text:
\section{Introduction}

Neutrinos play an essential role
in the mechanism of core-collapse supernovae
as a driving force in the dynamics
starting from the gravitational collapse of massive stars.
Transport of energy and lepton number via neutrinos
is particularly crucial to determine their outcomes, i.e. explosions
with an energy of 10$^{51}$ erg \citep{bet90,kot06,jan07}.
Electron-type
neutrinos are produced by electron captures
in the collapsing phase
and are trapped inside the central core
to give pressure support.
Pairs of neutrinos are created after bounce as
a part of the thermal
energy converted from the gravitational energy.
Emissions of the trapped neutrinos result in
an energy release of $\sim$10$^{53}$ erg
in the form of supernova neutrinos.
The release of this huge energy in neutrinos was indeed vindicated
in the observed events of SN 1987A \citep{hir87}.
The emitted neutrinos carry
valuable information of the mechanism of
explosion as well as the properties of
dense matter inside compact stars
and progenitors
(see \citet{nak13}, for example, and references therein).

Interactions of neutrinos with material
in the supernova core contribute to the energy transfer
in a significant size.
The shock wave generated by core bounce
stalls soon after the launch
and fizzles later in the case of strictly spherical geometry.
Without such an artificial restriction in geometry,
multi-dimensional hydrodynamical instabilities occur and
push the stalled shock so that it could hover at larger radii.
A portion of the emitted neutrinos is absorbed by the matter
behind the stagnant shock wave and may lead eventually to
its outward propagation again.
This so-called neutrino heating mechanism \citep{bet85}
combined with the multi-dimensional hydrodynamical instabilities
is one of the most promising ways to trigger the explosion
particularly under a marginal condition.
In fact, energy gain by the neutrino heating
amounts to $\sim$10$^{51}$ erg,
which is comparable to the observed explosion energy of
supernovae \citep{jan96}.
Whether the neutrino heating mechanism is
the essential cause of explosion or not
remains an unsolved issue, though \citep{jan12,bur13}.

In order to advance further
our understanding of the role of neutrinos
in the intrinsically multi-dimensional dynamics,
the evaluation of neutrino heating and cooling
should be as accurate as possible.
Neutrinos interact with matter very frequently
and diffuse outward only gradually inside the proto-neutron star
just born after the bounce.
Since the temperatures are high in the proto-neutron star,
neutrinos are coolant there.
On the other hand, a partial absorption of the emitted neutrinos
takes place outside the neutrino-sphere
below the stalled shock wave before they stream out of the star freely,
which heats the matter there.

Quantitatively accurate treatment of neutrino transfer is hence mandatory
to provide correct rates of cooling and heating since
the energy transport proceeds across the region,
where neither diffusion nor free-streaming approximation is applied.
The emergent fluxes and energy spectra of neutrinos
are determined by
the neutrino-matter interactions
around the energy-dependent neutrino-spheres.
We need to do
multi-energy group computations,
since the interactions
strongly depend on the neutrino energy.
In the evaluation of the heating rate,
on the other hand,
the local number density of neutrinos are important
in addition to the energy spectra.
It is hence
the angular distributions of neutrinos
in their momentum space that should be given precisely.
Although they are isotropic near the center and
become forward-peaked at large radii,
the angular distributions are highly non-trivial
in the transitional region, which includes the heating region.

The treatment of neutrino transfer should, therefore,
be as accurate as possible, particularly in the description of
the energy and angle distributions.
In most of multi-dimensional numerical simulations of supernovae,
however, some approximations are employed at the moment.
This is simply because the full description of neutrino distributions
is a six-dimensional
(three spatial coordinates and three momenta of neutrinos) problem,
which is formidable even for modern supercomputing resources.
While computations with the multi-energy groups are
becoming a standard nowadays,
angular distributions are still commonly approximated one way or another.
Since such approximations can be eliminated
in spherically symmetric simulations,
the influences of the approximations have been extensively studied
\citep{jan92,mes98,yam99,lie05}.
It is well known that the diffusion approximation
with a flux-limiter tends to overestimate the degree of forward-peak
and thus underestimate the heating rate \citep{jan92,mes98}.
Although the difference is not so drastic,
such a small change may be crucial, tipping the balance
for explosion particularly in the marginal conditions \citep{jan96}.

Validation of the approximate methods
in multi-dimensions has growing importance,
since the post-shock conditions obtained in multi-dimensional simulations
appear more marginal for successful explosion.
The diffusion approximations (including its variants such as IDSA)
and/or the ray-by-ray methods are frequently
used in the currently state-of-the-art
2D/3D simulations \citep{suw10,mul12b,bru13,tak12,han13,knak14a,suw14b,bru14}.
Although they have reported a considerable number of successful explosions
produced by the neutrino heating combined
with the neutrino-driven convection
and/or the standing accretion shock instability (SASI),
their approximate treatments of neutrino transfer introduce
a certain level of uncertainty.
The influences of these approximations should be examined carefully
before drawing any conclusion.

In the past years
we have indeed seen
the multi-dimensional treatment of neutrino transfer
progress substantially,
keeping pace with the rapid increase of supercomputing resources.
In two dimensions, multi-angle and multi-energy groups
neutrino radiation-hydrodynamics have been performed
to follow the post-bounce evolution of supernova cores
\citep{liv04,ott08,bra11}.
They adopted the discrete-ordinate (S$_n$) method
to fully describe angle distributions in 2D space \citep{liv04}.
\citet{ott08} reported the comparisons
with the diffusion approximation.
They demonstrated that the multi-angle transport is
needed to describe the neutrino distributions
in globally aspherical supernova cores.
\citet{bra11} analyzed further the neutrino heating
and showed that the multi-angle transfer
leads to less lateral variations in the neutrino distributions
owing to an averaging effect.
Such a detailed study has been limited to 2D.

In three dimensions, there has been a remarkable progress
in the multi-energy group transport,
but not
in the multi-angle group transport \citep{tak12,han13,tak14,tam14,mez14,hor14}.
These simulations commonly adopted the ray-by-ray approximations
although the radial transport was treated differently:
the isotropic diffusion source approximation (IDSA)
is employed by \citet{tak12,tak14}
whereas the moment equations are solved together
with a variable Eddington factor obtained from the solution of
the 1D model Boltzmann equations in \citet{han13,tam14}.
Results obtained by the two groups differ in many respects,
most notably on the success or failure of explosion
(see e.g. their 11.2M$_{\odot}$ model).
Note, however, that the origin of the differences
might be the employed neutrino reactions \citep{tak14b}.
Provided that the number of reported models is still small
and the employed physics and numerics are considerably different among them,
it is still premature to draw a firm conclusion
from these 3D simulations and
further studies should be done systematically
with the assessment of the influences
from the approximations in 3D neutrino transfer.

The purpose of this study is hence
to explore the basic features of
3D neutrino transfer in realistic supernova cores
with the multi-angle and multi-energy group Boltzmann solver
for the first time.
Adopting a set of profiles from 3D numerical simulations
by \citet{tak12,hor14,tak14b},
we run our newly developed code \citep{sum12}
to obtain stationary neutrino distribution functions
for those fixed matter configurations.
Taking advantage of the multi-angle treatment of our code,
we explore the non-radial (polar and azimuthal) transport
in the globally and locally aspherical matter distributions.
Note that in our previous paper \citep{sum12}
the background models were constructed
by artificially deforming spherically symmetric models.
We pay particular attention to
the behavior of the angle moments of neutrino distribution functions
in the transitional regime,
which will be useful for development of
numerical codes using the moment formalism.

Employing our Boltzmann code, we can study
the quality of the ray-by-ray approximation,
which is routinely used in many simulations
in 2D and 3D\footnote{Some simulations implemented
the ray-by-ray plus approximation, in which
the lateral coupling is partially taken into account.  }.
As a matter of fact, simply dropping
the polar and azimuthal advection terms,
we can emulate the ray-by-ray approximation
without changing other settings.
We examine the non-radial transport,
which will not be fully described in the ray-by-ray approximation.
As we will show, the contributions of polar and azimuthal
fluxes are significant even outside the optically thick region.
It is also found that
the ray-by-ray approximation tends to exaggerate
the contrast between different radial rays, which
may affect local neutrino-heating rates.

The paper is organized as follows.
We describe the methodology in \S \ref{section:3D-nu}.
The realization of the ray-by-ray approximation in our code
is also explained there.
The 3D profiles taken
from \citet{tak12,hor14,tak14b}
are presented in \S \ref{section:3D-hyd}.
We report in \S \ref{section:d-boltz} the basic features
of 3D neutrino transfer,
showing density, radial and non-radial fluxes
obtained by the 6D Boltzmann solver.
The corresponding results by the ray-by-ray approximation
are presented in \S \ref{section:d-ray} for comparison.
We investigate the heating rates obtained
by both the 6D Boltzmann solver and the ray-by-ray approximation
in \S \ref{section:h-boltz-ray}.
In \S \ref{section:h-boltz-ray.others}, we present
the results for other profiles and demonstrate
that our findings are generic.
The volume-integrated heating rates are examined
in \S \ref{section:h-total} to discuss the global
influences of 3D neutrino transfer on the supernova dynamics.
In \S \ref{section:moments},
we show the angle and energy moments of the neutrino
distribution functions.
Discussions and implications of our study
are given in \S \ref{section:discuss}
followed by the summary in \S \ref{section:summary}.

\section{Numerical settings}

\subsection{Neutrino transfer in 3D space}\label{section:3D-nu}

We adopt the numerical code
to solve the Boltzmann equation in 6D \citep{sum12}.
The code solves the time evolution of
the 6D neutrino distribution functions
in the spherical coordinate system
by using neutrino energy and two angles
to designate the neutrino momenta.
The 6D Boltzmann equation in the conservative form is
discretized in a finite-differenced method
on a spatial grid as wells as
multi-energy and multi-angle grids.
We employ the discrete-ordinate (S$_n$) method
to describe the angular distributions.
The implicit differencing is adopted
for the advance of time step
to solve stiff equations and
to ensure the stability for the equilibrium solution.
Further details of the numerical scheme
can be found in \citet{sum12}.

A basic set of the neutrino reactions is
implemented in the collision term \citep{bru85,sum05}.
The set includes the emission, absorption and scattering
of neutrinos via nucleons and nuclei.
The pair processes via electron-positron annihilation,
creation and the nucleon-nucleon bremsstrahlung
are also taken into account.
We treat the three neutrino species,
$\nu_e$, $\bar{\nu}_e$ and $\nu_\mu$.
The last one is actually a representative
of four species,
$\nu_{\mu}$, $\bar{\nu}_{\mu}$, $\nu_{\tau}$ and $\bar{\nu}_{\tau}$\footnote{This group is
often denoted by $\nu_x$ or $\nu_{\mu/\tau}$ and is
sometimes used to show the summed contributions of four species.  }.
It is to be noted that
the densities, fluxes and angle moments of $\nu_{\mu}$ is
for one species whereas the contributions of the four species
are summed for the cooling and heating rates.
The same setting has been used in our previous paper \citep{sum12}
for detailed comparisons
with the 1D results of core-collapse supernovae from \citet{sum05}.
These neutrino reactions are
the most important but certainly not sufficient.
The electron-neutrino scattering, for example, is omitted
in the current study \citep{tho03,bur06a}.
This is mainly due to the limitation of
numerical resources and modernization will be definitely needed \citep{len12a}.
The equation of state of \citet{lat91} are used
for all calculations in the current study
so that better consistency would be obtained
with the profiles
we adopt from the supernova simulations by \cite{tak12,hor14,tak14b}.

