Effects of Varying Mass Inflows on Star Formation in Nuclear Rings of Barred Galaxies

Our computational domain is a Cartesian cube with side $L=2048\,{\rm pc}$ surrounding a nuclear ring. We discretize the domain uniformly with $512^{3}$ cells with grid spacing of $\Delta x=4\,{\rm pc}$. The domain rotates at an angular frequency ${\bf\Omega}_{p}=36\,{\rm km\,s^{-1}\,kpc^{-1}}\,\hat{\bf z}$, representing the pattern speed of a bar. We use the Athena code (Stone et al., 2008) to integrate the equations of hydrodynamics in the rotating frame, coupled with self-gravity, heating and cooling, star formation, and SN feedback. The governing equations we solve read

where $\rho$ is the gas density, ${\bf v}$ is the gas velocity in the rotating frame, $P$ is the gas pressure, $\mathds{I}$ is the identity matrix, $n_{\rm H}=\rho/(1.4271m_{\rm H})$ is the hydrogen number density assuming the solar abundances, $n_{\rm H}^{2}\Lambda$ is the volumetric cooling rate, $n_{\rm H}\Gamma_{\rm PE}$ is the photoelectric heating rate, and $n_{\rm H}\Gamma_{\rm CR}$ is the heating rate by cosmic ray ionization. We assume that the cooling function $\Lambda$ depends only on the gas temperature $T$, and adopt the forms suggested by Koyama & Inutsuka (2002) for $T<10^{4.2}\,{\rm K}$ and Sutherland & Dopita (1993) for $T>10^{4.2}\,{\rm K}$.

The total gravitational potential $\Phi_{\rm tot}=\Phi_{\rm cen}+\Phi_{\rm ext}+\Phi_{\rm self}$ consists of the centrifugal potential $\Phi_{\rm cen}=-\frac{1}{2}\Omega_{p}^{2}(x^{2}+y^{2})$, the external gravitational potential $\Phi_{\rm ext}$ giving rise to the background rotation curve, and the self-gravitational potential $\Phi_{\rm self}$ that is related to $\rho$ and the newly-formed star particle density $\rho_{\rm sp}$ via

The background potential $\Phi_{\rm ext}$ arises from the central supermassive black hole modeled by the Plummer sphere

with mass $M_{\rm BH}=1.4\times 10^{8}\,M_{\odot}$ and the softening radius $r_{\rm BH}=20\,{\rm pc}$, and a spherical stellar bulge modeled by the modified Hubble profile

with the central density $\rho_{b0}=50\,M_{\odot}\,{\rm pc}^{-3}$ and the scale radius $r_{b}=250\,{\rm pc}$. The total stellar mass enclosed within the nuclear ring is $M_{b}\equiv\int_{0}^{R_{\rm ring}}4\pi r^{2}\rho_{b}(r)dr=6.7\times 10^{9}\,M_{\odot}$, where $R_{\rm ring}=600\,{\rm pc}$ is the ring radius²²2In our semi-global models, the ring radius is set by the velocity (or, more precisely, the specific angular momentum) of the inflows, as indicated by Equation (12) of Paper I.. The rotation curve resulting from $\Phi_{\rm ext}$ is similar to the observations of a prototypical nuclear-ring galaxy NGC 1097 (Onishi et al. 2015; see also Figure 1 of Paper I). With these parameters, the orbital period (in rotating frame) at $R_{\rm ring}$ is $t_{\rm orb}=18.4\,{\rm Myr}$.

