\documentclass[a4paper]{jpconf}
\usepackage{graphicx}
\begin{document}
\title{Gamma-Ray Burst Physics}

\author{Shigehiro Nagataki}

\address{Yukawa Institute for Theoretical Physics, Kyoto University,
 Oiwake-cho Kitashirakawa Sakyo-ku, Kyoto 606-8502, Japan}

\ead{nagataki@yukawa.kyoto-u.ac.jp}

\begin{abstract}
I have developed two-dimensional general
relativistic magnetohydrodynamic (GRMHD) code.
I have performed numerical
simulations of collapsars using these codes and realistic progenitor
models. In the GRMHD simulation, it is shown that a jet is
launched from the center of the progenitor.
We also performed simulations of collapsars with different 
Kerr parameters $a=$0, 0.5, 0.9, 0.95. It is shown that a more
rapidly rotating black hole is driving a more energetic jet. 
No jet is seen for the case of Schwartzschild black hole case,
while the total energy of the jet is as large as $10^{50}$
erg for a rapidly rotating Kerr black hole case ($a=0.95$).
In order to explain the high luminosity of a GRB, it is concluded 
that a rapidly rotating black hole is favored ('faster is better').
We also performed two-dimensional
hydrodynamic simulations in the context of collapsar model to
investigate the explosive nucleosynthesis happened there. It is found
that the amount of $^{56}$Ni is very sensitive to the energy
deposition rate. This result means that the amount of synthesized $^{56}$Ni
can be  little even if the total explosion energy is as large as
10$^{52}$ erg. Thus, some GRBs can associate with faint
supernovae. Thus we consider it is quite natural to detect no
underlying supernova in some X-ray afterglows.
\end{abstract}

\section{Introduction}

Gamma-Ray Bursts (GRBs; in this study, we consider only long GRBs, so
we refer to long GRBs as GRBs hereafter) have been mysterious
phenomena since their discovery in 1969. Last decade, observational
evidence for supernovae (SNe) and GRBs association has been
reported. Some of the SNe that associate with GRBs were very energetic
and blight. The estimated explosion energy was of the order 10$^{52}$
ergs, and produced nickel mass was about 
0.5 $M_{\odot}$. Thus they are categorized as a new type of SNe
(sometimes called as hypernovae). The largeness of the explosion
energy is very important, because it can not be explained by the
standard core-collapse SN scenario, and other mechanism should be
working at the center of the progenitors. 
In this paper, we investigate the
dynamics of collapsars using two-dimensional general
relativistic magnetohydrodynamic (GRMHD) code\cite{nagataki09}.

Also, we perform two-dimensional hydrodynamic simulations
in the context of collapsar model to investigate the explosive
nucleosynthesis happened there. We have to be careful to the point
that where and when $^{56}$Ni is synthesized in the collapsar is not
unclear. This is because the explosion mechanism is not known
well. One possibility is that $^{56}$Ni is synthesized in the jet region
like jet-like supernova explosion\cite{nagataki03,nagataki06}. Another possibility is that nucleosynthesis in the
accretion disk, and some fraction of the accreting matter escape from
the system\cite{macfadyen99}.  In this paper, we consider
the former possibility.  


\section{GENERAL RELATIVISTIC MHD SIMULATIONS}

\subsection{Formulation}

We have adopted a conservative,
shock-capturing scheme with Harten, Lax, and van Leer (HLL) flux term
with flux-interpolated constrained transport technique. We use a
third-order Total Variation Diminishing (TVD) Runge-Kutta method for
evolution in time, while monotonized central slope-limited linear
interpolation method is used for second-order accuracy in space. 2D
scheme (2-dimensional Newton-Raphson method) is usually adopted for
transforming conserved variables to primitive variables. 

When we perform simulations of GRMHD, Modified Kerr-Schild coordinate
is basically adopted with mass of the BH (M) fixed.
We use G =M = c = 1 unit.
The calculated region covers from r =1.8 to 3 $\times 10^4$ (that
corresponds to 5.3 $\times 10^5$ cm and 8.9 $\times 10^9$ cm in cgs units). 
We adopt the model 12TJ in Woosley and Heger (2006)\cite{woosley06}. We set the Kerr-parameter
of the black hole at the center is set to be 0.5 throughout of the simulation.
The initial, weak poloidal magnetic fields are put initially. The
minimum plasma beta in the simulation region is greater than 100
initially. 
 
