Modelling SN Type II: evolution up to collapse

From Woosley et al. (2002)

Woosley Lectures 11 and 12

Hydrogen Burning

Temperature sensitivity of nuclear reactions

The slowest reaction is 14N(p,)15O. For temperatures near 2 x 107 K.1/3

2 2

n

14 1

14 17 1-2

T n = 4.248 60.03 0.020

n =18

nuc

(More on nucleosynthesis later)

Nuclear Physics

In a low mass star

The 4 CNO cycles

CNO tri-cycle

3 4 5 6 7 8 9neutron number

C(6)

N(7)

O(8)

F(9)

Ne(10)

CN cycle (99.9%)

O Extension 1 (0.1%)

O Extension 2

O Extension 3

All initial abundances within a cycle serve as catalysts and accumulate at largest

Extended cycles introduce outside material into CN cycle (Oxygen, …)

Energy production

4.2 £ 1018 erg/g for X0 = 0.7

Helium Burning

Precursor reactions

14N(, )18F(e+)18O

Helium Burning

Helium burning is a two-stage nuclear process in which two alpha-particles temporarily form the ground state of unstable 8Be*.Occasionally the 8Be* captures a third alpha-particle before it fliesapart. No weak interactions are involved.

AFor binary reactions, N v

2 3 12 12αα 3α α αγ

122 3 12 12

α 3α α αγ

1612 12

α αγ

dY= -3ρ Y λ /6 - Y Y( C)ρ λ ( C)

dt

dY( C)= ρ Y λ /6 - Y Y( C)ρ λ ( C)

dt

dY( O)= Y Y( C)ρ λ ( C)

dt

12

12

12

16

For Y small or large

C

For Y large or small

O

Helium Burning

?

In a 15 solar mass star:

123( ) / 3C

Energy production

[5.85 + 2.86 X(16O)] £ 1017 erg/g

Woosley et al. (2002; RMP 74, 1015)

Neutrino loss mechanisms

Itoh et al. (1989; ApJ 339, 354)

Carbon Burning

Approximate initial conditions:

As we shall see, the temperature at which carbon burns in a massive stars is determined by a state of balanced power between neutrino losses by the pair process and nuclear energy generation.This gives 8 x 108 K for carbon core burning. Burning in a shell isusually a little hotter at each step, about 1.0 x 109K for carbon burning. Assuming that T3/scaling persists at the center, and that helium burned at 2 x 108 K and 1000 gm cm-3, this implies a carbonburning density around 105 gm cm-3. The initial composition is the ashes of helium burning, chiefly C and O in an approximate 1 : 4 ratio (less carbon in moremassive stars). There are also many other elements present in trace amounts:

• 22Ne, 25,26Mg from the processing of CNO elements in He-burning• The s-process• Traces of other heavy elements present in the star since birth• Up to ~1% 20Ne from 16O()20Ne during He-burning

Principal nuclear reaction

Many important secondary reactions: 20Ne()24Mg 23Na(,p)26Mg 26Mg(p,)27Al 23Na(p,)24Mg 23Na(p,)20Ne 25Mg(p,)26Al 22Ne(,n)25Mg 25Mg(,n)28Si 23Mg(n,p)23Na 25Mg(n,)26Mg and dozens (hundreds?) more

There are also some important weak interactions that can change the neutron excess

• The neutron branch of 12C + 12C itself makes 23Mg. At lower temperature this decays by 23Mg(e+)23Na. At higher temperature it is destroyed by 23Mg(n,p)23Na. The former changes ; the latter does not.

• 20Ne(p,)21Na(e+)21Ne

• 21Ne(p,)22Na(e+)22Ne

Together these reactions can add to the neutron excess that wascreated in helium burning by 14N()18F(e+)18O or, in stars of lowmetallicity they can create a neutron excess where no existed before.