At each timestep, we check for all cells whether (1) $\rho>\rho_{\rm LP}$, where we take the threshold $\rho_{\rm LP}\equiv 8.86c_{s}^{2}/(G\Delta x^{2})$ (for $c_{s}$ the sound speed) as indicative of runaway gravitational collapse, based on evaluation of the Larson-Penston asymptotic profile at radius half of the grid spacing $\Delta x$, (2) $\Phi_{\rm self}$ has a local minimum, and (3) the velocity is converging in all directions. If the above three conditions are met simultaneously in a given cell, we spawn a sink particle representing a star cluster and convert a portion of the gas mass in the surrounding 27 cells to the mass of the sink particle. The sink particles are allowed to accrete gas from their neighborhood until the onset of first SN explosion, which occurs at $t\sim 4\,{\rm Myr}$ after creation. We track the trajectories of the sink particles using the Boris integrator (Boris, 1970, see also Appendix of Paper I) which conserves the Jacobi integral very well.

Assuming that the sink particles fully sample the Kroupa (2001) initial mass function, we assign the far-ultraviolet (FUV) luminosities based on their mass and age using STARBURST99 (Leitherer et al., 1999). Treating the individual sink particles as sources of FUV radiation, we set the mean FUV intensity $J_{\rm FUV}$ based on radiation transfer in plane-parallel geometry, with an additional local attenuation in dense regions. We then scale the photoelectric heating rate $\Gamma_{\rm PE}$ in proportion to $J_{\rm FUV}$, while allowing for the metagalactic FUV background (a small contribution). The heating rate $\Gamma_{\rm CR}$ by cosmic ray ionization is taken to be proportional to the SFR averaged over a $40\,{\rm Myr}$ timescale, assuming that cosmic rays are accelerated in SN remnants.

Sink particles with age $\sim 4$–$40\,{\rm Myr}$ produce feedback representing type II SNe, with rates based on the tabulation in STARBURST99. The amount of the feedback energy or momentum depends on the gas density. If the gas density surrounding an SN is so low that the Sedov-Taylor stage is expected to be resolved, we inject the total energy $E_{\rm SN}=10^{51}\,{\rm erg}$, with $72\%$ and $28\%$ in thermal and kinetic forms, respectively. If the gas density is instead too high for the shell formation radius to be resolved, we inject the radial momentum of $p_{*}=2.8\times 10^{5}\,M_{\odot}\,{\rm km\,s^{-1}}\,(n_{\rm H}/{\rm cm}^{-3})^{-0.17}$, corresponding to the terminal momentum injected by a single SN (Kim & Ostriker, 2015a). Each SN also returns the ejecta mass of $M_{\rm ej}=10\,M_{\odot}$ from a sink particle back to the gas. The reader is referred to Kim & Ostriker (2017) for the full description of the sink formation and SN feedback.

In our semi-global models, the simulation domain is initially filled with rarefied gas with number density $n_{\rm H}=10^{-5}\,\exp\left[{-|z|/(50\,{\rm pc})}\right]\,{\rm cm^{-3}}$ and temperature $T=2\times 10^{4}\,\rm K$. Gas flows in to the domain through two nozzles located at the $y$-boundaries, mimicking bar-driven gas inflows along dust lanes. The mass inflow rate $\dot{M}_{\rm in}$ is controlled by varying the gas density inside the nozzles. Paper I focused on the cases with constant $\dot{M}_{\rm in}$ over time. In this paper we report on two additional suites of models to explore the effects of time variations and asymmetry in the mass inflow rates. Table 1 lists the models we consider.

In the first suite, we let the mass inflow rate vary sinusoidally with time in logarithmic scale as

