astro-ph/9503015 3 Mar 1995 Garching Germany lower luminosities even with neutrino uxes also spherically symmetricalrise mo dels of yield the energetic explosion explosions energy while as for a function of time in D and D mo dels loss of strongest heating while heated gas can quickly rise outward thus reducing its energy prompt energy dep osition by dimensional rst neutrino explosion One and two dimensional hydrodynamical simulations of the neutrino driven sup ernova group time of region number by However since to is not accreted into the low entropy neutronized surface layer of the neutron star only neutrino the M due Convection The On few a scale initial mass of
elements absence b ehind few leave sho ck The Turbulent activity around the protoneutron star is a transient phenomenon at to co ol uxes ms of ms re emission Department of Astronomy and Astrophysics University of Chicago luminosities e case a from matter emission but the and of the from dep end First after M Karl Schwarzschild Str signi cant primarily Max Planck Institut sho ck S Ellis gas do es ab out the
allowing co ol in the star sho ck of initial sensitively the Accretion is Second Here neutrino Max Planck Institut the neutrinos one Email Email are a down ows from the gains energy by not b ecause convection no strong explosions o ccur accretion formation protoneutron star and explosive nucleosynthesis of crucial protoneutron second p erformed Avenue Chicago Illinois p owerful H THOMAS JANKA b ecome thomas granta uchicago edu ewald mpa garching mpg de EWALD p ostsho ck matter on sphere ABSTRACT the help This f ur of after the while D Garching Germany explosions very AND overturn Astrophysik Convective for strength a shows for M interpretation and core star interactions with the ULLER the f urAstrophysik the Type I I neutron rich and its the sho ck explosions masses b ounce that phase explosion increases Sho ck can of to overturn Karl Schwarzschild Str the to U S A the b e p enetrate b etween in the is neutrino Systematic explosion obtained our Sup ernova supp orted develop the only heating Propagation in mo dels e ciency the in inward to core neutrino uxes entropy increases it the heating on only a energy stagnation zone variation neutrino heated For higher core by narrow are a in time the of loses lepton only during the the explosion neutrino di erent D window scale region of multi of ab out iron Due the the the of
ms after sho ck formation the turbulent shell decouples from the neutrino heated
zone and moves outward b ehind the sup ernova sho ck The sho ck shows deformation on
large scales and the inhomogeneities of temp erature density and velocity with contrasts
o o
of order unity on scales of to in the layer b ehind the sho ck can help to explain the
anisotropies and radial mixing observed in SN A When the sup ernova sho ck reaches
the entropy step of the Si O interface at km ab out ms after b ounce the
density inversion b etween the dense inhomogeneous shell b ehind the sho ck and the low
density hot bubble region around the neutron star b egins to steep en into a strong
reverse sho ck This reverse sho ck forms a sharp discontinuity in the neutrino driven
wind from the nascent neutron star The deceleration of the wind expansion might
trigger signi cant anisotropic matter fallback to the neutron star on a time scale of
several seconds
Sub ject headings sup ernovae general hydrodynamics sho ck waves convection
turbulence
INTRODUCTION
There is a variety of observational hints that large scale mixing pro cesses o ccurred and
played an imp ortant role in SN A even during the early phase of the explosion
X rays e g Dotani et al Sunyaev et al and rays e g Matz et al
Mahoney et al from the decay of Ni and Co and strongly Doppler shifted
infrared emission lines of iron e g Erickson et al Haas et al were observed
at a stage when Ni and Fe should have still b een obscured by the overlying expanding
material of the progenitor star This indicated that radioactive elements must have
b een mixed out with very high velocities from the place of their formation close to
the center of the sup ernova far into the stellar mantle and envelope The structure of
the infrared emission lines was interpreted as an indication that the iron p eak elements
were mixed outward macroscopically and inhomogeneously in form of ab out to
identical clumps Li et al
The smo othness of the light curve of SN A provided indirect evidence for the
existence and strength of the mixing pro cess Mixing of hydrogen towards the center
helps to explain the smo oth and broad light curve maximum by the time spread of
the lib eration of recombination energy Shigeyama Nomoto Mixing of heavy
elements into the hydrogen rich envelope homogenizes the opacity and again smo oths
the light curve Arnett et al and references therein Moreover the observation
of a large number of fast moving young pulsars Lyne Lorimer might indicate
the existence of violent non spherical pro cesses in the early moments of the sup ernova
