray} \frac{\partial (\Sigma \, \Omega r^2)}{\partial t} + \frac{1}{r}\frac{\partial (r v_r \Sigma \, \Omega r^2)}{\partial r} &=& - \frac{1}{2\pi r} \left(\frac{\partial T_{\rm visc}}{\partial r} - \frac{\partial T_{\rm disk}}{\partial r} \right)\nonumber\\ && + S(r,t) \, \Omega r^2, \end{eqnarray} \tag{ 9 }$$

where T_visc = −2πr³νΣ ∂Ω/∂r is the viscous torque with ν the viscosity. Taking into account that Ω (through M_enc) is not an explicit function of time, we combine Equations (8) and (9) to eliminate v_r, so that we are left with

$$\begin{equation} \frac{\partial \Sigma }{\partial t} = \frac{1}{2\pi r} \frac{\partial }{\partial r} \left[ \frac{\partial (r^2\Omega)}{\partial r} \right]^{-1} \left(\frac{\partial T_{\rm visc}}{\partial r} - \frac{\partial T_{\rm disk}}{\partial r} \right) + S(r,t). \end{equation} \tag{ 10 }$$

The radial motion of the star cluster is governed by the competing torques of dynamical friction and ring tides. Henceforth, the change of angular momentum of the star cluster with time is

$$\begin{equation} M_s\frac{\partial (\Omega _sr_s^2)}{\partial t} = T_{\rm df}- T_{\rm disk}. \end{equation} \tag{ 11 }$$

For the (bulge of the) spiral galaxy in which star cluster and nuclear ring are embedded, we adopt a density profile ρ ∝ r⁻² (see Section 5.1 for a discussion on the validity of such an isothermal profile). The corresponding mass profile is given by M_enc = M_enc,0(r/r₀), with M_enc,0 being the mass enclosed within radius r₀. Such a mass profile has the nice property that the orbital velocity, $v_{\rm orb}= \Omega r = \sqrt{G M_{\rm enc,0}/r_0}$, is constant, and thus ∂(Ω_sr²_s)/∂t = v_orb∂r_s/∂t in Equation (11). With this mass profile, the partial derivatives of the viscous and tidal torques reduce to (see also C08)

$$\begin{eqnarray} \frac{\partial T_{\rm visc}}{\partial r} & = & 2\pi v_{\rm orb}\frac{\partial (r \nu \Sigma)}{\partial r}, \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} \frac{\partial T_{\rm disk}}{\partial r} & \simeq & \beta \frac{G M_s^2 r_0}{M_{\rm enc,0}} \frac{r^3 \Sigma }{(r_s-r)^4}. \end{eqnarray} \tag{ 13 }$$

For the viscosity we follow C08, adopting ν = ν₀(Σ/Σ₀)² as suggested by Lin & Papaloizou (1986) and ν₀ = α(Q₀q_d)²/2r_d,0v_orb using the standard Shakura & Syunyaev (1973) α-disk with Toomre (1964) Q₀ parameter, and, as before, q_d = M_d/M_enc,0 the ratio of the nuclear ring (gas) mass to the enclosed mass.

Substituting these expressions and rescaling the surface mass density and time respectively as σ = Σ/Σ₀ and t' = t/t_df,0, with t_df,0 = M_enc,0/(Ω₀M_sln Λ), Equations (10) and (11) become

$$\begin{eqnarray} \frac{\partial \sigma }{\partial t^{\prime }} = \frac{\alpha ^{\prime }}{q^{\prime }} \, q_d\frac{r_{d,0}^2}{r} \frac{\partial ^2 (r\sigma ^3)}{\partial r^2} &+& \beta ^{\prime }q^{\prime }\, q_d\frac{r_{d,0}^3}{r} \frac{\partial }{\partial r} \frac{r^3\sigma }{(r_s-r)^4}\nonumber\\ &+& \gamma ^{\prime }q^{\prime }\, \frac{1}{q_d^{1/3}} f(r), \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \frac{\partial r_s}{\partial t^{\prime }} & = & \beta ^{\prime }\, 2 q_dr_{d,0}\int _{0}^{r_d} \frac{r^3\sigma }{(r_s-r)^4} \, {\rm d}r - \frac{r_{d,0}^2}{r_s}. \end{eqnarray} \tag{ 15 }$$

As before, q' = M_s/M_d is the ratio of the cluster mass to the nuclear ring (gas) mass, α' ≡ αQ²₀/(2ln Λ) and β' ≡ β/(2πln Λ) set the strength of the ring viscous and tidal torques relative to the dynamical friction torque, and the constant γ' controls the amount of inflow torque due to gas coming in along X₁ orbits that transitions to X₂ orbits. The dimensionless function f(r) defines the radial extent where this gas is deposited in the nuclear ring. We choose the form

$$\begin{equation} f(r) = \left\lbrace \begin{array}{ll} \frac{1}{2\Delta r} & {\rm if}\ r_{d,0}- \Delta r< r < r_{d,0}\\ 0 & {\rm otherwise} \end{array} \right., \end{equation} \tag{ 16 }$$

so that for γ = 1, the inflow torque and dynamical friction torque approximately cancel, corresponding to the scale of the mass inflow rate in Equation (7).

