SYSTEMATIC FEATURES OF AXISYMMETRIC NEUTRINO-DRIVEN CORE-COLLAPSE SUPERNOVA MODELS IN MULTIPLE PROGENITORS
Ko Nakamura\altaffilmark{1
We present an overview of two-dimensional (2D) core-collapse supernova simulations employing neutrino transport scheme by the isotropic diffusion source approximation. Studying 101 solar-metallicity progenitors covering zero-age main sequence mass from $10.8 \Msun$ to $75.0 \Msun$, we systematically investigate how the differences in the structures of these multiple progenitors impact the hydrodynamics evolution. By following a long-term evolution over 1.0 s after bounce, most of the computed models exhibit neutrino-driven revival of the stalled bounce shock at $\sim$ 200 - 800 ms postbounce, leading to the possibility of explosion. Pushing the boundaries of expectations in previous one-dimensional (1D) studies, our results confirm that the compactness parameter $\xi$ which characterizes the structure of the progenitors is a key to diagnose the properties of neutrino-driven explosions also in 2D. Models with high $\xi$ undergo high ram pressure from the accreting matter onto the stalled shock and it affects the shock expansion under the influence of neutrino-driven convection and the standing accretion-shock instability, as well as mass of protoneutron star left at the center. We show that the accretion luminosity becomes higher for models with high $\xi$, which makes the diagnostic energy higher and the synthesized nickel mass bigger. We find that these explosion characteristics tend to show a monotonic increase as a function of the compactness parameter $\xi$.
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\title{SYSTEMATIC FEATURES OF AXISYMMETRIC NEUTRINO-DRIVEN CORE-COLLAPSE SUPERNOVA MODELS IN MULTIPLE PROGENITORS
}
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\author{Ko Nakamura\altaffilmark{1}, 
Tomoya Takiwaki\altaffilmark{2}, 
Takami Kuroda\altaffilmark{3},
and Kei Kotake\altaffilmark{4,2}}

\altaffiltext{1}{Faculty of Science and Engineering, Waseda University,
Ohkubo 3-4-1, Shinjuku, Tokyo, 169-8555, Japan}
\altaffiltext{2}{Center for Computational Astrophysics, National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan}
\altaffiltext{3}{Department of Physics, University of Basel,
Klingelbergstrasse 82, 4056 Basel, Switzerland}
\altaffiltext{4}{Department of applied physics, Fukuoka University,  
8-19-1 Nanakuma Jonan, Fukuoka, 814-0180, Japan}

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\begin{abstract}
We present an overview of two-dimensional (2D) 
core-collapse supernova simulations employing neutrino transport 
scheme by the isotropic diffusion source approximation. 
Studying 101 solar-metallicity progenitors covering 
zero-age main sequence mass 
from $10.8 \Msun$ to $75.0 \Msun$, we systematically 
investigate how the differences in the structures of these multiple 
progenitors impact the hydrodynamics evolution.
By following a long-term evolution over 1.0 s after bounce, 
most of the computed models exhibit neutrino-driven revival of the 
stalled bounce shock at $\sim$ 200 - 800 ms postbounce, 
leading to the possibility of explosion. 
Pushing the boundaries of expectations in previous one-dimensional (1D) 
studies, our results confirm that 
the compactness parameter $\xi$ which characterizes the structure 
of the progenitors is a key to diagnose the properties of 
 neutrino-driven explosions also in 2D.
Models with high $\xi$ 
undergo high ram pressure from the accreting matter onto the stalled shock 
and it affects the shock expansion under the influence of neutrino-driven convection and 
the standing accretion-shock instability, 
as well as mass of protoneutron star left at the center. 
We show that the accretion luminosity becomes higher for models 
with high $\xi$, which makes the diagnostic 
energy higher and the synthesized nickel mass bigger. 
We find that these explosion characteristics 
tend to show a monotonic increase as a function of the 
compactness parameter $\xi$.
\end{abstract}
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\medskip