where $\Delta\tau_{\rm in}$ is the oscillation period. The dimensionless constants $A$ and $B$ parameterize the time-averaged inflow rate $\dot{M}_{\rm in,0}\equiv\int_{0}^{\Delta\tau_{\text{in}}}\dot{M}_{\text{in}}dt/\int_{0}^{\Delta\tau_{\text{in}}}dt$ and the oscillation amplitude ${\cal R}(\dot{M}_{\rm in})\equiv\max(\dot{M}_{\rm in})/\min(\dot{M}_{\rm in})$, such that $\dot{M}_{\rm in,0}/(M_{\odot}\,{\rm yr}^{-1})=I_{0}(B)e^{A}$ and ${\cal R}(\dot{M}_{\rm in})=e^{2B}$, where $I_{0}$ is the zeroth-order modified Bessel function of the first kind. In the first suite, we fix $\dot{M}_{\rm in,0}=0.5\,M_{\odot}\,{\rm yr}^{-1}$ and ${\cal R}(\dot{M}_{\rm in})=20$, corresponding to $A=-1.19$ and $B=1.50$, and consider three models P15, P50, and P100 with $\Delta\tau_{\rm in}=15$, $50$, and $100\,{\rm Myr}$, respectively. We also include model constant with ${\cal R}(\dot{M}_{\rm in})=1$ and $\Delta\tau_{\text{in}}=\infty$, which is identical to model L1 in Paper I. Note that $\dot{M}_{\rm in}(t)$ denotes the total inflow rate through the two nozzles: the mass inflow rate from one nozzle is $\dot{M}_{\rm in}(t)/2$ in the first suite.

In the second suite, we consider three models termed asym, off, and boost to investigate the effect of asymmetric inflows. In model asym, the mass inflow rates from the upper (at the positive $y$-boundary) and the lower (at the negative $y$-boundary) nozzles are $0.45\,M_{\odot}\,{\rm yr}^{-1}$ and $0.05\,M_{\odot}\,{\rm yr}^{-1}$, respectively, and do not vary in time throughout the simulation. The mass inflow rate in model off is initially $0.5\,M_{\odot}\,{\rm yr}^{-1}$ from each nozzle, but the lower nozzle is abruptly turned off at $t=150\,{\rm Myr}$, while the upper nozzle keeps supplying the gas with the same rate all the time. In model boost, the mass inflow rate from each nozzle is set to $0.05\,M_{\odot}\,{\rm yr}^{-1}$ at early time, but the upper nozzle suddenly boosts the inflow rate to $0.45\,M_{\odot}\,{\rm yr}^{-1}$ at $t=150\,{\rm Myr}$, while the inflow rate from the lower nozzle is kept unchanged. Note that the total mass inflow rate for all three models in the second suite is $0.5\,M_{\odot}\,{\rm yr}^{-1}$ after $t=150\,{\rm Myr}$.

We calculate the SFR using the sink particles that formed in the past $10\,{\rm Myr}$ as

where $M_{\rm sp}(t)$ is the total mass in the sink particles at time $t$. Figure 1 plots for models constant, P15, P50, and P100 the temporal histories of the SFR, the total gas mass $M_{\rm gas}$ in the computational domain, and the depletion time averaged over the past $10\,{\rm Myr}$,

We note that a depletion time $t_{\text{dep,40}}$ over $40\,{\rm Myr}$ (or other time) can be analogously defined. At early time, the gas streams from the nozzles collide with each other multiple times, increasing the SFR temporarily. A nuclear ring forms at $t\sim 100\,{\rm Myr}$, after which the evolution depends strongly on $\Delta\tau_{\text{in}}$. While models constant and P15 show weak fluctuations, the SFR in models P50 and P100 varies quasi-periodically with large amplitudes, even though the gas mass is almost the same. In all models with $\dot{M}_{\text{in,0}}=0.5\,M_{\odot}\,{\rm yr}^{-1}$, the ring mass is $M_{\text{gas}}\sim 4\times 10^{7}M_{\odot}$, weakly varying with time, with peaks in the mass phase-delayed relative to the peak in $\dot{M}_{\text{in}}$. For quantitative comparison, we define the fluctuation amplitude ${\cal R}$ by taking the ratio of the maximum to minimum values of $\dot{M}_{\text{SF,10}}$, $M_{\rm gas}$, and $t_{\text{dep,10}}$ during $t=200$–$300\,{\rm Myr}$. Table 2 gives ${\cal R}(\dot{M}_{\text{SF,10}})$, ${\cal R}(M_{\rm gas})$, and ${\cal R}(t_{\text{dep,10}})$ for all the models.

