A self-consistent analytical magnetar model: the luminosity of γ-ray burst supernovae is powered by radioactivity

We present an analytical model that considers energy arising from a magnetar central engine. The results of fitting this model to the optical and X-ray light curves of five long-duration gamma-ray bursts (LGRBs) and two ultralong GRBs (ULGRBs), including their associated supernovae (SNe), show that emission from a magnetar central engine cannot be solely responsible for powering an LGRB-SN. While the early afterglow (AG)-dominated phase can be well described with our model, the predicted SN luminosity is underluminous by a factor of 3-17. We use this as compelling evidence that additional sources of heating must be present to power an LGRB-SN, which we argue must be radioactive heating. Our self-consistent modelling approach was able to successfully describe all phases of ULGRB 111209A/SN 2011k1, from the early AG to the later SN, where we determined for the magnetar central engine a magnetic field strength of 1.1-1.3 x 10(15) G, an initial spin period of 11.5-13.0 ms, a spin-down time of 4.8-6.5 d, and an initial energy of 1.2-1.6 x 10(50) erg. These values are entirely consistent with those determined by other authors. The luminosity of a magnetar-powered SN is directly related to how long the central engine is active, where central engines with longer durations give rise to brighter SNe. The spin-down time-scales of superluminous supernovae (SLSNe) are of order months to years, which provides a natural explanation as to why SN 2011k1 was less luminous than SLSNe that are also powered by emission from magnetar central engines.