Modelling SN Type II: collapse and simple bounce

From Woosley et al. (2002)

Woosley Lectures 13 and 14

(Fry

er &

Kal

oger

a 20

01;

see

also

: B

urro

ws

1999

)

Eje

cted

“m

etal

s”

Eje

cted

“m

etal

s”

Eje

cted

“m

etal

s”

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

• He burning

• collapse and energy release

• + ! e+ + e-: 1 < 4/3

• Dynamical collapse, bounce, explosive burning (for M < 260 M¯)

• Dynamical collapse directly to black hole (for M > 260 M¯)

Pop. III stars, no mass loss

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 coregrows in discontinuous spurts. It approaches instability.

Pressure is predominantly due to relativistic electrons. As they become increasingly relativistic, the structural adiabatic indexof 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.

En

trop

y (S

/NAk

)

En

trop

y

Because of increasing degeneracy the concept of a ChandrasekharMass for the iron core is relevant – but it must be generalized.

0 implies degeneracy

The Chandrasekhar Mass

2

3 3 4 4 3 24/3

4 4

15 -24/3

Traditionally, for a fully relativistic, completely degenerate gas:

20.745

1.2435 10 dyne cm

1.457 M at 0.5

0

5

83

.

c c e

Ch

c c

e

eM Y

P K Y G M

K

Y

BUT

1) Ye here is not 0.50 (Ye is actually a functionof mass)

2) The electrons are not fully relativistic in the outer layers ( 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 Starspages 156ff. They find a critical density (entropy = 0).

Above this density the white dwarf is unstable to collapse. For Ye = 0.50this corresponds to a mass

2/3 1

1.415 M

in general, the relativistic correction to the Newtonian value is

M 0.50 9 4 310

M

2.87% 0.50

2.67% 0.45

Ch

e

e

e

M

Y

Y

Y

Coulomb Corrections

Three effects must be summed – electron-electron repulsion, ion-ionrepulsion and electron ion attraction. Clayton p. 139 – 153 gives a simplified treatment and finds, over all, a decrement to the pressure(eq. 2-275)

2/3 2 4/33 4

10 3Coul eP Z e n

Fortunately, the dependence of this correction on ne is the sameas relativistic degeneracy pressure. One can then just proceed to use a corrected

1/32 5/30 2/3

4/3 4/3

0 3 2/34/3

2 31

5

1 4.56 10

eK K Z

c

K Z

1/30 2 4/34 /3

3/ 2

Ch0 0Ch

2/30

Ch

where 34

Mand

M

hence

M 1 0.02266

A

Ch

cK N

K

K

ZM

2e

12Ch e

56

Putting the relativistic and Coulomb corrections together

with the dependence on Y one has

M 1.38 M for C (Y 0.50)

= 1.15 M for Fe (Y

e

e

260.464)

56 = 1.08 M for Fe-core with <Y 0.45

So why are iron cores so big at collapse (1.3 - 2.0 M ) and

why do neutron stars have masses 1.4 M

M¯

M¯

M¯

M¯

¯

2 2 24/30

4/3 2 2 4

1/31/3

For 3, 4 / 3, relativistic degeneracy

1 21 1 ...

30.01009

8

(Clayton 2-48)

c ee

Fe e

e

n

k TP K Y

x m c

phx n Y

mc m c

Finite Entropy Corrections

Chandrasekhar (1938)Fowler & Hoyle (1960) p 573, eq. (17)Baron & Cooperstein, ApJ, 353, 597, (1990)

In particular, Baron & Cooperstein (1990) show that

2

1/33

1/37

3/ 24/3

2

0

21 ...

3

3

8

1.11( ) MeV

and since a first order expansion gives

1

oF

eF F

F e

Ch

Ch ChF

kTP P

h np c

Y

M K

kTM M

And since early on we showed that

2

2

0

(relativistic degeneracy)

one also has

1 ...

ee

F

eCh Ch

e

kTYs

sM M

Y

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 importantfor 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 captureas 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 acceleratesto 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 NSEdistribution favors particles over bound nuclei. In fact, this transitionnever goes to completion owing to the large statistical weight afforded the excited states of the nuclei. But considerable energy is lost ina partial transformation.

56

1756

18 -1

13 "4 "

28.296 492.2629.65 10

4 56

1.7 10 erg g

nuc

photo

Fe n

q X

q

not really free neutrons.They stay locked insidebound nuclei that areprogressively 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., < 4/3. The collapse accelerates.

Photodisintegration also decreases se because at constant total entropy (the collapse is almost adiabatic), si increases since 14particleshave more statistical weight than one nucleus. The increase in si comesat 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 inturn makes the pressure at a given density smaller.

1/3

71.11 MeVF eY

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 intoone gigantic nucleus with 1057 nucleons – the neutron star.

Ye declines to about 0.37 before the core becomes opaque toneutrinos. (Ye for an old cold neutron star is about 0.05; Ye forthe 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.

HHe

O

FeSi

Stars of larger mass have thicker, more massive shells of heavy elementssurrounding the iron core when it collapses.

Note that the final masses of the 15 and 25 solar mass main sequence starsare nearly the same – owing to mass loss.

HHeO

Fe Si

Distribution of collapse velocity and Ye (solid line) in the inner2.5 solar masses of a 15 solar mass presupernova star. A collapsespeed 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.