Equations (14) and (15) are the governing equations governing our model. They are the same as, respectively, Equations (23) and (24) in C08, except for the extra source term at the right-hand side of Equation (14). In addition, our initial conditions and timescales of interest are different from the scenario studied in C08.

We give an order of magnitude estimate for the rate at which a newly formed star cluster will separate itself form the nuclear ring. For simplicity, we suppose that the gas in the nuclear ring does not react significantly to the tidal torque, i.e., Σ ≃ Σ₀ (σ ≃ 1) and r_d ≃ r_d,0 remain approximately constant. In this case, Equation (14) can be neglected, and ignoring the last term in Equation (15) due to dynamical friction, it simplifies to

$$\begin{equation} \frac{\partial r_s}{\partial t^{\prime }} \sim 2 \beta ^{\prime }q_dr_{d,0}\int \frac{r^3}{(r_s-r)^4} \, {\rm d}r. \end{equation} \tag{ 17 }$$

Since r_s − r ≪ r, the latter integral out to the edge of the ring r_d,0 can be approximated as [r_d,0/(r_s − r_d,0)]³/3. This leads to an ordinary differential equation in x ≡ (r_s − r_d,0)/r_d,0 of the form dx/dt' = (2β'q_d/3) x⁻³. The resulting solution, together with the Hill radius r_H,d defined in Equation (3) and r_s ≳ r_d,0 ≫ r_H,d, implies that

$$\begin{equation} \left(\frac{r_s-r_{d,0}}{r_{H,d}} \right)^4 \sim \frac{8\beta ^{\prime }}{3} \frac{r_{d,0}}{r_{H,d}} \frac{t}{t_{\rm df,0}}. \end{equation} \tag{ 18 }$$

Since the left-hand side and 8β'/3 are of order unity, the timescale for a star cluster to separate itself from the nuclear ring in which it was formed is of order t_sep ∼ (r_H,d/r_d,0)t_df,0. The observed range of q_d = 0.01 − 0.03 implies t_sep/t_df,0 ∼ 0.2 − 0.3. Furthermore, in units of the crossing time t_cross = t_df,0q'q_d2πln Λ, the separation time is about t_sep/t_cross ∼ (r_H,d/r_d,0)/q'. This means a few to tens of orbits for cluster-to-ring mass ratios from q' = 0.01 to q' = 0.1, or about 500 Myr to 50 Myr for a typical crossing time of t_cross = 20 Myr.

Since the time t in Equation (18) goes as a fourth power of the cluster-ring separation r_s − r_d,0, the initial separation is much faster than t_sep. This means that for a star cluster to reach a significant fraction of the Hill radius of the nuclear ring is very fast, i.e., typically of order less than 1% of the dynamical friction time. This implies a factor 20–30 quicker than t_sep, or, with t_cross = 20 Myr, less than 20 Myr for q' = 0.01 and less than 2 Myr for q' = 0.1. We come back to this separation in the next section when we (numerically) investigate the system in greater detail.

We solve Equations (14) and (15) using standard explicit finite-difference methods (Press et al. 1992). For good spatial resolution, we use 2000 grid points in r/r_d,0 between zero and 1.50, with the star cluster initially at r_s/r_d,0 = 1.01. We set q_d = M_d/M_enc = 0.01 similar to the observed ratios of (gas) mass in the ring to enclosed mass within the ring radius derived in Section 2.1. Furthermore, for the dimensionless parameters α' and β', representing the strength of the ring viscosity and tides relative to the dynamical friction, we choose β' = 1 and α' = 0.01. We first study the case without source term, i.e., γ' = 0, while in the next section Section 3.3, we include the effect of gas mass inflow.

