Download Conservation of energy

Stellar alchemy
• Energy sources
• Nuclear energy
• Nuclear reactions in
stars
• Internal structure of
stars
Energy sources
Age of the Sun
Solar luminosity ~ 4 × 1026 W
All electric plants together ~ 2 × 1012 W
Conservation of energy
Hermann von Helmholtz
energy stock
 lifetime ~
luminosity
→ search for the solar energy source
(1860s)
William Thomson, Lord Kelvin
Energy sources - 2
Chemical energy
M ~ 2 × 1030 kg
If Sun made of coal → lifetime ~ 5000 ans
→ ± compatible with Bible (Genesis ~ 4000 B.C.)
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
Energy sources - 3
Gravitational energy
GMm
Contraction of the Sun: E ~
requires a few dozen meters per
R
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
Energy sources - 4
Mass – energy equivalence
1905: Einstein discovers the equivalence of mass and energy
E  mc 2
→ 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
Nuclear energy
The atomic nucleus
A
Z
X : 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)
Ex: 73 Li : 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)
Nuclear energy - 2
Mass defect
Mass of nucleus < sum of masses of nucleons
Difference = mass defect ↔ binding energy: Δm = ΔE/c2
• increases from 1H to 56Fe
Binding energy per nucleus:
• decreases beyond 56Fe
Energy release by:
ΔE/A
56Fe
• fission of heavy nuclei
• fusion of light nuclei
(accompanied by
transmutation of neutrons into
protons)
1H
A
Nuclear energy - 3
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 - 4
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
A
Z
X 
A
Z 1
X  e
It concerns isotopes above the valley of stability
(excess of neutrons)
β+ process corresponds to the emission of a e+
(positon) by the nucleus
A
Z
X 
A
Z 1
X  e
(isotopes with excess of protons)
Henri Becquerel
Nuclear energy - 5
Natural radioactivity
The α process corresponds to the emission of a helium 4 nucleus
A
Z
X 
A4
Z2
X  42 He
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)
A
Z
X*  AZ X  γ
Marie Curie
Nuclear reactions in stars
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
ν = neutrino
• chargeless particle
(and massless?)
• necessary to ensure
conservation of energy
and momentum
(Δt ~ 109 years)
Nuclear reactions in stars - 2
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
(3) 3He + 3He → 6Be
(Δt ~ 106 years)
(3′) 6Be → 4He + 2 1H
The reaction rate is
dominated by the slowest
step, here (1)
Nuclear reactions in stars - 3
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)
U
coulombian repulsion (1/r)
E
0
It has some variants (pp2 and pp3)
that differ in the last qteps
strong interaction
r
Nuclear reactions in stars - 4
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 + γ
13N
→ 13C + e+ + ν
13C
+ 1H → 14N + γ
14N
+ 1H → 15O + γ
15O
→ 15N + e+ + ν
15N
+ 1H → 12C + 4He
(≈ 10% of solar energy)
Nuclear reactions in stars - 5
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
8Be
+ 4He → 8Be + γ
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 - 6
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 - 7
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 - 8
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 - 9
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
A
Z
Xn 
or:
etc…
A 1
Z
A 1
Z
X 
A 1
Z 1
Xn 
X  e
A2
Z
X 
A2
Z 1
X  e
These neutron captures form the basis of the production of all
chemical elements heavier than iron
Nuclear reactions in stars - 10
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 - 11
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
Internal structure of stars
Nuclear reactions in the stellar cores (for the Sun, this core extends
over 1/4 of the radius (1.6% of the volume)
Photosphere
Chromosphere
Convective zone
Core
Radiative zone
Internal structure of the Sun
Internal structure of stars - 2
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 - 3
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 - 4
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 - 5
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)
L’alchimie stellaire
• Sources d’énergie
• Énergie nucléaire
• Réactions nucléaires
dans les étoiles
• Structure interne
des étoiles
Fin du chapitre…