\keywords{hydrodynamics --- neutrinos --- supernovae: general}

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\vskip 1.3cm

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{INTRODUCTION}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Core-collapse supernova (CCSN) mechanism 
is essentially an initial value problem. This has been widely 
agreed among CCSN theorists recently 
\citep[e.g.,][]{Oconnor11,Ugliano12,couchott}. 
For low-mass progenitors with O-Ne-Mg core, the neutrino mechanism works 
successfully 
to explode in one-dimensional (1D) simulations because 
of the tenuous envelope \citep{kitaura}. For more massive progenitors 
with iron core, multi-dimensional (multi-D) effects such as 
neutrino-driven convection and the standing-accretion-shock-instability 
(SASI) are crucially important 
(e.g., \citealt{Burrows95,Janka96,thierry,Murphy13,rodrigo14} 
and references therein). 
Recently this has been confirmed by a number of self-consistent 
two-(2D) and three-dimensional (3D) simulations \citep[e.g.,][]{Buras06a,Ott08,marek,bruenn13,suwa10,suwa14,BMuller12b,BMuller13,Takiwaki12,taki14,Hanke13,dolence15}. 
Up to now, the number of these 
state-of-the-art models amounts to $\sim 40$
covering the zero-age main sequence (ZAMS) 
mass from $8.1 M_\odot$ \citep{BMuller12b} to  $27 M_{\odot}$ \citep{Hanke13}.

Based on stellar evolutionary calculations, on the other hand, 
 {\it hundreds} of CCSN progenitors are available now, depending 
 on a wide variety of the ZAMS mass, 
metallicity, rotation, and magnetic fields
\citep[e.g.,][]{Nomoto88,WW95,Woosley02,Woosley07,Heger00a,Heger05,Limongi06}. 
These huge sets of CCSN progenitors, aided as well as 
by development of high-performance computers and numerical 
schemes, make {\it systematic} numerical study of CCSNe possible.

By performing general-relativistic (GR) 1.5D simulations for over 100 presupernova models using a leakage scheme, 
 \citet{Oconnor11} were the first to 
 point out that the postbounce dynamics and the 
progenitor-remnant connections are predictable basically
by a single parameter, the compactness of the stellar core at bounce 
\citep[see also][]{oconnor13}.
Along this line, \citet{Ugliano12} performed 1D hydrodynamic simulations 
for 101 progenitors of \citet{Woosley02}.
By replacing the proto-neutron star (PNS) interior with an inner 
boundary condition, they followed an unprecedentedly 
long-term evolution over hours to days after bounce in 1D.
Their results also lent support to the finding by \citet{Oconnor11} that the 
compactness parameter is a good measure to diagnose
 the progenitor-explosion and 
 the progenitor-remnant correlation.

Joining in these efforts but going beyond the previous 1D approach,
 we perform neutrino-radiation hydrodynamics simulations in 2D 
 using the whole presupernova series (101 models) with 
 solar metallicity of \citet{Woosley02}.
Without the excision inside the PNS, we
 can self-consistently follow a long-term evolution 
 starting from the onset of core-collapse, bounce, 
neutrino-driven shock-revival, until the revived shock comes out of the iron core.
The goal of our 2D models is 
not to determine the very final fate of a massive star 
(which requires 3D-GR models 
 with detailed transport scheme), 
but to study the systematic dependence of the 
progenitors' structure on the shock revival 
time, diagnostic explosion energy, 
mass of remnant object, and nucleosynthetic yields.
To this end, this study is the first attempt in the multi-D context.

%-------------- figure-snap-1 -------------------
\begin{figure}[t]
\begin{center}
\includegraphics[bb=100 0 900 600,clip,scale=0.42]{Figure1.eps}
%\includegraphics[scale=0.6]{Figure1.eps}
\caption{Entropy distribution in unit of $k_{\rm B}$ per baryon 
for selected nine models at $t_{\rm pb}=400$ ms. 
Shown are models 
s11.2 (a)
to s24.0 (i), 
from top-left to bottom-right.
Note that each model presents a different scale as shown in the panel.}
\label{fig-entall1}
\end{center}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{NUMERICAL SETUP}\label{sec-num}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The employed numerical methods are essentially the same 
 as those in \citet{taki14}. 
Our 2D models are computed on a spherical polar grid 
of 384 non-equidistant radial zones from the center up to 5000 km and 
128 equidistant angular zones covering $0\leq \theta \leq \pi$.
We employ the equation of state by \citet{LS91} 
with a compressibility modulus of $K=220$ MeV. 
For the calculations presented here, self-gravity is computed by a Newtonian monopole approximation and our code is updated, from the ZEUS-MP \citep{Hayes06}, to use high-resolution shock capturing scheme with an approximate Riemann solver of \citet{Einfeldt88}.
As described in \citet{nakamura14}, we take into account 
 explosive nucleosynthesis and the energy feedback into hydrodynamics 
by solving a 13 $\alpha$-nuclei network including 
$^4$He, $^{12}$C, $^{16}$O, $^{20}$Ne, $^{24}$Mg, $^{28}$Si, 
$^{32}$S, $^{36}$Ar, $^{40}$Ca, $^{44}$Ti, $^{48}$Cr, $^{52}$Fe, and $^{56}$Ni.
The nuclear energy compensates for energy loss 
via endothermic decomposition of iron-like NSE nuclei to lighter elements
(see Appendix of \citet{nakamura14} for more details).

