Fuel-Supply-limited Stellar Relaxation Oscillations: Application to Multiple Rings around Asymptotic Giant Branch Stars and Planetary Nebulae

We describe a new mechanism for pulsations in evolved stars: relaxation oscillations driven by a coupling between the luminosity-dependent mass-loss rate and the H fuel abundance in a nuclear-burning shell. When mass loss is included, the outward flow of matter can modulate the flow of fuel into the shell when the stellar luminosity is close to the Eddington luminosity LEdd. When the luminosity drops below LEdd, the mass outflow declines and the shell is resupplied with fuel. This process can be repetitive. We demonstrate the existence of such oscillations and discuss the dependence of the results on the stellar parameters. In particular, we show that the oscillation period scales specifically with the mass of the H-burning relaxation shell (HBRS), defined as the part of the H-burning shell above the minimum radius at which the luminosity from below first exceeds the Eddington threshold at the onset of the mass-loss phase. For a stellar mass M* ~ 0.7 M☉, luminosity L* ~ 104 L☉, and mass-loss rate || ~ 10-5 M☉ yr-1, the oscillations have a recurrence time of ~1400 yr ~ 57τfsm, where τfsm is the timescale for modulation of the fuel-supply in the HBRS by the varying mass-loss rate. This period agrees very well with the ~1400 yr period inferred for the spacings between the shells surrounding some planetary nebulae. We also find the half-width of the luminosity peak to be ~0.39 times the oscillation period; for comparison, the observational shell thickness of ~1000 AU corresponds to ~0.36 of the spacing between pulses. We find oscillations only for models in which the luminosity of the relaxation shell is ~10%-15% of the total stellar luminosity and for which energy generation occurs through the p-p chain. We suggest this mechanism as a natural explanation for the circumnebular shells surrounding some planetary nebulae, which appear only at the end of the AGB phase.