Although the mass inflow rate for model constant is constant in time, its SFR shows random fluctuations with amplitude ${\cal R}(\dot{M}_{\text{SF,10}})=2.4$, due to the turbulence driven by the SN feedback, consistent with the results of Paper I. When the mass inflow rate oscillates with time, the fluctuation amplitudes increase with increasing $\Delta\tau_{\rm in}$. In model P100, for instance, the SFR varies by a factor of 17 while the gas mass varies by a factor of 2, resulting in a factor of 10 variations of the depletion time. The fluctuations become smaller in model P50, with a factor of 5 variations in the SFR and the depletion time. When the inflow rate oscillates very rapidly as in model P15, the fluctuation amplitudes are almost the same as those in model constant, as well as in the second suites of models asym, off, and boost in which the mass inflow rate is constant after $t=150\,{\rm Myr}$.

Figure 1($a$) shows that for models P100 and P50, the SFR varies coherently with the inflow rate with some time delay. To quantify the delay, we define the characteristic delay time $t_{\rm delay}$ as the time lag at which the cross-correlation between the SFR and the inflow rate is maximized, that is, $t_{\rm delay}\equiv\operatorname*{arg\,max}_{\tau}(\dot{M}_{\text{SF,10}}\star\dot{M}_{\rm in})(\tau)$. Our numerical results correspond to $t_{\rm delay}=25\,{\rm Myr}$ and $t_{\rm delay}=19\,{\rm Myr}$ for models P100 and P50, respectively. Multiple effects contribute to the time delay. First, it takes approximately $3\,{\rm Myr}$ for the inflowing gas to travel from the nozzle to the ring located at $R_{\text{ring}}$. Second, our SFR measurement introduces some delay, approximately $4\,{\rm Myr}$, because it counts all star formation events in the past $10\,{\rm Myr}$. Subtracting the above two effects, the delay time for models P100 and P50 reduces to $18\,{\rm Myr}$ and $12\,{\rm Myr}$, respectively. The resulting delay time can be interpreted as the timescale for the newly accreted gas first to enhance the overall mass in the ring, and then to produce local enhancements in the density above the (numerical) threshold for star formation. We note, from Figure 1($b$), that there are localized peaks in the ring gas mass that are earlier in phase by $\sim 10\,{\rm Myr}$ relative to the peak in SFR for both models P100 and P50. This phase offset of the peak in $M_{\text{gas}}$ is comparable to $\sim 0.7$ vertical oscillation periods associated with $\Phi_{\text{ext}}$. This appears to be the minimum time required for an enhancement in the SFR to develop via stochastic processes such as turbulent compression and gravitational contraction.

Unlike in models P100 and P50, there is no apparent correlation between the SFR and the inflow rate in model P15. This is because the SFR fluctuations due to the time variations of $\dot{M}_{\text{in}}$ are too weak to stand out against the feedback-driven fluctuations (see below).

To better understand what drives temporal variations of the SFR, Figure 2($a$) plots for model P100 the temporal histories of the mass inflow rate, the mean midplane density $\rho_{\text{mid}}$ of cold-warm gas with $T<2\times 10^{4}\,{\rm K}$, the SFR, the number of SN explosions per unit time $\dot{\mathcal{N}}_{\rm SN}$, and the gas scale height $H\equiv\left(\int\rho z^{2}\,dV/\int\rho\,dV\right)^{1/2}$. All the quantities are normalized by their respective time average over $t=100$–$300\,{\rm Myr}$. Overall, the time variations of $\dot{M}_{\text{in}}$ lead to the changes in, sequentially, $\rho_{\text{mid}}$, SFR, $\dot{\mathcal{N}}_{\rm SN}$, and $H$, with approximate delay times of $12$, $25$, $36$, and $47\,{\rm Myr}$, respectively. This makes sense since the mass inflows first enhance the ring density to promote star formation. The associated enhancement in SN feedback then inflates the ring vertically, increasing $H$. Both the mass inflows and feedback can affect the SFR by changing $\rho_{\text{mid}}$, with the latter being through $H$. All the quantities vary quasi-periodically with the dominant period of $100\,{\rm Myr}$.