To solve spectral transport of electron- ($\nu_e$) and anti-electron neutrinos 
(${\bar \nu}_e$), 
we employ the isotropic diffusion source approximation \citep[IDSA,][]{idsa}.
We take a ray-by-ray approach, in which the neutrino transport 
is solved along a given radial ray assuming that the 
hydrodynamic medium for the 
 direction is spherically symmetric \citep[e.g.,][]{Buras06a}. 
Although one needs to deal with the lateral 
  transport more appropriately \citep[e.g.,][]{sumi15,dolence15}, 
this approximation
 is useful because of the high computational efficiency in parallelization,
 which allows us to explore the more systematic progenitor survey
 based on the radiation-hydrodynamics models than ever in this study. 
Regarding heavy-lepton neutrinos 
($\nu_x = \nu_{\mu}, \nu_{\tau}, \bar{\nu}_{\mu}, \bar{\nu}_{\tau}$), we employ
 a leakage scheme to include the $\nu_x$ cooling via pair, 
photo and plasma processes (see \citealt{taki14} for more details).

The investigated progenitors with iron cores \citep{Woosley02} are given 
in $0.2\Msun$ steps between $10.8\Msun$ (s10.8) and $28.2\Msun$ (s28.2) 
and further from $30\Msun$ (s30) 
up to $75.0\Msun$ (s75) 
in 1.0 $M_{\odot}$ steps. 
The structure of these stars, such as density profiles and the pre-collapse 
masses have been already described in \citet{Ugliano12}. To induce 
non-spherical instability, we add initial 
 seed perturbations by zone-to-zone random density variations with 
 an amplitude less than 1\%.  


%-------------- figure-snap-2 -------------------
\begin{figure}[t]
\begin{center}
\includegraphics[bb=100 0 900 600,clip,scale=0.42]{Figure2.eps}
%\includegraphics[scale=0.6]{Figure2.eps}
\caption{Same as Figure \ref{fig-entall1} 
but for models 
s25.0 (a)
to s75.0 (i), 
from top-left to bottom-right.
}
\label{fig-entall2}
\end{center}
\end{figure}

%-------------- figure-(rshock+xi) -----------------------
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=1.0]{Figure3.eps}
\caption{Top left: average shock radii (thick solid lines) 
and mass accretion rate of the collapsing stellar core
 at 500 km (thin dashed lines) 
for some selected models. 
Bottom left: compactness parameters $\xi_M$ defined in Equation (\ref{eq-xi}) 
as a function of the ZAMS mass. 
The compactness parameters estimated at $M=1.5 \Msun$ at core bounce 
($\xi_{1.5,{\rm cb}}$, thin line with points) 
and at $M=2.5 \Msun$ ($\xi_{2.5}$  enhanced by a factor of three, thick line) 
are shown. 
To guide eyes, arrows 
are inserted in the bottom panel using the same color 
as in the other panels.
Right: diagnostic energy (top) and mass of protoneutron star (bottom) 
of the same models as in the top-left panel 
as a function of post-bounce time. 
}
\label{fig-rsh-xi}
\end{center}
\end{figure*}