In Figure 2, we show the (numerical) solution of Equations (14) and (15) for a star cluster with mass relative to the (gas) mass in the nuclear ring of q' = 0.01. The top panel shows the evolution of the radial position of the star cluster, r_s, relative to the initial outer radius of the nuclear ring, r_d,0, as a function of the time, t, in units of the initial dynamical friction time, t_df,0. The bottom panel shows snapshots of the surface density profile at the edge of the nuclear ring for various times, separated by Δt/t_df,0 = 0.06, with labels A–D as indicated in the top panel. The initial buildup of material at the edge of the ring is very fast and only after t/t_df,0 = 10⁻⁴ (point A) an enhancement of about 30% in the surface mass density is achieved just inside r_d,0. This rapid initial enhancement is due to the fourth-power radial dependence of the tidal torque in Equation (13), and may not develop in a realistic system. Nevertheless, the initial ring tidal torque will be very strong and pushes the star cluster outward beyond the initial edge of the ring. After a time t/t_df,0 = 0.06 (point B), the star cluster has moved out about 20% from its initial position. The surface mass density near the outer edge decreases now with respect to its initial value due to viscous spreading. Points C and D continue the trend of an outward moving star cluster and viscously spreading ring, albeit at a decreasing pace.

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In contrast, for the case of q' = 0.1 (a 10 times more massive star cluster), the star cluster does not continuously move outward, but rather stop and even moves inward as shown in Figure 3. As in the earlier case, an initial surface mass density enhancement is built at point A, though it is a factor of 10 larger than for the case q' = 0.01, i.e., 300% (q' = 0.1) as opposed to 30% (q' = 0.01) of its initial value. However, the subsequent evolution of this case differs from the earlier case. The star cluster still moves outward, but it reaches only about 15% of its initial position. Its maximum in r_s/r_d,0 is reached between point B and C, after which the star cluster "turns around" and starts moving inward with respect to the initial outer radius of the ring. The surface mass density at the edge of the ring monotonically moves inward, effectively being "shepherded" inward by the star cluster, similar to the scenario in C08.

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In both cases, the radial separation between the star cluster and the (edge of the) nuclear ring approach a constant value, which we expect from the order of magnitude estimate in Section 2.2 to be close to the Hill radius of the ring, r_H,d, as defined in Equation (3). Substituting q_d = 0.01 and using that r_s ≳ r_d,0 ≫ r_H,d, we estimate this Hill radius to be around one-fifth of the initial radius of the ring, i.e., r_H,d/r_d,0 ∼ 0.2. Combining the above shifts in the radial positions of the star cluster and nuclear ring, we find that the separations for both cases indeed converge to around one-fifth of r_d,0, although for q' = 0.01 the separation remains slightly smaller, while for q' = 0.1 it is somewhat larger. This is further illustrated in Figure 4, where we show the evolution of the radial separation r_s − r_d,edge normalized by the Hill radius r_H,d. We use two definitions of the radius of the disk edge, r_d,edge. For q' = 0.01, we define r_d,edge as largest r_d for which Σ/Σ₀ > 75%, while for q' = 0.1, r_d,edge is defined as the r_d for which Σ peaks. Despite these two different definitions, Figure 4 shows two important points. First, the separation between the disk and satellite is of order r_H,d, consistent with our expectations from Section 2.2. Second, the final separation is reached after about 20% of the dynamical friction time, consistent with the estimate of t_sep/t_df,0 ∼ r_H,d/r_d,0 ∼ 0.2 in Section 3.1 for q_d = 0.01. This figure also confirms that the initial separation is very quick, indeed reaching half of the final separation within a one percent of the dynamical friction time, also consistent with our earlier expectations.

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We now include the effect of a constant gas mass inflow on the dynamics of the cluster–ring systems. Figure 5 shows the results of (numerically) solving the same system with q' = 0.1 as in Figure 3, except that the source term in Equation (14) now contributes with γ' = 1. We estimated in Section 3.1 that the latter corresponds to an mass inflow rate of $\dot{M} \sim 0.1$ M_☉ yr⁻¹. Comparing the top panels of the two figures, we find that the star cluster moves (slightly) further out when γ' = 1 and remains constant at the maximum radial position, as opposed to the shrinking of r_s after having moved out for an initial period when γ' = 0. This is because in the case that γ' = 1, the inflow torque is balancing the torque due to dynamical friction, as we estimated in Equation (7). The bottom panels of the two figures are very similar, both showing an enhancement as well as inward migration of the surface mass density. The most significant difference is that in Figure 5 beyond the edge at r/r_d,0 ∼ 0.92 there is a long tail out to r/r_d,0 ∼ 1, due to mass inflow into this region. Another difference is that for γ' = 1 after the initial inward push of the edge, it remains fixed at r/r_d,0 ∼ 0.92, whereas for γ' = 0 it still moves inward, although at a decreasing rate. This is consistent with the above difference in the change in the radial position r_s of the star cluster. As a consequence, also in the case that γ' = 1, the separation between the star cluster and the (edge of the) nuclear ring converge to a constant value, which is again of the order of the Hill radius of the ring.

