Effects of Varying Mass Inflows on Star Formation in Nuclear Rings of
  Barred Galaxies
  Observations indicate that the star formation rate (SFR) of nuclear rings
varies considerably with time and is sometimes asymmetric rather than being
uniform across a ring. To understand what controls temporal and spatial
distributions of ring star formation, we run semi-global, hydrodynamic
simulations of nuclear rings subject to time-varying and/or asymmetric mass
inflow rates. These controlled variations in the inflow lead to variations in
the star formation, while the ring orbital period ($18\,{\rm Myr}$) and radius
($600\,{\rm pc}$) remain approximately constant. We find that both the mass
inflow rate and supernova feedback affect the ring SFR. An oscillating inflow
rate with period $\Delta \tau_\text{in}$ and amplitude 20 causes
large-amplitude (a factor of $\gtrsim 5$), quasi-periodic variations of the
SFR, when $\Delta \tau_\text{in} \gtrsim 50\,{\rm Myr}$. We find that the
time-varying ISM weight and midplane pressure track each other closely,
establishing an instantaneous vertical equilibrium. The measured time-varying
depletion time is consistent with the prediction from self-regulation theory
provided the time delay between star formation and supernova feedback is taken
into account. The supernova feedback is responsible only for small-amplitude (a
factor of $\sim 2$) fluctuations of the SFR with a timescale $\lesssim 40\,{\rm
Myr}$. Asymmetry in the inflow rate does not necessarily lead to asymmetric
star formation in nuclear rings. Only when the inflow rate from one dust lane
is suddenly increased by a large factor, the rings undergo a transient period
of lopsided star formation.
