In this section, we provide an overview of the core collapse and postbounce
evolution in our models for the 
M
and 
M stellar progenitors. These provide the physical
context for the code tests in section ![[*]](file:/usr/share/latex2html/icons/crossref.png){kind=icon}
.
A thorough discussion of supernova physics can be found in previous
reviews, e.g. in Bethe_90,Burrows_Young_00,Mezzacappa_Bruenn_00,Janka_Kifonidis_Rampp_01.
Earlier runs of the M model have been described
in Mezzacappa_et_al_01,Liebendoerfer_et_al_01 with Newtonian
and general relativistic gravity. Here, we add information on the
formation of the neutrino spectra and report on our first self-consistent
simulation running all the way from core collapse to the onset of
black hole formation in the case of the M
model. We have chosen progenitors on the light and massive side with
respect to the range of potential core collapse supernova progenitors.
This demonstrates the spread in the results. The results of our simulations
for intermediate mass progenitors are summarized in Messer_00,Liebendoerfer_et_al_01a,Liebendoerfer_et_al_02.
The latest was calculated with the finite differencing described in
this paper.
The most prominent characterization of a supernova explosion is the
trajectory of the shock position. We define the shock position as
the location with the maximum infall velocity (i.e. minimum in the
velocity profile). Before the shock is formed after bounce, the location
with maximum infall velocity coincides with the sonic point which
separates the causally connected inner core from the supersonically
infalling outer core. The pressure wave emerging from the center at
bounce turns at this transition point into a shock wave. The trajectory
of maximum infall velocity is therefore continuous across bounce as
shown in the magnifying window on the left hand side of Fig. ().
|
M
model with its position in the M model,
we find that they converge to the same point before bounce. For the
explanation, we recall that the sonic point depends on the Chandrasekhar
mass, which is determined by the electron fraction profile. The electron
fraction profile of the two runs converges due to a strong feedback
of the electron fraction on the free proton abundance Messer_et_al_03.
Within the ``standard'' input physics, it is assumed that electron
capture on nuclei with a full neutron f7/2 shell is Pauli-blocked
Bruenn_85. Under such conditions, our simulations allow only
electron capture on free protons. Now, if the electron fraction and/or
temperature in one model would only be slightly higher than in the
other model, this would result in a significantly higher free proton
abundance. It would cause a significantly higher number of electron
captures than in the other model. Hence, the differences between the
models are reduced. It has recently been shown that, due to the finite
temperature in the nuclei and due to correlations, electron capture
on nuclei dominates electron capture on free protons throughout core
collapse Langanke_et_al_03. It will be interesting to see
if the described feedback will to the same extent be at work with
these more realistic electron capture rates. We can only expect this
if the electron capture on low abundance nuclei would turn out to
be comparable to, or larger than, the electron capture on the most
abundant nuclei (the quantity to compare would of course always be
the product of the electron capture rate with the abundance of the
target). The right hand side of Fig. () shows the
shock trajectories over a longer time interval, up to one second after
bounce. The shocks recede in both models after 
ms.
The conditions for a shock revival deteriorate. The intensive neutrino
emission from the cooling region undermines the pressure support below
the heating region and the material is drained from the latter onto
the protoneutron star Janka_01. The transient stall in the
receding shock front in the M model at 
s after bounce is a reaction to an enhanced electron flavor luminosity.
It will be further analyzed below, together with the description of
the evolution of the luminosities. In the evolution of the
M progenitor, we encountered a numerical stability
problem after 
s. The adaptive grid created extremely
small mass zones such that the convergence radius of the Newton-Raphson
algorithm in the implicit hydrodynamics was severely reduced due to
truncation errors. We increased the artificial viscosity in two sequential
steps to widen the shock and continue the run. This numerical shock
widening is responsible for the outward steps in the shock position
s after bounce. Shortly after the collapse of the
protoneutron star to a black hole has set in, our code crashes unavoidably
because of the coordinate singularity in the comoving coordinates
at the Schwarzschild horizon. We will extensively use the hydrodynamic
profiles at 
s after bounce for testing in later subsections.
The neutrino luminosities and rms energies are shown in Fig. ().
|
erg/s. The collapse is halted at nuclear
densities and a bounce shock propagates outwards through neutrino
opaque material. The neutrino luminosity decays to a 
lower
level during this short period of 
ms duration. It has
been assigned to a decrease in the free proton fraction when the shock
is formed Thompson_Burrows_Pinto_03. Additionally, while the
shock is running out to the neutrinospheres, it condenses previously
still neutrino emitting material to even more neutrino opaque densities.
