GRB 111209A/SN 2011kl: Collapse of a supramassive magnetar with r-mode oscillation and fall-back accretion onto a newborn black hole

Ultra-long-duration gamma-ray burst GRB 111209A was found to be associated with a very luminous supernovae (SNe) SN 2011kl. The physics of GRB 111209A/SN 2011kl has been extensively studied in the literures, but does not settle down yet. By investigating in detail the characteristics of the X-ray light curve of GRB 111209A, coupled with the temporal and spectral features observed in SN 2011kl, we argue that a short-living supramassive magnetar can be responsible for the initial shallow X-ray emission. Then the electromagnetic extraction of spin energy from a black hole results in the steeply declining X-ray flux when the magnetar collapses into a black hole (BH). A fraction of the envelope materials falls back and activates the accretion onto the newborn BH, which produces the X-ray rebrightening bump at late times. During this process, a centrifugally driven baryon-rich quasi-isotropic Blandford \& Payne outflow from the revived accretion disk deposits its kinetic energy on the SN ejecta, which powers luminous SN 2011kl. Finally, we place a limitation on the magnetar's physical parameters based on the observations.