Following 
\citet{Oconnor11}, we estimate the compactness parameter 
as the ratio of mass $M$ and the enclosed radius $R(M)$,
\begin{equation}
\xi_M  \equiv \frac{M/M_{\odot}}{R(M)/1000{\rm km}}.
\label{eq-xi}
\end{equation}
The previous studies used $M=2.5 \, \Msun$ \citep{Oconnor11,Ugliano12} 
or $1.75 \, \Msun$ \citep{oconnor13} 
and estimated $\xi$ 
at the time of core bounce. 
On the other hand, 
the outer radius of our computational domain (5000 km) is too small to 
contain $2.5 \, \Msun$ for all models 
and even  $1.75 \, \Msun$ for some less massive models. 
In this Letter, we estimate $\xi$ at $M=2.5 \, \Msun$ ($\xi = \xi_{2.5}$) 
directly from pre-collapse data. 
It should be noted that 
our definition of $\xi$ gives almost the same compactness estimated 
at bounce, because the radius $R$ enclosing $2.5 \, \Msun$ is 
far from the center and the radial velocity $v_R$ there is very small 
(e.g., for s15.0 model $R=1.7 \times 10^{9}$ cm and 
$v_R=-6.8 \times 10^6 {\rm cm \, s}^{-1}$).
Actually $\xi_{2.5}$ of s15.0 model in our definition is 0.145, 
which is very close to the value (0.150) estimated by \citet{Oconnor11} at 
bounce. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{RESULTS}\label{sec-res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 For all the employed 101 models, the bounce shock stalls in a 
 spherically symmetric manner and only after that, we observe 
a clear diversity of the multi-D hydrodynamics evolution in
 the postbounce (pb) phase.
Figures \ref{fig-entall1} and \ref{fig-entall2} show a snapshot of entropy distribution 
for selected 18 models at $t_{\rm pb}=400$ ms.
For some less massive progenitors 
(e.g., model s11.2 in Figure \ref{fig-entall1}(a)),
the shock is reaching close to the outer boundary of the computational
 domain with developing 
 pronounced unipolar and dipolar shock deformations. At this time, 
the shock of the most massive progenitor 
(s75.0 in Figure \ref{fig-entall2}(i)) 
is reaching an average radius of $\langle r\rangle  \sim 1000$ km, whereas 
the shock of s24.0 (in Figure \ref{fig-entall1}i) still wobbles
 around at $\langle r\rangle \sim$ 200 km. This 
demonstrates that the ZAMS mass is not a good 
criterion to diagnose the possibility of explosion. 


This is more clearly visualized in the top left panel of 
Figure \ref{fig-rsh-xi}. Taking six models as an example 
in the mass range between $19.2$ $M_{\odot}$ and $24.0 \Msun$, 
the shock revival is shown to occur 
earlier for s20.0 (red line) and s22.0 (blue line) compared 
 to the lighter progenitor s19.2 (green line).
Comparing with the bottom left panel of Figure \ref{fig-rsh-xi},
 it can be seen that the compactness parameter (Equation (\ref{eq-xi}))
 is smaller for s20.0 (labeled by red arrow) and s22.0
 (by blue arrow) in the chosen mass range above\footnote{Note in Figure 3 that the correlation between the compactness 
and $t_{400}$ is rather weak, which is more clearly seen
 in Figure \ref{fig-all}(c).}. 
The smaller compactness is translated into smaller 
mass accretion rate onto the stalled bounce shock. 
 For model s20.0 (red line in the top panel),
 the relatively earlier shock revival  ($\sim 100$ ms postbounce)
 coincides with 
 the sharp decline of the accretion rate (dashed red line). 
 After that, the accretion rate gradually decreases to 
$\sim 0.1 \Msun \, {\rm s}^{-1}$ till $t_{400} =420$ ms at this time
 the revived shock has expanded to an average radius of
$\langle r \rangle = 400$ km. 
Here $t_{400}$ is a useful measure to qualify the vigor of the shock revival  \citep[e.g.,][]{hanke12}. 
On the other hand, high compactness (model s21.0, black 
 arrow in the bottom panel) leads to the high accretion rate 
(black dashed line in the top panel) 
and it takes $\sim$ 500 ms for the sloshing shock (black line 
 in the top panel) to gradually turn into a pronounced expansion 
later on ($t_{400}$ = 700 ms). 
The growing {\it diagnostic} explosion energy\footnote{
Following \citet{suwa10} and \citet{nakamura14}, 
we define the {\it diagnostic} energy 
that refers to the integral of the energy over all outward moving zones 
that have a positive sum of the specific internal, kinetic, and 
gravitational energy.}
(top right panel) and the almost converged mass of PNS 
(bottom right) are also shown as a function of post-bounce time. 
The PNS mass has a clear correlation with $\xi_{2.5}$. 
Regarding the diagnostic energy, 
the increasing rate ($E_{\rm dia}/(t_{\rm fin} - t_{400})$ 
in unit of $10^{51}\,{\rm erg \, s}^{-1}$) 
tends to become higher for models with high $\xi_{2.5}$ 
(0.754, 1.12, and 1.44, for s22.0, s23.0 and s24.0, respectively).
This also indicates that the 
diagnostic energy of the high-$\xi$ models might become higher 
later on. 
Note that in previous 1D studies with simplified 
neutrino heating and cooling scheme \citep{Oconnor11} 
or with the excision inside the PNS \citep{Ugliano12},
the relation between the compactness and 
these explosion properties 
could not be determined in a self-consistent manner.