explosion when the neutron star is formed The large dense explosion fragments seen
on ROSAT X ray images outside of the sho ck front in the Vela sup ernova remnant
Aschenbach et al might p ossibly also originate from such early instabilities
Numerical simulations of Rayleigh Taylor instabilities at the comp osition interfaces
metal He He H in the stellar mantle and envelope after sho ck passage e g Arnett
et al Den et al Herant Benz could neither explain the extent of
the required mixing nor could they account for the observed high velocities and the
large scales of inhomogeneities and anisotropies M uller et al This inspired
sup ernova mo dellers to p erform multi dimensional calculations of the early phases of
the sup ernova explosion i e the core collapse sho ck formation and neutrino heating
phases Spherically symmetrical mo dels had indeed shown that convectively unstable
layers are present in the collapsed stellar core after formation and propagation of the
prompt sho ck Burrows Lattimer Theoretical considerations con rmed the
p ossible imp ortance of overturn and mixing in these layers for the evolution of the
sho ck Bethe
MULTI DIMENSIONAL MODELS OF THE EXPLOSION
Herant et al rst demonstrated by a hydrodynamical simulation that strong
turbulent overturn o ccurs in the neutrino heated layer outside of the protoneutron star
and that this helps the stalled sho ck front to start re expansion as a result of energy
dep osition by neutrinos Although the existence and fast growth of these instabilities
was con rmed by Janka M uller and M uller Janka the results of
their simulations in D and in D indicated a very strong sensitivity to the conditions
at the protoneutron star and to the details of the description of neutrino interactions
and neutrino transp ort Since the knowledge ab out the high density equation of state in
the nascent neutron star and ab out the neutrino opacities of dense matter is incomplete
Ra elt Seckel Keil et al the in uence of a contraction of the neutron
star and of the size of the neutrino uxes on the evolution of the explosion has to b e
tested by systematic studies
In the following we rep ort the main results of a set of calculations of D and D
Newtonian mo dels with di erent core neutrino luminosities and with varied temp oral
contraction of the inner b oundary The latter is placed somewhat inside the neutrino
sphere and is used instead of simulating the evolution of the very dense inner core of
the nascent neutron star This gives us the freedom to set the neutrino uxes to chosen
values at the inner b oundary and also enables us to follow the D simulations until
ab out one second after core b ounce with a reasonable number O of time steps
and an acceptable computation time i e several h on one pro cessor of a Cray YMP
with a grid of zones and a highly e cient implementation of the microphysics
Doubling the angular resolution multiplies the computational load by a factor of
Our simulations are started at ab out ms after sho ck formation using an initial mo del
that was evolved through core collapse and b ounce by Bruenn The b oundary
luminosities of all neutrino kinds are set to values near those obtained by Bruenn
and in Newtonian computations by Bruenn et al For details and information
ab out the numerical treatment see Janka M uller a
Our approach and aims di er from those chosen by other groups who recently
p erformed D simulations of sup ernova explosions Herant et al Burrows et
al These groups attempted a self consistent treatment that includes the central
high density part of the neutron star up to a certain stage in the evolution typically
ms after core b ounce but they did not reveal the dep endence of their results
on uncertain asp ects of the input physics or its numerical description
RESULTS
One Dimensional Mo dels
The evolution of the stalled prompt sup ernova sho ck in D mo dels turns out to b e
extremely sensitive to the size of the neutrino luminosities and to the corresp onding
strength of neutrino heating exterior to the gain radius Increasing the core neutrino
uxes from low to higher values has the e ect that the sho ck is pushed further and
further out to reach a successively larger maximum radius during a phase of ab out
ms of slow expansion Nevertheless it nally recedes again to b ecome a standing
accretion sho ck at a much smaller radius For a su ciently high threshold luminosity
however neutrino heating is strong enough to drive the sho ck front outwards and to
cause a successful explosion For even higher neutrino uxes the explosion develops
faster and gets more energetic In case of our M star with a M iron core
Woosley et al we nd that explosions o ccur for electron neutrino and antineutrino
luminosities in excess of erg s in case of a contracting inner b oundary to
mimic the shrinking protoneutron star but of only erg s when the radius of
the inner b oundary is xed
The transition from failure to explosion requires the neutrino luminosities to b e
