deletions | additions
diff --git a/untitled.tex b/untitled.tex
index c6fb1e8..416f683 100644
--- a/untitled.tex
+++ b/untitled.tex
...
Neutron stars typically have magnetic fields of $\sim 10^{12} \rm \ G$. However, if a neutron star's rotation period is comparable to the convective overturn time, magnetic fields can be amplified by helical motion in a
mechanism called mean field dynamo. These highly magnetized neutron stars, magnetars, are born with short periods of $\sim$ 1 ms, which allow them to support an efficient $\alpha-\Omega$ dynamo, resulting in large magnetic dipole fields of $10^{14}-10^{15} \rm \ G$ \cite{Duncan_1992}.
%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}
E_{\rm rot} = I \omega^2 /2 = \dfrac{1}{5}MR^2 \left(\dfrac{2\pi}{P}\right)^2 \simeq 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
\begin{equation}
\label{eqn:timescale}
\tau \simeq 0.6 B_{15}^{-2} (P/1 \ {\rm ms})^2 \ \rm hr
\end{equation} \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 %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.
\
Around the same time that magnetars were proposed, a realization had emerged that GRBs could have cosmological origin. This would require a total energy of $\sim 10^{51} \ \rm erg$ to explain the observed flux assuming isotropic emission (e.g. \citealp{Paczynski_1991}). Interestingly,
\citet{Usov_1992} proposing the first proposal of highly magnetized neutron stars as central engines of these
bursts bursts, \citet{Usov_1992}, was published only a day after \citet{Duncan_1992}. The proposed scenario is such that a magnetar forms from a white dwarf via accretion induced collapse (AIC). The WD magnetic field of $\sim 10^9 \ \rm G$ is amplified to $10^{15} \rm \ G$ by magnetic flux conservation. The rotational period of $\sim 1 \ \rm ms$ is a result of angular momentum conservation.
\footnote{\citet{Duncan_1992} also briefly mentioned this scenario as a possible explanation for cosmological GRBs.}
The newly formed neutron star then loses its rotational energy quickly due to electromagnetic torque, generating electric fields that accelerate particles to ultra-relativistic energies, which eventually give out $\gamma$-ray. The
timescale of energy
released time scale release due to magnetic dipole luminosity and gravitational wave emission of $\sim 20 \ \rm s$ for a typical magnetar is consistent with the timescale of long-duration GRBs \cite{Usov_1992}. Near the magnetar's surface out to the light cylinder, the optical depth to this radiation due to Compton scattering, absorption, and pair-cration is large. The radiation has to propagate out to a photosphere radius of $\sim 10^8 \ \rm cm$ before it is released. The typical radiated $\gamma$-ray energies of $0.1-1 \ \rm MeV$ is also consistent with those observed from GRBs \cite{Usov_1992}.
\
...
\
The requirement of collimated relativistic flows leading to bi-polar jets in GRBs limit the central engines to only magnetars having minimal periods of $1 \ \rm ms$ and magnetic fields of $\sim 10^{15} \ \rm G$. However, their less extreme population can still power quite fantastic cosmic fireworks. \cite{Duncan_1992} noted that the spin down timescale given in \ref{eqn:timescale} is shorter than the SN shock breakout time, making SNe that create magnetars brighter than usual. These subclasses of brighter Type II SNe are indeed observed (e.g. \citealp{Richardson_2002}). Recently, a number of rare superluminous SNe (SLSNe) are discovered, emitted the total radiation energy of $\sim 10^{51} \ \rm erg$ (e.g. SN\,2005ap \cite{Quimby_2007}; SN\,2008es \cite{Miller_2008}).
\citet{Kasen_2010} showed that magnetars with $B \sim 10^{14} \ \rm G$ and initial period $P \sim 2-20 \ \rm ms$ can release rotational energy via magnetic dipole radiation on the spin-down timescale comparable to the effective diffusion time of the ejecta. They showed that these magnetars can enhance the peak luminosity to what given by Equation (4) in \citet{Kasen_2010}, which is exceeding $10^{43} \rm \ erg/s $, brighter than normal core-collapse Type II-P SNe.