%----------- figure-pol ------------
\begin{figure*}[ht]
\begin{center}
\includegraphics[scale=1.]{Figure4.eps}
\caption{
Top: entropy along the north and south polar axis
as a function of post-bounce time for models s11.2 (left) and s15.0 (right). 
Bottom: time evolution of the anisotropic velocity $v_{\rm aniso}$ ({\it upper})
and the normalized pressure perturbation $\Delta p$ ({\it lower}) 
in the post-shock region (see the text for details).
}
\label{fig-pol}
\end{center}
\end{figure*}




We investigate the postbounce shock evolution in more details
for two specific models, s11.2 and s15.0. 
The density profile of model s11.2 falls rapidly off with radius and 
the compactness parameter 
$\xi_{2.5}$=0.005 is one of the smallest values 
among the 101 progenitors, whereas model s15.0 has moderate 
compactness 
($\xi_{2.5}$=0.145) 
casting more extended envelope out to the iron core.
Color-coded panels in Figure \ref{fig-pol} show the differences 
in the postbounce evolution for s11.2 ({\it left}) and s15.0 
({\it right}), respectively. 
Model s11.2 explodes rather early 
($t_{400}$ = 150 ms, top panels) 
and convective activity as well as the oscillations of the shock is moderate 
before the onset of the explosion 
(see the bottom left panel).
Note in the bottom panels that 
the anisotropic velocity $v_{\rm aniso}$ ({\it upper}), and the 
normalized  pressure perturbation $\Delta p$ ({\it lower}),
is defined respectively as $v_{\rm aniso} = \sqrt{\langle 
\rho [(v_r - \langle v_r \rangle)^2 + v_{\theta}^2]\rangle
/\langle \rho \rangle} $ and $\Delta p = \sqrt{ \langle p^2 \rangle 
- \langle p \rangle^2} /\langle p\rangle$,
where $\langle A \rangle$ represents the angle average of 
quantity $A$ \citep[e.g.,][]{Takiwaki12,kuroda12}. 

