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III. Stellar Evolution, Thermonuclear Reactions,

and Stellar Nucleosynthesis

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1. Stellar Evolution The Hertzsprung- Russell diagram

M, R, L and Te do not vary

independently.

Two major relationships

L with T

L with M

The first is known as the

Hertzsprung-Russell (HR)

diagram or the colour-

magnitude diagram.

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Basic parameters to compare theory and observations:

• Mass (M)

• Luminosity (L)

– The total energy radiated per second i.e. power (in W)

• Radius (R)

• Effective temperature (Te)

– The temperature of a black body of the same radius as the star that would radiate the same amount of energy. Thus

L= 4R2 Te4

where is the Stefan-Boltzmann constant (5.67 10-8 W m-2 K-4)

Observable properties of stars

3 independent quantities

L Ld0

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Sun

1,2. Why are Stars as they are?

Trying to understand the physical

concept of stellar structure!

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)r(L)r(Tr

)r()r(

dr

)r(dT R

3264

3

2r

)r()r(GM

dr

)r(dP

)r(rdr

)r(dM 24

T(r))ρ(r) ε(r,πrdr

dL(r)

Material functions of stellar matter:

Equation of state P = P (, T, chemical composition)

Opacity R = R(, T, chemical composition)

Energy production = (, T, chemical composition) WS 2019/20 CCE.III

The equations of stellar structure We have four differential equations, which govern the

structure of stars (note – in the absence of convection).

r = radius

P = pressure at r

M = mass of material within r

= density at r

L = luminosity at r (rate of energy flow across sphere of radius r)

T = temperature at r

R = Rosseland mean opacity at r

= energy release per unit mass per unit time

Figure 12.1 Hydrostatic Equilibrium

Internal structure of the Sun

Internal models

shown for ZAMS

Sun (subscript z)

and for present

day Sun (radius

reaching out to

1.0, subscript )

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2. Nuclear Reactions

How much nuclear energy can

be gained from mass deficit

by means of fusion reactions?

Hence the only known way of producing sufficiently large amounts of

energy is through nuclear reactions. There are two types of nuclear

reactions, fission and fusion. Fission reactions, such as those that

occur in nuclear reactors, or atomic weapons can release ~ 510-4 of

rest mass energy through fission of heavy nuclei (uranium or

plutonium).

But also fusion reactions are known!

Task: How many fusion reactions of e.g. H He4 are necessary to release enough

energy to power the Sun (atomic weight of H=1.008172 and He4=4.003875)?

Given the limits on P(r) and T(r) that we have just obtained - are the central

conditions suitable for fusion ?

Mm

cmE

lost

426

2

1010 kg

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The kinetic energy of particles at central stellar temperature Tc = few 106 – 107 K is 3/2 kTc ≈ 1 keV

At large T, the Maxwellian v2

distribution decreases as

At Tc gas is fully ionized and positively charged nuclei with number of protons in each nucleus = z1, z2 act with repulsive energy of ~ 1 MeV (e= electron charge = 1.6 · 10-19 C, 0 = permittivity of free space

= 8.85 · 10-12 C2 N-1 m-2 )

To fuse nuclei, must surmount the Coulomb barrier. There is a finite probability for a particle to penetrate This barrier: “tunnel” effect

kT

mv

e 2

2

2.1. Fusion Reactions

r

ezz

0

2

21

4

kT

mv

evkT

m)v(f 22

23 2

24

:distr. Maxwell

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The Gamow peak

Quantum effect discovered by George Gamow (1928). Penetration probability (calculated by Gamow) is given as: Schematically this is plotted, and the fusion most likely occurs in the energy window defined as the Gamow Peak. The Gamow peak is the product of the Maxwellian distribution and tunnelling probability. The area under the Gamow peak determines the reaction rate:

.

vh

eZZ

e 0

221

kT

vm

vh

eZZ

ee)fusion.(probab 2

2

0

221

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2.2.The binding energy of atomic nuclei

The higher the electric charges of the interacting nuclei, the greater

the repulsive force, hence the higher Ekin and T before reactions

occur.

Highly charged nuclei are obviously the more massive, so reactions

between light elements occur at lower T than reactions between

heavy elements

In any nuclear reaction the following must be conserved:

1. The baryon number – protons, neutrons and their anti-particles

2. The lepton number – light particles, electrons, positrons,

neutrinos, and anti-neutrinos.