We follow the time evolution of the neutrino distribution functions
by the 6D Boltzmann solver for the fixed background profiles
until stationary states are reached (typically in $\sim$20 ms)
from arbitrarily set initial conditions, in which essentially no neutrino exists.
We ignore fluid motions although the original numerical data include such information.
We neglect all the velocity-dependent terms in the current study
and do not distinguish the inertial frame from the fluid-rest frame,
since
our aim here is to understand the basic 3D features of the neutrino transfer.
It should be emphasized, however, that these
velocity-dependent terms in the Boltzmann equation
cannot be neglected in more realistic simulations
as demonstrated by \citet{len12b}.
As a matter of fact, the implementation
of all the velocity-dependent effects
to the current Boltzmann solver
has been already done and its application to
2D neutrino-radiation hydrodynamics simulations
is currently under way \citep{nag14}.

In addition,
we perform approximate calculations by a ray-by-ray prescription
in our framework
by dropping off the advection terms
for spatial $\theta$- and $\phi$-derivatives
as well as neutrino-angle $\phi_{\nu}$-derivative
in the Boltzmann equation.
We compute the stationary states of the 3D neutrino transfer
with the same setting for profiles and reactions.
We compare the numerical results by the two methods
and examine the differences due to the approximation.

The numbers of spatial grid points employed
for neutrino distribution functions are
$N_r=256$, $N_{\theta}=64$, $N_{\phi}=32$
for our baseline 3D models.
The numbers of neutrino angle ($\theta_{\nu}$, $\phi_{\nu}$)
and energy ($\varepsilon_{\nu}$) grids are
$N_{\theta_{\nu}}=6$, $N_{\phi_{\nu}}=12$ and
$N_{\varepsilon_{\nu}}=14$, respectively.
The energy grid points are placed logarithmically from 0.9 MeV
to 300 MeV with additional grid points for high-energy tails.
Computations with this mesh using 1.6TB memory
are the largest scale available on 8 nodes of HITACHI SR16000 at KEK.
In order to obtain one configuration of the model
by the time evolution in several tens of milliseconds,
we need a run of $\sim$4$\times10^{4}$ steps
using the MPI-parallel code of the 6D Boltzmann solver.
It typically takes $\sim$200~hours of wall-clock time
on 8 nodes of SR16000 by 128 MPI processes.

Although
the current setting of
the angle and energy grids is sufficient to obtain
the reasonable results
as validated by the convergence test in \citet{sum12},
we have additionally studied the convergence
by increasing the number of grid in some sample models.
For this purpose, we perform the numerical simulations
for the inner part of the supernova core ($\le$~360~km)
using $N_r=128$ and increase the angle resolution.
When we increase $N_{\theta_{\nu}}$ from 6 to 8,
we find typical
differences within 2~$\%$ in the neutrino densities,
fluxes and heating rates.
In the case of increasing $N_{\phi_{\nu}}$ from 12 to 16,
the differences are within $\sim$0.8~$\%$.
Appearance of large differences of the heating rates
is limited near the gain radius
due to very small values close to zero.

\subsection{3D supernova cores}\label{section:3D-hyd}

We utilize the post-bounce matter profiles taken from the 3D numerical simulations
of core-collapse supernovae by \citet{tak12,hor14,tak14b}, who followed
the evolutions of 11.2M$_{\odot}$ and 27.0M$_{\odot}$ stars of \citet{woo02}.
Their computations employed
the extended ZEUS-MP code for hydrodynamics \citep{hay06,iwa08}
and the IDSA scheme \citep{lie09} for neutrino transport
with the ray-by-ray approximation.
We use the 11.2M$_{\odot}$ model (11M) as the baseline model
for detailed analysis.
The profiles adopted from this model
represent the typical situations, in which the revived
shock wave is expanding toward an explosion.
On the other hand, the 27.0M$_{\odot}$ model (27M) \citep{hor14,tak14b},
which were computed with the improved code
and details will be reported elsewhere,
does not lead to shock revival
and represents the situations in such prolonged stagnations.
It is to be noted that
since the profiles we use in this study
are obtained in the simulations with the ray-by-ray approximation,
our results may be somewhat influenced indirectly by the approximation.

We adopt the 3D profiles of density, temperature and
electron fraction (but not velocity) from the dynamical simulations
and remap them
onto our numerical mesh.
We use the same radial (logarithmically spaced) grid
up to 2600 km
as those employed in the simulation of 11.2M$_{\odot}$.
To cover the whole solid angle in space,
we deploy the same number of uniform azimuthal
grid points as in the dynamical simulation
while a different (non-uniform) polar grid is used
so that it would fit our Boltzmann solver.
We take the same spacial mesh also for 27M,
interpolating the density, temperature and electron fraction.
Other thermodynamical quantities are derived from
the equations of state by \citet{lat91}
with the incompressibility of 180 MeV
for 11M and 220 MeV for 27M, respectively,
just as in the original simulations.

In Figure \ref{fig:3d-entropy},
we show the 3D profiles of supernova core
for the 11.2M$_{\odot}$ star
at 100, 150 and 200 ms after the bounce.
They correspond to
the phase, in which the shock wave expands gradually
from $\sim$300 km to $\sim$600 km
in a non-spherical manner.
The revival of shock wave occurs earlier (at $\sim$50 ms)
due to the neutrino heating
and its outward propagation is assisted by
the enhanced heating
through the hydrodynamical instabilities.
This 3D model leads to a successful explosion.

We display in Fig. \ref{fig:3db-xyslice}
the color maps of the core (11M) on the xy-plane (z=0)
at 150 ms after the bounce.
The shock wave is located around 400--500 km
with an elongated shape
at $\phi \sim45^\circ$ and $\phi \sim225^\circ$.
The low entropy region prevails
along $\phi \sim135^\circ$ and $\phi \sim315^\circ$,
which correspond to
down flows of proton-rich matter.
This nearly dipolar configurations may have arisen
from the weak $\pi/2$ periodicity in $\phi$,
which was planted accidentally in the perturbation
imposed initially in the dynamical simulation\footnote{
\citet{tak12} indeed added a perturbation pointwise-randomly.
The resultant perturbation, however, had a $\pi/2$ periodicity
weakly in the azimuthal direction by accident,
which was not mentioned in their paper.  }.
The density and temperature distributions
of the central object (proto-neutron star)
are almost spherical
with only the outer layer being deformed.
The configurations at other times
are qualitatively similar but with elongations at different $\phi$-angles.

Figures \ref{fig:3db-phi.slice.iph05}
and \ref{fig:3db-phi.slice.iph13} show
the color maps of entropy and mass fractions of nucleons
on the meridian slices at $\phi$=51$^\circ$ and 141$^\circ$, respectively.
Mushroom-like shapes of high entropy regions
reflect convective motion and the associated shock dynamics.
On the equator, the high entropy region
extends up to the shock front at $\phi$=51$^\circ$
and there is a narrow channel of low entropy matter
at $\phi$=141$^\circ$.
Free neutrons are abundant close to and
inside the proto-neutron star
whereas free protons populate rather abundantly
at larger radii where the entropy is high.
Nucleons behind the shock wave
act as chief absorbing material
in the neutrino heating.

In Figure \ref{fig:3d-entropy.3d.s27}, we show the
3D snapshots for the 27.0M$_{\odot}$ model
at 150 and 200 ms after the bounce.
In the supernova simulation, the shock wave
stalls and is hovering
around 200 km up to 700 ms.
There is no sign of shock revival.
Figures \ref{fig:3d.s27-xyslice-entropy}
and \ref{fig:3d.s27-phi.slice.iph05}
present the color maps of entropy
at 150 and 200 ms, respectively, on the xy-plane (z=0) and
the meridian slice at $\phi$=51$^\circ$.
The shape of the stalled shock wave is nearly spherical
with a slight deformation due to the convection in the heating region.
High-entropy blobs grow and move upward and
low-entropy matter falls downward
in channels formed between the mushroom-like shapes.
The convective motion with the down flows leads to
the non-uniform composition
in the central core as in
Figs. \ref{fig:3db-phi.slice.iph05}
and \ref{fig:3db-phi.slice.iph13} for the 11M model.
We note that there is no periodic pattern in the azimuthal
direction in this model.

\section{Neutrino distributions}

\subsection{Density and flux by Boltzmann evaluation}\label{section:d-boltz}

We show the 3D features of the neutrino distributions and
its associated moments obtained
for the snapshots of the 3D supernova core.
We mainly display the numerical results
for the snapshot of 11M at 150 ms after the bounce.
The 3D characteristics of the neutrino transfer shown below
are common also for the other snapshots.

We show in Fig. \ref{fig:3db-density.3d.in2}
the iso-surfaces of the density of
electron-type anti-neutrinos ($\bar{\nu}_e$)
with arrows of its flux vector.
The distribution of neutrinos is spherical at center,
but deformed in the surrounding region,
reflecting the convective high entropy region.
The corresponding fluxes are non-radial
according to the gradients of the deformed distributions.
The distribution further out is again spherical
due to the outward neutrino fluxes at large radii.
These 3D features are seen also for the densities
and fluxes for electron-type neutrinos ($\nu_e$) and
$\mu$-type neutrinos ($\nu_{\mu}$).

Figure \ref{fig:3db-density.xyslice} displays
the color maps of
the density of neutrinos for three species
($\nu_e$, $\bar{\nu}_e$ and $\nu_{\mu}$)
on the xy-plane (z=0).
The $\nu_e$-distribution is focused
inside the proto-neutron star
with high chemical potential of neutrinos.
The $\bar{\nu}_e$- and $\nu_{\mu}$-distributions
are concentrated in the off-center region,
where the degeneracy is low due to high entropies.
Among the three species,
the $\bar{\nu}_e$-distribution exhibits
the most non-spherical shape.
The $\nu_e$-distribution is spherical at center
and slightly deformed in the surrounding region.
They are affected by the non-spherical shape of
the degeneracy of neutrinos ($\eta_{\nu}=\mu_{\nu}/T$).
The $\nu_{\mu}$-distribution is entirely spherical,
reflecting the spherical shape of temperature distribution
as seen in Fig. \ref{fig:3db-xyslice}.

The 6D Boltzmann solver handles the advection in 3D space
and, therefore, describes the non-radial advection
as well as the radial advection.
Figure \ref{fig:3db-flux.xyslice} reveals
the three components of the flux of $\bar{\nu}_e$
on the xy-plane (z=0)
corresponding to Fig. \ref{fig:3db-density.xyslice}.
There are significant contributions of polar ($\theta$)
and azimuthal ($\phi$) fluxes
due to the deformed $\bar{\nu}_e$-distribution.
The direction of the flux follows roughly
the gradient of neutrino distribution in the central region.
The magnitude of the non-radial component of the flux
reaches more than 50 $\%$ of the radial component
around 20--50 km inside the proto-neutron star.

