Modelling SN Type II: collapse and simple bounce

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Modelling SN Type II collapse and simple bounce
From Woosley et al. (2002) Woosley Lectures 13
and 14
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Most distant explosion smashes previous
recordNASA NEWS RELEASEPosted September 12,
2005http//www.spaceflightnow.com/news/n0509/12sw
ift/ This powerful burst was detected September
4. It marks the death of a massive star and the
birth of a black hole. It comes from an era soon
after stars and galaxies first formed, about 500
million to 1 billion years after the Big Bang.
The September 4 burst, named GRB 050904, has a
redshift of 6.29, which translates to a distance
of about 13 billion light-years from Earth. The
Universe is thought to be 13.7 billion years old.
The previous most distant gamma-ray burst had a
redshift of 4.5. The most distant quasar known is
at a redshift of 6.4. This burst was also very
long, lasting more than 200 seconds, whereas most
bursts last only about 10 seconds. The detection
of this burst confirms that massive stars mingled
with the oldest quasars.
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(Fryer Kalogera 2001 see also Burrows 1999)
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Ejected metals
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Ejected metals
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Ejected metals
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8 11 M uncertain situation

• M lt M1 ' 8 M No C ignition
• M gt M2 ' 12 M Full nondegenerate burning
• In between ????

?

• Degenerate off-centre ignition
• Possibly O-Ne-(Mg?) white dwarfs (after some
  additional mass loss)
• With sufficient O-Ne core mass continued
  burning and core collapse

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Pair-instability supernovae
Pop. III stars, no mass loss

• He burning
• collapse and energy release
• g g ! e e- G1 lt 4/3
• Dynamical collapse, bounce, explosive burning
  (for M lt 260 M)
• Dynamical collapse directly to black hole (for M
  gt 260 M)

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Possibly observed SN 2006gy
Smith et al. (2007 ApJ 666, 1116)
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Normal core collapse
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Gravitational Binding Energy of the Presupernova
Star
solar
low Z
This is just the binding energy outside the iron
core. Bigger stars are more tightly bound and
will be harder to explode. The effect is
more pronounced in metal-deficient stars.
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As silicon shells, typically one or at most
two, burn out, the iron core grows in
discontinuous spurts. It approaches
instability. Pressure is predominantly due
to relativistic electrons. As they become
increasingly relativistic, the structural
adiabatic index of the iron core hovers
precariously near 4/3. The presence of
non- degenerate ions has a stabilizing influence,
but the core is rapidly losing entropy to
neutrinos making the concept of a Chandrasekhar
Mass relevant. In addition to neutrino
losses there are also two other important
instabilities

• Electron capture since pressure is dominantly
  from electrons, removing them reduces the
  pressure.
• Photodisintegration which takes energy that
  might have provided pressure and uses it
  instead to pay a debt of negative nuclear
  energy generation.

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Entropy (S/NAk)
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Entropy
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Because of increasing degeneracy the concept of a
Chandrasekhar Mass for the iron core is relevant
but it must be generalized.
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The Chandrasekhar Mass
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Effect on MCh
BUT

• Ye here is not 0.50 (Ye is actually a
  functionof mass)
• The electrons are not fully relativistic in the
  outer layers (g is not 4/3 everywhere)
• General relativity implies that gravity is
  stronger than classical and an infinite central
  density is not allowed (there exists a critical
  r for stability)
• The gas is not ideal. Coulomb interactions
  reduce the pressure at high density
• Finite temperature (entropy) corrections
• Surface boundary pressure (if WD is inside a
  massive star)
• Rotation

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Relativistic corrections, both special and
general, are treated by Shapiro and Teukolsky in
Black Holes, White Dwarfs, and Neutron
Stars pages 156ff. They find a critical density
(entropy 0).
Above this density the white dwarf is unstable to
collapse. For Ye 0.50 this corresponds to a mass