When the shock reaches the electron neutrinosphere, an electron neutrino
burst with a peak height of
erg/s is launched
by copious electron capture. As the neutrinos escape quickly, the
freed phase space is refilled with new neutrinos and the matter deleptonizes
rapidly. This phase is very similar in both models because, after
core collapse, the structure of the inner core is so similar. Differences
appear later, when the accretion luminosity dominates over the core
diffusion luminosity with a ratio of about 
to 
.
The higher densities in the outer layers of the more massive progenitor
produce considerably higher accretion luminosities when they settle
in the gravitational potential on the surface of the protoneutron
star. The rising electron neutrino rms energies before bounce reflect
the conditions in the compactifying material at infall. After the
neutrino burst, the rms energies adjust to the spectrum set by the
shock heated mantle and reflect the conditions at the location of
decoupling. However, we have to keep in mind that the location of
decoupling strongly varies for individual neutrino flavors and neutrino
energies. Thus, the rms energy rather reflects an emission-weighed
sampling of conditions at different locations. Quite generally, the
and 
neutrinos decouple deeper because of
the insensitivity of these neutrinos to charged current reactions.
The electron antineutrinos decouple deeper than the electron neutrinos
because of the smaller proton than neutron abundance.
In order to investigate the origin of the neutrino luminosities in
more detail, we introduce in appendix
radius- and energy-dependent attenuation coefficients,
,
that express the probability that a neutrino with energy 
emitted at radius 
escapes from the computational domain without
a reaction that changes its type or energy. The attenuation coefficients
carry information that is similar to the optical depth,
:
.
However, their evaluation according to Eq. ()
is fully consistent with the finite differencing of the Boltzmann
equation and takes the variations in the local flux factors into account.
Assume for example, we investigate a reaction 
that produces
at radius neutrinos in the energy interval 
with
an energy emissivity
. With the help of
the attenuation coefficients, we can quantify the contribution of
a given volume element 
to the total luminosity,
over energy
and position for all reactions that contribute, 
that
the attenuation coefficients in Eq. ()
represent the total luminosity in the simulation accurately. In the
following figures, we cumulatively plot the quantity
in units of erg/s/km. It is natural to choose 
in accordance
with the width of the
energy groups we used
in the simulations. At the bottom of the figure we start with the
electron or positron capture reaction (depending on the neutrino type).
We draw
for
the lowest energy group and enclose it by a black line. On top of
it, we add
for
the next energy group, again enclosed by a black line. The shading
of the enclosed areas indicates the energy group according to the
legend at the bottom of the figure. Energy groups that do not contribute
collapse to a single line with zero enclosed area. On top of
we continue with
for the pair creation reaction. We enclose this and the following
area elements by a white line to distinguish them from the electron
or positron capture reactions. After the addition of
,
we continue with the contributions from neutrino-electron scattering,
i.e.
to
.
We use again black lines as separators for the neutrino-electron scattering.
The figures become intuitively accessible as soon as one realizes
that the total shaded area is proportional to the total luminosity
of the star. The total area of a specific energy shading is proportional
to the contribution to the total luminosity of neutrinos from the
corresponding energy group. The total area of a specific reaction
is proportional to the contribution to the total luminosity of neutrinos
with this last inelastic interaction before the escape. A cross section
through the shaded area at a given radius tells about the spectrum
of the neutrinos escaping from that region, and about the probability
of the reaction type they had at that position before the escape.
In order to characterize also the extent of isoenergetic scattering
of neutrinos off nucleons and nuclei, we mark the neutrinospheres
for the energy groups at the top of the figure. The energies are rising
from the left to the right according to the legend at the bottom of
the figure. For the interpretation of the figures it is also useful
to know the thermodynamical conditions at the locations the neutrinos
are emitted. For each density decade we set a marker at the bottom
of the figure. Additionally, we include the electron fraction in the
graph with the electron neutrino analysis, and the entropy in the
graph with the electron antineutrino analysis. The solid line represents
the profile in the simulation, the dashed line represents the equilibrium
value that would be achieved by infinitely long exposure of the stationary
fluid element to the prevailing neutrino abundances. Finally, the
graph with the - and -neutrino analysis obtains
profiles with the temperature (dashed line) and electron chemical
potential (solid line).