The 6D Boltzmann solver enables
the seamless description of non-radial fluxes
in the wide region
while the diffusion approximation is applicable
only under the optically thick condition.
The non-radial fluxes remain significant
up to large radii, going out the optically thick region.
The neutrino transport is diffusive well inside a radius of 50 km
and transitional to the free-streaming around 100 km
as we will see that
the second angle moment, $\langle \mu_{\nu}^2 \rangle$,
is less than 0.35 inside 50 km
in Figs. \ref{fig:3db-mom.mu.iph05.slice} and
\ref{fig:3db-mom.mu.iph13.slice}.

Figure \ref{fig:3db-density.iphxx.slice} shows
the neutrino densities for three species
in the central region
on the meridian slices at $\phi$=51$^\circ$ and 141$^\circ$.
The deformed shapes of density distributions
are seen for $\nu_e$ and $\bar{\nu}_e$
while the shape is spherical for $\nu_{\mu}$
as we saw in Fig. \ref{fig:3db-density.xyslice}.
The $\bar{\nu}_e$ and $\nu_{\mu}$ are abundant
in the off-center region due to their origin
of thermal production and
the neutrino degeneracy for the former.
There is a bump around the equator
in the case of $\phi$=141$^\circ$
for $\nu_e$ and $\bar{\nu}_e$.
This corresponds to the narrow channel
with the proton-rich and low entropy condition
seen in Fig. \ref{fig:3db-phi.slice.iph13}.
The production of neutrinos is amplified
by local enhancement of thermodynamical quantities,
which we discuss later.

Prominent non-radial fluxes of $\nu_e$ and $\bar{\nu}_e$
can be seen in Figs. \ref{fig:3db-flux.iph05.slice.inx}
and \ref{fig:3db-flux.iph13.slice.inx},
where the three components of flux are shown on the meridian slices.
The polar component exhibits fine structure
with alternating directions
and the azimuthal component widely extends.
It is remarkable to see that
the directions of flux for $\nu_e$ and $\bar{\nu}_e$
are opposite each other.
This is because
the neutrino chemical potential and temperature
(and their gradient) affect the distributions of two species
in an opposite way.
(See Fig. \ref{fig:2d.phslice-etanu.3db.iph13} and discussion below)
The flux for $\nu_{\mu}$ is almost
spherically symmetric (not shown here)
and its radial flux is concentrated
around its off-center density distribution.
The magnitude of the non-radial fluxes
is comparable to that of radial flux in some regions.
Its ratio to the magnitude of the radial flux reaches more than 50 \%
around the central object up to $r \sim40$ km.
The non-radial fluxes extend up to 100 km for $\nu_e$ in the case
of $\phi$=141$^\circ$.
This corresponds to the narrow region
near the equator noted above.

As we have seen, the neutrino distributions are
affected by the neutrino degeneracy.
While the temperature distribution is almost spherical,
the chemical potential (i.e. degeneracy) shows
non-uniform features in the central region inside the neutrino-sphere
and the surrounding region up to the location of the shockwave.
We show in Fig. \ref{fig:2d.phslice-etanu.3db.iph13} the temperature,
neutrino chemical potential and neutrino degeneracy
on the meridian slices corresponding to
Figs. \ref{fig:3db-density.iphxx.slice},
\ref{fig:3db-flux.iph05.slice.inx} and
\ref{fig:3db-flux.iph13.slice.inx}.
Having a deformed shape in the profile of electron fraction,
the distribution of the neutrino chemical potential is non-spherical.
We overlay the location of the neutrino-sphere
for the neutrinos with an energy bin of $E_{\nu}=$ 34 MeV
for three species.
Here the neutrino-sphere is defined by the location
where the optical depth is $2/3$ along the radial coordinate.
For this purpose,
we evaluated the optical depth by using the opacities
for the neutrinos in the forward angle bin.

In the outer layer, the neutrino distributions become rather
spherical due to an averaging effect by the integration of neutrino fluxes
from various angles
while the neutrino distributions are inhomogeneous in the central region.
This is the characteristics of the 6D Boltzmann solver
to describe the advection in 3D space.
This finding is in accord with the result found in 2D space by \citet{ott08}.
We will demonstrate characteristics of the ray-by-ray approximation in 3D
through comparisons with the 6D Boltzmann results in the following sections.

The deviations from the spherical shape even in the outer layer occur
in the case of $\phi$=141$^\circ$ due to a spot in the narrow channel
near the equator.
In this region, the neutrino-spheres of $\nu_e$ and $\bar{\nu}_e$
pass through the degenerate regime,
i.e. low temperature and high neutrino chemical potential.
This spot effect is evident in densities and fluxes
for $\nu_e$, modest for $\bar{\nu}_e$
and not discernible for $\nu_{\mu}$.
This order corresponds to the positions of neutrino-sphere
for three species from outside toward the center.
In the case of $\phi$=51$^\circ$, variations of neutrino degeneracy
around the neutrino-spheres are not drastic, therefore,
leading to rather spherical distributions in the outer layer
besides non-uniform distributions at center.
We see these two types of behavior in non-uniform situations
depending on positions and timing in the models.
The spot effect may cause enhanced/reduced neutrino distributions
and associated changes of heating rates as we will examine below.

\subsection{Density and flux by ray-by-ray evaluation}\label{section:d-ray}

In this section, we demonstrate
by comparison between the full Boltzmann transport and the ray-by-ray approximation
that the latter tends to exaggerate asymmetries outside the neutrino-sphere.
The relative difference of a quantity $Q$ between the two methods
is hereafter denoted by
$\delta_Q = ( Q_{RbR} - Q_{Boltz} ) / Q_{Boltz} $,
where $Q_{RbR}$ and $Q_{Boltz}$ are the values of $Q$
obtained by the ray-by-ray and Boltzmann evaluations, respectively.
As already mentioned,
the background models employed in the current study
were obtained with the ray-by-ray approximation and
the following results may be somewhat affected indirectly.

In Fig. \ref{fig:rbr.3db-density.xyslice}, we display
the densities of $\nu_e$ and $\bar{\nu}_e$ on the xy-plane
obtained for the 11M model at 150 ms by the two methods.
The relative difference between them
is also shown in the bottom panels.
The distribution is not spherical even in the outer region
($>$ 100 km) for the ray-by-ray evaluation.
This is the artifact of the ray-by-ray approximation.
The neutrino fluxes are mainly determined
by the environment around the neutrino-sphere.
In the ray-by-ray approximation,
the local enhancement (or suppression) of neutrino fluxes
at a particular point on the neutrino-sphere
due to variations of temperature and/or chemical potential
are propagated along the radial ray
that emanates from the point alone.
In reality, however, neutrinos move also in other directions
and affect the densities and fluxes on different radial rays.
In the absence of such a mixing among different radial rays,
the ray-by-ray approximation exaggerates the contrast
among different rays
as shown in the bottom panels.

The same features can be consistently seen
in the behavior of fluxes
shown in Fig. \ref{fig:rbr.3db-flux.xyslice}.
The distributions of fluxes in the ray-by-ray approximation
have non-spherical features in the outer region
due to enhancement or reduction along the radial rays.
In some cases, the difference of $\nu_e$ flux
has anti-correlation with that of $\bar{\nu}_e$ flux.
When the neutrino chemical potential is high
at the neutrino-sphere in one ray,
the $\nu_e$ flux is enhanced along the ray
but $\bar{\nu}_e$ flux is suppressed.
Although the neutrino fluxes for two species may depend
on the location of neutrino-spheres,
the environment and other factors,
we find that these features often appear in many profiles
of our models as we will show.

We demonstrate that the anisotropy in the ray-by-ray approximation
can be large enough to affect the densities and fluxes at large distances,
depending on the position.
Fig. \ref{fig:rbr.3db-density.iphxx.slice} shows the neutrino densities
obtained by the ray-by-ray approximation on the meridian slices,
which corresponds to the ones in Fig. \ref{fig:3db-density.iphxx.slice}.
The neutrino densities of $\nu_e$ and $\bar{\nu}_e$ are not spherical
at $\sim$100~km in contrast to the spherical shape found in the 6D Boltzmann evaluation.
In the case of $\phi$=141$^\circ$, the anisotropy around the equator
is enhanced more than that in the 6D Boltzmann evaluation.
We display the relative difference between the two methods
in Fig. \ref{fig:ratio.3db-density.iphxx.slice}.
Depending on the polar direction, there are regions of overestimation
or underestimation of densities for $\nu_e$ and $\bar{\nu}_e$
in the outer region.
The enhancement (or reduction) at the emission region persists along the ray
due to the averaged radial propagation of neutrinos.
There is anti-correlation between the two species ($\nu_e$ and $\bar{\nu}_e$)
in the differences along the ray.
This is again because the emission is determined by the neutrino degeneracy,
which enhances and reduces the flux of $\nu_e$ and $\bar{\nu}_e$, respectively.

The anisotropy of densities arises from angular variations of
the fluxes of neutrinos emitted from the region of neutrino spheres.
We show the behavior of neutrino fluxes in Figs. \ref{fig:rbr.3db-flux.iph05.slice.inx}
and \ref{fig:rbr.3db-flux.iph13.slice.inx} for the two meridian slices
at $\phi$=51$^\circ$ and 141$^\circ$, respectively.
The relative differences of the ray-by-ray evaluation from the 6D Boltzmann evaluation
are also displayed for each profile in the right panels.
One can clearly see the overestimation or underestimation of the fluxes
along the radial ray, depending on the polar angle.
The anti-correlation between the two species appears apparently
on the two slices as seen in the profiles on the xy-plane
in Fig. \ref{fig:rbr.3db-flux.xyslice}.
The pattern of differences in the flux corresponds to that in the densities
shown in Fig. \ref{fig:ratio.3db-density.iphxx.slice}.
In the case of $\phi$=141$^\circ$, the differences are large
for both species in a wide region.
The region around the equator has a significant deviation
due to the narrow channel.

We further examined the variations of the neutrino distributions
and the anti-correlation between the two species.
We show the polar angle dependence of quantities at the fixed radius
on the meridian slice at $\phi$=51$^\circ$
in Figs. \ref{fig:1d.polar.ir040.3db.iph05}
and \ref{fig:1d.polar.ir060.3db.iph05}.
The neutrino densities are well determined by the local environment
(temperature and neutrino chemical potential) in the optically thick region
at radius of 54~km in Fig. \ref{fig:1d.polar.ir040.3db.iph05}.
We see the neutrino densities follow
the variation of the neutrino chemical potential
while the anti-neutrino densities have the anti-correlated variation.
$\mu$-type neutrinos have a constant distribution
since the temperature stays almost constant.
Both the two evaluations
provide the neutrino distributions simply reflecting the environment.
The ray-by-ray evaluation shows slightly stronger variations
than the 6D Boltzmann evaluation.