For model s15.0 with higher compactness parameter, 
sloshing motions of the shock are 
more clearly visible (right panels of Figure \ref{fig-pol}). 
These features regarding the dominance of the SASI over neutrino-driven
convection for high-$\xi$ models are quantitatively consistent with those 
observed in \citet{BMuller12b}.
Near at the bottom of the gain region, the accreting flows receive an 
abrupt deceleration, below which the regions are convectively 
stable. A strong pressure perturbation forms there 
(seen, in the lower part of the bottom-right panel in Figure \ref{fig-pol}, 
as a boundary between the regions colored by white and deep blue at a 
radius of $r \lesssim 100$ km after $t_{\rm pb} \sim 100$ ms). 
Subsequently the pressure perturbations propagate outward 
before they hit the shock, 
which is seen as a up-going stripe in the $\Delta p$ plot.
This leads to the formation of the next vortices (e.g.,
 yellow regions behind the shock in the $v_{\rm aniso}$ plot).
 These features, as previously identified, are natural outcomes of
the SASI and neutrino-driven convection (e.g., \citealt{rodrigo14} and references therein). 
 When the residency timescale becomes enough long, due to these multi-D 
 effects, compared to the 
neutrino-heating timescale in the gain region,
 the runaway shock expansion initiates 
at $t_{\rm pb} \sim$ 500 ms ($t_{400} \sim 600$ ms) for this model. Such 
 features have been also presented in previous 2D simulations with 
more detailed transport scheme \citep[e.g.,][]{BMuller12b,bruenn13}.
Here we emphasize that the use of the leakage scheme, together
 with the omission of inelastic neutrino scattering on electrons 
and GR effects in the present scheme, is likely to facilitate 
 artificially easier explosions (see discussions in \citealt{taki14} 
for more details).
 Discrepancies between our Newtonian 
models and GR models with detailed transport \citep[e.g.,][]{BMuller12b}
 become remarkable especially for progenitors with high compactness 
(e.g., compare model s27.0 
in Figure \ref{fig-entall2}b 
with the bottom 
right panel of Figure 1 in \citealt{BMuller13}). 
For models with moderate compactness parameter, on the other hand, 
we like to note that some key hydrodynamic features in the 
postbounce phase (such as the onset time of shock 
revival $t_{400}$) are rather similar (surprisingly) 
between ours versus the Garching models 
($t_{400}$ = 640 vs. 580 ms for model s18.4, 
320 vs. 400 ms for s19.6, 
540 vs. 560 ms for s21.6, 
460 vs. 460 ms for s22.4, 
420 vs. 440 ms for s27.0, 
respectively, e.g., Janka et al., TAUP Conference, 2013).  

%-------------- figure-all -------------------
\begin{figure*}[t]
\begin{center}
\includegraphics[scale=1.]{Figure5.eps}
\caption{
Resultant supernova properties from our 101 simulations 
as a function of compactness parameter $\xi_{2.5}$.
Left: 
Mass accretion rate (a), 
and electron neutrino luminosity (b), 
estimated at time of shock revival $t_{400}$ (c).
Right: 
diagnostic energy (d), mass of proto-neutron star (e), 
and mass of nickel in outgoing unbound material (f), 
at $t=t_{\rm fin}$.  
Failed models which cannot carry the shock to the outer boundary 
during our simulation time 
are excluded from these panels.
}
\label{fig-all}
\end{center}
\end{figure*}



In Figure \ref{fig-all}, we plot various quantities to summarize 
our results for the 101 simulations as a function of $\xi_{2.5}$.
In the left panels,
the mass accretion rate $\dot{M}$ at 500 km, 
electron-type neutrino luminosity $L_{\nu_{\rm e}}$, 
and the shock revival time $t_{400}$ are shown.
Three panels in the right column show 
a {\it diagnostic} explosion energy $E_{\rm dia}$, 
a mass of central remnants $M_{\rm PNS}$, 
and mass of nickel in the ejected material $M_{\rm Ni}$, 
at the final time $t=t_{\rm fin}$ of our simulations. 
Here the PNS is 
defined by the region where the density $\rho > 10^{11}~{\rm g\, cm}^{-3}$.
For most of our models $t_{\rm fin}$ is defined as 
the time when the maximum shock radius arrived 
at the outer boundary. %(89 models). 
%In the remaining 12 models, the shock of model s25.0 cannot reach the outer boundary within our simulation time ($t_{\rm sim} = 1.5$ s), although it is still expanding,  and the other eleven models are stopped during $t_{\rm sim}$  because the density near the outer boundary of the computation domain goes below the lowest value covered by the EOS table. These incomplete models are not shown in Figure \ref{fig-all}.