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Coulomb Corrections
Three effects must be summed electron-electron
repulsion, ion-ion repulsion and electron ion
attraction. Clayton p. 139 153 gives a
simplified treatment and finds, over all, a
decrement to the pressure (eq. 2-275)
Fortunately, the dependence of this correction on
ne is the same as relativistic degeneracy
pressure. One can then just proceed to use a
corrected
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Finite Entropy Corrections
Chandrasekhar (1938) Fowler Hoyle (1960) p
573, eq. (17) Baron Cooperstein, ApJ, 353, 597,
(1990)
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In particular, Baron Cooperstein (1990) show
that
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And since early on we showed that
The entropy of the radiation and ions also
affects MCh, but much less. This finite entropy
correction is not important for isolated white
dwarfs. Theyre too cold. But it is very
important for understanding the final evolution
of massive stars.
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• But when Si burning in this shell is complete
• The Fe core is now 1.3 M
• se central 0.4
• se at edge of Fe core 1.1
• hence average se ' 0.7
• MCh now about 1.34 M (uncertain to at least a
• few times 0.01 M

Neutrino losses farther reduce se. So too do
photodisintegration and electron capture as we
shall see. And the boundary pressure of the
overlying silicon shell is not entirely
negligible.
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Electron
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The collapse begins on a thermal time scale
and accelerates to a dynamic implosion as other
instabilities are encountered. Photodisintegratio
n As the temperate and density rise, the
star seeks a new fuel to burn, but instead
encounters a phase transition in which the
NSE distribution favors a-particles over bound
nuclei. In fact, this transition never goes to
completion owing to the large statistical weight
afforded the excited states of the nuclei. But
considerable energy is lost in a partial
transformation.
not really free neutrons. They stay locked
inside bound nuclei that are progressively more
neutron rich.
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What happens? As the density rises, so
does the pressure (it never decreases at the
middle), but so does gravity. The rise in
pressure is not enough to maintain hydrostatic
equilibrium, i.e., G lt 4/3. The collapse
accelerates. Photodisintegration also
decreases se because at constant total entropy
(the collapse is almost adiabatic), si increases
since 14 a-particles have more statistical weight
than one nucleus. The increase in si comes at the
expense of se.
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Electron capture
The pressure predominantly comes from
electrons but as the density increases, so does
the Fermi energy, eF. The rise in eF means more
electrons have enough energy to capture on nuclei
turning protons to neutrons inside them. This
reduces Ye which in turn makes the pressure at a
given density smaller.
By 2 x 1010 g cm-3, eF 10 MeV which is above
the capture threshold for all but the most
neutron-rich nuclei. There is also briefly a
small abundance of free protons (up to 10-3 by
mass) which captures electrons.
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But the star does not a) photodisintegrate
to neutrons and protons then b) capture
electrons on free protons and c) collapse to
nuclear density as a free neutron gas as some
texts naively describe. Bound nuclei
persist, then finally touch and melt into one
gigantic nucleus with 1057 nucleons the neutron
star. Ye declines to about 0.37 before the
core becomes opaque to neutrinos. (Ye for an old
cold neutron star is about 0.05 Ye for the
neutron star that bounces when a supernova occurs
is about 0.29). The effects of a)