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In Figure 6, we show the evolution of the radial position of the star cluster relative to the initial outer edge of the ring r_s − r_d,0, in units of the latter Hill radius r_H,d, for four different constant mass inflow rates. From the bottom to the top the curves correspond to γ' = 0, 0.3, 1 and 3. This clearly shows the transition around γ' = 1: after a similar initial outward migration, the motion of the star cluster goes from infalling for γ' ≲ 1 to outgoing for γ' ≳ 1. This is fully consistent with our estimated scale for the mass inflow rate $\dot{M}$ in Equation (7), case of single star cluster with q' = 0.1.

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From our analysis in Section 2, we expect that a newly formed star cluster moves outward until it is about a Hill radius separated from (the edge of) the nuclear ring. In Section 2.2, we derived a range in r_H,d from about 50 pc to 500 pc, with 150 pc as a typical value. For a host galaxy at distance of ∼10 Mpc, this corresponds to about 1'' to 10'', and typically 3'', which should be readily observable. However, while the position of the (young) star clusters with respect to the center of their host galaxy is straightforward to measure, the radial extent of the gas in the ring is often difficult to determine. It requires high spatial resolution CO measurements to trace the underlying molecular gas. Also, part of the gas (at the edge) of the ring may be severely disturbed due to winds that clear out the gas and dust from the newly formed star clusters and/or ionized by the massive, bright stars in the young star cluster (but see Section 5.3 and Section 5.4). Still, the dust seems to be well mixed into the gas (e.g., Brunner et al. 2008), indicating that dust provides a good tracer of the extent of the gas in the ring. Apart from the contact points where the bar dust lanes connect with the nuclear ring, the dust is indeed found on the inside of the distribution of the star clusters (see also Section 2.2).

Still, using the (peak in the) dust distribution as an estimate of the edge of the ring, the (unique) measurement of a separation is complicated by the spread in radii of the star clusters. Part of this spread is because of the dependence of the dynamical friction time on the mass of the star cluster: t_df/t_cross ≃ 1/q'. Even though we found that it takes less than 1% of the dynamical friction time to reach about half of the Hill radius, with a typical crossing time of t_cross = 20 Myr, this means only 2 Myr for the most massive star clusters (q' = 0.1), but as much as 200 Myr for the least massive clusters (q' = 0.001). Since t_sep/t_df∼ 0.2–0.3, to reach the full separation takes a factor 20–30 longer. This means that only the most massive clusters reach at least half of the Hill radius separation before all bright, massive stars die. The less massive clusters have already faded significantly, or even dissolved. So, we then might expect a smooth radial gradient with the most massive clusters on the outside, which in turn can be used to establish the Hill radius.

Unfortunately, we found that the more massive clusters indeed rapidly reach a maximum separation, but after remaining for a while at this radius (depending on the mass inflow rate), they move inward again by pushing the ring inward (compare, for example, the top panels of Figures 2 and 3). Hence, over time the less massive clusters, if they survive, may catch up with the more massive clusters. We conclude that only among the (very) young (few Myr) clusters we might expect a radial gradient with the more massive clusters to the outside, from which the Hill radius can be estimated. Later on, if a radial gradient is visible at all, it might even be opposite, with the more massive clusters on the inside. Since the more massive clusters have higher chance of survival, the latter "turn around" may result in at most a mild radial gradient in age (with the older clusters on the inside). Mazzuca et al. (2008) found only for two (out of 22) nuclear ring galaxies, a radial age gradient with indeed (slightly) older clusters on the inside.

Unlike the radial age gradient, Mazzuca et al. (2008), as in other detailed studies (e.g., by Böker et al. 2008), do find signatures of an azimuthal age gradient in most of the nuclear rings, ranging from clean bipolar age gradients to a correlation between the youngest star clusters and the contact points of the bar dust arms with the nuclear ring. The natural explanation is that at (or very close to) these contact points the inflowing gas shocks and forms stars, after which the resulting star cluster travels in the ring away from the contact points. Indeed, for a typical bar pattern speed of ∼50 km s⁻¹ kpc⁻¹ (e.g., Gerssen 2002) the corresponding time for the contact points to rotate is ∼120 Myr, significantly shorter than the local crossing time t_cross ∼ 20 Myr. Alternatively, if the star formation occurs due to (local) gravitational instability of the gas in the ring, the star clusters are expected to form throughout the ring without an azimuthal age gradient. Instead of associating this instability with an ILR (Elmegreen 1994), it is a natural consequence of shepherding: The gas in the ring loses angular momentum to the star cluster(s), which in turn lose angular momentum to the host galaxy through dynamical friction. This means that gas at the edge of the ring moves inward, resulting in an increase in the gas surface density at the edge (see also Figures 2, 3 and 5), until its becomes unstable against self-gravity.

