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Samaporn Tinyanont edited untitled.tex
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%They were first proposed by \cite{Duncan_1992} to explain Soft Gamma Repeaters (SGRs).
A magnetar of mass $1.4 M_{\odot}$, $R = 10 \ \rm km$, and $P = 1 \ \rm ms$ has a rotational energy of
\begin{equation} \nonumber
E_{\rm rot} = I \omega^2 /2 = \dfrac{1}{5}MR^2 \left(\dfrac{2\pi}{P}\right)^2 \simeq
4.4 4 \times 10^{51} \ \rm erg
\end{equation}
assuming the solid sphere moment of inertia as an approximation. The strong magnetic stresses and torques damped the rotation and release a large fraction of $E_{\rm rot}$ on the time scale of $\tau \simeq 0.6 B_{15}^{-2} (P/1 \ {\rm ms})^2 \ \rm hr$ \cite{Duncan_1992}. This amount of energy is similar to that released in $\gamma$-ray bursts and superluminous supernovae (SLSNe), thus making magnetar an attractive candidate for the central engine that power these explosions. In the past two decades, a connection has been made between supernovae and at least some subclass of long-duration $\gamma$-ray bursts (GRBs), supporting the case that they are powered by the same mechanism.