exceeding the Chandrasekhar mass, b)
photodisintegration and c) electron capture
operate together, not independently.
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He
Fe
H
Si
O
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He
Fe
Si
H
O
Stars of larger mass have thicker, more massive
shells of heavy elements surrounding the iron
core when it collapses. Note that the final
masses of the 15 and 25 solar mass main sequence
stars are nearly the same owing to mass loss.
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Ye
vcollapse
Distribution of collapse velocity and Ye (solid
line) in the inner 2.5 solar masses of a 15
solar mass presupernova star. A collapse speed of
1000 km/s anywhere in the iron core is a working
definition of presupernova. The cusp at about
1.0 solar masses is the extent of convective
core silicon burning.
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Different weak interaction rates (FFN vs LM) a
few years ago gave a smaller value of Ye in
essentially the same star.
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Core Collapse
Once the collapse is fully underway, the time
scale becomes very short. The velocity starts at
108 cm s-1 (definition of the presupernova link)
and will build up to at least c/10 30,000 km
s-1 before we are through. Since the iron core
only has a radius of 5,000 to 10,000 km, the next
second is going to be very interesting.
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Neutrino Trapping
Trapping is chiefly by way of elastic neutral
current scattering on heavy nuclei. Freedman,
PRD, 9, 1389 (1974) gives the cross section
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From this point on the neutrinos will not freely
stream but must diffuse. Neutrino producing
reactions will be inhibited by the filling of
neutrino phase space. The total lepton number
YL Ye
Yn will be conserved, not necessarily the
individual terms. At the pointwhere trapping
occurs YL Ye 0.37. At bounce Ye 0.29 Yn
0.08.
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Bounce
Up until approximately nuclear density the
structural adiabatic index of the collapsing star
is governed by the leptons the electrons and
neutrinos, both of which are highly
relativistic, hence nearly G4/3. As nuclear
density is approached however, the star first
experiences the attactive nuclear force and G
goes briefly but dramatically below 4/3. At
still higher densities, above rnuc, the repulsive
hard core nuclear force is encountered and
abruptly G gtgt 4/3.
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at about point b) on previous slide
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The collapse of the iron core continues
until densities near the density of the atomic
nucleus are reached. There is a portion of the
core called the homologous core that collapses
subsonically (e.g., Goldreich Weber, ApJ, 238,
991 (1980) Yahil ApJ, 265, 1047 (1983)). This
is also approximately equivalent to the sonic
core. This part of the core is called
homologous because it can be shown that within
it, vcollapse is proportional to radius. Thus the
homologous core collapses in a self similar
fashion. Were G 4/3 for the entire iron core,
the entire core would contract homologously, but
because G becomes significantly less than 4/3,
part of the inner core pulls away from the outer
core. As the center of this inner core
approaches and exceeds rnuc the resistance of the
nuclear force is communicated throughout its
volume by sound waves, but not beyond its edge.
Thus the outer edge of the homologous core
is where the shock is first born. Typically, MHC
0.6 0.8 solar masses. The larger MHC and
the smaller the mass of the iron core, the
less dissipation the shock will experience on its
way out.
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Factors affecting the mass of the homologous
core