To examine whether the quasi-periodic cycles of the SFR are really driven by $\dot{M}_{\text{in}}(t)$ rather than the SN feedback, we restart model P100 from $t=150\,{\rm Myr}$ by fixing the inflow rate to $\text{max}(\dot{M}_{\text{in}})=1.36\,M_{\odot}\,\text{yr}^{-1}$ thereafter, which we term model P100_restart. Figure 2($b$) plots the resulting time histories of $\dot{M}_{\text{in}}$, $\rho_{\text{mid}}$, SFR, $\dot{\mathcal{N}}_{\text{SN}}$, and $H$ of model P100_restart, normalized by the time-averaged values of model P100 for direct comparison. With fixed $\dot{M}_{\text{in}}$, the system reaches a quasi-steady state at $t\sim 200\,{\rm Myr}$ in which the SFR and the other quantities do not vary much with time. The short-term ($\lesssim 40\,{\rm Myr}$) fluctuations in the averaged quantities are due purely to turbulence driven by the SN feedback, which is active for $4$–$40\,{\rm Myr}$ after the star formation. The corresponding fluctuation amplitude in the SFR is ${\cal R}(\dot{M}_{\text{SF,10}})=2.4$ in model P100_restart, which is 7 times smaller than that in model P100, but comparable to the fluctuation amplitude in model constant. This demonstrates that the large-amplitude, quasi-periodic variations of the SFR in model P100 are caused by $\dot{M}_{\text{in}}(t)$, while the stochastic SN feedback is responsible for small-amplitude (a factor of $\sim 2$), short-term fluctuations of the SFR with timescale $\lesssim 40\,{\rm Myr}$. Of course, longer-term variations in the SN feedback rate induced by variations in the inflow rate and SFR are also dynamically important, as noted above (see also Section 3.3)

Figure 3 plots the histories of the various quantities for models P50, P15, off, and boost. Model P50 behaves qualitatively similarly to model P100, in that the oscillating inflows drive long-term ($\sim 50\,\rm Myr$), large-amplitude variations of the SFR, while short-term, small-amplitude fluctuations are due to the SN feedback. Note however that the oscillation amplitudes of the SFR decreases with decreasing $\Delta\tau_{\text{in}}$. This is because in our simulations about 80% of the inflowing gas turns to stars (Paper I), so that neglecting the effect of the SN feedback changing the scale height, the SFR is roughly given by

where $t_{\text{delay}}$ is the delay time between the mass inflow and star formation mentioned above. Since our SFR is proportional to the gas mass accreted over a $10\rm\,Myr$ interval, its amplitude should be an increasing function of $\Delta\tau_{\text{in}}$ even though the oscillation amplitude of $\dot{M}_{\text{in}}$ is the same. Note that the averaging interval of $10\,\rm Myr$ is likely a lower limit, considering that the time taken for an inflowing gas parcel to turn into stars is not unique but distributed around $t_{\rm delay}$. When $\Delta\tau_{\text{in}}\lesssim 10\rm\,Myr$, the effect of the temporal variations of $\dot{M}_{\text{in}}$ on the SFR would be smoothed out almost completely. In addition, small-amplitude fluctuations of the SN feedback are present in all models, tending to reduce the effect of the time-varying inflow rate for small $\Delta\tau_{\text{in}}$. Considering the SN timescale of $\sim 40\,{\rm Myr}$ (combined with the short vertical dynamical time in galactic center regions) it is very likely that the star formation can follow the variations imposed by the inflow when $\Delta\tau_{\text{in}}\gtrsim 40\,{\rm Myr}$, as is indeed seen in models P50 and P100. In model off or boost, where the inflow rate drops by a factor of 2 or increases by a factor of 5, respectively, at $t=150\,{\rm Myr}$, the SFR decreases or increases by a similar factor and undergoes feedback-induced, small-amplitude fluctuations with ${\cal R}(\dot{M}_{\text{SF,10}})=2.5$, similarly to that in model constant.

