Download The Physics of Energy sources Stellar fusion

The Physics of Energy sources
Stellar fusion
B. Maffei
[email protected]
Stellar Fusion
1
Introduction
!   Many sources of renewable energy rely on the Sun
!   From radiation directly
•  Solar cells
•  Solar water heating
!   From radiation indirectly
•  Wind created from radiative heating
•  Photosynthesis
•  Waves
!   We will then see where the Sun gets its energy from.
!   Fusion: link between real fusion application  renewable energies
!   But actually the real source of energy is gravity
•  Fusion is a consequence
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Quick summary of stellar formation
!   Due to gravity a cloud of interstellar gas contracts
!   The interstellar matter contains mainly hydrogen
!   About 90% is hydrogen
!   About 9% helium
!   1% is the rest
!   The density increases and so is the temperature
!   Gravitational energy  Kinetic energy
!   The star emits radiation ~ like a blackbody
!   The temperature reaches ~ 107K
!   The temperature is high enough to start thermonuclear reaction
•  First reaction to take place is fusion of hydrogen into helium
!   Equilibrium reached
!   After a while the energy released by fusion counterbalance gravity
!   The star stops collapsing and reach a stable diameter (at least for a
while…)
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Proton-Proton chain
!   Fusion of hydrogen into helium results from a sequence of reactions
!   This sequence is known as the proton-proton chain
The first reaction is the combination of 2 protons to form the only stable two-nucleon system:
the deuteron.
Note: simple fusion of 2 protons is impossible because diproton is unstable.
So we have β+ decay (proton  neutron + positron)
(1)
p + p → d + e + +ν
or
1
H +1H →2H + e + +ν
Q=0.93MeV
The probability of β decay is low making the cross-section of this reaction very small
σ~ 10-33b at keV energies and ~ 10-23b at MeV energies
Even with a high proton density at the centre of the Sun (~7.5x1025 protons.cm-3)
the reaction rate is still fairly low.
What makes it work is the enormous number of protons in the Sun (~1056)
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Proton-Proton chain (2)
!   The previous reaction is then quickly followed by:
(2)
p + d →3He + γ
or
1
H + 2H →3He + γ
Q=5.49MeV
This is the most likely reaction to happen after formation of deuteron.
D-D reaction is very unlikely due to the small number of deuterons compared to protons
The previous reaction is very slow  1 deuteron formed /sec for every 1018 protons.
Once formed the lifetime of deuteron is 2sec.
 Much more probable to collide with proton than another deuteron
 Deuterons are transformed into 3He almost as soon as they are formed
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Proton-Proton chain (3)
!   Reaction of 3He with proton is not possible
!   The isotope 4Li is unbound and breaks up as soon as it is formed
3
He+1H →4Li →3He+1H
!   It is unlikely for 3He to react with 2H
!   Density of 2H is very low and transformed into 3He very rapidly
3He
(3)
3
Stellar Fusion
He+ 3He " # + 2 p
!
will have to react with another 3He
or
3
He+ 3He"4 He +21H
Q=12.86MeV
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Summary of p-p chain
p + p "d + e + + #
3
1
H+1H "2 H + e + + #
Q=0.93 MeV
H+ 2H "3 He + $
Q=5.49 MeV
p + d" He + $
1
3
3
He+ 3He " % + 2 p
Figure from wikipedia
He+ 3He"4 He + 2 1H
x2
Q=12.86 MeV
The complete process is the conversion of 4 protons to helium
4 p → α + 2e + +!2ν + 2γ
41H →4He + 2e + + 2ν + 2γ
Q=25.7 MeV
Note: reactions with nuclear particles
In order to find the total Q we need to take into account the electrons:
!  We have a plasma, mix of protons and electrons, neutral overall
!  We add 4 electrons on each side of the equation
!  4 neutral H atoms give a neutral He atom and the 2 remaining e- annihilate the 2 e+
Calculation of Q gives 26.7MeV
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Nucleosynthesis continues
!   Once a star has exhausted its hydrogen, helium fusion starts
!   Hydrogen in stellar core becomes depleted  not enough energy
produced to counter-balance gravity
!   Gravitational contraction resume until T reaches ~ 108K
!   Overcome the Coulomb barrier between α particles  helium burning
!   Several reactions are possible using 4He
Triple α process
4
He+ 4 He"8 Be
4
He+ 8Be"12 C + #
4Be
is short lived but can react
with α particle to give carbon
α process
!
Stellar Fusion
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He+12C"16 O + #
4
He+16O"20 Ne + #
4
He+ 20Ne"24 Mg + #
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What next?
