Fuel-Supply-Limited Stellar Relaxation Oscillations: Application to Multiple Rings around AGB 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 $L_{\rm Edd}$\mathrm{.}Whentheluminositydropsbelow$L_{\rm Edd}$, the mass outflow declines and the shell is re-supplied 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 $M_\sim 0.7\Msun$,luminosity$L_\sim 10^4\Lsun$,andmasslossrate$|\dot M|\sim 10^{-5}\Msun$yr$^{-1}$,theoscillationshavearecurrencetime$\sim 1400$years$\sim 57\tau_{\rm fsm}$,where$\tau_{\rm fsm}$ is the timescale for modulation of the fuel supply in the HBRS by the varying mass-loss rate. This period agrees with the $\sim$ 1400-year period inferred for the spacings between the shells surrounding some planetary nebulae, and the the predictied shell thickness, of order 0.4 times the spacing, also agrees reasonably well.