\subsection{Results}


\begin{figure}[h]
\begin{minipage}{18pc}
\includegraphics[width=18pc]{Fig1.ps}
\caption{\label{label1}Contours of rest mass density at the central region in
logarithmic scale at t = 180000 (that corresponds 
to 1.773 sec), in which cgs units are used assuming that the
gravitational mass of the BH is 2 $M_{\odot}$. The length unit in the
vertical/horizontal axes corresponds to 2.95 times10$^5$ cm.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=18pc]{Fig2.ps}
\caption{\label{label1-2}Contour of the plasma beta (pgas/pmag) at t = 180000 in logarithmic scale.\vspace*{1.8cm}}
\end{minipage} 
\end{figure}



%\begin{figure} [pb]
%\centerline{\psfig{file=Fig5.eps,width=4.7cm}\psfig{file=Fig6.eps,width=4.7cm}}
%\vspace*{8pt}
%  \caption{Left: Contour of the plasma beta (pgas/pmag) at t = 180000
%  in logarithmic scale. Right: Contours of the energy flux per unit rest-mass flux at t =
%  180000 that represent the bulk Lorentz factor of the invischid fluid
%  element when all of the internal and magnetic energy are converted
%  into kinetic energy at large distances. The contours are written in
%  logarithmic scale.}
%\end{figure}

%\begin{figure}
%\includegraphics[height=.25\textheight]{Fig6}
%  \caption{Contours of the energy flux per unit rest-mass flux at t =
%  180000 that represent the bulk Lorentz factor of the invischid fluid
%  element when all of the internal and magnetic energy are converted
%  into kinetic energy at large distances. The contours are written in
%  logarithmic scale.}
%\end{figure}



In Figure 1, contours of rest mass density at the central region
are shown. Contours represent the density in units of g cm$^{-3}$ in
logarithmic scale. The length r = 200 corresponds to 5.9 times 10$^7$
cm. The time unit corresponds to 9.85 times 10$^{-6}$ sec. 
Figure 2 shows contours of the plasma beta (pgas/pmag) in
logarithmic scale at t = 180000. As expected, the plasma beta is low
in the jet region while it is high in the accretion disk region.




\section{Dependence on the Kerr Parameter}\label{initial}

Next, we studied the dependence of dynamics on the Kerr Parameter.
The initial condition is also same with the previous section, but for 
different Kerr Parameters. 
In the previous section, the Kerr parameter, $a$, was assumed to be 0.5,
but in this study we perform simulations for $a=$ 0, 0.5, 0.9, 0.95,
respectively (we name them as Model A,B,C,D, respectively). 

In Figure 3, contours of rest mass density
in logarismic scale for all models at the same time-slice $t=160000$ (that
corresponds to 1.5760 sec) are shown.     
Cgs units are used for the rest mass density, while the length
in the vertical/horizontal axes is written in $G=M=c=1$ unit.
$r=1$ and 4000 corresponds to 2.95 $\times 10^5$ cm and 1.18 $\times
10^9$ cm, respectively.
These results are projected on the (r sin$\theta$, r cos $\theta$)-plane.
Upper left panel shows the state of Model A ($a=0$), upper right panel
shows the one of Model B ($a=0.5$), lower left panel shows the one
of Model C ($a=0.9$), and lower right panel shows the one of Model D 
($a=0.95$). It is clearly shown that the rotating black hole drives the
jet (the Schwartzschild black hole cannot drive a jet (Model A),
while a more rapidly rotating black hole is driving a stronger jet). 