Ye

vcollapse

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 becomesvery 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 beforewe are through. Since the iron core only has a radius of 5,000 to10,000 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 crosssection

22 2 44 20

22 44 2 -1

coh

219 2 2 -1

0

220 2 -1

2 4 20

1.5 10 cmMeV

hence

1.5 10 cm gmMeV

5.0 10 cm gm56 MeV

2.6 10 cm gm56 MeV

if one takes sin ( ) (0.229) 0

coh

o A

coh

W

a A

a A N

Aa

A

a

.0524

20

W

sin ( ) where

is the "Weinberg

angle", a measure of the

importance of weak

neutral currents

Wa

19 2 6 11 -3

Therefore neutrino trapping will occur when

~1

10 10 10 ~1 ~ 4 10 g cm

(for 56)

R

A

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 pointwhere trapping occurs YL = Ye ~ 0.37. At bounce Ye~ 0.29; Y~ 0.08.

1/37

11 -3

1.11( ) MeV

~ 30MeV at

=10 g cm

F eY

Bounce

Up until approximately nuclear density the structural adiabaticindex of the collapsing star is governed by the leptons – the electrons and neutrinos, both of which are highly relativistic,hence nearly =4/3.

As nuclear density is approached however, the star first experiencesthe attactive nuclear force and goes briefly but dramaticallybelow 4/3.

At still higher densities, above nuc, the repulsive hard corenuclear force is encountered and abruptly>> 4/3.

at about point b) on previous slide

The collapse of the “iron” core continues until densities nearthe 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 homologouscore collapses in a self similar fashion. Were = 4/3 for the entire ironcore, the entire core would contract homologously, but because becomessignificantly 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 resistanceof the nuclear force is communicated throughout its volume by sound waves,but not beyond its edge. Thus the outer edge of the homologous core iswhere 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 lessdissipation 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 -particles

56 17

18 -1

51

492.26 MeV( 26 ,30 ) 9.65 10

56

8.5 10 erg gm

1.7 10 erg/0.1 M

nucq Fe p n

Neutrino losses

Especially as the shock passes to densities below 1012 g cm-3, neutrinolosses from behind the shock can rob it of energy. Since neutrinos oflow energy have long mean free paths and escape more easily, reactionsthat degrade the mean neutrino energy, especially neutrino-electron scatteringare quite important. So too is the inclusion of andflavored neutrinos

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

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 fartherout with less matter to photodisintegrate and less neutrino losseson its way out.

Collapse and bounce in a 13 solar mass supernova.Radial velocity vs. enclosedmass at 0.5 ms, +0.2 ms,and 2.0 ms with respect tobounce. The blip at 1.5 solar masses is due to explosive nuclear burningof oxygen in the infall(Herant and Woosley 1996).

Colgate and White, (1966), ApJ, 143, 626

see also Arnett, (1966), Canadian J Phys, 44, 2553 Wilson, (1971), ApJ, 163, 209

The explosion is mediated by neutrino energy transport ....

Myra and Burrows, (1990), ApJ, 364, 222

Woosley et al. (1994), ApJ,, 433, 229

Neutrino luminosities of order 10Neutrino luminosities of order 1052.552.5 are are maintained for several seconds after an maintained for several seconds after an initial burst from shock break out.initial burst from shock break out.

At late times the luminosities in each flavorAt late times the luminosities in each flavorare comparable though the are comparable though the - and - and --neutrinos are hotter than the electron neutrinos.neutrinos are hotter than the electron neutrinos.

Wilson20 M-sun

K II 2140 tons H2OIMB 6400 tons “

Cerenkov radiation from

(p,n)e+ - dominates (e-,e-) - relativistic e all flavors

Hirata et al. (1987; Phys. Rev. Lett. 58, 1490)

2

16 2 -1

14 -3

6 2

10

1~

~10 cm gm for 50 MeV (next page)

~ 3 10 gm cm ~ 30 cm R ~ 20 km

(2 10 )~ ~ 5 sec

30 3 10

Diff

Diff

Rl

l c

l

Time scale

Very approximate

Neutrino Burst Properties:

2

tot

53

e

3E ~ M = 1.5 M

5

~ 3 10 erg R = 10 km

emitted roughly equally in , , , , , and e

GM

R

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.

Scatteri2

20 2 -1

a

ng: 1.0 10 cm gmMeV

Absorption: 4

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 ~

s

s

E

16 2 -110 cm g

bounce = 5.5 x 1014 g cm-3

Explosion energy at 3.6 s 3 x 1050 erg

20 Solar Masses

Mayle and Wilson (1988)

Infall

Accretion Shock

radius

Velocity

13000 skm

gain radiusN

eutr

inos

pher

e

Inside the shock, matter is in approximate hydrostatic equilibrium.Inside the gain radius there is net energy loss to neutrinos. Outsidethere is net energy gain from neutrino deposition. At any one time thereis about 0.1 solar masses in the gain region absorbing a few percentof the neutrino luminosity.

Energy deposition here drives convectionBethe, (1990), RMP, 62, 801

(see also Burrows, Arnett, Wilson, Epstein, ...)

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

Neutron Star +

Rotation

Black Hole+

Rotation

Colgate and White (1966)ArnettWilsonBetheJankaHerantBurrowsFryerMezzacappaetc.

Hoyle (1946)Fowler and Hoyle (1964)LeBlanc and Wilson (1970)Ostriker and Gunn (1971)Bisnovatyi-Kogan (1971)MeierWheelerUsovThompsonetc

Bodenheimer and Woosley (1983)Woosley (1993)MacFadyen and Woosley (1999)Narayan (2004)

All of the above?

10 20 35 M

Gravitational Binding Energy of the Presupernova Star

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 morepronounced in metal-deficient stars.

solar

low Z

; mass cut at Fe-core

(after fall back)

Above 35 M

black holes form

in Z=0 stars