Principal Nucleosynthesis in carbon burning:

20,21Ne, 23Na, 24,25,26Mg, (26),27Al, and to a lesser extent, 29,30Si, 31P

The 16O initially present at carbon ignition essentially survivesunscathed. There are also residual products from helium burning – the s-process, and further out in the star H- and He-burning continue.

A typical composition going into neon burning – majorabundances only would be

70% 16O, 20% 20Ne, 5% 24Mg

D. Energy Generation

Suppose we make 20Ne and 24Mg in a 3:1 ratio (approximately solar)

12 20 24

17 -1 -1inuc i

20 12 24 12

122 12

16,16

1217

7 C 3 Ne + Mg

dYε = 9.65 × 10 BE erg g s

dt

dY( Ne) 3 dY( C) dY( Mg) 1 dY( C) = - = -

dt 7 dt dt 7 dt

dY( C)= - 2ρ Y ( C)λ /2

dt

3 1 dY( C)9.65× 10 - (160.646) - (198.258) +1(92.160)

7 7 dtnuc

1217

18 2 12 -1 -112,12

12,12

dY( C)(9.65 10 )(5.01)

dt

4.84 10 ( ) erg g s

where was given previously.nuc Y C

12,12/2

12,12 ' 3.9 £ 10-11 T928 cm3 g-1 s-1

BEi : Binding energy (MeV)

17

12 12

17

20 12 12

24 12

17 -112

9.65 10 ( )

1

12

3 9.65 10(5.01)

7 121

7

4.03 10 erg g

nuc i i

nuc

nuc

q Y BE

Y X

Y Y q X

Y Y

q X

The total energy released during carbon burning is

Since X12 ~ 0.25 << 1, this is significantly less than helium burning

actually should use a smallerradius here a longer lifetime.

No centralconvectivecarbon burning!

Burning Stages in the Life of a Massive Star

0

Neon Burning

Following carbon burning, at a temperature of about 1.5 x 109 K,neon is the next abundant nucleus to burn. It does so in a novel“photodisintegration rearrangement” reaction which basicallyleads to

20 16 242( Ne) O + Mg + energy

The energy yield is not large, but is generally sufficient to power a brief period of convection. It was overlooked earlyon as a separate burning stage, but nowadays is acknowledgedas such.

The nucleosynthetic products resemble those of carbon burningbut lack 23Na and have more of the heavier nuclei, (26),27Al, 29,30Si,and 31P.

The composition following carbon burning is chiefly 16O, 20Ne, 24Mgbut 16O is not the next to burn (influence of Z = N = 8 = magic)

Species S(MeV) energy required to remove

an -particle.16O 7.1620Ne 4.7324Mg 9.32

Before the temperature becomes hot enough for oxygen to fuse(T9 = 2.0 as we shall see), photons on the high energy tail of the Bose-Einstein distribution function begin to induce a new kind of reaction -

20Ne()16O

The -particle “photo-disintegrated out of 20Ne usually just adds back onto 16O creating an “equilibrated link” between 16O and 20Ne.Sometimes though an captures on 20Ne to make 24Mg. When this happens the equilibrium between 16O and 20Ne quickly restores the that was lost.

16 20 20 20O Ne Ne Ne

20 16 24

20

16 20

The net result is that 2 Ne O + Mg at a rate

that is determined by how fast Ne captures alpha particles

from the equilibrium concentration set up by O and Ne.

Other secondary reactions:

24Mg()28Si 27Al(,p)30Si 25Mg(.n)29Si 30Si(p,)31P 26Mg(,n)30Si etc.

Products:

some more 16O and 24Mg, 29,30Si, 31P, 26Al and a small amount of s-process.