!   Low mass stars (< 0.5 solar masses) will not go beyond H fusion
!   After burning all their hydrogen they will finish as red dwarfs
!   Many mid-sized stars will end when He burning is complete
!   We get a white dwarf star or a giant red
!   For much more massive stars (several times solar masses)
!   Several more burning stages can occur
!   Each preceded by gravitational contraction  increase in the core temperature
T > 5x108K  carbon fusion
T > 2x109K  oxygen fusion
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C +12C →20Ne+ 4He
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C +12C →23Na +1H
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O+16O→28Si + 4He
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O+16O→32S + γ
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C +12C →23Mg + n
16
O+16O→31P +1H
......
T > 3x109K  silicon fusion for very massive star (~10 solar masses)
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Si + 28Si →56Ni But the Coulomb barrier is very high so most probably
α process occurs at the same time
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O→20Ne→24Mg →28Si
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Stellar Fusion
Si →32S →36Ar →40Ca →44Ti →48Cr →52Fe→56Ni
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Nucleosynthesis in massive stars
Sequence of stellar burning stops when core is
mainly composed of elements with A~56
Fe, Ni
Above A=60, no energy to be gained by fusion
Last elements to be formed are Fe and Ni
The various fusion reaction stages will
not be exactly sequential:
!
!
!
!
  ressure and T will vary across the star
P
 The highest T will be in the core
 More advanced fusion will start in the core
 Creation of layers of different elements
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CNO cycle
!
!
!
!
  Most of the interstellar material is hydrogen but stars are synthesising heavier atoms
  These heavier elements will then be released when stars explode
  Second generation stars will already contain heavier elements
  This can lead to other reactions such as CNO cycle at the same time than H burning
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C +1H →13N + γ
1.95 MeV
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N →13C + e + + ν
2.22 MeV
C +1H →14N + γ
7.54 MeV
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14
N +1H →15O + γ
7.35 MeV
O →15N + e + + ν
2.75 MeV
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15
N +1H →12C + 4He
4.96 MeV
Heavy elements stay the same but:
Fusion of hydrogen
Energy released
Creation of 4He
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Energy emitted
6000K
While the star is consuming its hydrogen,
its luminosity increases with time

Main sequence represented by
Hertzsprung-Russell diagram
I(λ,T)
5500K
5000K
Wavelength (m)
The star surface being like a plasma, it will radiate like a
blackbody at temperature T (lower than core temperature)
Blackbody emission = Planck function
2hν 3
I (ν , T ) =
⎛ hν ⎞
⎜
⎟
⎝ kT ⎠
in W .m − 2 .Hz −1.sr −1
⎛
⎞
⎜
c e
− 1⎟
⎜
⎟
⎝
⎠
2hc 2
I (λ , T ) =
in W .m − 2 .m −1.sr −1
⎛ hc ⎞
⎛ ⎜ ⎟ ⎞
5 ⎜ ⎝ λkT ⎠
λ e
− 1⎟
⎜
⎟
⎝
⎠
2
Stellar Fusion
h : Planck constant
k : Boltzmann constant
ν : frequency
λ : wavelength
T : temperature
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Energy from the Sun
!   The Sun has a surface temperature of ~ 5500K- 5700K
!   The integration of the Planck function at T~5600K over the full spectral domain,
over the area of the Sun and over the whole solid angle gives the solar luminosity
(ref to lecture 1)
L =
***
2hc 2
# #% hc &( &
"5 %%e$ "kT ' )1((
$
'
d+dAd" = 3.83 , 10 26 W
Knowing the luminosity and the mass of the Sun (~2x1030kg), taking into account hydrogen
fusion only, we can then estimate how long the Sun will burn (see example on website)
How much!energy do we get from the Sun?
Average Sun-Earth distance re = 1.5x1011m
Power per surface unit (irradiance) delivered on the Earth by the Sun :
re
L
3.83 # 10 26
$2
F =
=
=
1.35kW.m
4 " re2 4 " (1.5 # 1011 ) 2
If we suppose that 70% (average atmospheric transmission) of the
radiation reaches the Earth s surface, we get ~ 950W per square meter
Stellar Fusion
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References
!   Most of the material of this lecture is coming from
!   Lilley, J. Nuclear Physics Principles and Applications (Wiley 2006)
!   Kenneth S. Krane, Introductory Nuclear Physics (Wiley 1988)
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Summary
!   Could you explain what is the source of energy of a Star?
!   What is the proton-proton chain?
!   Why can it work?
!   What is the slowest process in this chain?
!   What is the CNO cycle?
!   What are the last elements to be formed though fusion in
massive stars?
!   How would you derive the radiative power emitted by the
Sun and received on Earth?
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