Download Power-point slides for Lecture 5

Modelling SN Type II: collapse and
simple bounce
From Woosley et al. (2002)
Woosley Lectures 13 and 14
(Fryer & Kalogera 2001;
see also: Burrows 1999)
Ejected “metals”
Ejected “metals”
Ejected “metals”
8 – 11 M¯: uncertain situation
• M < M1 ' 8 M¯: No C
ignition
• M > 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
?
Pair-instability supernovae
Pop. III stars, no mass loss
• He
burning
• collapse and energy release
• g + g ! e+ + e-: G1 < 4/3
• Dynamical collapse, bounce, explosive burning (for
M < 260 M¯)
• Dynamical collapse directly to black hole (for M >
260 M¯)
Possibly observed: SN 2006gy
Smith et al. (2007; ApJ 666, 1116)
Normal core collapse
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 nondegenerate 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.
Entropy (S/NAk)
Entropy
Because of increasing degeneracy the concept of a Chandrasekhar
Mass for the iron core is relevant – but it must be generalized.
  0 implies degeneracy
The Chandrasekhar Mass
Traditionally, for a fully relativistic, completely degenerate gas:
Pc3

4
c

K 43/ 3 c4 Ye4
c4
G3M 2

20.745
K 4 / 3  1.2435  1015 dyne cm -2
M Ch  5.83 Ye2
 1.457 M ¯ at Ye  0.50
BUT
1) Ye here is not 0.50 (Ye is actually a function
of mass)
2)
The electrons are not fully relativistic in the
outer layers (g is not 4/3 everywhere)
3)
General relativity implies that gravity is stronger
than classical and an infinite central density is not
allowed (there exists a critical  for stability)
4)
The gas is not ideal. Coulomb interactions reduce
the pressure at high density
5)
Finite temperature (entropy) corrections
6)
Surface boundary pressure (if WD is inside a
massive star)
7)
Rotation
Effect on MCh
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
M Ch  1.415 M¯
in general, the relativistic correction to the Newtonian value is
2/3 



M
0.50
  9  4  310 
 
M

 Ye  

  2.87%
Ye  0.50
  2.67%
Ye  0.45
1
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)
PCoul  
3  4  2 / 3 2 4 / 3

 Z e ne
10  3 
Fortunately, the dependence of this correction on ne is the same
as relativistic degeneracy pressure. One can then just proceed to
use a corrected
K4/ 3
1/ 3
2
5/ 3

e
2
3
  
0
2/3
 K 4 / 3 1  Z
  
c
5
   

 K 40/ 3 1  4.56 103 Z 2 / 3 
0
4/3
c
2 1/ 3

3  N A4 / 3

4
where
K
and
M Ch
 K 

 0
M 0Ch
K 
3/ 2
hence
0
M Ch  M Ch
2/3

Z 
1  0.0226   
 6  

Putting the relativistic and Coulomb corrections together
with the dependence on Ye2 one has
M Ch  1.38 MM¯
M¯
= 1.15 M
M¯
= 1.08 M
for 12C (Ye  0.50)
26
 0.464)
56
for Fe-core with <Ye   0.45
for 56 Fe (Ye 
So why are iron cores so big at collapse (1.3 - 2.0 M ¯ ) and
why do neutron stars have masses 1.4 MM¯
Finite Entropy Corrections
Chandrasekhar (1938)
Fowler & Hoyle (1960) p 573, eq. (17)
Baron & Cooperstein, ApJ, 353, 597, (1990)
For n  3, g  4 / 3, relativistic degeneracy
Pc  K
0
4/3


1  2 2 k 2T 2 
4/3
 1  ...  Ye 
1  2 
2 4

 x  me c

1/ 3
h  3 
x
ne 

mc  8 
pF
1/ 3

 0.01009  Ye 
me c
(Clayton 2-48)
In particular, Baron & Cooperstein (1990) show that
 2   kT 2

P  Po 1  
 ... 

 3  F 



1/ 3
 3h ne 
 F  pF c  

8



 F 1.11( 7 Ye )1/ 3 MeV
3
and since M Ch  K 43// 32 a first order expansion gives
   kT 2 
0
1  

M Ch  M Ch

  F  


And since early on we showed that
 2 kTYe
se 
F
(relativistic degeneracy)
one also has
0
M Ch  M Ch
  s 2

1   e   ... 
   Ye 



The entropy of the radiation and ions also affects MCh, but much less.
This finite entropy correction is not important for
isolated white dwarfs. They’re too cold. But it is very important
for understanding the final evolution of massive stars.
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.
The collapse begins on a thermal time scale and accelerates
to a dynamic implosion as other instabilities are encountered.
Photodisintegration:
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.
56
Fe 13a  "4 n "
 28.296 492.262 
qnuc  9.65 10 