higher than some threshold value for a quantitative analysis see Janka in preparation
Yet this is not su cient High neutrino energy dep osition has to b e maintained for a
longer p erio d of time to ensure a successful explosion If the decay of the neutrino
uxes is to o fast e g if a signi cant fraction of the neutrino luminosity comes from
neutrino emission by spherically accreted matter b eing shut o when the sho ck starts
to expand then the outward propagation of the sho ck may break down again and the
mo del zzles To drive a continuous sho ck expansion a su ciently high push from the
neutrino heated matter must b e maintained until the material b ehind the sho ck has
achieved escap e velocity and do es not need pressure supp ort to make its way out for a
detailed discussion see Janka in preparation
This contradicts a recent suggestion by Burrows Goshy that the explosion
can b e viewed at as a global instability of the star that once excited inevitably leads to
an explosion The analysis by Burrows Goshy may allow one to estimate the radius of
sho ck stagnation The start up phase of the explosion however can hardly b e describ ed
by steady state assumptions b ecause the times scales of sho ck expansion of neutrino
co oling and heating b etween protoneutron star and sho ck and the corresp onding time
scales of temp erature and density changes in the p ostsho ck region are all of the same
order although they are long compared to the sound crossing time scale and may b e
short compared with the characteristic times of luminosity changes or variations of the
mass accretion rate into the sho ck In particular due to the high sound sp eed and
rather slow sho ck expansion the propagation of the sho ck is very sensitive to changes of
the conditions in the neutrino heated layer A contraction of the neutron star enhanced
co oling of the gas inside the gain radius which is a pro cess that also accelerates the
advection of matter through the gain radius and reduces the time the matter b ehind
the sho ck gets heated or even a mo derate decay of the neutrino uxes can therefore b e
harmful to the outward motion of the sho ck
Two Dimensional Mo dels
In spherical symmetry the expansion of the neutrino heated matter and of the sho ck
can o ccur only when also the overlying material is lifted in the gravitational eld of
the neutron star In the multi dimensional case this is di erent Blobs and lumps
of heated matter can rise by pushing colder material aside and cold material from
the region b ehind the sho ck can get closer to the zone of strongest neutrino heating
to readily absorb energy Also when buoyancy forces drive hot matter outward the
energy loss by re emission of neutrinos is signi cantly reduced This overturn of low
entropy and high entropy gas results in an increase of the e ciency of neutrino energy
dep osition external to the gain radius and leads to explosions in D already for lower
neutrino uxes than in the spherically symmetrical case Our mo dels however do not
show the existence of a convective cycle or convective engine Herant et al
that transp orts energy from the heating region into the sho ck The matter b etween
protoneutron star and sho ck is sub ject to strong neutrino heating and co oling and our
high resolution calculations reveal a turbulent unordered and dynamically changing
pattern of rising and sinking lumps of material with very di erent thermo dynamical
conditions and with no clear indication of in ows of co ol gas and out ows of hot gas at
well de ned thermo dynamical states
FIG Explosion energies as functions of time after the start of the simulations
ab out ms after core b ounce for the set of explo ding D mo dels dotted lines
and D mo dels solid lines The numbers indicate the size of the initial and
e e
luminosities in units of erg s
D mo dels explo de for core neutrino luminosities which cannot pro duce explosions
in D There is a window of neutrino uxes with a width of ab out of the threshold
luminosity for explosions in D where convective overturn b etween the gain radius and
the sho ck is a signi cant help for sho ck revival For lower neutrino uxes even convective
overturn cannot ensure strong explosions but the explosion energy gets very low We do
not nd a continuous accumulation Herant et al of energy in the convective
shell until an explosion energy typical of a Type I I sup ernova is reached For neutrino
uxes that cause p owerful explosions already in D turbulent overturn o ccurs but is
not crucial for the explosion In fact in this case the e ects asso ciated with the fast rise
of bubbles of heated material lead to a less vigorous start of the explosion and to the
saturation of the explosion energy at a somewhat lower level Fig The explosion
energy de ned as the net energy of the expanding matter at in nity do es not exceed
erg earlier than after ab out ms of neutrino heating This is the characteristic
time scale of neutrinos to transfer an amount of energy to the material that is roughly