\citet{Ugliano12} showed that these quantities are not a monotonic 
function of ZAMS mass. 
We confirm it in our 2D simulations 
and find that these values are nearly in the linear 
correlation with the compactness parameter $\xi_{2.5}$. 
This can be interpreted as follows: 
the core of high-$\xi$ models is surrounded by high-density Si/O layers 
and the mass accretion rate therefore remains high 
long after the stalled shock has formed 
(Figures \ref{fig-all}(a) and 3). 
This makes the PNS mass of the high-$\xi$ models heavier
(Figure \ref{fig-all}(e)). 
Due to the high accretion rate, 
the accretion neutrino luminosities become higher for models 
 with high $\xi$ (Figure \ref{fig-all}(b), 
see also \citealt{oconnor13}). 
As a result, 
we obtain the high diagnostic explosion energy (Figure \ref{fig-all}(d)), 
as well as outgoing unbound material rich in nickel (Figure \ref{fig-all}(f)) 
for high-$\xi$ models. 
On the other hand, the time of shock revival 
($t_{400}$, Figure \ref{fig-all}(c)) shows a large 
scatter and weaker correlation with $\xi_{2.5}$. This may come from 
the stochasticity of the initial random perturbations that should affect the 
onset of an explosion.
It should be noted that 
the diagnostic energy of almost all models, 
which ranges in $\sim 0.2$ - $0.7 \times 10^{51}$ erg, 
is still increasing at the final time of our simulation.
% (see also bottom right panel of Figure \ref{fig-rsh-xi}).
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CONCLUSIONS AND DISCUSSION}\label{sec-con}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have conducted a systematic study of neutrino-driven CCSNe in 2D 
to explore the difference that the structure of progenitors makes 
on the explosion characteristics, 
such as mass of central remnants. 
We find, as \citet{Ugliano12} did in 1D, that 
these characteristics exhibit a non-monotonic variability 
as a function of the progenitor ZAMS mass, 
which is a consequence of the non-monotonicities of the progenitor structure. 
Thus, we adopted the compactness parameter $\xi$ 
introduced by \citet{Oconnor11} 
to characterize the progenitor structure. 
%Compared to models with low $\xi$, models with high $\xi$
% undergo high ram pressure 
%from the accreting matter onto the stalled 
%shock and it takes longer time before the shock expansion
% is initiated under the influence of neutrino-driven convection and 
%the standing accretion-shock instability.
We find that the accretion luminosity becomes higher for models
 with high $\xi$, which makes the diagnostic 
 energy higher and the synthesized nickel mass bigger. 
We point out that these explosion characteristics
 tend to show a monotonic increase as a function of the 
compactness parameter $\xi$.


%{\bf The compactness parameter $\xi_{2.5}$ estimated directly from pre-collapse data 
%has the clear correlations with 
%all quantities displayed in Figure \ref{fig-all} except $t_{400}$.
%This means that our definition to estimate $\xi_M$ 
%is not bad in characterizing the accreting matter onto a shock, 
%although it is usually estimated at core bounce 
%in the previous studies 
%(\citealt{Oconnor11,Ugliano12}, see also \citealt{Sukhbold14}).}

Our simulations are limited in space ($r < 5000$ km) and time ($t \leq 1.5$ s). 
The simulations are terminated before the diagnostic energies are 
saturated. Later on neutrino energy deposition would get smaller with time
as the neutrino luminosity as well as post-shock density becomes smaller. 
Further global simulation, 
taking account of gravitational energy of an envelope and 
nuclear energy released via recombination process behind the shock, 
is necessary to determine the final explosion energy and the quantities
 shown in the right panels of Figure \ref{fig-all}.
Moreover, the finding of this study should be reexamined
 by 3D models \citep{hanke12,dolence12,couch13,couch14,nakamura14_2},
 in which neutrino transport is appropriately solved (see, e.g.,  
\citet{Hanke13,taki14,nagakura14,tony14}
for different levels of sophistication).
To get a more accurate amount of the synthesized nickel and other isotopes,
a post-process calculation with a larger nuclear network is 
needed. 
After all, long-term 3D full-scale simulations 
 are needed to unambiguously clarify the critical 
 $\xi$ parameter, below or above which neutron stars or black holes 
will be left behind.


In this work we have reported our results 
of %only solar-metallicity 
progenitors from one modeling group. 
Currently we are conducting the same sort of CCSN simulations 
for sets of progenitors from the other groups including 
a variety of metallicity, rotation, and magnetic fields, 
which will be presented in the forthcoming work.
We hope that the compilation of our results would be helpful to many 
researchers to study nucleosynthetic yields from CCSNe and 
its impacts on chemical and dynamical evolution of galaxies, and also
detectability of neutrino and gravitational-wave signals by 
next-generation detectors. %surely being on line in the decade to come.