In the case of shepherding we do expect, as observed, a mixture in azimuthal age gradients. Initially, when the gas surface density in the ring is below the critical value for star formation, star clusters only form in episodes at the two contact points, they move together with the remaining gas along the ring, resulting in a bipolar age gradient. The newly formed star clusters move outward, while at the same time "pushing" the gas inward, causing an overdensity buildup at the edge of the ring. When the gas surface density somewhere along the edge of the ring reaches the critical value, it becomes unstable and forms stars. The more massive the star cluster (or group of star clusters acting together), the higher the overdensity and the sooner the critical value is reached. Generally, the most massive clusters and hence the star formation might be anywhere along the ring. However, depending on the mass inflow rate, there might be relatively more gas near the contact points to start with, so that first instability-driven star clusters still form near the contact points. Therefore, an azimuthal age gradient might still exist over part of the ring, but eventually the gradient is expected to flatten.

We found in Section 3.3 that for a low enough mass inflow rate, the (most massive of the) existing star clusters move inward by dynamical friction, while "pushing" and increasing the surface mass density of the gas in the outer part of the nuclear ring. As a result, the gas might reach the critical surface mass density and form new stars (in clusters) all along the edge of the ring. At the same time, near the contact points, which also move inward along with the nuclear ring, star formation from inflowing and shocking gas might be ongoing. We argue below in Section 4.3 that the nuclear ring in NGC 4314 is an example of this mechanism. Since even in the case of a high mass inflow rate, eventually enough (massive) star clusters are formed for their (combined) dynamical friction to "push" to ring inward, the corresponding gas and bright star clusters migrate inward from larger to smaller X₂ orbits. In other words, nuclear rings will be located between the so-called outer and inner ILR, although strictly speaking resonances are only applicable to systems with a weak bar. Depending on the details of the mass inflow rate and star formation efficiency over time, we thus expect a variety in sizes of nuclear rings with respect to the resonance radii, as well as a significant diversity in their star cluster distribution. Hydrodynamic simulations indeed show the inward migration of nuclear rings (e.g., Regan & Teuben 2003; Fukuda et al. 2000), and observations of nuclear rings confirm the diversity in their star cluster distribution (e.g., Mazzuca et al. 2008)

NGC 4314 is a nearby (D ∼ 10 Mpc, 1'' ≃ 48.5 pc) galaxy with a large-scale bar out to a radius of ≃65'' (3.15 kpc). It hosts a prominent circumnuclear ring of star formation that peaks at a radius of ≃70 (340 pc), which is visible in Hα, radio continuum as well as optical color maps. High spatial resolution CO observations by Benedict et al. (1996) clearly show that the molecular gas peaks at a smaller radius of ≃55 (270 pc).

The circular velocity curve flattens at ∼7'' to a value of ∼190 km s⁻¹ (Combes et al. 1992), corresponding to a (spherically) enclosed mass M_enc ∼ 2.8 × 10⁹ M_☉. The total molecular gas mass within 7'' is ∼15 × 10⁷ M_☉. However, this includes gas that is still flowing in along the dust lanes at the contact points, as well as a significant amount in the center where a strong peak in the CO coincides with a peak in ${\rm H}\alpha +[\textsc {N}\,\textsc {II}]$ consistent with the LINER definition. Hence, we estimate the ring-to-enclosed mass to be q_d ∼ 0.03 (see also Benedict et al. 1996). The Hill radius of the ring is thus r_H ∼ 105 pc or r_H ∼ 22. The observed difference of ≃15 between the peaks in star formation and CO flux is smaller than the predicted Hill radius, which is expected based on the discussion in Section 4.1. Although the peak in CO flux nicely corresponds with the peak in the dust distribution and thus likely traces the edge of the ring, the star clusters have not yet reached the Hill radius. Given a separation time of t_sep/t_df ∼ q^1/3_d ∼ 0.3, a dynamical friction time of t_df/t_cross ≃ 1/q', a crossing time of t_cross ∼ 11 Myr, and an average age of the star clusters of t ∼ 15 Myr, we expect, based on Equation (18), that the star clusters have reached a separation of about Δr ∼ 1.5 q'^1/4 r_H,d. For star clusters with a (combined) mass relative to the gas mass in the ring from q' = 0.01 to q' = 0.1, this means a separation from about 10 to 18, bracketing the observed difference.