\begin{figure}[h]
\includegraphics[width=25pc]{Fig3.eps}\hspace{2pc}%
\begin{minipage}[b]{10pc}\caption{\label{label2}Contours of rest mass density in logarismic scale for all
models at the same time-slice $t=160000$ (that corresponds to 1.5760 sec).  
Cgs units are used for the rest mass density, while the length
in the vertical/horizontal axes is written in $G=M=c=1$ unit.
$r=1$ and 4000 corresponds to 2.95 $\times 10^5$ cm and 1.18 $\times
    10^9$ cm, respectively.
These results are projected on the (r sin$\theta$, r cos $\theta$)-plane.
Upper left panel shows the state of Model A ($a=0$), upper right panel
    shows the one of Model B ($a=0.5$), lower left panel shows the one
    of Model C ($a=0.9$), and lower right panel shows the one of Model
    D ($a=0.95$).}
\end{minipage}
\end{figure}


In Figure 4, plots of the jet energy for all models
at $t=160000$ are shown.
The definition of the jet energy is:
\begin{equation}
E_{\rm Jet} = 2 \times  2\pi \int_{r_+}^{\infty} dr
\int_{0}^{\theta} d \theta \sqrt{-g} (T^t_{t} - \rho u^0 u_0 ),
\label{GRB3-1}
\end{equation}
where $T^t_{t}$ is the ($t$,$t$) component of total energy-momentum tensor
and integration is done only for the region where $u^r$ (radial
component of 4-velocity of fluid) is positive.
It is noted that the contribution of the rest mass energy is subtracted.
Factor 2 is coming from the symmetry of the system with respect to the
equatorial plane.
Dot curve represents the jet energy within the opening angle
    $\theta = 5^{\circ}$, while dashed curve represents the one within
    $\theta = 10^{\circ}$.  
The unit of vertical axis is $10^{48}$ erg. 
It is clearly seen that a more rapidly rotating black hole is driving
a stronger 
jet. The total energy of the jet for Model D ($a=0.95$) is as large as
$10^{50}$ erg. 

\begin{figure}[h]
\includegraphics[width=25pc]{Fig4.eps}\hspace{2pc}%
\begin{minipage}[b]{10pc}\caption{\label{label4}
Plots of the jet energy (see text for the definition in
    detail) for all models at $t=160000$ (that
    corresponds to 1.5760 sec). The unit of vertical axis is $10^{48}$
    erg. Dot curve represents the jet energy within the opening angle
    $\theta = 5^{\circ}$, while dashed curve represents the one within
    $\theta = 10^{\circ}$.  
}
\end{minipage}
\end{figure}





\section{EXPLOSIVE NUCLEOSYNTHESIS IN COLLAPSAR}

\subsection{Formulation}

%\begin{figure}
%\includegraphics[height=.3\textheight]{Fig7}
%  \caption{Density structure and velocity field of model E51 at 1.5 sec.}
%\end{figure}


\begin{figure}[h]
\begin{minipage}{18pc}
\includegraphics[width=18pc]{Fig5.ps}
\caption{\label{label3}Positions of the ejected test particles at 0 sec that meet
  the condition that the mass fraction of $^{56}$Ni becomes greater than
  0.3 as a result of explosive nucleosynthesis for model E51. The
  total ejected mass of $^{56}$Ni is 0.0439 $M_{\odot}$.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=18pc]{Fig6.ps}
\caption{\label{label}Same
  with left, but for model E52. The total ejected mass of $^{56}$Ni is
  0.23 solar masses.\vspace*{1.6cm}}
\end{minipage} 
\end{figure}

%\begin{figure}
%\includegraphics[height=.3\textheight]{Fig9}
%  \caption{Same as Figure 8, but for model E52. The total ejected
%  mass of $^{56}$Ni is 0.23 solar masses.} 
%\end{figure}

We have done two-dimensional hydrodynamic simulations taking into
account self-gravity and the gravitational potential of the central
point mass. The calculated region corresponds to a quarter of the
meridian plane under the assumption of axisymmetry and equatorial
symmetry. The spherical mesh with 250 by 30 grid points is used for
all the computations. The radial grid is nonuniform, extending from 
2 $\times 10^7$ to 3 $\times 10^{11}$ cm with finer grids near the
center, while the polar grid is uniform.  
We adopt the collapsar model of MacFadyen \& Woosley (1999)\cite{macfadyen99}.

