A self-consistent analytical magnetar model: The luminosity of $\gamma$-ray burst supernovae is powered by radioactivity

Abstract

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 (LCs) of five long-duration $\gamma$-ray bursts (LGRBs) and two ultra-long 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 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 2011kl, from the early afterglow to the later SN, where we determined for the magnetar central engine a magnetic field strength of $1.1-1.3\times 10^{15}$ 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 $1.2-1.6\times 10^{50}$ 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 timescales of superluminous supernovae (SLSNe) are of order months to years, which provides a natural explanation as to why SN 2011kl was less luminous than SLSNe that are also powered by emission from magnetar central engines.

14 pages, two figures, two tables. Accepted with minor revision at MNRAS (this is the newly submitted, revised version)

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