On the formation of hydrogen-deficient low-mass white dwarfs

Context. Two of the possible channels for the formation of low-mass (M-star less than or similar to 0.5 M-circle dot) hydrogen-deficient white dwarfs are the occurrence of a very-late thermal pulse after the asymptotic giant-branch phase or a late helium-flash onset in an almost stripped core of a red giant star.Aims. We aim to asses the potential of asteroseismology to distinguish between the hot flasher and the very-late thermal pulse scenarios for the formation of low-mass hydrogen-deficient white dwarfs.Methods. We computed the evolution of low-mass hydrogen-deficient white dwarfs from the zero-age main sequence in the context of the two evolutionary scenarios. We explore the pulsation properties of the resulting models for effective temperatures characterizing the instability strip of pulsating helium-rich white dwarfs.Results. We find that there are significant differences in the periods and in the period spacings associated with low radial-order (k less than or similar to 10) gravity modes for white-dwarf models evolving within the instability strip of the hydrogen-deficient white dwarfs.Conclusions. The measurement of the period spacings for pulsation modes with periods shorter than similar to 500 s may be used to distinguish between the two scenarios. Moreover, period-to-period asteroseismic fits of low-mass pulsating hydrogen-deficient white dwarfs can help to determine their evolutionary history.