Neutrino distributions become different from the variation of local environment
as we leave the the optically thick region and go further away
from the region of the neutrino-sphere.
At radius of 94 km in Fig. \ref{fig:1d.polar.ir060.3db.iph05},
the neutrino densities by the 6D Boltzmann evaluation become
more flat than the variations in the neutrino chemical potential.
This is due to the averaging effect by the non-radial fluxes.
In contrast, the neutrino densities by the ray-by-ray evaluation
show clear variations due to the degeneracy,
keeping the anisotropy produced deep inside the core.
The anti-correlation between the two species remains similar
to the situation at the inner position.
As shown in these examples,
the non-uniform distribution of neutrino chemical potential
inside the core is important to determine the neutrino distributions
in the outer region.
Note that the electron fraction is a key factor
to determine the neutrino chemical potential
even when the distributions of density and temperature are
spherical inside the core.

\section{Evaluation of energy transfer rates}\label{section:heating}

In this section,
we examine the heating rates in the supernova core
by the 6D Boltzmann solver.
Since the heating rate is a crucial quantity
to explore the neutrino heating mechanism,
we investigate also the heating rates obtained by the ray-by-ray approximation
so that we can assess possible differences due to the approximation
popularly used in multi-dimensional simulations.
We show that differences of the heating rate occur along the radial ray
due to the ray-by-ray approximation
in the same way as those found above.
The size of differences depends on the environment around the neutrino-sphere,
which provides the differences in the densities and fluxes.

\subsection{Heating rates by Boltzmann and ray-by-ray evaluations}\label{section:h-boltz-ray}

We describe here the behavior of the heating rates by the two methods
in the supernova core of 11M at 150 ms after the bounce.
We display the heating rates on the xy-plane in Fig. \ref{fig:rbr.2d.xyslice-heating.3db}
by the 6D Boltzmann (top) and ray-by-ray (middle) evaluations.
The relative differences of the ray-by-ray evaluation
with respect to the 6D Boltzmann evaluation
are also shown in the bottom panel.
While the profiles of heating region in red are similar each other,
there are two spots with enhanced heating rates in the ray-by-ray evaluation.
Along the two directions, the ray-by-ray approximation leads to
the overestimation of $\sim20\%$.
This enhancement corresponds to the enhanced densities and fluxes
shown in Figs. \ref{fig:rbr.3db-density.xyslice} and \ref{fig:rbr.3db-flux.xyslice}
and is caused by
the inhomogenous distributions of neutrino degeneracy
with the {\it hot spot} for neutrino emissions as discussed above.
There are also regions of
minor differences along some radial directions.
Note that large differences near the gain radius in the central part
appear since the energy transfer rate is very small there.

Figure \ref{fig:rbr.2d.phslice-heating.3db.iphx} displays
the profiles for the neutrino-heating rate
on the meridian slices at $\phi$=51$^\circ$ and 141$^\circ$
corresponding to Fig. \ref{fig:rbr.2d.xyslice-heating.3db}.
The heating rates by
the 6D Boltzmann (left) and ray-by-ray (middle) evaluations
are shown along with the relative differences of the ray-by-ray evaluation (right).
In the case of $\phi$=51$^\circ$, there is no noticeable difference
in the heating rates between the two methods
except for a slight underestimation along two directions.
A significant overestimation
is seen around the equator in the case of $\phi$=141$^\circ$.
This corresponds to the one of the azimuthal directions with overestimation
seen in Fig. \ref{fig:rbr.2d.xyslice-heating.3db}.
The overestimation in the ray-by-ray approximation occurs
due to the enhanced neutrino fluxes in the narrow channel.
There are also wide areas with overestimation
and a radial direction with underestimation in the heating region.

While the differences of the {\it total} heating rate appear
in the limited directions,
those of each heating rate for major reactions
generally occur in the whole region.
In Figure \ref{fig:rbr.2d.phslice-heating.3db.iph05.reactions},
we show the heating rates and their relative differences between the two methods
for $\nu_e$ and $\bar{\nu}_e$ absorptions on nucleons
on the meridian slice at $\phi$=51$^\circ$.
One can see the pattern of enhancement or reduction of the individual heating rate
just in the same way as those in the densities and fluxes.
On the one hand,
significant differences with an alternate sign
appear along the radial ray
for the $\nu_e$ absorption in the upper panel.
This pattern arises from the differences of $\nu_e$ fluxes
seen in Fig. \ref{fig:rbr.3db-flux.iph05.slice.inx}.
On the other hand,
differences of the similar size for the $\bar{\nu}_e$ absorption
appear in the same pattern but with a different sign.
This pattern reflects the corresponding pattern for $\bar{\nu}_e$
in Fig. \ref{fig:rbr.3db-flux.iph05.slice.inx}.
Due to the production mechanism of the fluxes for two species,
the overestimation and underestimation of the heating rates
arise in the opposite way.
Therefore, the total heating rates do not have
large differences in most of the regions due to cancellation.
This mechanism of the cancellation of the two heating rates
is generally seen in our analyses of the models.

The anisotropy of heating rates
in the ray-by-ray approximation may have influences
on the hydrodynamics in multi-dimensions.
The enhancement of heating beamed along a certain direction
may lead to the modification of shock dynamics.
In contrast, the 6D Boltzmann evaluation provides
mildly anisotropic distributions of the neutrino heating
by the integration from various angles.
We remark here that the anisotropy in the ray-by-ray approximation
is found in the fixed hydrodynamics.
It is necessary to examine whether these features persist
for a long time scale in dynamical situations
by neutrino-radiation hydrodynamics (See also \S \ref{section:h-total.time}).

\subsection{Heating rates in other profiles}\label{section:h-boltz-ray.others}

We explore generic characteristics of the heating rates by examining the profiles
at different timing and models.
We demonstrate that the characteristic differences along the radial ray
due to the ray-by-ray approximation
occur also in the other cases.
Their magnitude and sign, however, vary depending on the environment.

\subsubsection{Different snapshots of 11M}\label{section:h-boltz-ray.others.11M}

The heating rates at 100 ms after the bounce by the 6D Boltzmann evaluation
are shown in
Figs. \ref{fig:2d.xyslice-heating.3da}
and \ref{fig:rbr.2d.phslice-heating.3da.iphx}
together with the relative differences of the ray-by-ray evaluation.
One can see, in Fig. \ref{fig:2d.xyslice-heating.3da},
the deformed shape of the heating region and
the underestimation of the heating rate
in the ray-by-ray evaluation
widespread on the xy-plane,
being independent of $\phi$-direction,
with a fluctuating size.
In Fig. \ref{fig:rbr.2d.phslice-heating.3da.iphx},
the overestimation of the heating rate
occurs around the z-axis within a wide polar angle
while the underestimation is confined around the equator.

The pattern of differences looks differently,
depending on the environment.
We show the case
at 200 ms in Figs. \ref{fig:2d.xyslice-heating.3dc}
and \ref{fig:rbr.2d.phslice-heating.3dc.iphx}.
The deviations alternately appear
depending on the $\phi$-direction on the xy-plane
as seen in Fig. \ref{fig:2d.xyslice-heating.3dc}.
They depend on the polar angle and
there are large variations on the meridian slice
in Fig. \ref{fig:rbr.2d.phslice-heating.3dc.iphx}.
While the underestimation with a large magnitude
appears in the limited region along a certain direction
in the case of $\phi$=51$^\circ$,
significant overestimation appears
at two polar directions
on the meridian slice at $\phi$=141$^\circ$.
These beam-like structures resemble the situation
in the case of $\phi$=141$^\circ$ at 150 ms.
In fact, the neutrino degeneracy around the neutrino-sphere
is locally high at the two spots, which lead to high neutrino
fluxes and, therefore, heating rates.
These spots are located around a narrow channel
between the two mushroom-like flows
like the case for 150 ms.

\subsubsection{Snapshots of 27M}\label{section:h-boltz-ray.others.27M}

We examine the heating rate for the 27M models
in order to find the generic characteristics of the ray-by-ray approximation.
In contrast to the 11M models considered so far,
the shock wave in this model remains stagnated and
does not show any hint of shock revival
until the end of the simulation ($\sim$700 ms),
which is consistent with other group's results \citep{han13,tam14}.
This model is a representative of those models,
which seem to be most difficult for the neutrino heating to produce explosions,
and will hence serve our purpose here.

We found that
differences between the ray-by-ray approximation and
the full Boltzmann treatment are no less significant
in this case.
We show the heating rates
as well as the relative differences
at 150 and 200 ms after the bounce
in Figs. \ref{fig:2d.xyslice-heating.3d.s27}
and \ref{fig:rbr.2d.phslice-heating.3d.s27.iph05}.
The upper panels display the heating rates
obtained by the 6D Boltzmann solver on the xy-plane
and the meridian slice at $\phi$=51$^\circ$.
The lower panels present
the relative differences of the ray-by-ray evaluation.
One can see
that the differences are conserved along radial rays,
i.e., over- and underestimations occur as radial beams
just as in the 11M models.
The deviations are substantial in the heating region,
particularly near the gain radius,
but remain beyond the shock wave located at $\sim$200 km.
We confirm that this behavior
is generic and is observed for other slices.
When there is a hot spot on the neutrino-sphere that
the down flow of proto-rich matter hits, then
the ray-by-ray approximation tends to
propagate the local enhancement in the neutrino density
along the radial ray that emanates from the spot.
These generic features of the ray-by-ray approximation
may have non-negligible influences on the revival of the stalled shock wave
and should be carefully assessed further.

\subsection{Volume-integrated heating rates}\label{section:h-total}

Since we found that the heating rates are locally different,
it is of great importance to examine the total amount of
the heating rate
by the 6D Boltzmann solver and the ray-by-ray approximation.
We evaluate the volume-integrated heating rates in the two evaluations
to check this point.
The specific heating rates,
which we have discussed in the previous sections,
are defined by
\begin{equation}
q_{\nu} = - \int d\Omega E_{\nu}^2 dE_{\nu}~E_{\nu}
\left( \frac{\delta f}{\delta t} \right)_{collision} ~,
\label{eq:spec.heat}
\end{equation}
with the collision term in the Boltzmann equation.
The total amount of the heating rate is calculated by the integral
over the volume in the heating region,
\begin{equation}
Q_{\nu} = \int dV q_{\nu} \rho ~,
\label{eq:tot.heat}
\end{equation}
where $\rho$ is the mass density.
The range of the integral covers the heating region,
where the energy transfer is positive.
We set the inner boundary of the heating region
at the gain radius, where $q_{\nu}$ becomes zero along the radial coordinate.
We define the outer boundary by the location of the shock wave
when the entropy per baryon becomes
5.5 in the unit of the Boltzmann constant, $k_B$.

\subsubsection{Snapshots}\label{section:h-total.snapshots}

In Table \ref{table:heating},
we compare the volume-integrated heating rates
for the 11M and 27M models
in the two evaluations.
The volume-integrated heating rates in the ray-by-ray approximation
are slightly smaller than the values in the 6D Boltzmann evaluation.
The difference is within $\sim$2$\%$ for 11M and less than 1$\%$ for 27M.
One may wonder the reason why
the volume-integrated heating rate by the ray-by-ray approximation
is not significantly different from the one
by the 6D Boltzmann solver.
This is because the overestimation and underestimation of the heating rates
in the local regions along the radial ray
cancel out by the integral over the solid angle.
The current analysis suggests that
the local deviations in the ray-by-ray approximation
are more influential than the total deviations.
It should be noted, however, that
even a slight difference in the heating rate
can be crucial to the revival of the shock wave
in the marginal situation \citep{jan96}.
It is necessary to study this influence further
by examining the hydrodynamical evolution.