Fig. (),
|
ms before bounce in the
collapse of the M progenitor. Neutrinos
are escaping from the range between 
km and 
km
radius, roughly corresponding to a density range of 
g/cm
to 
g/cm. The electron fraction
profile (solid line) reflects the deleptonization that has already
occurred in the more interior regions where 
approaches
values around . The equilibrium value (dashed line) is still
lower, indicating that the deleptonization is ongoing and that the
deleptonization time scale is slightly slower than the dynamical time
scale. The pair process does not contribute in this collapse phase
of high electron degeneracy. The corresponding area collapses to one
white line in the graph that separates the electron capture contributions
below it from the neutrino-electron scattering contributions above
it. Almost no neutrinos escape directly after an electron capture
at a radius as small as 
km. Neutrinos escaping from this
region are thermalized by scattering off electrons. They escape with
quite low energies. Around 
km radius we find that about
half of the escaping neutrinos stem directly from electron capture,
while the other half has scattered off an electron. At 
km 
of the neutrinos escape without further electron-scattering.
Among the neutrinos produced by electron capture, the neutrinos with
higher energies escape from smaller radii than the neutrinos with
lower energies. In this collapse phase, the ``standard'' input
physics only includes electron captures on free nucleons. The Q value
of the reaction is small and very few low energy neutrinos are directly
produced if the electron chemical potential is large (e.g. 
MeV at km radius in this time slice). The escaping low
energy neutrinos have scattered off electrons. However, the larger
Q value of more realistic electron capture rates on neutron-rich nuclei
may shift the energy of directly escaping neutrinos to lower values
such that the count of direct escapes is increased Langanke_et_al_03.
Finally, we remark that the neutrinos in a given energy group are
produced at a significantly smaller radius than the location of the
corresponding transport sphere. This is the result of the dominance
of the isoenergetic scattering cross section in the collapse phase.
The diffusive propagation of the neutrinos extends to much lower densities
than neutrino-electron scattering or neutrino absorption.
We switch to the next interesting phase: the electron neutrino burst
at 
ms after bounce (Fig. ).
|
km) in the burst phase.
In the neutrino burst, the electron neutrinos escape directly from
electron capture. Some neutrino-electron scattering does occur in
front of the shock in an earlier stage and inelastic scattering of
burst neutrinos on nuclei in front of the shock are possible Bruenn_Haxton_91,
but not included in the standard input physics. Only the energy groups
at 
MeV and 
MeV contribute significantly to the
burst. This is in good agreement with the position of the corresponding
transport sphere in the upper part of the figure (fourth and fifth
from the left, respectively). The density is of order 
g/cm. The trapped electron neutrinos inside the region
of main emission are in weak (and thermal) equilibrium with the fluid.
This is evident in the congruence of the electron fraction profile
(solid line) with the equilibrium (dashed line).
The situation at ms after bounce is shown in Fig. ().
|
km radius, the reaction time
scales are comparable to, or larger than, the infall time scale. Graph
(b) shows the origin of the electron antineutrino luminosity. The
electron antineutrinos are emitted from a slightly smaller radius
at densities exceeding g/cm (while they
were similar to g/cm for the electron neutrinos).
The electron antineutrinos decouple at smaller radii because of the
higher neutron than proton abundance. The contribution of electron-scattered
neutrinos (above the white line) is slightly larger than in the case
of the electron neutrinos. The pair production process does not noticeably
contribute to the luminosity at this stage. The equilibrium entropy
(dashed line) is increasing with increasing radius. An infalling fluid
element changes its entropy rather slowly by electron capture until
it hits the accretion shock. At the shock front, most of its kinetic
energy is converted into heat by shock dissipation. The heavy nuclei
are dissociated, mainly into free nucleons which are good neutrino
absorbers. However, before the stage at ms after bounce,
the entropy of an infalling fluid element has already reached or exceeded
the equilibrium entropy, alone by shock dissipation. Only cooling
is then possible during the continued infall. After 
ms, however, the equilibrium entropy is not reached by the shock dissipation
and the fluid element indeed increases its entropy towards the equilibrium
by neutrino absorption during its flight through the heating region
(solid line in graph (b) 
km radius). As the fluid
element also tends to decrease the electron fraction (graph (a)),
electron antineutrinos are preferentially absorbed. Once the equilibrium
is reached, cooling becomes unavoidable if the fluid element is still
infalling because the reaction rates at these densities are faster
than the dynamical infall time. Convection in the heating region is
expected to increase the neutrino heating efficiency (see e.g. the
parameter study of Janka_Mueller_96), but not included in
our simulations. We did not find any sign of shock revival in spherical
symmetry with the included input physics. Graph (c) shows the origin
of the - and -luminosities. The -
and -neutrinos are produced at an average density slightly
larger than g/cm. Almost no neutrinos escape
directly from pair creation (area enclosed by white lines at the bottom
of the figure). Most of the neutrinos have scattered off electrons
before their escape (area enclosed by black lines). Pair production
is not the dominant source of - and -neutrinos.