3. Charge

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The total mass of a nucleus is known to be less than the mass of the

constituent nucleons. Hence there is a decrease in mass if a

companion nucleus is formed from nucleons, and from the Einstein

mass-energy relation E = m c2

the mass deficit is released as energy. This difference is known as the

binding energy of the compound nucleus. Thus if a nucleus is

composed of Z protons and N neutrons, it’s binding energy is

For our purposes, a more significant quantity

is the total binding energy per nucleon:

And we can then consider this number

relative to the hydrogen nucleus

2c)N,Z(mNmZm)N,Z(Q np

A

)N,Z(Q

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Binding energy per nucleon

Peak binding energy: 56Fe

Nuclei with A > 56 form via p, s, r processes.

More stable if A is a multiple of 4:

4He 8Be 12C 16O 20Ne 24Mg 28Si …. 56Fe

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The variation of binding energy per nucleon with baryon number A

• General trend is an increase of Q with atomic mass up to A= 56 (Fe).

Then slow monotonic decline.

• Steep rise from H through 2H, 3He, to 4He fusion of H to He should

release larger amount of energy per unit mass than say fusion of He

to C.

• Energy may be

gained by fusion

of light elements

to heavier, up to

iron Fe.

• Or from fission of

heavy nuclei into

lighter ones down

to Fe.

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2.3. p-p chain The most important series of fusion

reactions are those converting H to He

(H-burning). This dominates ~90% of

lifetime of nearly all stars.

• Fusion of 4 protons to give one 4He is

completely negligible.

• Reaction proceeds through steps –

involving close encounter of 2 part.s.

• We will consider the main ones:

the pp-chain and the CNO cycle

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Solar Composition Change

Figure 12.2

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Relative importance

of PPI and PPII chains

(branching ratios)

depend on conditions

of H-burning (T, ,

abundances). The

transition from PPI to

PPII occurs at

temperatures in

excess of 1.3107 K.

Above 3107 K the

PPIII chain dominates

over the other two,

but another process

takes over in this

case.

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Further p-He paths

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PPI Chain

1 p + p d + e+ + e

2 d + p 3He +

3 3He + 3He 4He + 2p

PPII Chain

3' 3He + 4He 7Be +

4' 7Be + e 7Li + e

5' 7Li + p 4He + 4He

PPIII Chain

4'' 7Be + p 8B +

5'' 8B 8Be + e+ + e

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or

Bethe-Weizäcker cycle

2.4. CNO Cycle

At birth present stars

contain a small (2%)

mix of heavy elements,

some of the most

abundant of which are

carbon, oxygen, and

nitrogen (CNO).

These nuclei may

induce a chain of H-

burning reactions in

which they act as

catalysts.

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Timescale of the CNO Cycle

1 12C + p 13N +

2 13N 13C + e+ + e

3 13C + p 14N +

4 14N + p 15O +

5 15O 15N + e+ + e

6 15N + p 12C + 4He

In the steady state case, the abundances of isotopes must take values such that the isotopes which react more slowly have higher abundance. The slowest reaction is p capture by 14N. Hence most of 12C is converted to 14N.

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pp chain and CNO Cycle: T dependence

The two processes have very different temperature dependences.

The rate of energy production in each:

64

0

2

0

.

HppT

TX

716

601025

.

NCNOHCNO

TfXX

K 112

1

7

501071

.

CN

H

X

X.T

Below this temperature the pp

chain dominates, and above it

the CNO Cycle, that is in stars

slightly more massive than the

Sun e.g. 1.2-1.5M

.

Equating these two gives the T

at which they produce the

same rate of energy:

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Red Giant state Figure 12.4

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• Stage 8 - Subgiant branch

• Stage 9 - Red Giant branch

• H depleted at center, He core grows

• Core pressure decreases,

gravity not

• He core contracts, H shell

burning increases

• Star’s radius increases,

surface cools, lumin. increases

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2.5. He Burning: Triple- Process

Simplest reaction in a helium gas should be the fusion of two

helium nuclei. There is no stable configuration with A=8. For

example the beryllium isotope 8Be has a lifetime of only 2.610-16 s.

4He + 4He 8Be But a third He nucleus

can be added to 8Be

before decay,

forming 12C by the

“triple-” reaction 4He + 4He 8Be 8Be + 4He 12C +

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• Red Giant core contracts (no

nuclear burning there)

• Central temperature reaches 108 K

• Fusion of He starts abruptly -

Helium flash for a few hours

• Star re-adjusts over 105 years

from stage 9 to 10

• H and He burning with C core -

horizontal branch

Figure 12.5

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Horizontal Branch stage 10: He Fusion

Reascending the Giant Branch

• C core contracts (no

nuclear burning there)

• Gravitational heating

• H and He burning

increases

• Radius and luminosity

increases

Figure 12.7

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Evolution of a Sun-like Star

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Maeder & Meynet (1989)

When they die their nucleosynthe-sis products are fixed. Depending on their mass loss, at least part of fresh elements are released to the interstellar medium.