• YL the lepton number, the sum of neutrino
  and electron more numbers after
  trapping. Larger YL gives larger MHC
  and is more conducive to explosion. Less
  electron capture, less neutrino escape, larger
  initial Ye could raise YL.
• GR General relativistic effects decrease MHC,
  presumably by strengthening gravity.
  In one calculation 0.80 solar masses
  without GR became 0.67 with GR. This may be
  harmful for explosion but overall GR
  produces more energetic bounces and
  this is helpful.
• Neutrino transport how neutrinos diffuse out
  of the core and how many flavors are
  carried in the calculation.

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Relevant Physics To Shock Survival
Photodisintegration
As the shock moves through the outer core, the
temperature rises to the point where nuclear
statistical equilibrium favors neutrons and
protons over bound nuclei or even a-particles
Neutrino losses
Especially as the shock passes to densities below
1012 g cm-3, neutrino losses from behind the
shock can rob it of energy. Since neutrinos
of low energy have long mean free paths and
escape more easily, reactions that degrade the
mean neutrino energy, especially
neutrino-electron scattering are quite important.
So too is the inclusion of m- and t-flavored
neutrinos
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The Equation of State and General Relativity
A softer nuclear equation of state is
springier and gives a larger amplitude bounce
and larger energy to the initial shock. General
relativity can also help by making the bounce go
deeper.
Stellar Structure and the Mass of the Homologous
Core
A larger homologous core means that the shock
is born farther out with less matter to
photodisintegrate and less neutrino losses on its
way out.
The Mass of the Presupernova Iron Core
Unless the mass of the iron core is
unrealistically small (less than about 1.1 solar
masses) the prompt shock dies
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Collapse and bounce in a 13 solar mass
supernova. Radial velocity vs. enclosed mass at
0.5 ms, 0.2 ms, and 2.0 ms with respect
to bounce. The blip at 1.5 solar masses is due
to explosive nuclear burning of oxygen in the
infall (Herant and Woosley 1996).
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It is now generally agreed (despite what you
may read in old astronomy text books), that the
so called prompt shock mechanism worked on
extensively by Bethe, Brown, Baron, Cooperstein,
and colleagues in the 1980s does not work.
The shock fails and becomes in a short time (lt 10
ms) an accretion shock. It will turn to
neutrinos to actually blow up the star. But the
success of the neutrino model will depend, in
part, upon the conditions set up in the star by
the failure of the first shock. How far out did
it form? What is the neutrino luminosity? Does
convection occur beneath the neutrinosphere? So
all the factors listed on the previous pages are
still important.
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The explosion is mediated by neutrino energy
transport ....
Colgate and White, (1966), ApJ, 143, 626 see
also Arnett, (1966), Canadian J Phys, 44,
2553 Wilson, (1971), ApJ, 163, 209
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Wilson 20 M-sun
Myra and Burrows, (1990), ApJ, 364, 222
Neutrino luminosities of order 1052.5 are
maintained for several seconds after an initial
burst from shock break out. At late times the
luminosities in each flavor are comparable though
the m - and t - neutrinos are hotter than the
electron neutrinos.
Woosley et al. (1994), ApJ,, 433, 229
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K II 2140 tons H2O IMB 6400 tons Cerenkov
radiation from n (p,n)e - dominates
n(e-,e-)n - relativistic e
all flavors n
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Hirata et al. (1987 Phys. Rev. Lett. 58, 1490)
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Neutrino Burst Properties
Time scale
Very approximate
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Temperature
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20 Solar Masses Mayle and Wilson (1988)
rbounce 5.5 x 1014 g cm-3
Explosion energy at 3.6 s 3 x 1050 erg
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Energy deposition here drives convection Bethe,
(1990), RMP, 62, 801 (see also Burrows,
Arnett, Wilson, Epstein, ...)
Velocity
gain radius
radius
Neutrinosphere
Infall
Accretion Shock
Inside the shock, matter is in approximate
hydrostatic equilibrium. Inside the gain radius
there is net energy loss to neutrinos.
Outside there is net energy gain from neutrino
deposition. At any one time there is about 0.1
solar masses in the gain region absorbing a few
percent of the neutrino luminosity.
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Colgate (1989 Nature 341, 489)
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Burrows (2005)
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Beneficial Aspects of Convection

• Increased luminosity from beneath the
  neutrinosphere
• Cooling of the gain radius and increased
  neutrino absorption
• Transport of energy to regions far from the
  neutrinosphere (i.e., to where the shock is)

Also Helpful

• Decline in the accretion rate and accompanying
  ram pressure as time passes
• A shock that stalls at a large radius
• Accretion sustaining a high neutrino luminosity
  as time passes (able to continue at some
  angles in multi-D calculations even as the
  explosion develops).

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Challenges

• Tough physics nuclear EOS, neutrino opacities
• Tough problem computationally must be 3D
  (convection is important). 6 flavors of
  neutrinos out of thermal equilibrium
• (thick to thin region crucial). Must be
  followed with multi-energy group and
  multi-angles
• Magnetic fields and rotation may be important
• If a black hole forms, problem must be done
  using relativistic (magnto-)hydrodynamics
  (general relativity, special relativity,
  magnetohydrodynamics)

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When Massive Stars Die, How Do They Explode?
Neutron Star Neutrinos
Neutron Star Rotation
Black Hole Rotation
Colgate and White (1966) Arnett Wilson Bethe Janka
Herant Burrows Fryer Mezzacappa etc.
Bodenheimer and Woosley (1983) Woosley
(1993) MacFadyen and Woosley (1999) Narayan (2004)
Hoyle (1946) Fowler and Hoyle (1964) LeBlanc and
Wilson (1970) Ostriker and Gunn
(1971) Bisnovatyi-Kogan (1971) Meier Wheeler Usov
Thompson etc
All of the above?
10 20
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Gravitational Binding Energy of the Presupernova
Star
solar
low Z
This is just the binding energy outside the iron
core. Bigger stars are more tightly bound and
will be harder to explode. The effect is
more pronounced in metal-deficient stars.
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mass cut at Fe-core
(after fall back)
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