It has been shown, that the production from bremsstrahlung Thompson_Burrows_Horvath_00
and from electron flavor neutrino annihilation Buras_et_al_03a
exceeds the pair process production rate (both reactions are not included
in our simulations). The latter reference finds that the -
and -neutrino luminosities show differences of 
in the first ms after bounce and converge to the standard
luminosities afterwards. The spectra are not significantly different.
This finding is also supported by our graph (c): The production site
of the - and -neutrinos is at a much smaller
radius than their transport sphere. Hence, the neutrino luminosity
is set by the (though energy-dependent) diffusivity between the location
of neutrino production and the transport sphere. Moreover, graph (c)
shows that the majority of escaping neutrinos scattered off electrons
in their last reaction. Differences in the production-spectrum are
likely to be washed out during the thermalization the neutrinos are
experiencing while they are diffusing outwards to the transport sphere.
Finally, we present the situation at 
ms after bounce in
Fig. ().
|
ms after bounce.
But there are not much qualitative changes. The neutrinos with larger
energies still escape from larger radii because of the corresponding
staggering of the transport spheres in the steep density gradient
at the surface of the protoneutron star. The shock has receded to
a radius of 
km. Graph (b) shows the origin of the electron
antineutrino luminosity. The infalling fluid elements are now crossing
the heating region that rapidly (with several thousand km/s) that
there is no time for significant neutrino heating. This can be seen
in the flat top of the entropy curve between 
km and
km radius (solid line). Even the cooling sets in with a slight delay
and thermal balance is only reached at densities larger than
g/cm. Graph (c) shows the origin of the - and
-luminosities. Still very few neutrinos escape directly
from pair creation (area enclosed by white lines at the bottom of
the figure). Most of the neutrinos have scattered off electrons before
their escape (area enclosed by black lines). The continued cooling
by - and -neutrino emission becomes now clearly
visible in the entropy profile at a radius of 
km, where
an entropy dip develops. At larger radii, however, between 
km and the shock position, the temperature is rising and the electrons
have become nondegenerate. The overlap of this material with the emission
region of electron antineutrinos in graph (b) lets the emission of
higher energy antineutrinos shift to lower densities than before.
Figure ()
|
M
model, the luminosities decrease as a consequence of the declining
accretion rate and continued deleptonization of the core. The electron
flavor luminosities reach very similar values because the lifted electron
degeneracy in a large part of the cooling region (see Fig. (bc))
allows the electrons and positrons to be captured from similar chemical
potentials. The luminosities are higher than the luminosities of the
- and -neutrinos because the latter do not
have an accretion luminosity component. The rms energies show the
usual hierarchy at the beginning, but after 
s, the
rms energy of the - and -neutrinos falls below
the rms energy of the electron antineutrino. This is also understood
if one looks again at Fig. (bc). While the emission
of high energy electron antineutrinos is aided by shock-heated material
settling at the base of the cooling region with moderate electron
degeneracy, the layers around km radius, i.e. where the
energy spectra of the - and -neutrinos are
set, are barely affected by the continued accretion on the still not
very massive protoneutron star. This domain just slowly cools by neutrino
emission (compare the entropy profiles in Fig. (b)
and (b)). The result are decreasing luminosities
and rms energies of the - and -neutrinos. The
more massive M model shows qualitatively
different features. In order to understand them, we first discuss
Fig. ().
|
M model. Nothing special happens there,
the mass flux generally decreases in accordance with the decreasing
density in the outer layers. Moreover, the contraction of the stiff
core, which is far from its maximum mass, is minimal. A similar evolution
is visible during the first 
ms in the M
model. However, if we examine the massflux in the ms time
slice in Fig. () more closely, we find a variation
in the density profile around 
km that falls in (from 
km at ms after bounce). The corresponding variation in
the massflux or accretion rate leads to a step in the electron flavor
neutrino luminosities between ms and 
ms after
bounce. The increase is of order 
. The slope in their rms
energies flattens slightly. More independent of the details of the
progenitor model, however, might be that the protoneutron star in
the M model approaches its maximum mass
much more rapidly because a high accretion rate is maintained when
the outer layers fall in. They have a significantly larger density
in comparison to the M model. The fast mass
accumulation in the M model becomes evident
in Table (), which lists the enclosed mass
at km radius for different time slices in both models.
|
M protoneutron
star gets closer to the maximum mass, the protoneutron star starts
to contract faster by the general relativistic enhancement of the
effective gravitational potential (an effect absent in Newtonian calculations).