B. Photodisintegration Reactions

At high temperatures, the inverse reaction to radiative capture, [(n,),(p,),( becomes important as there exists an appreciableabundance of -rays out on the tail of the Bose-Einstein distributionthat have energy in excess of several MeV. The reactions these energetic photons induce are called photodisintegration reactions – the major examples being (,n),(,p), and (,)

Consider

+

and

I j L

L I j

3/ 2 3/ 2

2

In equilibrium, the abundances must obey the Saha equation

For the reaction

2 exp( / )

(deriveable from considerations of entro

I j I j I jj

L L L A

I j L

n n g g A A kTQ kT

n g A h N

3/ 2

33 3/ 29 9

py and the

chemical potential and the fact that the chemical potential

of thephoton is zero). Thus

5.942 10 exp( 11.60485 / )I j I jj

L L

n n A AT Q T

n A

9

I

3/ 2

11.6048 /33 3/ 29

9 3/ 29

Equilibrium in the reaction also implies

Y ( )

and since

5.942 10( )

( ) ( ) 9.868 10

j

j j L j

ii

A

Q TI j A I j j A I j

L L j L

I j Ij j

L

I j L

Y I Y

nY

N

n n N Y Y N A AT e

n Y I A

g g A AL I T

g

3/ 2

9exp( 11.6048 / )jj

L

Q TA

339 5.942 10

where 9.686 10AN

YI

Nucleosynthesis

The principal nuclei with major abundances at the endof neon burning are 16O and 24Mg. Most of the neutron excess resides in 25,26Mg. Most of the 16O has in factsurvived even since helium burning.

In terms of major production of solar material, important contributions are made to

[16O], 24,25,26Mg, (26),27Al, 29,30Si, and 31P

Oxygen Burning:

After neon burning the lightest nucleus remaining with appreciableabundance is 16O. This not only has the lowest Coulomb barrier but because of its double magic nature, has a high -particle separationenergy. It is the next to burn. Because of its large abundance and the fact that it is a true fusionreaction, not just a rearrangement of light nuclei, oxygen burningreleases a lot of energy and is a very important part of the late stagesof stellar evolution in several contexts (e.g., pair-instability supernovae). It is also very productive nucleosynthetically. It’s chief products being most of the isotopes from 28Si to 40Ca as well as (part of)the p-process.

16 16 32 * 31

30

31

28

( ) 1.45MeV 5%

2.41MeV 5%

7.68MeV 56%

9.59MeV 34%

O O S S n

P d

P p

Si

Initial composition:

16O, 24Mg, 28Si

Nuclear reactions:

The deuteron, d, is quickly photodisintegrated intoa free neutron and proton.

proceeds through the 32S compound nucleus with a high density of resonances. Very like carbon burning.

Nucleosynthesis

28Si, 32,33,34S, 35,37Cl, 36,38Ar, 39,41K, 40,42Ca, some p-process

Element-wise: Si, S, Ar, Ca in roughly solarproportions.

Destruction of the s-process

Increasing neutronization, especially right afteroxygen disappears from the center.

Production factors only in inner solar mass of 25solar mass star near oxygen depletion (5%).

s25a28

main products Si, S, Ar, Ca, Cl, K

p-process

too big

Whole star production factors near oxygen depletion (5%)in a 25 solar mass star.

s25a28

Silicon Burning

Silicon burning proceeds in a way different from any nuclear processdiscussed so far. It is analogous, in ways, to neon burning in that itproceeds by photodisintegration and rearrangement, but it involves manymore nuclei and is quite complex.

The reaction 28Si + 28Si (56Ni)* does not occur owing to the large Coulomb inhibition. Rather a portion of the silicon (and sulfur, argon, etc.)“melt” by photodisintegration reactions into a sea of neutrons, protons, and alpha-particles. These lighter constituents add onto the silicon and heavierelements, gradually increasing the mean atomic weight until species in the iron group are produced.

Initial Composition

The initial composition depends on whether one is discussing the inner core or locations farther out in the star. It is quite different,e.g., for silicon core burning in a presupernova star and the explosive variety of silicon burning we will discuss later that goeson in the shock wave and gets ejected.