As Figure 1 shows, the gas depletion time in model P100 varies by about an order of magnitude, while the ring gas mass is almost constant. In this subsection, we shall show that this is consistent with the results of the self-regulated star formation theory (Ostriker et al., 2010; Ostriker & Shetty, 2011) provided that the time delay between star formation and the ensuing SN feedback is properly considered (see Figure 2$a$).

According to the self-regulation theory, an (unmagnetized) disk in vertical dynamical equilibrium should obey

where $P_{\rm mid}$ is the total (thermal plus turbulent) midplane pressure averaged over the ring area $A_{\rm ring}$ (see below), $\mathcal{W}$ is the weight of the overlying gas above (or below) the midplane, $\Sigma_{\text{gas}}$ is the mean gas surface density within the ring area, and $\left<g_{z}\right>$ is the density-weighted mean vertical gravity defined as

Here, the integration is performed over the annular region between $R_{\rm min}=400\,\text{pc}$ and $R_{\rm max}=800\,{\rm pc}$, where $A_{\rm ring}=\pi(R_{\rm max}^{2}-R_{\rm min}^{2})$, and over the upper (or lower) half of the computational domain in the vertical direction. Equation (11) assumes that the pressure above the gas layer is small compared to $P_{\mathrm{mid}}$, i.e. $\Delta(P+\rho v_{z}^{2})\approx P_{\mathrm{mid}}$. When the gravitational potential is dominated by the stellar bulge (Equation 6), it can be shown that

where $f$ is a dimensionless parameter of order unity that depends on the spatial gas distribution: e.g., $f=1$ for a thin ring with a Gaussian density distribution in the vertical direction. We find $f=0.64$ and $0.72$ for model P100 and constant. Figure 4 plots the relationship between $P_{\text{mid}}$ and $\mathcal{W}$ for models P100 and constant, showing that the two quantities agree with each other within $\sim 12\%$ and $\sim 8\%$, respectively: other models also show a good agreement between $P_{\text{mid}}$ and $\mathcal{W}$. This demonstrates that the system maintains an instantaneous vertical equilibrium, while undergoing (quasi-periodic) long-term oscillations in response to changes in the inflow rate. We also note that the weights from the analytic (Equation 13) and numerically measured $\left<g_{z}\right>$ (Equation 12) are practically the same, indicating that Equation (13) is a good approximation for the vertical gravitational field in our simulations.

The self-regulation theory further asserts that the midplane pressure is sustained by feedback. In the present case, SN feedback supplies both thermal (through hot bubbles) and turbulent pressures, although other forms of feedback or other pressure contributions (e.g., magnetic pressure) may be important more generally (Ostriker et al., 2010; Ostriker & Shetty, 2011; Kim & Ostriker, 2015b). Assuming that the midplane pressure is proportional to the SFR surface density $\Sigma_{\text{SFR}}\equiv\dot{M}_{\rm SF}/A_{\rm ring}$, one can write $P_{\rm mid}=\Upsilon_{\rm tot}\Sigma_{\text{SFR}}$, where $\Upsilon_{\rm tot}$ is the total feedback yield (see also, Kim et al. 2011, 2013; Kim & Ostriker 2015b Paper I). We find $\Upsilon_{\rm tot}=340\,{\rm km\,s^{-1}}$ from the time averaged $P_{\rm mid}$ and $\Sigma_{\text{SFR}}$ for model P100 (with $\sim 2\%$ differences for other models). Identifying the pressure predicted from dynamical equilibrium (Equation 11) with the pressure predicted from star formation feedback then gives the prediction for the gas depletion time (Equation 9) as

Figure 5 compares the measured $t_{\text{dep,10}}$ and the predicted $t_{\text{dep}}^{\text{pred}}$ for model P100. Although $t_{\text{dep}}^{\text{pred}}$ oscillates in time in a similar fashion to the directly-measured $t_{\text{dep,10}}$ as the ring repeatedly shrinks and expands vertically to change $H$, the amplitude and phase are not well matched.