\subsubsection{Time variations}\label{section:h-total.time}

So far we have discussed the features of stationary neutrino distributions
obtained for snapshots that are mutually separated in time by 50 ms.
One may argue that rapid temporal changes in matter distributions might erase
some of those features by averaging.
In order to check this, we perform another series of simulations.
We adopt consecutive snapshots from the period of 145 ms to 155 ms
with an interval of 1 ms
from the same supernova simulation of 11.2M$_{\odot}$ star;
there are hence 11 snapshots including the one at 150 ms,
which we have already discussed;
we then follow the time evolution of neutrino distributions
until they become stationary for each time slice and
take the average of the results over these 11 simulations.
In this way we can judge how sensitive our findings are
to the short-time variations of the background.

We show the temporal evolution of the total heating rates
during the 10 ms in the two evaluations
in Fig. \ref{fig:deviation-heating.tslice}.
In both cases,
there is a gradual decrease owing to
the change in hydrodynamical conditions.
One can see that
the ray-by-ray approximation provides systematically smaller rates
than the 6D Boltzmann evaluation over this period.
The deviation of the ray-by-ray evaluation
stays around $-2\%$.
This result suggests that
the underestimation by the ray-by-ray approximation is
not an artifact of the fixed-background models,
but a generic feature originated from
the nature of the approximation itself.
In fact, we find that the local deviations over 20$\%$,
which are seen in Figs. \ref{fig:rbr.2d.xyslice-heating.3db}
and \ref{fig:rbr.2d.phslice-heating.3db.iphx},
are also observed in this case and last for $\sim$10 ms.

\section{Angle moments}\label{section:moments}

Various moments of the neutrino distribution function are important
to reveal the 3D features of the neutrino transfer.
They are also useful to develop new methods for the closure
relation in the moment formalism (see \citet{shi11} for example).
In principle, we can derive various moments with respect to angle and energy
from the obtained neutrino distribution function in six dimensions.
As an example, we show the contour maps of
the angle moments for three species of neutrinos
in Figs. \ref{fig:3db-mom.mu.iph05.slice} and \ref{fig:3db-mom.mu.iph13.slice}
on the meridian slices at $\phi$=51$^\circ$ and $\phi$=141$^\circ$, respectively,
for the 11M model at 150 ms after the bounce.
The spatial distributions of both the first angle moment or
the flux factor, $\langle \mu_{\nu} \rangle$,
and the second moment, $\langle \mu_{\nu}^2 \rangle$,
are almost spherical with a slight deformation in the central part.
Some bumps are seen in the narrow region near the equator
on the meridian slice at $\phi$=141$^{\circ}$.
The asymptotic behavior of these angle moments at large radii
is consistent with those obtained in
the previous studies on the flux factor \citep{mes98,yam99}
and on the basic validation of the 6D Boltzmann code \citep{sum12}.

We further examine the angle moments
obtained by the ray-by-ray approximation
through comparisons with the 6D Boltzmann evaluation.
In Figs. \ref{fig:ratio.3db-mom.mu.iph05.slice} and
\ref{fig:ratio.3db-mom.mu.iph13.slice},
we show the relative differences of the angle moments
evaluated by the ray-by-ray approximation
with respect to the 6D Boltzmann evaluation
shown in Figs. \ref{fig:3db-mom.mu.iph05.slice} and \ref{fig:3db-mom.mu.iph13.slice}.
Appreciable deviations of the angle moments arise in some regions
where non-radial fluxes are large in the neutrino distributions.
The deviations become large up to $\sim10\%$ and
is comparable to those of the other quantities we have seen.
The deviations are very small within $\sim3\%$ for $\langle \mu_{\nu}^2 \rangle$
on the meridian slice at $\phi$=51$^{\circ}$ where the distribution is almost spherical.
On the meridian slice at $\phi$=141$^{\circ}$,
the deviation is significant around the narrow channel near the equator
due to the hot spot effect.
Alternate deviations of underestimation and overestimation
appear depending on the radial direction.

It is profitable to note that
we can examine all (diagonal and non-diagonal) elements
of the Eddington tensors by the 6D Boltzmann solver.
Components of the Eddington tensor show
interesting features with non-spherical shapes in some cases.
The diagonal elements of the Eddington tensor
for the energy bin ($E_{\nu}=$ 34 MeV)
of two species of neutrinos ($\nu_e$ and $\bar{\nu}_e$) are shown
in Figs. \ref{fig:3db-eddi.iph05.slice.inx}
and \ref{fig:3db-eddi.iph13.slice.inx}
on the two meridian slices
for the 11M model at 150 ms.
In the case of $\phi$=51$^{\circ}$,
the distributions of the three components exhibit deformed shapes
in the outer region ($\sim$250 km).
In the central part,
they have spherical shapes for $\nu_e$
and non-spherical ones for $\bar{\nu}_e$,
reflecting their distributions.
In the case of $\phi$=141$^{\circ}$,
the non-spherical features of the distribution are prominent
in a wide region for the two species.
The bumpy structure reflects the non-uniform shape of the central core,
which has the narrow channel between the two mushrooms.

In usual situations with nearly spherical shapes,
only the diagonal elements are dominant
with zero non-diagonal elements.
We found that there are some regions
where the non-diagonal elements are non-negligible.
In Fig. \ref{fig:3db-eddi.subd.iph13.slice.in1},
we show, as an example, the color maps of the non-diagonal
($r \theta$), ($r \phi$) and ($\theta \phi$)-components
for the energy bin ($E_{\nu}=$ 34 MeV) of $\nu_e$
on the meridian slice at $\phi$=141$^{\circ}$ in the 11M model at 150 ms.
Their magnitude amounts to $\sim$0.1 and is not negligible
around the channel along the equator.

These non-trivial properties of the Eddington tensors are
valuable information to express the anisotropy of neutrino distribution functions
in 3D supernova cores.
Since the neutrino distribution function is symmetric
around the radial coordinate in the ray-by-ray approximation,
only the three diagonal components are non-zero and
the $rr$-component determines
the other two $\theta\theta$ and $\phi\phi$-components.
Hence, the ray-by-ray approximation loses the information
of the non-trivial structure of the Eddington tensor.
Further studies on the detailed features of the Eddington tensors
obtained by the 6D Boltzmann solver
are underway to develop new closure relations for the 3D problems
as a separate article.

\section{Discussions}\label{section:discuss}

We discuss here some possible implications
of our finding for
the supernova mechanism and associated phenomena.

The 6D Boltzmann solver provides
a seamless description of the neutrino transfer
from optically thick to thin regions
in 3D space.
Non-radial neutrino transfer occurs in general
up to large distances from the center beyond the neutrino-sphere,
which can have influences
on the convective motions of neutrino-rich matter
inside the proto-neutron star
as well as the emission properties
around the neutrino-sphere.
These may in turn affect the neutrino heating
in the gain region and ultimately the revival of shock wave.

Although the diffusion approximation can describe
the non-radial transfer of neutrinos in the optically thick regime,
the transition to the transparent regime is handled
by a prescribed flux limiter,
which is not easy to justify \citep{jan92,mes98}.
The non-radial neutrino flux
at large radii in the semi-transparent and transparent regimes,
which we have seen above,
can not be properly described by the diffusion-based approximation,
since the flux in these regimes is
not determined by the local gradient of neutrino density
but is obtained non-locally
as a superposition of neutrinos flying in different directions
from distant points.
The 6D Boltzmann treatment is a unique solution
to treat such circumstances rigorously.
It is, of course, more desirable to conduct
the neutrino-radiation hydrodynamics simulations,
which will be a future work, though.

A recent study by the multi-energy group flux-limited diffusion approximation in 2D
reported non-explosions for 12--25M$_{\odot}$ stars \citep{dol14},
being different from the successful 2D simulations
by other groups \citep{suw10,mul12b,bru13,knak14a,suw14b}.
This example again casts a light on uncertainties
due to different approximations in neutrino transfer,
which might lead to totally different outcomes.
The 6D Boltzmann code
will be helpful to decipher the discrepancies
among them.
(See \citet{lie09} for the validation of IDSA in 1D.  )

The comparison of the ray-by-ray approximation
with the 6D Boltzmann evaluation
is beneficial for the supernova research,
since this approximation is routinely used
for the currently state-of-the-art numerical simulations
in 2D and 3D spatial dimensions
but its validation is rather scarce.
While the ray-by-ray approximation is an efficient way
to handle the neutrino transfer in multi-dimensions,
it inevitably neglects some of the non-radial neutrino transport
(See \citet{bura06} for partial coupling)
and, as a consequence, tends to enhance artificially
the contrast among different radial rays.
These features found by the 6D Boltzmann solver in this paper
are in accord with the comparison in 2D
between the Boltzmann transport and the flux-limited
diffusion approximation
by \citet{ott08} and \citet{bra11}
(See also \citet{dol14} for the discussions with a formal solution).
It remains to be seen if this effect persists
in 3D dynamical simulations.

If the artificial enhancement of neutrino anisotropy
lasts for a long period, it may have impact
on the hydrodynamics.
We have actually seen that it lasts at least for $\sim$10 ms.
A larger neutrino-heating rate
along a certain radial ray will give a
strong push to the stalled shock wave locally in that direction.
The opposite situation
can arise in another direction,
where neutrino heating is suppressed erroneously.
The ray-by-ray approximation,
therefore, may enhance the deformation of shock wave
and might induce stronger hydrodynamical instabilities.
In the 6D Boltzmann treatment, such anisotropies tend to be
smoothed by the non-radial transport.
It is advisable to pay an appropriate attention
to this effect when one studies
the anisotropy of neutrino distributions produced by
local matter inhomogeneities.

It remains to be seen whether the improvement by the 6D Boltzmann treatment
indeed changes
the behavior of hydrodynamical instabilities
such as SASI \citep{blo07a,iwa08,nor10,han12,tak12,han13}.
Very recently \citet{tam14} reported with the ray-by-ray approximation
a new instability named
Lepton-number Emission Self-suitained Asymmetry (LESA),
which produces self-sustained, globally dipolar configurations.
Note, however, that \citet{dol14} have found no such instability
with the flux-limited diffusion approximation.
LESA may hence be an artifact of the approximation.
In fact, since it is claimed that the global coupling of anisotropy
between the neutrino-sphere and shock wave is the key to the instability,
accurate evaluations of the neutrino anisotropy may be crucial.
It is hence interesting to see what the Boltzmann transport would produce.
It may be that eventually
only by performing full neutrino-radiation hydrodynamics
simulations with Boltzmann transport can we disentangle this new twist.