In the center of the star, one typically has, after oxygen burning, anda phase of electron capture that goes on between oxygen depletion and silicon ignition:

30Si, 34S, 38Ar and a lot of other less abundant nuclei

Farther out one has:

28Si, 32S, 36Ar, 40Ca, etc.

Historically, Si burning has been discussed for a 28Si rich composition

Quasi-equilibrium

This is a term used to describe a situation where groups of adjacent isotopes, but not all have come into equilibrium with respect to the exchange of n, p, , and .

20 16 This began in neon burning with Ne + O + and

continued to characterize an increasing number of nuclei during

oxygen burning. In silicon burning it becomes the rule

rather than the exception.

28 29 30 31 32 28

28 28n

Si Si Si P S Si

n

A typical quasiequilibrium group might inclu

n p p

de the

equilibrated reactions:

By which I mean Y( Si) Y ( Si)n

29 29n

30 29 30 30n n

Y( Si) ( Si)

Y( Si)Y ( Si) Y( Si) ( Si)

etc.n

24 45 46 60 (at least)A A

Late during oxygen burning, many isolated clusters grow and mergeuntil, at silicon ignition, there exist only two large QE groups

Reactions below 24Mg, e.g., 20Ne()24Mg and 12C()16O are, in general,not in equilibrium with their inverses (exception, 16O()20Ne whichhas been in equilibrium since neon burning).

Within the groups heavier than A = 24, except at the boundaries,the abundance of any species is related to that of another by successive application of the Saha equation.

40 32 36 40

28 28 32 36

28 32 36

32 36 40

3

( ) ( ) ( ( ). .,

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( , , ) .

Y Ca Y S Y Ar Y Cae g

Y Si Y Si Y S Y Ar

Y Si Y S Y Ar

S Ar Ca

f T Q Y etc

289

35

40

( ) ( , , ) ( )

Z-14where largest integer

214 2

14 2 . .,

1 1 2

2 1 3

p nA Ap n

n

p

p n

p n

Y Z C Z T Y Si Y Y Y

N

Z e g Cl

K

In the group that contains 28Si, one can write any abundance

Need 6 parameters: Y, Yp, Yn andY(28Si) plus T and .

32S

31P

28Si 29Si 30Si

32 30 2

30 28 2

32 28

32 32 302 2

28 30 28

For constant and T

( ) ( )

( ) ' ( )

( ) '' ( )

( ) ( ) ( )

( ) ( ) ( )

p

n

p n

Y S const Y Si Y

Y Si const Y Si Y

Y S const Y Si Y

Y S Y S Y SiY Y Y

Y Si Y Si Y Si

33 3 3 9 / 2

9 9

2 29

1(5.94 10 ) exp(328.36 / )

2

where 328.36 = BE( )/kT

( , ) n p

C T T

Y C T Y Y

This reduces the number of independent variables to 5, butwait …

Moreover there exist loops like:

p

p

n n

The situation at the end of oxygen burning is that there are two large QE groups coupled by non-equilibratedlinks near A = 45.

Early during silicon burning these two groups merge and the only remaining non-equilibrated reactions are forA < 24.

45A

24 45A

24Mg

The non-equilibrated linkhas to do with the doubleshell closure at Z = N = 20

QE

24 A 60

28Si

24Mg+

20Ne+

16O+

12C+

3

7

289, , , ( ) QET Y Si

The cluster evolves at a rate given by 24Mg()20Ne

The photodisintegration of 24Mgprovides ’s (and n’s and p’s sinceYa =CaYn

2Yp2) which add onto

the QE group gradually increasingits mean atomic weight. As a result the intermediate mass group,Si-Ca gradually “melts” into the iron group.