To simulate the jet-induced explosion, we deposit only thermal energy
at a rate 10$^{51}$ ergs/s homogeneously within a 30 degree cone around
the rotation axis for 10 sec.
We name this model E51.
For comparison, we perform a calculation in which total
explosion energy (10$^{52}$ ergs) is put initially with the same
deposition region as model E51. We name these models E52.

Since the chemical composition behind the shock wave is not in nuclear
statistical equilibrium, the explosive nucleosynthesis has to be
calculated using the time evolution of density and temperature, and a
nuclear reaction network, which is called post-processing. The nuclear
reaction network contains 250 species\cite{nagataki97,nagataki00}. 
   
\subsection{Results}

We deposited thermal energy to launch a jet from the central region of
the collapsar. The density structure for models E51 shows
a sharp, narrow jet propagates along the rotation axis in model E51.
On the other hand, in the case of E52, a broad,
deformed shock wave propagates in the progenitor (see also Nagataki et
al. (2006)\cite{nagataki06}.

In Figure 5, positions of the ejected test particles for model E51 at
0 sec are shown that satisfy the condition that the mass fraction of
$^{56}$Ni becomes greater than 0.3 as a result of explosive
nucleosynthesis. The total ejected mass of $^{56}$Ni becomes 0.0439 $M_{\odot}$, which is much smaller than the observed values of
hypernovae. In Figure 6, the same values are shown as in
Figure 5, but
for models E52. The total ejected mass of $^{56}$Ni is 0.23 $M_{\odot}$,
which is comparable to the observed values of hypernovae.  


Thus we can conclude that the resulting amount of $^{56}$Ni is very
sensitive to the energy deposition rate. This result means that the
amount of synthesized $^{56}$Ni can be little even if the total explosion
energy is as large as 10$^{52}$ erg. Thus, some GRBs can associate with
faint supernovae. Thus we consider it is quite natural to detect no
underlying supernova in some X-ray afterglows such as GRB060614.  

\section{Summary and Conclusion}
I have performed numerical
simulations of collapsars using these codes and realistic progenitor
models. In the GRMHD simulation, it is shown that a jet is
launched from the center of the progenitor.
We also performed simulations of collapsars with different 
Kerr parameters $a=$0, 0.5, 0.9, 0.95. It is shown that a more
rapidly rotating black hole is driving a more energetic jet. 
No jet is seen for the case of Schwartzschild black hole case,
while the total energy of the jet is as large as $10^{50}$
erg for a rapidly rotating Kerr black hole case ($a=0.95$).
In order to explain the high luminosity of a GRB, it is concluded 
that a rapidly rotating black hole is favored ('faster is better').
We also performed two-dimensional
hydrodynamic simulations in the context of collapsar model to
investigate the explosive nucleosynthesis happened there. It is found
that the amount of $^{56}$Ni is very sensitive to the energy
deposition rate. This result means that the amount of synthesized $^{56}$Ni
can be little even if the total explosion energy is as large as
10$^{52}$ erg. Thus, some GRBs can associate with faint
supernovae. Thus we consider it is quite natural to detect no
underlying supernova in some X-ray afterglows such as GRB060614. 


\section*{References}
\begin{thebibliography}{9}

\bibitem{nagataki09} S. Nagataki, {\it Astrophys. J.} {\bf 704} (2009)
937.

\bibitem{nagataki03} S. Nagataki, A. Mizuta, S. Yamada, H. Takabe,
K. Sato, {\it Astrophys. J.} {\bf 596} (2003) 401.

\bibitem{nagataki06} S. Nagataki, A. Mizuta, K. Sato, {\it
Astrophys. J.} {\bf 647} (2006) 1255. 

\bibitem{macfadyen99} A. I. MacFadyen, S. E. Woosley, {\it
Astrophys. J.} {\bf 524} (1999) 262. 

\bibitem{woosley06} S. E. Woosley, A. Heger {\it
Astrophys. J.} {\bf 637} (2006) 914. 

\bibitem{nagataki97} S. Nagataki, M. Hashimoto, K. Sato, S. Yamada,
{\it Astrophys. J.} {\bf 486} (1997) 1026.

\bibitem{nagataki00} S. Nagataki, {\it Astrophys. J. Suppl.} {\bf 127} (2000)
141.


\end{thebibliography}

\end{document}