2.6. Stellar lifetimes

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from binary stars

tMS (yrs) ≈ 1010 (M/M )-2.8 o .

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3. Stars as Manufactories of Elements

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3.1 Nucleosynthesis by intermediate-mass Stars

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65 Intermediate-mass stars (M 1 ... 8-10 M

) are characteristic

for their He, C and N production and enrichment. CCE.III WS 2019/20

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Thermal pulses during the AGB phase lead

to convection and by this to a mixing of

different burning layers.

Lattanzio (2002)

Lattanzio & Karakas (2001)

Typical s-process elements: 14N, 22Na, 25Mg, 26Al, Ba, Xe

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H shines in red

[OIII] 4959 Å, 5007 Å in the green

blue color from [NeIII] 3869 Å u.a.

Rosetta Nebula Ring Nebula

Intermediate-mass Stars

end-up by PNe and

release Elements

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3.2. Nucleosynthesis by Massive Stars

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Massive stars release their

nucleosynthesis products

by peeling of the outer-

most layers by stellar

winds and final typeII

supernova explosions.

The most abundant

elements produced and

released by massive stars

are: 16O, 20Ne, 24Mg, 28Si, 32S

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3.3. Wolf-Rayet Bubbles are hot

NGC 6888

HII regions are used to determine ISM abundances: unaffected by released elements?

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Maeder & Meynet (1987)

WR winds peel off burning shells – Release of C, N, O - Influence on abundance determination?

Massive stars produce heavy elements up to 56Fe. The most abundant elem.s are produced by subsequent α-burning shells and released winds and as supernovae II: 16O, 20Ne, 24Mg, 28Si, 32S

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Pre-Supernova “Onion Skin” Structure

(g cm-3) T (108 ºK) 102 0.2

104 2

105 5

106 10

106 20

107 40

Core collapses at M ~ 1.4M

Fe-Ni

core

Prominent

constituents

H, He, 2% CNO, 0.1% Fe

He, 2% 14N

12C, 16O, 2% 22Ne

16O, 20Ne, 23Na, 24Mg

16O, 24Mg, 23Na

28Si

Heavy elements

settle into layers.

Shell burning at

interfaces.

Composition of

layers dominated by

more stable nuclei

(a multiple of 4)

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3.4. Major nuclear burning processes

Common feature is release of energy by consumption of nuclear fuel.

Rates of energy release vary enormously. Nuclear processes can also

absorb energy from radiation field, we shall see consequences.

Nuclear

Fuel

Process Tthreshold

106K

Products Energy per nucleon

(Mev)

H PP ~4 He 6.55

H CNO 15 He 6.25

He 3 100 C,O 0.61

C C+C 600 O,Ne,Ma,Mg 0.54

O O+O 1000 Mg,S,P,Si ~0.3

Si Nuc eq. 3000 Co,Fe,Ni <0.18

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Carbon burning: 12C + 12C 24Mg + γ

23Na + p

20Ne + α

16O + 2α

Neon burning: 20Ne + α 24Mg + γ

20Ne + γ 16O + α

Photodesintegration above 1.5 109 K

Oxygen burning: 16O + 16O 32S + γ

31S + n

31P + p

28Si + α

24Mg + 2α

16O + α 20Ne + γ at T~109 K

Silicon burning: 28Si + 28Si 56Ni + γ

56Ni 56Co + e+ + νe

56Co 56Fe + e+ + νe

CCE.III WS 2019/20

Periodic Table

16O + 16O 32S + energy 4He + 16O 20Ne + energy

Light Elements Heavy Elements

4 (1H) 4He + energy 3(4He) 12C + energy 12C + 12C 24Mg + energy 4He + 12C 16O + energy 28Si + 7(4He) 56Ni + energy 56Fe C-N-O Cycle

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Stellar Nucleosynthesis

p

n

Add protons or

neutrons,

then beta decay back

to valley of stability.

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Many low-mass stars. Long lives (> Hubble time).

A few high-mass stars: Quickly go Supernova (SN),

enrich the ISM with metals. M* > 8 M

SN => enrichment of ISM

M* < 8 M

retain most of their metals, => little enrichment of ISM

Intermediate mass stars: ( 1 < M*/M < 8 )

Make He, … C, N, O, …, Fe, but no SN.

Some ISM enrichment ( stellar wind, planetary nebulae )

but most metals stay locked up in collapsed remnant

( ~ 0.5-0.8 M

WD ).