equal to its gravitational binding energy and it is also the time scale that the convective
overturn b etween gain radius and sho ck needs to develop to its full strength It is
not p ossible to determine or predict the nal explosion energy of the star from a short
p erio d of only ms after sho ck formation Typically the increase of the explosion
energy with time levels o not b efore ms after core b ounce followed by only a
very slow increase due to the much smaller contributions of the few M of matter
blown away from the protoneutron star in the neutrino wind Fig Since the wind
material is heated slowly and can expand as so on as the internal energy p er nucleon
roughly equals its gravitational binding energy the matter do es not have a large kinetic
energy at in nity
Although the global evolution of p owerful explosions in D i e the increase of the
explosion energy with time the sho ck radius as a function of time or even the amount of
Ni pro duced by explosive nucleosynthesis is not much di erent from energetic explo
sions of spherically symmetrical mo dels the structure of the sho ck and of the thick layer
of expanding dense matter b ehind the sho ck clearly show the e ects of the turbulent
activity The sho ck is deformed on large scales and its expansion velocity into di erent
directions varies by ab out The material b ehind the sho ck reveals large scale
inhomogeneities in density temp erature entropy and velocity these quantities showing
contrasts of up to a factor of The typical angular scale of the largest structures is
o o
b etween ab out and We do not nd indications that the turbulent pattern tries
to gain p ower on the largest p ossible scales and to evolve into the lowest p ossible mo de
l Herant et al Turbulent motions are still going on in the extended
dense layer b ehind the sho ck when we stop our simulations at ab out second after
core b ounce We consider them as the origin of the anisotropies inhomogeneities and
non uniform distribution of radioactive elements which were observed in SN A The
contrasts found b ehind the sho ck in our mo dels are ab out an order of magnitude larger
than the arti cial p erturbations that were used in the hydrodynamical simulations to
trigger the growth of Rayleigh Taylor instabilities in the stellar mantle and envelope
From Core Bounce to Second
Convective overturn outside of the protoneutron star develops within ab out ms
after sho ck formation Ab out ms after b ounce neutrinos have dep osited a siz
able amount of energy in the material b elow the sho ck front The turbulent layer b egins
to move away from the region of strongest neutrino heating and to expand outward b e
hind the accelerating sho ck This is the time when we nd turbulent activity around
the protoneutron star to come to an end Our mo dels do not give an extended phase of
convection and accretion outside of the protoneutron star The in ows of low entropy
proton rich gas from the p ostsho ck region towards the neutrino heated zone are not ac
creted onto the protoneutron star Although the gas loses lepton number while falling
in it do es not get as neutron rich as the material inside the gain radius In addition
neutrino heating and mixing with the surrounding high entropy gas increase the en
tropy in the down ows Both high electron proton concentration and high entropy
have a stabilizing e ect and prevent the p enetration of the gas through the gain radius
into the co oler and more neutron rich surface layer of the protoneutron star
FIG Formation of the reverse sho ck in a D mo del with an explosion energy close
to erg The evolution of D mo dels lo oks very similar when angle averaged
quantities are plotted but the dense shell b ehind the sho ck is inhomogeneous and
turbulent The dash dotted line is the entropy pro le in k nucleon at the start of
B
the simulation t s ab out ms after sho ck formation The dotted lines show the
density pro les at times t ms and t ms and the solid lines display
the velocity in cm s at times t and ms
resp ectively
At ab out ms the protoneutron star has b ecome quite compact already and
the density outside of it has dropp ed appreciably This indicates the formation of the
high entropy low density hot bubble region Bethe Wilson and the phase of
small mass loss from the nascent neutron star in the neutrino driven wind accompanied
by slowly increasing entropies The wind expansion accelerates as a consequence of the
steep ening of the density decline b etween the shrinking neutron star and the evacuating
bubble region The push from b elow creates a density inversion b etween the dense
slowly moving and inert shell b ehind the sho ck and the low density hot bubble region
At around the time the outgoing sup ernova sho ck reaches the entropy and comp osition
discontinuity of the Si O interface at ab out km this density inversion steep ens into
a strong reverse sho ck that forms a sharp discontinuity in the neutrino wind slowing
down the wind expansion from more than cm s to a few cm s Fig We