There is no evidence for an azimuthal age gradient, but a slight tendency for a radial age gradient with the few clusters outside the ring begin all older than 15 Myr. This is consistent with the shepherding scenario in which the gas in the ring has been "pushed" inward until a critical surface density is reached and all along the edge of the nuclear ring star clusters are formed, which then move outward. The (group of) star cluster(s) responsible for the inward migration of the ring likely have faded too much to be visible against the old underlying stellar population of the host galaxy. However, further support for the inward migration of the ring comes from two blue stellar spiral arms that seem to connect the outermost X₂ orbits (also referred to as the outer ILR) with the nuclear ring (Wozniak et al. 1995; Benedict et al. 2002). Subtracting a model of the red host galaxy, Benedict et al. (2002) found that the stars in the arm close to the nuclear ring are ∼20 Myr old, with an indication of an increase in age beyond ∼100 Myr going outward in the arms. Whereas during the inward migration the gas surface density along (the edge of) the ring was insufficient to form stars, shocks at the contact points might still have allowed the formation of stars. While the contact points moved inward together with the ring, these stars stayed behind and created the blue arms.

Benedict et al. (1996) derive an outer ILR at ∼13'' and inner ILR at ∼4'' for an assumed bar pattern speed Ω_p ∼ 72 km s⁻¹ kpc⁻¹ of the large-scale bar. The latter is likely an upper limit because typically the corotation radius is beyond the ∼3 kpc radius of the bar, so that with a 190 km s⁻¹ circular velocity, Ω_p ≲ 60 km s⁻¹ kpc⁻¹. In turn this implies that the above outer and inner ILR are probably lower and upper limits, respectively. This clearly shows that it is critical to obtain an independent measurement of the bar pattern speed, using for example the Tremaine & Weinberg (1984) method. Even so, the nuclear ring is in between the outer and inner ILR, in line with the shepherding scenario, in which the ring migrates inward until the increasing gas surface density at the edge reaches the critical value for star formation. While the newly formed star clusters move (slightly) outward peaking at ∼70, the gas in the ring is "pushed" further inward peaking at ∼55. Note that Benedict et al. (1993) detected a nuclear bar within ∼4'' that might provide a mechanism to transport part of the gas further inward toward the presumed black hole in the center (see also Benedict et al. 2002).

In deriving the equations for our model in Section 2.3, we have adopted a density profile ρ ∝ r⁻² for the (bulge of the) spiral galaxy in which the star cluster and nuclear ring are embedded. At the radii where nuclear rings are present (0.2–1.7 kpc; Mazzuca et al. 2008), the density of galaxies indeed seems to be not too far from such an isothermal profile, as indicated, for example, by studies that combine stellar dynamics with strong gravitational lensing (e.g., Koopmans et al. 2006; van de Ven et al. 2008). A more straightforward way is to consider that the surface brightness profiles of (bulges of) galaxies are well fitted by a Sérsic (1968) profile I(R) ∝ exp [ − (R/R_e)^1/n], with index n and the effective radius R_e enclosing half of the total light. The (numerically) deprojection is well approximated by the density profile of Prugniel & Simien (1997), of which the (negative) slope is given by

$$\begin{equation} \gamma = p_n + \frac{b_n}{n} \left(\frac{r}{R_e}\right)^{1/n}, \end{equation} \tag{ 19 }$$

with p_n = 1.0 − 0.6097/n + 0.05463/n² and b_n = 2 n − 1/3 + 4/405 n + 46/25515 n² (Ciotti & Bertin 1999). For an average bulge of a spiral galaxy n ∼ 2 and R_e ∼ 1 kpc (e.g., Hunt et al. 2004), so that at the typical radius r ∼ 0.5 kpc of a nuclear ring, we indeed find that the slope of the density γ ∼ 2 is close to isothermal.

The star cluster that is being formed in the nuclear ring is already subjected to dynamical friction while it is still a gas clump, as long as the material that makes up the (proto) star cluster is bound enough relative to the background stellar population of the host galaxy. Since this gas clump contains (significantly) more mass than the final cluster of only stars, the initial dynamical friction is (much) stronger. However, the clearing of gas (and dust) from the newly formed star cluster is very fast and efficient, since even the youngest visible star clusters are only mildly reddened (e.g., Maoz et al. 2001).

We assume that the gas in the nuclear ring is uniform, i.e., no gas clouds (or clumps) aside from those which turn into star clusters. Since such gas clouds would also be subject to dynamical friction from the background bulge stars and thus move inward, dynamical friction on the nuclear ring itself has also been proposed as an explanation of the inward migration of the nuclear ring in for example NGC 4314 (Combes et al. 1992).