3.5. Role of Supernovae

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Exploding Supernova Ejecta

Remnant

NS or BH

explosive O-burning (s-process?, p-

process?)

explosive C-burning (s-process?)

explosive He-burning (r-process?)

explosive H-burning (p-process?)

explosive Si-burning

Stellar surface

Core collapse (few ms),

bounces at nuclear density,

outward shock wave (few

hours), ignites shells and

expels the stellar envelope

(V>104 km/s).

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3.6. Neutron Capture In stars and in explosions free neutrons are captured by heavy

elements: AX + n A+1X + γ

Rapid neutron capture leads to r-process elements with large N, i.e.

n-capture faster than β-decay n p + e- + νe.

Slow (s-)process isotopes decay to larger p numbers. CCE.III WS 2019/20

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Origin of Heavy Elements

Problem: Coulomb repulsion too high for fusion beyond Fe Solution: Neutrons

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Neutron Capture Processes

r-process

s-process • Capture several n’s

• beta decay

• Repeat • Capture one n

• beta decay

• Wait for next n

• Repeat

i-process? WS 2019/20 84 CCE.III

Neutron capture processes: classical picture

neutrons

pro

tons

N=50

N=82

dn~1023cm-3

T~109K

r (rapid) (n, ) (,n) b

dn~106cm-3

T~107K

s (slow)

s process, close to stability, slow

neutron captures

r process, far from stability, rapid

neutron captures

- Sn : location of (n,g)- (g,n) equil.

- T1/2 : accumulation time, increase Z

- Pn smoothing of abundance curve

- sn going further from stability

- fission when reaching highest Z.

s r r s

Solar abundance

Mass Number

Log

(Ab

un

dan

ce)

Sneden & Cowan 2003

135 131 134 132 133

133 135 134

137 138 136 134 135

136

Xe

Cs

Ba

136

r only

r process

s only

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r-process

s-process

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Neutron Capture Processes

• Main site: Late stages of stars

of 1-3 M☉ during AGB phase.

• Require > 1 Gyr from birth.

• Should not contribute in the

early Galaxy

• Nuclear physics inputs are well

constrained by experiments.

s-process r-process

• NS-NS and NS-BH mergers

(GW170817!)

• CCSN: neutrino-wind Z≲50,

Jets/MR CSSN?

• Associated with massive stars

• Problem: Considerable

theoretical uncertainties

Early Galaxy: Only r-process should

operate in the early Galaxy

All heavy element pattern should

look like Solar-r 87

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Neutron Capture: Speed Matters

r-process

departs from

valley of stability

s-process

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Elemental breakdown in r and s components

143 144

s r r s

Mass Number

Log

(So

lar

abu

nd

ance

)

Sneden & Cowan 2003

In the solar system:

Eu is a pure r element

Ba is mainly an s

element

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r-process: Rapid neutron capture:

High n0 flux: absorb many n0s before b emission.

n

Fe56

26 Fe60

26

 

27

61Co

b

n

n

Fe59

26….

n

These processes require energy. Occur only at high r & T :

Core & shell burning: p & s process

Supernovae: p & r process

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Nucleosynthesis Sites and Production Timescales

Massive stars (M > 10 M

) and SNe II: synthesis of most of the

nuclear species from oxygen through zinc, and of the r-process heavy

elements ( < 108 years)

Red Giant Stars (1 < M < 10 M

): synthesis of carbon and of the

heavy s-process elements ( > 109 years)

SNe Ia: synthesis of the 1/2-2/3 of the iron peak nuclei not produced

by SNe II ( > 1.5-2 x109 years)

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Slow Neutron Capture: s-Process

Main neutron source: 13C(α,n)16O

22Ne(α,n)25Mg

Typical s-process elements: 14N, 22Na, 25Mg, 26Al, Ba, Xe

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Intermediate mass AGB > 3 M Higher temperature 3 108 K 14N(,)18F(b)18O(,)22Ne(,n) Flux 1013 n cm-3

Duration 10 y, A ≈ 60-80 Weak component

Low mass AGB stars < 3 M Low temperature 108 K 12C(p,)13N(b)13C(,n) Flux 108 n cm-3

Duration 105 y, A > 80 Main component

o . . o

Time

Mas

s

Neutron sources in the s process

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s process examples

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p-process: Proton capture:

s-process: Slow neutron capture:

Absorb n0, then … later … emit e- (b-particle).

Repeat. Progress up the valley of stability.

n

Ouf!

Fe56

26 Fe57

26

 

27

57Co

b

p 56Fe

Ouf!

57Co

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Solar s and r patterns

Sneden et al 2008

[Ba/Eu]r = -0.8

[Ba/Eu]s = 1.6

Very different

abundance patterns

B

a E

u

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Sneden et al. 2003

Robust r-process pattern

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