Our results suggest that the errors by
the ray-by-ray approximation may be moderate
for most of the cases.
This will be a good news
for the supernova research in 3D,
since the current supercomputing resources are limited
and allow only the 3D simulations with an approximate neutrino transfer.
In future, however, it is necessary to remove
even such moderate uncertainties.
In fact, even a $\sim10\%$ error in the heating rate
may have an impact on the revival of shock wave particularly
when it is marginal.

The neutrino distributions
in the transparent regions
are determined
not locally but globally as a superposition of neutrinos
flying in different directions from various distant points.
The Boltzmann solver can fully take this into account
in sharp contrast to other approximate methods.
The neutrino spectra
are also affected by such non-local features in neutrino transfer,
reflecting thermodynamical states not just at a single point
but in a wide region on the neutrino-sphere.
It may hence be possible that the Boltzmann solver reveals
new features in the neutrino spectra.
It would then be also interesting to examine
its influences
on the nucleosynthesis of heavy elements
via neutrino processes.

This will be particularly the case when
the proto-neutron star is strongly deformed
due to rapid rotation.
Since the core is globally asymmetric,
the resulting fluxes are intrinsically non-radial.
Such an effect was pointed out by \citet{ott08}
in the case of a rapidly rotating 2D supernova core.
Its observational consequences were studied \citep{bra11}.
In the case of 3D
the neutrino-sphere may be globally non-axisymmetric
if the core rotates very rapidly.
It would be interesting then to study
its possible observational implications with
the 6D Boltzmann transport.

The main limitation of the current Boltzmann code is
the angular resolution
in describing the forward-peaked neutrino distribution functions
in the optically thin region.
A large number of grid points would be required
if one were to fully resolve such configurations,
which is simply unaffordable.
Fortunately, the area, which we
are most concerned with and where the Boltzmann solver is most needed,
is the region from beneath the neutrino-sphere
up to the stalled shock wave and
the neutrino distribution function is not so forward-peaked yet there;
the moderate angular resolution we can afford
will be acceptable particularly
for the evaluations of heating and cooling rates
although detailed convergence tests are certainly necessary.
If one wants to obtain the neutrino luminosities and spectra
at different viewing angles quantitatively,
one may need a finer angular mesh, which may
require prohibitive computing resources
in the current approach.
(See, however, \S 4 in \citet{kot12} for the scaling of computational load.  )
It would then be necessary to develop alternative techniques
such as the variable angle mesh \citep{yam99}
or
the Monte Carlo method \citep{abd12}.

Although
the current study employs only
the snapshots from the 3D supernova simulations
with the ray-by-ray IDSA \citep{tak12,tak14,hor14,tak14b},
it is desirable to investigate
other background models, since
the neutrino transfer
may be somewhat affected
indirectly by the approximation employed in the supernova simulations.
It will be also preferable to
study the influences of the numerical grid employed
(ex. spherical-polar vs 3D-cartesian as well as its resolution)
and progenitor models.
World wide collaborations will be indispensable
in order to fully understand the 3D neutrino transfer.
It may become possible then
to choose suitable tool for a given problem
(See \citet{ili06} for the cosmological radiative transfer codes,
for example).
The validation of the approximate methods currently used
in 3D supernova simulations will be
a necessary first step in that direction.

The Boltzmann solver can be combined with the magneto-hydrodynamical code
with no problem to study magneto-hydrodynamics scenarios of core-collapse supernovae
in more realistic settings.
The application will not be limited to the supernova.
The neutrino transfer in the collapsar,
which originates from more massive stars, for example, will be
important for the jet formation in the gamma ray burst.
The Boltzmann transport is particularly suitable
to study such highly anisotropic and non-local problem.
In particular information
on the angle moments of neutrino distribution functions
will be very helpful to develop a closure relation
for the moment formalism, another approximate method.
These issues are now under investigation
with the 6D Boltzmann solver and will be reported elsewhere.

Improvements of
the current 6D Boltzmann code
are certainly needed and may somewhat change the results.
Among other things, the implementation of
the modern reaction rates
instead of the conventional set is mandatory
to investigate the post-bounce dynamics quantitatively further
as demonstrated in the previous papers \citep{bura06,len12a}.
The modifications of the emission and absorption rates
affect directly the cooling and heating rates.
Addition of inelastic scatterings of neutrinos will
modify their energy spectra and, as a result, anisotropy as well.
Even the electron captures on nuclei
affect the post-bounce dynamics by setting the initial stage.

The proper account of relativistic corrections is another issue
toward the ultimate realism \citep{lie04,sum05,bura06,len12b}.
Implementations of the special relativistic effects
such as the Doppler shift and aberration
have been actually done recently \citep{nag14}.
Combined with a Newtonian hydrodynamics code,
it is now applied to the 2D supernova simulation and
the results will be reported elsewhere.
The incorporation of general relativistic corrections to the present code
is also currently underway (See also \citet{car13b,shi14}).
Its coupling to a general relativistic hydrodynamics code and
a solver of the Einstein equation in 3D will be a grand challenge.

\section{Summary}\label{section:summary}

We have explored the 3D properties of the neutrino transfer
in supernova cores by solving the Boltzmann equation
in 6 dimensions (3 in space and 3 in momentum space).
We have performed the numerical simulations to obtain
the stationary state of neutrino distribution functions
in 6D phase space
for the fixed backgrounds of the 3D supernova core.
We used the typical profiles of the supernova core
taken from the 3D supernova simulations of
11.2M$_{\odot}$ and 27.0M$_{\odot}$ stars
by \citet{tak12,hor14,tak14b}.
The numerical code to solve the Boltzmann equation in 6D
handles the neutrino transfer
in multi-energy and multi-angle group for three neutrino species.
The three dimensional propagation of neutrinos is described
by the S$_n$ method.
The standard set of neutrino reactions is implemented
in the collision term together with the table of equation of state.

This is the application of
the 6D Boltzmann solver to the realistic profiles
of the 3D supernova cores for the first time.
The 6D Boltzmann equation directly
provides the energy and angle distributions
of neutrinos in the whole region of the 3D supernova cores.
Most importantly, the transport of neutrinos
in the multi-angle description
provides the information of the non-radial
(polar and azimuthal) directions as well as the radial one.
We have demonstrated that the 6D Boltzmann solver describes
the 3D features of neutrino transfer,
removing the approximations often used
in the currently state-of-the-art simulations.
The numerical results show that the non-radial fluxes
are generally seen inside the proto-neutron star
and can extend over the wide region beyond the optically thick region.

We find that the ray-by-ray approximation, which is a popular
approximation in the current supernova simulations, provides
enhanced directional variations in neutrino distributions.
We have computed the 3D neutrino transfer in the ray-by-ray approximation
by using our numerical code,
dropping off the non-radial transport terms.
We have examined the characteristics of the ray-by-ray approximation
for the same 3D profiles
through the comparison with the 6D Boltzmann evaluation.
In the ray-by-ray approximation,
the non-spherical profiles of neutrino densities and fluxes
tend to prevail from center to large distances.
This is in marked contrast to the 6D Boltzmann evaluation
in which the integral from many directions provides
the nearly spherical distribution
outside the proto-neutron star.

Our analysis has revealed that the ray-by-ray evaluation
provides noticeable overestimation or underestimation
of densities and fluxes along the radial ray.
By examining the relative differences of the ray-by-ray evaluation
with respect to the 6D Boltzmann evaluation,
the deviations of $\sim20\%$ was found to
develop depending on the environment.
Once the neutrino emission is enhanced due to a local
hot spot of the neutrino chemical potential,
this modification continues along the radial ray.
The reduced neutrino emission is similarly maintained.
Therefore, we find alternate distributions of
the rays with underestimation and overestimation
outside the proto-neutron star.
This behavior is seen commonly
for $\nu_e$ and $\bar{\nu}_e$ distributions,
which reflect the non-uniform distribution
of the neutrino degeneracy.
The sign of the deviations for the two species
is opposite each other
because of the opposite effects on the emission
through the degeneracy.

We have also found that
the heating rates evaluated by the ray-by-ray approximation
have deviations along the radial ray.
The relative differences from the 6D Boltzmann evaluation amount to
$\sim20\%$ in some regions but smaller than
the case of the densities and fluxes in wide regions.
This is because the overestimation and underestimation
of the heating rates by $\nu_e$ and $\bar{\nu}_e$
cancel each other due to their opposite deviations
in many cases.
Significant deviations occur in the regions
where the neutrino degeneracy
is prominent from the average distribution, for example,
in the narrow channels between the two mushroom-like outflows.
This hot spot provides the enhanced heating rate
along the particular radial ray.
The hot spot effect may last for $\sim$10 ms
as found in our study using the time series of the profile.
We have found that
the total amount of heating in the whole heating region
in the ray-by-ray evaluation
is slightly smaller than that in the 6D Boltzmann evaluation.
These features of the ray-by-ray characteristics are
common for the adopted profiles from
the 11.2M$_{\odot}$ and 27.0M$_{\odot}$ stars,
which represent the exploding and stalled shock situations.
Our study calls for careful assessment of the angle-dependent
heating effects which may affect the hydrodynamics.
We remark that these features are found
in the fixed background profiles and
further studies in neutrino-radiation hydrodynamics
are necessary.

We have examined the basic features of the angle moments
and the Eddington tensor in the 3D supernova cores.
We show that the angle moments of the neutrino
distribution functions obtained by the 6D Boltzmann solver
have rather spherical profiles.
The relative differences of the ray-by-ray evaluation
appear in some regions, but limited within the inner part.
The components of the Eddington tensor
exhibit deformed distributions in more wide regions
than the case of angle moments.
Since we have the 6D neutrino distribution functions,
we can compute all components of the Eddington tensor
and clarify the behavior of the diagonal and non-diagonal elements.
We have demonstrated that
there are regions
with appreciable non-diagonal components
in one of the snapshots as an example.
Studies of
these basic quantities will be helpful to develop
new closure relations for the moment formalism
in largely deformed profiles such as collapsars.

The current study is a step toward the complete treatment
of neutrino-radiation hydrodynamics
implementing the full list of neutrino reactions
in the general relativistic framework.
It is of great interest how the 6D Boltzmann treatment
affects the neutrino heating mechanism
in hydrodynamical instabilities.
The coupling of the 6D Boltzmann solver
with the hydrodynamics is made recently \citep{nag14}
and numerical studies
to explore the dynamical outcome for explosions
are currently underway.

\section*{Acknowledgments}

We are grateful to K. Kotake, H. Suzuki
for the collaborations on supernova simulations.
K. S. thanks to H. Nagakura, W. Iwakami,
K. Nakazato, S. Furusawa, Y. Sekiguchi, K. Kiuchi and N. Ohnishi
for collaborative works and profitable discussions.
K. S. expresses his thanks to T.-H. Janka, C. Ott and A. Mezzacappa
for valuable comments
on the numerical methods of neutrino transfer in 2D and 3D.
We thank A. Imakura and T. Sakurai
for the project of computational science
for sparse matrix solvers
and the supporting team of HITACHI
for tuning the numerical code on SR16000.