We observe in Fig. () that, after an initial
decrease, the mass flux in the inner core is increasing again. This
is by no means a ``sudden'' change on the short dynamical time
scale of the protoneutron star. The contraction is a hydrostatic adaption
to the accumulated mass in the gravitational potential. The change
is, however, sudden on the time scale of the variations in the neutrino
properties shown in Fig. (). The -
and -neutrino luminosities and rms energies rise steeply.
This happens because, by the contraction of the protoneutron star,
electron-nondegenerate shock-heated material is condensed to densities
where the main emission of heavy neutrinos occurs. We investigate
the conditions at the locations of the main neutrino emission in more
detail in Fig. ().
|
). Instead of
the average radius, we calculate here the average density (graph (a)),
temperature (graph (b)), and electron fraction (graph (c)). These
represent the typical conditions where an escaping neutrino makes
its last energy-changing interaction with the matter. The conditions
at the origin of electron neutrino emission are traced with a solid
line, the conditions at the origin of electron antineutrino emission
with a dashed line, and the conditions at the origin of -
and -neutrino emission with a dash-dotted line. We discuss
first the M model (thin lines). After an
initial decrease up to the time of maximum neutrino heating around
ms after bounce, the average density at the sites of neutrino
production increases steadily. This goes along with a temperature
increase for all neutrino types. At least in the case of the very
temperature-sensitive pair production rates for the - and
-neutrinos, however, the argumentation should be turned
around: Because the temperature at a given density is slowly decreasing
on the long time scale, the emission from lower density regions decreases
and the average emission conditions shift to increasingly deeper layers,
where the temperatures are higher. Graph (c) shows that the electron
fraction at the place of emission is increasing for the electron flavor
neutrinos. This is due to the rather high electron fraction in the
shock-heated material behind the shock front, where the electron degeneracy
is gradually lifted. The - and -neutrinos escape
from the floor in the electron fraction profile, which is steadily
decreasing by continued deleptonization. This is reflected in the
declining electron fraction at the location of the production of heavy
neutrinos in graph (c). The contraction of the protoneutron star in
the M model causes a qualitatively different
characterization of the regions of main neutrino emission. The temperature
increase by adiabatic compression is no longer fully balanced by neutrino
cooling. The temperature at a given density starts to increase. The
domain of the shock-heated material where the electrons are non-degenerate
reaches down to deeper layers close to the place where the -
and -neutrinos are produced. The temperature increase
makes these regions to significantly more efficient - and
-neutrino emitters such that the average emission density
starts to decrease. In spite of this steep density decrease in graph
(a), the temperature in graph (b) still increases at the average emission
condition. The electron fraction in graph (c) rises dramatically as
the mean neutrino emission region moves out of the trough
in the electron fraction profile. As the production site of heavy
neutrinos moves outwards to lower densities, the fraction of directly
escaping neutrinos (without neutrino-electron scattering) also increases.
Additionally, the forming ``density cliff'' shortens the escape
path for few high energy - and -neutrinos that
may leave the star without thermalization. It can be expected that
the not included reactions of - and -flavor
neutrino production by bremsstrahlung and electron flavor neutrino
annihilation will more significantly influence the - and
-neutrino spectra in this less opaque density regimes.
Unfortunately, we cannot follow the evolution beyond the collapse
of the protoneutron star because of the coordinate singularity at
the formation of the Schwarzschild horizon. However, the neutrino
luminosities are expected to decay on a short timescale (see e.g.
Baumgarte_et_al_96).
s is the consequence of a physical change in the luminosities. The
two perturbations after
s are
the result of an artificial increase of the numerical shock width
we had to apply in order to run the simulations to the end.
km
to account for the (approximate) propagation
time to the sampling radius. The two progenitors show a comparable
neutrino burst with a peak height of
km radius. The density markers
at the bottom of the graph indicate the position of density decades,
, where 
is given in g/cm
.
Their locations, marked in the upper part of the figure, statistically
indicate the positions of the last interaction of the neutrinos with
matter, isoenergetic scattering included. The energies rise from the
left to the right according to the legend. The transient ondulations
in the luminosity contributions in this phase are a numerical artefact
caused by the low resolution of the Fermi surface in the energy dimension
of the momentum phase space (see discussion in subsection 
km radius. The density markers
at the bottom of the graph indicate the position of density decades,
km radius. However, the equilibrium
entropy is declining towards smaller radii such that cooling becomes
unavoidable once the fluid entropy has joined the equilibrium entropy
in a still infalling state. Graph (c) shows the origin of the 
s after bounce,
the rms energy of the 
km in Fig.
(
s, 
s, 
s after bounce. While the accreted mass
piles up on the small protoneutron star in the 
s. An incoming variation in the accretion rate is visible outside
of 
s time window so far explored with Boltzmann neutrino transport. The
outer layers in the 