60 24

60 24

1

( )

i iA

i i iA

AY

N Z Y

The large QE cluster that includes nuclei from A = 24 through at least A = 60 contains most of the matter (20Ne, 16O, 12C, and are all small), so we have the additional two constraints

The first equation can be used to eliminate one more unknown, say Yp, and the second can be used to replaceYn with an easier to use variable. Thus 4 variables now specify the abundances of all nuclei heavier than magnesium . These are

, T9, , and Y(28Si)

Reaction rates governing the rate at which silicon burns:

Generally speaking, the most critical reactions will be those connecting equilibratednuclei with A > 24 (magnesium) with alpha-particles. The answer depends ontemperature and neutron excess:

Most frequently, for small, the critical slow link is 24Mg()20Ne

The reaction 20Ne()16O has been in equilibrium with 16O()20Ne ever since neon burning. At high temperatures and low Si-mass fractions,20Ne()24Mg equilibrates with 24Mg()20Ne and 16O()12C becomesthe critical link.

However for the values of actually appropriate to silicon burning ina massive stellar core, the critical rate is 26Mg(p,)23Na(20Ne

Nucleosynthesis

Basically, silicon burning turns the products of oxygen burning(Si, S, Ar, Ca, etc.) into the most tightly bound nuclei (in the irongroup) for a given neutron excess,

The silicon-burning nucleosynthesis that is ejected by a super-nova is produced explosively, and has a different composition dominatedby 56Ni and will be discussed later.

The products of silicon-core and shell burning in the core are both so neutron-rich ( so large) that they need to be left behind in a neutron star orblack hole. However, even in that case, the composition and its evolutionis critical to setting the stage for core collapse and the supernova explosion that follows.

Following Si-burning at the middle of a 25 solar mass star:

54Fe 0.48758Ni 0.14756Fe 0.14155Fe 0.07157Co 0.044

Neutron-rich nuclei in the iron peak. Ye = 0.4775

Following explosive Si-burning in a 25 solar mass supernova, interestingspecies produced at Ye = 0.498 to 0.499.

44Ca 44Ti47,48,49Ti 48,49Cr 51V 51Cr 55Mn 55Co50,52,53Cr 52,53Fe54,56,57Fe 56,57Ni59Co 59Cu58,60,61,62Ni 60,61,62Zn

product parent

Silicon burning nucleosynthesis

44Ti and 56.57Ni are importanttargets of-ray astronomy

Nuclear Statistical Equilibrium

24 24 24 20

16 20 20 16

12 16 16 12

12 12

( , ) ( , )

( , ) ( , ) (for a long time already)

( , ) ( , )

3 ( , )2

Ne Mg Mg Ne

O Ne Ne O

C O O C

C C

As the silicon abundance tends towards zero (though it never becomes microscopically small), the unequilibratedreactions below A = 24 finally come into equilibrium

Then every isotope is in equilibrium with every other isotope by strong, weak, and electromagnetic reactions(but not by weak interactions)

28 7

9

A9

1

9

In particular, Y( ) ( , ) with the result that now

only 3 variables, , T ,and now specify the abundances

of everything

Y( ) ( , , )

( , , )

A N Zn p

AAA

Si f T Y

Z C Z T Y Y

C Z T N C

'9

' 199

33 3/ 29

9

A9

( , )

( , )( , ) exp ( ) /

2

5.943 10

( , ) is the temperature-dependent

partition function.

At low T

G( , ) (

A

AA A A

A

A

o

Z T

G Z TC Z T BE Z kT

T

G Z T

Z T g

A ( )9

-1

2 1)

At high T

G( , )6

MeV9

o

a kT

J

Z T eakT

Aa

56

54

The most abundant nuclei are those with large binding energy

per nucleon and "natural" values of . Fo

= 0 Ni

0.03

r example,

7 Fe

5

56

6

In general, the abundance of an isotope peaks at its

natural value for .

0.071

E. g.,

Fe

30 - 26( Fe) =

et

0.071456

.

c

The resultant nucleosynthesis is most sensitive to

Woosley et al. (2002)

Woosley et al. (2002)