The numerical computations in this work were mainly performed
on the supercomputers
at High Energy Accelerator Research Organization (KEK)
under the support of its Large-scale Simulation Program
(Nos.12/13-05, 13/14-10, 14/15-17).
K. S. acknowledges also the extensive usage of the supercomputing resources at
Yukawa Institute for Theoretical Physics (YITP) in Kyoto University,
The University of Tokyo, K-computer
and
Research Center for Nuclear Physics (RCNP) in Osaka University.
This research is supported by the computational resources of the K computer
provided by the RIKEN Advanced Institute for Computational Science
through the HPCI System Research project (Project ID:hp130025, hp140211).
Large-scale storage of numerical data
is supported by the JLDG constructed over the SINET4 of NII.

This work is partially supported by
the Grant-in-Aid for Scientific Research on Innovative Areas (Nos. 20105004, 20105005, 24103006)
and
the Grant-in-Aid for the Scientific Research (Nos. 19104006, 21540281, 22540296, 24244036, 26870823)
from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) in Japan.

This numerical study on core-collapse supernovae using the supercomputer facilities
is supported by the HPCI Strategic Program of MEXT, Japan.

\bibliographystyle{apj}

redpajama_set_name: RedPajamaArXiv


text:
Visitors Zebra Mussels
What are Zebra Mussels?
Zebra Mussels are an aquatic invasive species originating from Eastern Europe and Western Asia. Zebra Mussels are impossible to get rid of once they have established themselves in a body of water, which why it is vital that we prevent them from entering our waters. One female Zebra Mussel lays around one million eggs per year, and Zebra Mussels have an average lifespan of 2-5 years. Adult Zebra Mussels are between 1-3cm large and attach themselves to anything submerged in water. They can survive for 7-30 days outside of water depending on temperature and humidity. Larval Zebra Mussels, otherwise known as veligers, are water dependent and move through bodies of water with the underwater currents.
What are Aquatic Invasive Species (AIS)?
An aquatic invasive species (AIS) is an organism that is not native to the local environment and spreads rapidly due to high reproductive rates and a lack of natural predators or diseases that lives in either freshwater or marine environments. Most of the species of concern to Manitoba are freshwater invasive species. Manitoba currently has 17 aquatic invasive species.
In Manitoba, Zebra Mussels are designated as an Aquatic Invasive Species (AIS). It is illegal to:
Possess a member of an AIS in Manitoba
Bring a member of an AIS into Manitoba, or cause one to be brought into Manitoba
Deposit or release an AIS in Manitoba, or cause one to be deposited or released into Manitoba
Transport a member of an AIS into Manitoba
How to Stop the Spread of Zebra Mussels
When transporting any watercraft between bodies of water, it is crucial that you CLEAN the watercraft thoroughly, DRAIN any standing water left inside the watercraft, and DRY your watercraft completely before introducing it to another body of water. DISPOSE of any unwanted bait in the trash, rather than dumping it in the water. Manitoba operates several AIS Watercraft Inspection Stations, and all watercrafts must stop at these stations when they are open. If you have recently brought your watercraft into water that is known to contain Zebra Mussels, it is the law that you must stop and have it decontaminated at one of the decontamination stations.
Why Do We Need to Be Concerned About Zebra Mussels and Aquatic Invasive Species?
Zebra Mussels are highly adaptable and pose a significant environmental and economic concern to Manitoba. They disrupt ecosystems and decrease the numbers of our native marine life; they attach to other mussels and smother them. They are also known to attach to boats and motors thus reducing efficiency, and their shells attach to rocks and ladders and can cut the feet of swimmers.
Are Zebra Mussels in Our Waters?
So far, Zebra Mussels have not been found in our municipality's lakes and rivers. However, with the recent discovery of Zebra Mussels spreading to more Manitoba lakes, it is more important than ever that we continue to be cautious and thoroughly clean, drain, and dry anything that comes out of or into our waters. A list of invaded lakes and rivers in Manitoba can be found Here. For more information, please visit https://www.gov.mb.ca/stopais/zebra-mussels.html
Follow the guide below to practice safe watercraft transport.
Don't Move a Mussel.pdf

redpajama_set_name: RedPajamaCommonCrawl


text:
← Ex-Hot Rods hurler gets win in first MLB start
2016 Snappers pitcher wins MiLB Gold Glove Award →
Former Snappers infielder Bostick traded to Pirates
Posted on September 26, 2016	by Craig Wieczorkiewicz
Chris Bostick lays down a bunt for the Beloit Snappers during the 2013 season. (Photo by Dave Baker)
The Pittsburgh Pirates acquired former Beloit Snappers infielder Chris Bostick from the Washington Nationals for catcher Taylor Gushue and cash considerations Monday.
Bostick played 129 games for the 2013 Snappers, batting .282 with 14 HR, 89 RBI and 25 SB. He split this season between Washington's Double-A and Triple-A affiliates, cumulatively batting .250 with 8 HR, 51 RBI and 11 SB in 135 games.
Gushue did not play in the Midwest League. He batted .226 with 8 HR and 38 RBI in 90 games with Pittsburgh's High-A affiliate this season.
Torres makes MLB debut
Former Snappers reliever Jose Torres made his major-league debut for the San Diego Padres on Thursday, limiting the San Francisco Giants to one hit in a scoreless 9th inning. He struck out two batters.
Torres pitched for the Snappers in 2015, going 4-5 with a 2.69 ERA, eight saves, 80 strikeouts and 23 walks in 73.2 innings (44 games).
Torres is the 200th former Beloit player to reach the major leagues. The 2016 season was Beloit's 35th year in the Midwest League.
This entry was posted in Beloit Snappers, Midwest League. Bookmark the permalink.


If you have an up-to-date computer, a high-speed Internet connection, a dedicated landline, a quiet place to work and a pleasant personality, you may have what it takes to work as an online customer service representative. Depending on your employer, you'll be answering calls and taking orders, or tracking existing orders. You may be asked to troubleshoot, solve technical problems, and even respond to online chat sessions or e-mails.
While many of these customer service jobs were outsourced in the past decade, the good news is many positions are coming back to the United States as companies realize there is an affordable, knowledgeable employee base here. Some companies hire directly, while others use virtual call center operators who hire and manage a team of at-home workers [source: Hannon, AARP].


They can survive in just about any weather conditions.
Learn how to grow your own organic kale so you can harvest leaves right through the winter months.
When life gives you lemons, make trees! Even during winter and autumn, a productive lemon tree can be growing inside of your home.
Not sure what to plant in your autumn veggie patch? Why not give some snow peas a go!
Are your cabbages being demolished by caterpillars and your strawberries eaten by slugs? We have you covered with our DIY guide to natural pest control.


text:
Exchanges settle patent suit
Kelly Quigley
The Chicago Mercantile Exchange and the Chicago Board of Trade each have agreed to pay $15 million to New York-based eSpeed Inc., an online bond trader, to resolve a long-standing patent infringement lawsuit.
A subsidiary of Cantor Fitzgerald L.P., eSpeed claimed that the Chicago exchanges violated its patent for an electronic trading platform with a computerized system that instantly matches bids and offers. ESpeed acquired the patent this spring from Dallas-based Electronic Trading Systems Corp., and in doing so assumed the litigation filed in 1999.
"We made a business decision to settle the case and move on," Chicago Mercantile Exchange (CME) Chairman Terry Duffy said in a joint statement from the exchanges.
The out-of-court settlement announced Tuesday releases the Chicago exchanges from legal claims without admitting liability.
CME said it will recognize the eSpeed settlement, equivalent to about 28 cents per share, in the third quarter of 2002. It will make an initial payment of $5 million in the third quarter, and five subsequent annual payments of $2 million.
The Chicago Board of Trade (CBOT) also said it will recognize the cost of the settlement in the third quarter, but did not provide details on its payment plan to eSpeed.


text:
Q: How to communicate between JS and Python? I have made a Discord bot in which I wanted to add the Oxford Dictionary API.
So,if you give input "!find word-to-search" the bot will return the meaning of the word.
I have done almost everything, I have a python script that takes a string input and returns the meaning of it.
And, I have a JS file that takes user input and then responds accordingly.
What I want to do is, take user input in JS file, send it to the python file, which returns the meaning of the word.
I am using Heroku to host by bot
I need your help, to let me know how to send the input string from bot.js to trying.py and then return an array of string from trying.py to bot.js
The code of both the files is :
trying.py
import requests
import json

app_id = 'my-app-id'
app_key = 'my-app-key'

language = 'en-gb'
word_id = 'Hello'
fields = 'definitions'
strictMatch = 'false'

url = 'https://od-api.oxforddictionaries.com:443/api/v2/entries/' + language + '/' + word_id.lower() + '?fields=' + fields + '&strictMatch=' + strictMatch;

r = requests.get(url, headers = {'app_id': app_id, 'app_key': app_key})

theListOfMeanings = r.json()

if 'error' in theListOfMeanings:
    print("Sorry, I couldn't find ",word_id," in the dictionary\nPlease check the spelling")
else:
    counter=1
    print("The different meanings of",word_id," are -")
    for j in theListOfMeanings['results']:
        for x in j['lexicalEntries']:
            for i in (x['entries'][0]['senses']):
                print(counter,". "," ".join(i['definitions']))
                counter+=1

Bot.js
const Discord = require('discord.js');
const client = new Discord.Client();
const auth = require('./auth.json') //TEMPORARY

client.on('ready', () => {
  console.log(`Logged in as ${client.user.tag}!`);
});

client.on('guildMemberAdd', member => {
  member.guild.channels.get('channelID').send("Welcome");
});

client.on('message', msg => {
  theMessage = msg.content
  if (msg.content === '!hello') {
    msg.reply('Hii ! :grin:');
  }
  else if (theMessage.slice(0, 5) === '!find'){
    msg.reply('Hi, my name is Norm :neutral_face:')
    //theMessage.pop(0)
    theMessage = theMessage.substring(5);
    msg.reply(theMessage);

  }
});
client.login(auth.token);

I agree that recoding the python script in JS  will be a good idea.
But, the problem is, the node.js code available at Oxford returns an output, that I don't know how to manage.
The node.js code (replacement for trying.py) is available in the above link, it's output for me is this :

If someone can tell me how I can use the returned JS code, I'll be very grateful.
OXFORD API CODE (Node.js(replacement of trying.py))
const http = require("https");

const app_id = "my_app_id"; // insert your APP Id
const app_key = "my_app_key"; // insert your APP Key
const wordId = "ace";
const fields = "pronunciations";
const strictMatch = "false";

const options = {
  host: 'od-api.oxforddictionaries.com',
  port: '443',
  path: '/api/v2/entries/en-gb/' + wordId + '?fields=' + fields + '&strictMatch=' + strictMatch,
  method: "GET",
  headers: {
    'app_id': app_id,
    'app_key': app_key
  }
};

http.get(options, (resp) => {
    let body = '';
    resp.on('data', (d) => {
        body += d;
    });
    resp.on('end', () => {
        let parsed = JSON.stringify(body);

        console.log(parsed);
    });
});

A: Problem solved in the comments, by recoding the program in JS using the node-fetch npm module.