The temporal offset between $t_{\text{dep,10}}$ and $t_{\text{dep}}^{\text{pred}}$ implies that one has to consider the time delay between star formation and SN feedback: while $\dot{M}_{\text{SF,10}}$ and $t_{\text{dep,10}}$ only accounts for the stars formed in the past $10\,\text{Myr}$, the SN feedback that sustains the midplane pressure also depends on previous star formation, occuring in star particles with age up to $40\,\text{Myr}$ (Leitherer et al., 1999; Kim & Ostriker, 2017). Since the scale height $H$ (and $P_{\text{mid}}$, implicitly) in Equation (14) is responsive to SNe (Figure 2$a$), the corresponding predicted depletion time is sensitive to a longer-term average of the SFR. This motivates us to compare $t_{\text{dep}}^{\text{pred}}$ with the depletion time $t_{\text{dep,40}}$ averaged over $40\,\text{Myr}$ instead of $t_{\text{dep,10}}$. Figure 5 shows $t_{\text{dep,40}}$ agrees with $t_{\text{dep}}^{\text{pred}}$ much better than $t_{\text{dep,10}}$, indicating that the self-regulation theory predicts the time-varying depletion time averaged over the timescale associated with the dominant feedback process, which is $\sim 40\,\text{Myr}$ for SNe in our current models. The slight mismatch in the amplitude is due to the secondary effect of time-varying $\Upsilon_{\text{tot}}$, which we do not consider in this work. The phase offset and larger fluctuation of $t_{\text{dep,10}}$ compared to $t_{\text{dep,40}}$ is because the SN feedback is delayed behind $\dot{M}_{\text{SF,10}}$. We note that Equation (11) implies $M_{\text{gas}}=\Sigma_{\text{gas}}A_{\text{ring}}\propto P_{\text{mid}}/\left<g_{z}\right>\propto\dot{\mathcal{N}}_{\text{SN}}/H$, which is roughly constant since $H$ is correlated with $\dot{\mathcal{N}}_{\text{S}N}$ (see Figure 2$a$), in agreement with Figure 1($b$)³³3Since most of the gas in our simulations is contained in the ring region, $M_{\text{gas}}$ agrees with $\Sigma_{\text{gas}}A_{\text{ring}}$ within $\sim 8\%$.. The above analyses suggest that the self-regulation theory is applicable even when the ring star formation is time-varying, as long as feedback time delays and appropriate temporal averaging windows are taken into account.

We now explore how asymmetry in the mass inflows affects the spatial distributions of star particles in the rings. Figure 6 plots the projected distributions in the $x$–$y$ plane of gas and young star clusters with age younger than $10\,{\rm Myr}$, for models P100, asym, off, and boost from left to right at four selected epochs from $t=151\,{\rm Myr}$ to $210\,{\rm Myr}$. Note that the inflow rate in models off and boost changes abruptly at $t=150\,{\rm Myr}$. In model P100, the fading gas streams over time manifest the continuous decrease of the inflow rate during this time span. The time-varying kinetic energy of the streams makes the ring more eccentric and promotes its precession. With the symmetric mass inflow rate in this model, young star clusters are distributed more-or-less uniformly across the whole length of the ring.