We thank F.K Thielemann, M. Liebend\"{o}rfer, R. M. Cabez\'on, 
M. Hempel, and  K.-C. Pan for stimulating discussions and for their kind hospitality during our research stay at Basel in this February.
We are also grateful to S.Yamada and Y.Suwa for informative discussions. 
Numerical computations were carried out in part on 
XC30 and general common use computer system at the center for 
Computational Astrophysics, CfCA, 
the National Astronomical Observatory of Japan,
Oakleaf FX10 at Supercomputing Division in University of Tokyo, and on 
SR16000 at YITP in Kyoto University.
 This study was supported in part by the Grants-in-Aid for the Scientific 
Research from the Ministry of Education, Science and Culture of Japan 
(Nos. 23540323, 23340069, 24103006, 24244036, and 26707013) and by HPCI Strategic Program of Japanese MEXT.



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\end{document}

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[Burrows et al.(1995)]{Burrows95 Burrows, A., Hayes, J., \& Fryxell, B.~A.\ 1995, \apj, 450, 830
1406.2415.extracted_bib
[Couch(2013)]{couch13 Couch, S.~M.\ 2013, \apj, 775, 35
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[Couch \& Ott(2013)]{couchott Couch, S.~M., \& Ott, C.~D.\ 2013, \apjl, 778, L7
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[Dolence et al.(2014)]{dolence15 Dolence, J.~C., Burrows, A., \& Zhang, W.\ 2014, arXiv:1403.6115
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[Einfeldt(1988)]{Einfeldt88 Einfeldt, B.\ 1988, SIAM Journal on Numerical Analysis, 25, 294
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[Hanke et al.(2012)]{hanke12 Hanke, F., Marek, A., M{\"uller, B., \& Janka, H.-T.\ 2012, \apj, 755, 138
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[Hayes et al.(2006)]{Hayes06 Hayes, J.~C., Norman, M.~L., Fiedler, R.~A., et al.\ 2006, \apjs, 165, 188
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[{{Heger {et~al.(2000){Heger, {Langer, \& {Woosley]{Heger00a {Heger, A., {Langer, N., \& {Woosley, S.~E. 2000, \apj, 528, 368
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[Janka \& Mueller(1996)]{Janka96 Janka, H.-T., \& Mueller, E.\ 1996, \aap, 306, 167
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[{{Janka(2012)]{Janka12 ---. 2012, Annual Review of Nuclear and Particle Science, 62, 407
1406.2415.extracted_bib
[Kitaura et al.(2006)]{kitaura Kitaura, F.~S., Janka, H.-T., \& Hillebrandt, W.\ 2006, \aap, 450, 345
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[Kuroda et al.(2012)]{kuroda12 Kuroda, T., Kotake, K., \& Takiwaki, T.\ 2012, \apj, 755, 11
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[Murphy et al.(2013)]{Murphy13 Murphy, J.~W., Dolence, J.~C., \& Burrows, A.\ 2013, \apj, 771, 52
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[Mezzacappa et al.(2014)]{tony14 Mezzacappa, A., Bruenn, S.~W., Lentz, E.~J., et al.\ 2014, arXiv:1405.7075
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[Nagakura et al.(2014)]{nagakura14 Nagakura, H., Sumiyoshi, K., \& Yamada, S.\ 2014, arXiv:1407.5632
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[{{Nakamura {et~al.(2014){Nakamura, {Takiwaki, {Kotake, \& {Nishimura]{nakamura14 {Nakamura, K., {Takiwaki, T., {Kotake, K., \& {Nishimura, N. 2014, \apj, 782, 91
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[Nakamura et al.(2014)]{nakamura14_2 Nakamura, K., Kuroda, T., Takiwaki, T., \& Kotake, K.\ 2014, arXiv:1403.7290
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[{{Nomoto \& {Hashimoto(1988)]{Nomoto88 {Nomoto, K., \& {Hashimoto, M. 1988, \physrep, 163, 13
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[Sumiyoshi et al.(2014)]{sumi15 Sumiyoshi, K., Takiwaki, T., Matsufuru, H., \& Yamada, S.\ 2014, arXiv:1403.4476
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[Suwa et al.(2010)]{suwa10 Suwa, Y., Kotake, K., Takiwaki, T., et al.\ 2010, \pasj, 62, L49
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[Suwa et al.(2014)]{suwa14 Suwa, Y., Yamada, S., Takiwaki, T., \& Kotake, K.\ 2014, arXiv:1406.6414
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[{{Woosley \& {Weaver(1995)]{WW95 {Woosley, S.~E., \& {Weaver, T.~A. 1995, \apjs, 101, 181