A: Here is a working Javascript Snippet that fetches the definitions for you using node-fetch.
Outputs
hello
1. [interjection] used as a greeting or to begin a phone conversation
1. [noun] an utterance of 'hello'; a greeting
1. [verb] say or shout 'hello'
world
1. [noun] the earth, together with all of its countries and peoples
2. [noun] a particular region or group of countries
3. [noun] human and social interaction

const fetch = typeof window !== 'undefined'?window.fetch:require("node-fetch")

const app_id = "f84663ce"; // insert your APP Id
const app_key = "9d0cc4ee0694e65386b9bfd69cba3aba"; // insert your APP Key
const fields = "definitions";
const strictMatch = "false";

async function lookupDefs (wordId = 'ace') {
  const url = 'https://od-api.oxforddictionaries.com/api/v2/entries/en-gb/' + wordId + '?fields=' + fields + '&strictMatch=' + strictMatch;
  const options = {
    method: "GET",
    headers: {
      'app_id': app_id,
      'app_key': app_key
    }
  };

  const response = await fetch(url, options);
  return response.json();
}

(async () => {
    for (const word of ['hello', 'world']) {

      const res = await lookupDefs(word);
      console.log (word);
      for (const result of res.results) {
        for (const entry of result.lexicalEntries) {
          for (let i=0; i<entry.entries[0].senses.length; i++) {
          const sense = entry.entries[0].senses[i];
          const cat = entry.lexicalCategory.id;
          console.log(`${i+1}. [${cat}] ${sense.definitions.join(', ')}`);
        }
      }
    }
  }
})()

A: There are couple of ways you can go, but I will suggest you only flask since I believe it will require minimal amount of efforts for you to start.
You need to create Flask application, place your javascript code as static assets which talks to flask view. The flask view internally execute logic you wrote in python script and return json response to the javascript code, take a look on Flask quickstart and Flask by Example – Project Setup as your starting point.
When you complete the application on local machine here is guides on deploying python application on Heroku and Deploy Python Flask App on Heroku
.

A: To answer your initial question, here is how you could let the two communicate. This assumes that both the JavaScript script and Python script are located on the same machine and can communicate through the stdin/stdout channels.
# script.py
import sys

word = sys.stdin.read().strip()
print('definition of ' + word)

// script.js
const { spawn } = require('child_process');

// assuming script.py is in the same folder
const python = spawn('python', ['script.py']);

python.stdout.on('data', data => {
  const definition = data.toString().trim();
  console.log(definition);
});

const userInput = 'Foo';
python.stdin.end(userInput);

When you run:
node script.js

It will print:
definition of Foo

redpajama_set_name: RedPajamaStackExchange


text:
No U.S. Weapons for Countries with Child Soldiers (Oh, You Have a Waiver?)
Abigail R. Hall Blanco • Thursday January 22, 2015 5:53 AM PST •
As the world observes conflicts in Iraq, Syria, the Ukraine, and elsewhere, the impact on human life is undeniably tragic. Nowhere is this tragedy more pronounced than in the impact of these conflicts have on children.
According to the Syrian Observatory for Human Rights director Rami Abdel, more than 10,000 children have died in the Syrian conflict. In addition to these immediate threats against life and limb, war has other consequences for children. Conflict disrupts a child's education and grinds economic activity around the conflict zone to a halt. This delay in education and stunted economic activity perversely impacts young lives both now and in the future. It is estimated that some 40 million children are out of school in conflict-affected countries throughout the world.
As if this damage were not enough, some children bear an even greater burden. They are soldiers.
There are an estimated 250,000 child soldiers in world today. These children, often abducted from their homes by armed forces, are desirable as soldiers because they are easier to control than adults, have an underdeveloped sense of danger, and use fewer resources than a comparable adult. Children are often sent into battle as infantry in order to draw fire away from adult combatants.
In response to this problem, members of the international community have taken steps to curtail the practice throughout the globe. In the United States, the Child Soldiers Prevention Act of 2008 (CSPA) was meant to assist the goal of ending child soldiering by banning the U.S. from sending military assistance to countries with "governmental armed forces or government–supported armed groups, including paramilitaries, militias, or civil defense forces, that recruit and use child soldiers."
In 2010, President Obama signed into law an additional bill, the Lord's Resistance Disarmament and Northern Uganda Recovery Act, and pledged the U.S. would help end the "killing, raping...and brutalizing [of] children" in Joseph Kony's army of "Invisible Children."
So did the U.S. government fulfill its supposed goal of snuffing out child soldiering by withholding aid? Not by a long shot. Just five months after his 2010 pledge, Obama granted several waivers to the Child Soldier Prevention Act, allowing funding to continue to four countries known to use child soldiers. Chad, Congo, Sudan, and Yemen each received a reprieve for a variety of reasons including their assistance in fighting terrorism. Congo was exempted because of American programs in the country aimed at training the military to "be more professional." During the Obama administration, Congo, Yemen, and Chad would each receive millions in weapons despite their humanitarian records and the known use of child soldiers.
The consequences of such actions are tragic. As I've discussed in other posts, there are a variety of problems with using weapons as aid. Even if we assume that the military equipment given to these countries is initially sent for legitimate purposes, it is practically impossible to control the flow of these weapons after they reach foreign soil. These weapons allow conflicts to grow, perpetuate, and provide the arsenal for future conflicts. These arms fuel the very humanitarian crises the U.S. government supposedly wants to combat. Those who pay the highest price for these policies may be the most innocent.
Abigail R. Hall Blanco is a Research Fellow at the Independent Institute and an Associate Professor of Economics at Bellarmine University in Louisville, Kentucky.
Posts by Abigail R. Hall Blanco | Full Biography and Publications
Afghanistan Africa Child Soldiers Children Civil Liberties Defense Family Iraq Lord's Resistance Army Middle East Military Slavery Terrorism U.S. Foreign Policy War Weapons Weapons and Foreign Aid


text:
Inaugural Singapore Writers Festival Access Programme
Inaugural Singapore Writers Festival Access Programme made possible by British Council
British Council lends support to Singapore Writers Festival to make the festival accessible to Deaf and Hard of Hearing individuals
Singapore, 4 November 2016 – For the first time, Deaf and Hard of Hearing communities will be able to participate in the Singapore Writers Festival (SWF) organised by the National Arts Council.
The British Council announced their support towards the SWF's first-ever Access Programme. The SWF Access Programme will introduce sign language interpretation for Deaf and hearing impaired participants at 12 selected SWF events. This pilot initiative hopes to increase arts access for local beneficiaries, and enable opportunities for them to be a part of the Festival.
The Access Programme was initiated by the British Council as part of the Council's wider push to create a more inclusive arts sector. In March 2016, the British Council, in conjunction with the National Arts Council and Singapore International Foundation launched the inaugural Arts & Disability Forum, aimed at addressing disability and inclusivity in the arts and cultural sector. Access to arts venues and activities were some pressing issues discussed at the forum.
The British Council hopes to expand disability access to more festivals, events and arts venues in Singapore.
Speaking about the partnership, Dr Sarah Meisch Lionetto, Director of Arts, British Council Singapore said that, "Inclusivity, diversity and equality are very much a part of the British Council's core ethos. By pioneering this programme, we are excited to support people with disabilities by enabling them access to equal opportunities to participate in the arts. Creating basic access to arts programmes allows a starting point for discussing an inclusive society and cultivating disabled artists. Through our ongoing art programmes, we hope to contribute to an even more dynamic and inclusive cultural scene in Singapore."
"We have made efforts to implement inclusive programming to widen our reach, so that everyone can have the opportunity to attend and enjoy our events. In this way, every festival-goer can be exposed to the wonderful and exciting ideas being shared over the next 10 days," said Mr Yeow Kai Chai, the director of SWF.
The SWF will run from 4 to 13 November 2016 at the Civic District.
Deaf and hearing impaired participants will each be given complimentary festival passes to the selected events. Sign language interpreters will be present at the selected programmes.
Each event has reserved seating for up to eight Deaf and hearing impaired attendees.
As venue capacity is limited, and to assist us in planning, registration is required athttp://bit.ly/SWFsignlanguage
For the two ticketed lectures, attendees will need to purchase a separate ticket to attend. Festival Pass holders will be entitled to a 20% discount.
More information on the SWF Accessibility Programme can be found at https://www.britishcouncil.sg/programmes/arts/arts-and-disability/access-programme-singapore-writers-festival. Registration for programmes with sign language interpretation can be done via the SWF website, https://www.singaporewritersfestival.com.
Download the full press release below.
Name & Designation: Joanne Tay, Programme Manager, Cultural Skills and Arts
Email: joanne.tay@britishcouncil.org.sg
The British Council is the UK's international organisation for cultural relations and educational opportunities. We create friendly knowledge and understanding between the people of the UK and other countries. Using the UK's cultural resources we make a positive contribution to the countries we work with – changing lives by creating opportunities, building connections and engendering trust.
We work with over 100 countries across the world in the fields of arts and culture, English language, education and civil society. Each year we reach over 20 million people face-to-face and more than 500 million people online, via broadcasts and publications.
Founded in 1934, we are a UK charity governed by Royal Charter and a UK public body. The majority of our income is raised delivering a range of projects and contracts in English teaching and examinations, education and development contracts and from partnerships with public and private organisations. Eighteen per cent of our funding is received from the UK government.
Press Release - Inaugural Singapore Writers Festival Access Programme made possible by British Council (Adobe PDF 172KB)
Access Programme at the Singapore Writers Festival
Singapore Writers Festival Official Website


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Mike McGuirl Nijel Pack NCAA Men's Division I Basketball Championship Men's college basketball College sports College basketball Basketball Men's basketball Sports Men's sports
Gibson hits 7 3s, scores 29; Oklahoma beats Kansas St. 78-71
Kansas State guard Mike McGuirl (00) and Oklahoma guard Marvin Johnson, top, go after the ball during the first half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Mar. 05, 2022 05:06 PM EST
Oklahoma guard Umoja Gibson (2) shoots a 3-point basket as Kansas State guard Nijel Pack (24) defends during the first half of an NCAA college basketball game on Saturday, March 5, 2022 in Manhattan, Kan. (AP Photo/Colin E. Braley)
Oklahoma guard Umoja Gibson (2) scores against Kansas State guard Nijel Pack (24) during the first half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Kansas State guard Selton Miguel (3) scores against Oklahoma forward Jacob Groves (34) during the second half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Oklahoma forward Tanner Groves (35) grabs a rebound as Kansas State forward Davion Bradford (21) defends during the first half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Oklahoma guard Umoja Gibson, center, passes the ball as Kansas State forward Davion Bradford, left, defends during the first half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Kansas State guard Nijel Pack, front right, knocks the ball out of the hands of Oklahoma forward Jalen Hill, left, during the first half of an NCAA college basketball game on Saturday, March 5, 2022 in Manhattan, Kan. (AP Photo/Colin E. Braley)
Oklahoma forward Ethan Chargois (15) attempts to get past Kansas State forward Davion Bradford, top, during the first half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Kansas State guard Mark Smith, right, attempts to score against Oklahoma guard C.J. Noland (22) during the second half of an NCAA college basketball game on Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)
Kansas State head coach Bruce Weber reacts during the first half of an NCAA college basketball game against Oklahoma, Saturday, March 5, 2022, in Manhattan, Kan. (AP Photo/Colin E. Braley)


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