• Energy sources

• Nuclear energy

• Nuclear reactions in stars

• Internal structure of stars

Stellar alchemy

Age of the Sun

Solar luminosity ~ 4 × 1026 W

All electric plants together ~ 2 × 1012 W

Conservation of energy

→ search for the solar energy source

(1860s)

Energy sources

Hermann von Helmholtz

William Thomson, Lord Kelvin

luminositystockenergy ~lifetime

Chemical energy

M ~ 2 × 1030 kg

If Sun made of coal → lifetime ~ 5000 ans → ± compatible with Bible (Genesis ~ 4000 B.C.)

Energy sources - 2

But Darwin’s theory of species evolution through natural selection requires at least hundreds of millions of years

→ search for other energy sources Charles Darwin

Gravitational energy

Contraction of the Sun: requires a few dozen meters per year

Contraction from the orbit of Mercury to the actual radius→ age ~ 30 millions years

→ hardly compatible with evolution of species

→ Kelvin criticizes Darwin’s theory

End of the Century: geologists estimate age of Earth to be at least 700 millions years

→ gravitational contraction insufficient

RGMmE ~

Energy sources - 3

Mass – energy equivalence

1905: Einstein discovers the equivalence of mass and energy

→ potential age reaches several billion years

→ energy stock amply enough

→ no more age problem

But new question: by which mechanism do the Sun (and other stars) transform masse into energy?

Albert Einstein

Energy sources - 4

The atomic nucleus

: atom with a nucleus made of Z protons and (A−Z) neutrons

Z = atomic number (determines type of atom and chemical properties)

A = mass number = number of nucleons (determines isotope)

Nuclear energy

XAZ

Ex: : main isotope of lithium (3p, 4n)

Protons: positive electrical charge

Neutrons: no electrical charge

→ electrostatic repulsion between protons

Nucleons bound by strong nuclear force (very intense but short range)

Li73

Mass defect

Mass of nucleus < sum of masses of nucleons

Difference = mass defect ↔ binding energy: Δm = ΔE/c2

Binding energy per nucleus: • increases from 1H to 56Fe

• decreases beyond 56Fe

Nuclear energy - 2

ΔE/A

A1H

56FeEnergy release by:

• fission of heavy nuclei

• fusion of light nuclei

(accompanied by transmutation of neutrons into protons)

Solar lifetime

M ≈ 2 × 1030 kg

Composed essentially of hydrogen 1H (~90% in number of atoms)

Nuclear fusion: 4 1H → 4He + energy

MHe = 3.9726 MH → ΔM = 0.0274 / 4 per 1H nucleus

→ ΔE ≈ 6 × 1014 J/kg

The Sun is able to convert ~10% of its hydrogen into helium:

→ ΔE ≈ 0.1 × 6 × 1014 × 2 × 1030 ≈ 1044 J

→ Δt ≈ ΔE / L ≈ 1044 / 3 × 1026 ≈ 3 × 1017 s ≈ 10 billion years

Nuclear energy - 3

Stability of nuclei

A given atom can have several isotopes

Stable isotopes have a number of neutrons:

• ≈ equal to the number of protons (light nuclei):

N = A−Z ≈ Z

• in excess of the number of protons (heavy nuclei):

N = A−Z > Z

They follow the valley of stability in the N,Z diagram Valley of stability

Nuclear energy - 3

Natural radioactivity

1896: Becquerel fortuitously discovers natural radioactivity

Several processes are idetified:

β− process corresponds to the emission of an e− by the nucleus, accompanied by transmutation of a neutron into a proton

It concerns isotopes above the valley of stability (excess of neutrons)

β+ process corresponds to the emission of a e+ (positon) by the nucleus

(isotopes with excess of protons) Henri Becquerel

eX X A

1ZAZ

eX X A

1ZAZ

Nuclear energy - 4

Natural radioactivity

The α process corresponds to the emission of a helium 4 nucleus

He X X 42

4A2Z

AZ

The remaining nucleus is generally left in an excited state

It gets back to the fundamental state, of minimum energy, by emetting a high energy photon (γ ray)

γ X X AZ

*AZ

Marie Curie

Nuclear energy - 5

The proton–proton chain

The simultaneous encounter of 4 protons is highly improbable

→ fusion of hydrogen into hélium proceeds by steps

(1) 1H + 1H → 2H + e+ + ν (Δt ~ 109 years)

Nuclear reactions in stars

ν = neutrino

• chargeless particle (and massless?)

• necessary to ensure conservation of energy and momentum

The proton–proton chain

One could have: 2H + 2H → 4He + γ

But 1H is much more numerous than 2H and the dominant reaction is

(2) 2H + 1H → 3He + γ(Δt ~ 1 s)

One could have: 3He + 1H → 4He + e+ +… but it does not work

Nuclear reactions in stars - 2

(3) 3He + 3He → 6Be (Δt ~ 106 years)

(3′) 6Be → 4He + 2 1H

The reaction rate is dominated by the slowest step, here (1)

The proton–proton chain

The pp chain needs a temperature T > 107 K in order for protons to be able to overcome the coulombian repulsion and fuse

This is eased by a quantum effect: the tunnel effect (wavefunction → nonzero probability to cross a potential barrier)

The pp chain is the dominant reaction in the solar core (T ~ 15 × 106 K)