Suppose that the gas is made up of N gas clouds of the same mass M_c = M_d/N, then about N = 1/q'² clouds are needed to have a combined dynamical friction torque of the same order as that of a single star cluster of mass M_s = q'M_d. Adopting as before the two cases of q' = 0.01 and q' = 0.1, this respectively means N = 10⁴ and N = 10² gas clouds of mass M_c = 5 × 10³ M_☉ and M_c = 5 × 10⁵ M_☉ for a typical nuclear ring mass of M_d = 5 × 10⁷ M_☉. From Section 2.1 we have that the dynamical friction time in units of the crossing time is approximately given by t_df/t_cross ≃ M_d/M_s, so that with M_s = M_c = M_d/N for a gas cloud we obtain t_df ≃ Nt_cross. Given a typical crossing time of t_cross ∼ 20 Myr, the timescales of dynamical friction for the latter two cases are about t_df ∼ 200 Gyr and t_df ∼ 2 Gyr, respectively. We thus conclude either nearly no inward migration (q' ∼ 0.01), or gas that is very clumpy with quite large gas clouds (q' ∼ 0.1).

Moreover, since the gas clouds will span a range in mass with different corresponding dynamical friction timescales, they migrate inward at different rates, leading to a (radial) diffusion of the nuclear ring, especially with respect to the gas that is not in clouds and stays behind. Although in some galaxies, such as M 83, which are claimed to harbor a (partial) nuclear ring, the corresponding morphology is rather clumpy and chaotic, most of the nuclear rings are quite smooth and well defined, including the nuclear ring in NGC 4314. Collisionless star clusters are also less prone to destruction than collisional gas clouds, in particular in the presence of shear stresses from differential speeds. As a result, at least the most massive star clusters will sink faster inward than the gas clouds, and appear to be on the inside of the gas (clouds) in the nuclear ring, which typically is not observed.

We have assumed that the star cluster form just exterior of the nuclear ring. This assumption simplifies the physics and the numerical computation, though it is not absolutely necessary. If the star cluster forms near the outer edge of the nuclear ring, we do not expect the resulting dynamics to be significantly different. By near, we demand that the separation between the star cluster and the outer edge is within one Hill radius of the nuclear ring. In this way, the amount of mass which participates in the tidal evolution of the cluster-ring system exterior to the star cluster is less then interior to the star cluster. As long as this condition is fulfilled, the evolution should proceed along the lines of our calculation.

There are a few reasons to believe that the star clusters (preferably) form near the outer edge of the nuclear ring. The supply of gas from the bar merges with the ring at the outermost radius. It is unknown how the gas spreads radially over the rest of the ring, but if it is due to viscous processes, for instance the magneto radial instability (MRI; Balbus & Hawley 1991), then the surface density is maximal at the outer edge. As a result, the gas in the ring is most likely to be (come) unstable near the outer edge and hence preferentially fragment there.

While many clusters will form near the outer edge (and hence be susceptible to shepherding) for the reasons given above, clusters are also likely to form inside of the outer edge. For instance, once massive star clusters are formed, their interaction with the gas ring may result in large and fairly symmetric enhancements in the gas surface mass density (see for example Figure 3), so that new, but significant smaller, star clusters can be formed around these enhancements. For the relatively small star clusters formed on the inside, we expect the inward migration to follow that of forming proto planets (see, e.g., Ward 1997). In addition, if the gas flowing along the bar is very clumpy, possess a different vertical scale height, or a different inclination than the gas ring, it may not (only) merge at the outermost radius. In this way, even rather large star clusters might form on the inner edge of the nuclear ring. They then migrate further inward (relatively faster than cluster formed on the outer edge as they encounter less of the gas ring), possibly along an inward extension of the inspiraling gas. Whereas these are possible explanations for the observation that some star clusters appear more on the inside of the nuclear ring (see also Section 2.2), alternatively they might form already inside of the ring as the result of the local interaction of a nonaxisymmetric structure such as a nuclear bar or spiral with the ISM (see also Section 5.5).

Winds originating from (massive) stars and/or supernovae in the star clusters might not only clear out the remaining gas in the star cluster itself, but also push away part of the gas in the nuclear ring. This creates a separation between the cluster and the ring that might mimic the outward migration of the star cluster. However, since the density of the ISM is far less than that of the gas in the ring, the winds will escape in the vertical direction like a fountain arising from the (equatorial) plane of the ring. Therefore, even if the cluster is embedded in the ring itself, the winds are expected to clear out at most a region that is equal to scale height of the disk, h_d, which is significantly smaller than the Hill radius of the ring, r_H,d.

