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Both the long-duration gamma-ray bursts (LGRBs) and the Type I superluminoussupernovae (SLSNe~I) have been proposed to be primarily powered by centralmagnetars. A correlation, proposed between the initial spin period ($P_0$) andthe surface magnetic field ($B$) of the magnetars powering the X-ray plateausin LGRB afterglows, indicates a possibility that the magnetars have reached anequilibrium spin period due to the fallback accretion. The correspondingaccretion rates are inferred as $\dot{M}\approx10^{-4}-10^{-1}$ M$_\odot$s$^{-1}$, and this result holds for the cases of both isotropic and collimatedmagnetar wind. For the SLSNe~I and a fraction of engine-powered normal type Icsupernovae (SNe~Ic) and broad-lined subclass (SNe~Ic-BL), the magnetars couldalso reach an accretion-induced spin equilibrium, but the corresponding $B-P_0$distribution suggests a different accretion rate range, i.e., $\dot{M}\approx10^{-7}-10^{-3}$ M$_\odot$ s$^{-1}$. Considering the effect of fallbackaccretion, magnetars with relatively weak fields are responsible for theSLSNe~I, while those with stronger magnetic fields could lead to SNe~Ic/Ic-BL.Some SLSNe~I in our sample could arise from compact progenitor stars, whileothers that require longer-term accretion may originate from the progenitorstars with more extended envelopes or circumstellar medium.

A unified accreting magnetar model for long-duration gamma-ray bursts and some stripped-envelope supernovae