It has some variants (pp2 and pp3) that differ in the last qteps

U

r

0

coulombian repulsion (1/r)

strong interaction

E

Nuclear reactions in stars - 3

The CNO cycle

At temperatures T > 15 × 106 K, hydrogen can fuse into helium following a reaction cycle that uses carbon nuclei already present in the star (produces of preceding generations)

12C + 1H → 13N + γ13N → 13C + e+ + ν13C + 1H → 14N + γ14N + 1H → 15O + γ15O → 15N + e+ + ν15N + 1H → 12C + 4He

(≈ 10% of solar energy)

Nuclear reactions in stars - 4

The triple alpha process

Fusion of heavier nuclei requires higher temperatures to overcome the Coulomb repulsive force

→ core of more massive stars

If T > 108 K: fusion of helium into carbon 4He + 4He → 8Be + γ8Be is highly unstable: 8Be → 4He + 4He in 10−16 s

However, from time to time, it will collide before disintegrating 8Be + 4He → 12C + γ

→ production of carbon, which is the basis of life on Earth

Nuclear reactions in stars - 5

Alpha captures by carbon and oxygen

At temperatures allowing fusion of helium into carbon, carbon nuclei can also capture an α particle: 12C + 4He → 16O + γ

Oxygen itself can also capturer an α particle: 16O + 4He → 20Ne + γ

As Z increases, higher and higher temperatures are necessary to overcome the Coulomb barrier

In stars similar to the Sun, nuclear fusion stops here

In stars of more than 8 M , additional reactions follow

Nuclear reactions in stars - 6

Combustion of carbon and oxygen

If T ~ 6 × 108 K: 12C + 12C → 20Ne + 4He 12C + 12C → 23Na + 1H 12C + 12C → 24Mg + γ

+ other reactions, some of them endothermal

If T > 109 K: 16O + 16O → 28Si + 4He 16O + 16O → 31P + 1H 16O + 16O → 31S + n

+ other reactions, some of them endothermal

Nuclear reactions in stars - 7

Combustion of silicon

If T > 3 × 109 K: 28Si + 4He

+ 4He

+ 4He

… → 56Fe

56Fe = most stable nucleus → the star cannot produce energy by fusion of Fe with other nuclei

→ reactions producing elements heavier than iron participate to nucleosynthesis but not to energy production

Nuclear reactions in stars - 8

Nucleosyntheis of heavy elements

Some of the previous reactions produce neutrons

Those neutrons can be captured by nuclei to form heavier isotopes

If these isotopes are unstable, they transmute into the following element by β− disintegration

or:

etc…

These neutron captures form the basis of the production of all chemical elements heavier than iron

eX X n X 1A1Z

1AZ

AZ

eX X n X 2A1Z

2AZ

1AZ

Nuclear reactions in stars - 9

Abundances of chemical elements

Nuclear reactions in stars are responsible for the production of the vast majority of chemical elements heavier than hydrogen and helium (+ Li, Be, B) → starting from carbon

The chemical composition of the primitive solar system can be determined by the analysis of some meteorites as well as of the solar spectrum

It is representative of what is usually found in the Univers (cosmic abundances) within a common scale factor for carbon and heavier elements

This factor is called metallicity

Nuclear reactions in stars - 10

Brown dwarfs

If M < 0.08 M and M > 0.013 M = 13 MJup

→ the core temperature never reaches the value required for valeur requise hydrogen fusion

→ gravitational contraction until R ~ RJup Tsurf ~ 1000 K

• Brief episode of deuterium fusion (allows to define the limit between brown dwarfs and planets)

• gradual cooling → L ~ 10−6 L

→ very hard to detect

First detection in 1994: Gliese 229B, double system with a main sequence star

Nuclear reactions in stars - 11

Nuclear reactions in the stellar cores (for the Sun, this core extends over 1/4 of the radius (1.6% of the volume)

Internal structure of stars

Internal structure of the Sun

Chromosphere

Convective zone

Core

Radiative zone

Photosphere

Stability of the stellar nuclear reactor

Most stars radiate in a very stable way because their energy production is `autoregulated´

If energy production is reduced

→ central pressure is reduced

→ the core contracts because of gravity

→ pression increases

→ temperature increases

→ energy production increases

Et conversely… → energy production is stabilised at the right level to prevent gravitational collapse

Internal structure of stars - 2

Energy transport

3 mechanisms:

• conduction: not efficient in gases → marginal in most stars

• radiation: the more transparent matter is, the more efficent is the energy transport by photons; in stars, numerous absorptions – re-emissions

• convection: when matter is too opaque, energy accumulates at the bottom of the opaque zone → appearance of convection currents, energy is transported by matter in motion

Internal structure of stars - 3

Internal structure

How can we determine the physical state (temperature, pressure,…) in the stellar interiors?

A star is a rather simple structure (in 1st approximation)

= sphere of gas in equilibrium under its own gravity

→ solve a system of equations:

• hydrostatic equilibrium: pressure ↔ weight of upper layers

• mass conservation

• energy production

• transport (and conservation) of energy

• equation of state (ex: perfect gas)

Internal structure of stars - 4

Tests of models

Compare predictions with observations (surface conditions)

• HR diagrams HR of clusters (assemblies of stars with same age and same chemical composition)

• detection of neutrinos (very low interaction with matter → come directly from the core)

• helio and asteroseismology (study of oscillations)

Internal structure of stars - 5

L’alchimie stellaire

Fin du chapitre…

• Sources d’énergie

• Énergie nucléaire

• Réactions nucléaires dans les étoiles

• Structure interne des étoiles