do not know yet how this reverse sho ck develops on a longer time scale Since the
velocities of the wind and of the layer b ehind the sho ck decrease with time it is very
likely that the fallback of a signi cant fraction of the matter that was blown out in the
neutrino wind will b e triggered Once the infall of the outer wind material is initiated
and the pressure supp ort of the gas further out vanishes the inward acceleration might
even enforce the fallback of the more slowly moving parts of the dense shell b ehind the
sup ernova sho ck
Fallback of a signi cant amount of matter b etween and M has to b e
p ostulated to solve two ma jor problems in the current sup ernova mo dels On the one
hand due to the fast development of the explosion and the lack of an extended phase of
accretion onto the collapsed stellar core the protoneutron star formed at the center of
the explosion has quite a small initial baryonic mass only ab out M in case of our
M star with the M iron core On the other hand the explosive nucleosynthesis
yields of iron p eak elements are incompatible with observational constraints for Type I I
sup ernovae as deduced from terrestrial abundances and galactic evolution arguments
In case of p owerful explosions with explosion energies of erg ab out M
of material are heated to temp eratures ab ove K and are ejected b ehind the
sho ck during the early phase of the explosion Only roughly half of this matter
M has an electron fraction Y and will end up with Ni as the dominant
e
nucleosynthesis pro duct In that resp ect the mo dels seem to match the observations
quite well Yet only some part ab out M of the matter that is sho ck heated
to T K has an electron fraction Y and will end up with relative
e
abundance yields in acceptable agreement with solar system values The amount of Ni
pro duced in neutrino driven explosions turns out to b e correlated with the explosion
energy In case of more energetic explosions the sho ck is able to heat a larger mass to
su ciently high temp eratures
DISCUSSION AND CONCLUSIONS
Turbulent overturn b etween the zone of strongest neutrino heating and the sup ernova
sho ck aids the re expansion of the stalled sho ck and is able to cause p owerful Type
II sup ernova explosions in a certain although rather narrow window of core neutrino
uxes where D mo dels do not explo de The turbulent activity outside and close to
the protoneutron star is transient and b etween and ms after core b ounce the
essentially spherically symmetrical neutrino wind phase starts and the turbulent shell
moves outward b ehind the expanding sup ernova sho ck Our D simulations do not
show a long lasting p erio d of convection and accretion after core b ounce Only very
little of the co ol low entropy matter that ows down from the sho ck front to the zone
of neutrino energy dep osition is advected into the protoneutron star surface Since the
matter is proton rich and its entropy increases quickly due to neutrino heating it stays
in the heated region to gain more energy by neutrino interactions and to start rising
again The strong large scale inhomogeneities and anisotropies in the expanding layer
b ehind the outward propagating sho ck front will probably help to explain the e ects of
macroscopic mixing observed in SN A and can account for mo derately high recoil
velocities of the neutron star for details see Janka M uller b
Although the D mo dels develop energetic explosions for su ciently high neutrino
luminosities and pro duce an amount of Ni that is in go o d agreement with observational
constraints the initial mass of the protoneutron star is clearly on the low side of the
sp ectrum of measured neutron star masses Moreover the mo dels eject ab out
M of material with Y which implies an overproduction of certain elements
e
in the iron p eak by an appreciable factor compared with the nucleosynthetic comp osition
in the solar system The fallback of a signi cant fraction of this matter to the neutron
star at a later stage of the evolution would ease these problems It is p ossible that the
reverse sho ck which develops in our mo dels around the time when the sup ernova sho ck
passes the entropy discontinuity of the Si O interface will trigger this fallback on a time
scale of seconds Due to the strong inhomogeneities in the dense layer b ehind the sho ck
this fallback could happ en with considerable anisotropy and impart an additional kick
to the neutron star Janka M uller b
ACKNOWLEDGEMENTS
We are very grateful to S W Bruenn for providing us with the data of his core collapse
calculations which we used to construct the initial mo dels for our simulations This
work was supp orted in part by the National Science Foundation under grant NSF AST
by the National Aeronautics and Space Administration under grant NASA
NAG and by an Otto Hahn Postdoctoral Scholarship of the Max Planck Society
H Th J The computations were p erformed on the CRAY YMP of the Rechen
zentrum Garching
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