 X 56
4
56


q photo  1.7 1018 erg g -1
17
not really free neutrons.
They stay locked inside
bound nuclei that are
progressively more
neutron rich.
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 < 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.
Electron capture
The pressure predominantly comes from electrons but as the
density increases, so does the Fermi energy, F. The rise in F
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.
 F  1.11  7Ye 
1/ 3
MeV
By 2 x 1010 g cm-3, F= 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.
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.
Fe
He
Si
O
H
Fe
Si
O
He
H
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.
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.
Different weak interaction rates
(FFN vs LM) a few years ago gave a
smaller value of Ye in essentially the
same star.
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.
Neutrino Trapping
Trapping is chiefly by way of elastic neutral current scattering
on heavy nuclei. Freedman, PRD, 9, 1389 (1974) gives the cross
section
  
44
2
1.5

10
cm

 MeV 
2
 coh  a02 A2 
hence
a0  sin 2 (W ) where
 W is the "Weinberg
angle", a measure of the
importance of weak
neutral currents
  
2
 coh  ao2 A N A 
44
2
-1
1.5

10
cm
gm

MeV


 A    
2
2
-1
 5.0 1019 a02   
cm
gm

 56   MeV 
 A   
 coh  2.6 1020      cm 2 gm -1
 56   MeV 
if one takes a02  sin 4 (W )  (0.229) 2  0.0524
2
 F 1.11( 7Ye )1/ 3 MeV
Therefore neutrino trapping will occur when
  R ~1
10 10   10  ~1
19
2
6
~ 30 MeV at
 =1011 g cm-3
  ~ 4 1011 g cm -3
(for A  56)
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 +Y
will be conserved, not necessarily the individual terms. At the point
where trapping occurs YL = Ye ~ 0.37. At bounce Ye~ 0.29; Y~ 0.08.
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 G=4/3.
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 nuc, the repulsive hard core
nuclear force is encountered and abruptly G >> 4/3.
at about point b) on
previous slide
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 nuc 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.
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.
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
 492.26 MeV 
qnuc (56 Fe  26 p,30n)  9.65 1017 

56


 8.5 1018 erg gm-1
1.7 1051 erg/0.1 M
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
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
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).
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
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
K II 2140 tons H2O
IMB 6400 tons “
Cerenkov radiation from
 (p,n)e+ - dominates
(e-,e-) - relativistic e
all flavors 
Hirata et al. (1987; Phys. Rev. Lett. 58, 1490)
Neutrino Burst Properties:
E tot
3 GM 2
~
5 R
~ 3 1053 erg
M = 1.5 M
R = 10 km
emitted roughly equally in  e ,  e ,  m ,  m ,  t , and  t
Time scale
 R2 
1
t Diff ~  
l
 
l c
 ~1016 cm 2 gm -1 for   50 MeV (next page)
 ~ 3 1014 gm cm -3
t Diff

l ~ 30 cm
 (2 106 ) 2 
~
~ 5 sec
10 
 30 3 10 
R ~ 20 km
Very approximate
At densities above nuclear, the coherent scattering
cross section (see last lecture) is no longer appropriate.
One instead has scattering and absorption on individual
neutrons and protons.
2
Scattering:  s 1.0 1020
 E 
2
-1
cm
gm


MeV


Absorption:  a  4  s
The actual neutrino energy needs to be obtained from a simulation
but is at least tens of MeV. Take 50 MeV for the example here.
Then  ~ 1016 cm 2 g -1
20 Solar Masses
Mayle and Wilson (1988)
bounce = 5.5 x 1014 g cm-3
Explosion energy at 3.6 s
3 x 1050 erg
Energy deposition here drives convection
Bethe, (1990), RMP, 62, 801
Velocity
Neutrinosphere
(see also Burrows, Arnett, Wilson, Epstein, ...)
gain radius
radius
 3000 km s 1
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.
Colgate (1989; Nature 341, 489)
Burrows (2005)
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).
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)
When Massive Stars Die,
How Do They Explode?
Neutron Star
+
Neutrinos
Colgate and White (1966)
Arnett
Wilson
Bethe
Janka
Herant
Burrows
Fryer
Mezzacappa
etc.
10
Neutron Star
+
Rotation
Hoyle (1946)
Fowler and Hoyle (1964)
LeBlanc and Wilson (1970)
Ostriker and Gunn (1971)
Bisnovatyi-Kogan (1971)
Meier
Wheeler
Usov
Thompson
etc
20
Black Hole
+
Rotation
Bodenheimer and Woosley (1983)
Woosley (1993)
MacFadyen and Woosley (1999)
Narayan (2004)
All of the above?
35 M
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.
; mass cut at Fe-core
(after fall back)
Above 35 M
black holes form
in Z=0 stars