To show this, we start from Toomre (1964) Q parameter for the gas in the ring: Q = c_sκ/πGΣ (Binney & Tremaine 1987). Here, the epicycle frequency κ is defined as κ² = (4 + dln Ω/dln r)Ω², so that $\kappa = \sqrt{2} \Omega$ for the assumed (isothermal) density profile ρ ∝ r⁻². Adopting c_s = h_dΩ, we obtain for the scale height of the ring $h_d/r_d\simeq (M_d/M_{\rm enc}) \, Q/\sqrt{2}$. On the other hand, the Hill radius of the ring r_H,d/r_d ≃ (M_d/M_enc)^1/3. Assuming Q ≃ 1 and substituting the mean observed ring-to-enclosed mass ratio M_d/M_enc ≃ 0.02, we find that h_d/r_H,d ≃ 0.05. This indeed shows that winds might clear out a region that is at most a few per cent of the Hill radius of the nuclear ring.

A second possible stellar feedback effect, is due to the (UV) radiation coming from the massive, bright stars in the (young) star cluster, which might (photo)ionize the (molecular) gas in the nuclear ring. We can estimate the radial extent of this ionization from the radius r_ion of the Strömgren sphere. We start from $\dot{N}_{\rm ion} \simeq \alpha \, n_g^2 \, 4\pi r_{\rm ion}^3/3$, with a cross section α ≃ 3 × 10⁻¹³ cm s⁻¹. We assume that bright stars radiate roughly at 10% of the Eddington luminosity in the UV (⩾10 eV), to obtain an ionization rate of $\dot{N} < 10^{48} \, (M_\star /{\rm M}_\odot)$ photons per second. With the above estimate of the scale height h_d, we find for the gas mass density $\rho _g \simeq M_{\rm enc}\sqrt{2}/\pi r_d^3 Q$. Substituting M_enc ∼ 2.5 × 10⁹ M_☉, r_d ∼ 0.5 kpc and Q ≃ 1, and dividing by the proton mass, we find for the number density of the gas n_g ∼ 4 × 10² cm⁻³. In this way, we estimate that a (UV) radiating star of mass M_⋆ is able to ionize gas out to a (maximum) radius of r_ion < 3(M_⋆/M_☉)^1/3 pc.

For a typical O star of M_⋆ ∼ 40 M_☉, this means r_ion < 10 pc, which is significantly smaller than the typical Hill radius of r_H,d ∼ 150 pc. The combined (and focused) ionizing radiation of thousands of massive young stars with a lifetime of only a few to ten Myr is needed to reach the latter radius. Even then, the dust, which is hardly effected by the radiation, will still provide a good tracer of the gas in the ring, and as such can be used to establish the separation between star clusters and the nuclear ring.

For sufficiently massive star clusters and/or for a sufficiently large number of them, it is in principle possible for these star clusters to shepherd the nuclear ring to smaller radii. As in C08, we assumed the gas ring sits in a simple spherically symmetric potential. However, the gas ring arises as a result of the stellar bar, which is a nonaxisymmetric feature in the potential, driving the gas inward toward the X₁ − X₂ orbit transition. In addition, the gas ring is self-gravitating with Toomre's (1964) Q near unity. We now speculate on the effect of this nonaxisymmetry and self-gravity on the gas shepherding scenario.

First, as the gas ring is being shepherded inward, it moves from one X₂ orbit to a smaller one. Eventually, it may move inward of the smallest nonintersecting X₂ orbit in a barred potential (e.g., Regan & Teuben 2003). The gas which is pushed beyond this point will find itself in intersecting orbits, losing angular momentum on each orbit as it shocks against gas on the same orbit or on neighboring orbits. The high(er) density regions resulting from the shocks, might give rise to the observed (chaotic) spurs of gas and dust in the centers of (late-type) galaxies (e.g., Elmegreen et al. 1998; Martini et al. 2003).

Second, the effect of self-gravity in the shepherded gas ring is nontrivial. As we stated above in Section 5.3, the effect of an enhanced surface density at the edge of the ring results in Q ≲ 1, if the initial Q ≃ 1. In addition, this local enhancement in the surface density may give rise to the gaseous analogy of stellar self-gravitating groove and edge modes (see also Lovelace & Hohlfeld 1978; Sellwood & Kahn 1991; Papaloizou & Savonije 1991). This may result in the formation of nuclear bars (e.g., Laine et al. 2002; Erwin & Sparke 2002; Englmaier & Shlosman 2004) or nuclear spirals (e.g., Englmaier & Shlosman 2000; Pogge & Martini 2002), providing a means for inward transport of gas from the nuclear ring region in analogy to the "bars within bars" scenario (Shlosman et al. 1990).