The ring of model asym is more circular since the inflowing gas has constant kinetic energy. While the inflow rate from the upper nozzle is $9$ times higher than that from the lower one in this model, star clusters with age $\lesssim 10\,{\rm Myr}$ are still distributed almost uniformly throughout the ring, as in model P100. This is presumably because the depletion time ($\sim 100\,{\rm Myr}$) is longer than the ring orbital time ($\sim 20\,{\rm Myr}$), allowing the gas from the upper and lower streams to be well mixed before turning into stars. The inflow rate in model off becomes asymmetric after $t=150\,{\rm Myr}$, due to the cessation of the lower stream. Nevertheless, the continued inflow from the upper nozzle smoothly lands on the ring without causing large deformation of the ring. The distribution of star particles in model off is relatively symmetric despite the asymmetric inflow rate, similarly to model asym.

Unlike the other models, however, model boost shows lopsided distributions of star particles for a few tens of ${\rm Myr}$ after the boosted inflow, due mainly to the enhanced SFR in the lower part of the ring marked by the white boxes in the second and third rows of the last column in Figure 6. In this model, the boosted inflow from the upper nozzle has such large inertia that it is almost unhindered when it hits the ring at $(x,y)\sim(-500,400)\,{\rm pc}$ on a nearly ballistic orbit. The inflowing streams converge and collide with the ring at the opposite side, triggering star formation at $(x,y)\sim(0,-600)\,{\rm pc}$. This star formation, induced directly by the boosted inflow, makes the overall distribution of star particles lopsided. However, this phase of asymmetric star formation persists only for a few orbital periods as the ring gradually adjusts its shape and size corresponding to the boosted inflow. Gas from the boosted inflow then smoothly joins the ring at the near side and spreads along the ring over the depletion time, returning to distributed star formation again.

To quantify the degree of the lopsided star formation, we divide the ring into two parts by a straight line $y=\tan(\phi)x$, where $\phi$ is the position angle of the dividing line measured counterclockwise from the positive $x$-axis. We calculate the SFR separately in each part using the star particles with age younger than $10\,{\rm Myr}$, and then average it over 21 snapshots taken from $161$ to $181\,{\rm Myr}$ at $1\,{\rm Myr}$ interval. We define the asymmetry parameter ${\cal A}(\phi)\,(\geq 1)$ as the (higher-to-lower) ratio of the averaged SFRs from the two parts for a given $\phi$. We then repeat the calculations by varying $\phi$ to find the maximum value ${\cal A}_{\rm max}=\max_{\phi}{\cal A}(\phi)$. Note that the position angle $\phi_{0}$ corresponding to ${\cal A}_{\rm max}$ differs for all models.

Table 3 lists the maximum asymmetry parameter ${\cal A}_{\rm max}$ and the corresponding position angle $\phi_{0}$. Model constant has ${\cal A}_{\rm max}=1.2$ due to the randomness of star-forming positions under the symmetric inflows. Models P15, P50, and P100 have similar or slightly higher ${\cal A}_{\rm max}$ than model constant due probably to perturbations introduced by time variability of the inflow rate. Models asym and off have ${\cal A}_{\rm max}=1.8$ and $1.6$, respectively, which are higher than the asymmetry parameter of model constant but still quite small considering a large asymmetry in the inflow rate. In contrast, model boost has ${\cal A}_{\rm max}=4.9$, that is, the boosted mass inflow from the upper nozzle makes the SFR in the lower right side of the dividing line with $\phi_{0}=80^{\circ}$ higher by a factor of about 5 than in the opposite side, which is caused by the new star clusters in the boxed regions in Figure 6. The similar asymmetry parameter for gas mass is almost unity for all models, because the newly accreted gas mass during one orbit $\dot{M}_{\text{in}}t_{\text{orb}}$ is smaller than the existing ring gas mass $\dot{M}_{\rm SF}t_{\text{dep}}\approx\dot{M}_{\text{in}}t_{\text{dep}}$, quickly spreading along the whole length of the ring within $t_{\text{orb}}$. Our results suggest that asymmetric inflows alone are unable to create lopsided star formation in the rings. Rather, asymmetry in the ring star formation in our models requires a large (and sudden) boost in the inflow rate from one nozzle.