Download Nuclear fusion in stars

Nuclear fusion in stars
Paul Hickson
October 19, 2016
Kelvin-Helmholtz mechanism
In the late 19th century Lord Kelvin and
Hermann von Helmholtz proposed a
mechanism to explain the source of the
energy radiated by the Sun.
They reasoned that as the Sun radiated
energy, it would cool, lowering the gas
pressure (which according to the ideal gas
law is proportional to T ).
This would result in a slow contraction of
the Sun, which would maintain the
temperature and pressure needed for
hydrostatic equilibrium, allowing the Sun to
continue to radiate energy.
With this mechanism, the source of the
energy is gravitational potential energy.
Kelvin-Helmholtz timescale
Let’s estimate how long this mechanism could power the Sun.
The internal gravitational potential energy of a body is defined as
the energy needed to separate all the atoms and move them away
to infinity, against their mutual gravitational attraction. It is
negative because we have to do work to separate the atoms.
For a homogeneous sphere of mass M and radius R, the internal
gravitational energy is
U “´
3GM 2
.
5R
If we divide this energy by the luminosity (the present rate at
which energy is radiated), we get a characteristic time, the
Kelvin-Helmholtz time
tKH “
|U |
GM 2
„
.
L
RL
Powering the Sun
Lets evalute the Kelvin-Helmholtz time for Sun,
tKH »
6.671 ˆ 10´11 ˆ p1989 ˆ 1030 q2
“ 9.88 ˆ 1014 s “ 31 Myr.
6.955 ˆ 108 ˆ 3.838 ˆ 1026
This is much shorter than the geological age of the Earth, so there
must be another source of energy.
It is easy to show that chemical energy is insufficient. For example
the reaction O ` 2H Ñ H2 O produces 1.26 eV of energy
(2.02 ˆ 10´19 J) per oxygen molecule.
The number of oxygen atoms in the Sun is about
0.00954Md {16mH “ 7.09 ˆ 1054 , so the total energy available is
1.43 ˆ 1035 J. This would only power the Sun for
1.43 ˆ 1035 J
“ 3.7 ˆ 108 s “ 11.8 yr.
3.85 ˆ 1026 J/s
Nuclear fusion
The problem was solved by the discovery of
nuclear fusion in the 20th century.
The idea that fusion powers the stars was
first proposed by the English astrophysicist
Arthur Stanley Eddington in 1920.
In 1939, German physicist Hans Bethe
discovered the two main stellar fusion
reaction sequences, for which he received a
Nobel prize.
Before examining these, lets review some
properties of particles and light nuclei,
Properties of particles and light nuclei
Particle
electron
positron
neutrino
photon
proton
neutron
deuteron
tritium
helium-3
helium
Symbol
e
e`
νe
γ
p
n
d, 2 H
3H
3 He
4 He, α
Charge{e
-1
1
0
0
1
0
1
1
2
2
Mass{mp
0.000544
0.000544
0.000000
0
1.000000
1.001378
1.999552
2.993718
2.993154
3.972600
where mp “ 1.672622 ˆ 10´27 kg is the mass of a proton.
Fusion energy
The rest-mass energy associated with mass m is given by
Einstein’s equation
E “ mc2 .
From the table, we see that four protons are less massive than a
4 He nucleus (an α particle).
Therefore, if we could somehow combine four protons to make a
4 He nucleus, there would be a release of energy equal to
∆E “ p4 ´ 3.972600qmp c2 “ 4.119 ˆ 10´12 J “ 25.71 Mev.
This is more than 10 million times greater than chemical energy!
We know that a fraction X “ 0.747 of the Sun’s mass is hydrogen.
If it were all converted to helium, this could provide enough energy
to power the Sun for
t“
XMd ∆E
E
“
“ 2.3 ˆ 1018 s » 74 Gyr.
L
4mp L
Proton-proton chain
The principal reaction that occurs in the Sun and lower-mass stars
is the proton-proton chain (pp I).
p`pÑ
2
H ` e` ` νe
pˆ2q
H`pÑ
3
He ` γ
pˆ2q
He ` 3 He Ñ
4
He ` p ` p
2
3
The net result of this is
4p Ñ
4
He ` 2e` ` 2νe ` 2γ.
The first step in this chain is the most critical. When two protons
collide they form a “diproton” which is unstable and quickly splits
into two protons.
Bethe realized that a small fraction of the time, one of the protons
could decay into a neutron, positron and neutrino before the
diproton can split, thus producing a deuteron.
Proton-proton chain
Several other branches also occur, for example the pp II branch is
dominant at temperatures between 14 and 23 MK.
3
He ` 4 He Ñ
7
7
´
Be ` e Ñ
Li ` p Ñ
4
7
Be ` γ
7
Li ` νe
He ` 4 He
And the pp III branch dominates at temperatures above 23 MK,
3
He ` 4 He Ñ
7
8
Be ` p Ñ
BÑ
8
8
7
Be ` γ
8
B`γ
`
Be ` e ` νe
Be Ñ
4
He ` 4 He
The pp IV (Hep) branch occurs very rarely,
3
He ` p Ñ 4 He ` e` ` νe
Proton-proton chain
The neutrinos carry about 2% of the released energy. They interact
with matter only very rarely, and generally escape from the Sun.
The positrons annihilate with electrons, producing two 511 keV
gamma-ray photons each
e` ` e Ñ 2γ.
All the photons have energies in the gamma-ray region of the
spectrum. They interact strongly with free electrons by the process
of electron scattering.
Absorbed photons transfer energy to the gas, which re-emits
photons with the Planck energy distribution, according to the local
temperature.
At the photosphere of the Sun, the temperature has dropped to
about 5780 K. Photons emitted here can escape, producing the
radiation that we see.
CNO cycle
Stars having central temperature higher than about 18 MK
produce most of their energy by the CNO cycle,
12
13
N`γ
NÑ
13
C ` e` ` νe
C`pÑ
14
N`γ
N`pÑ
15
O`γ
OÑ
15
N ` e` ` νe
N`pÑ
12
C `4 He
C`pÑ
13
13
14
15
15
The net result of this is
4p Ñ
4
He ` 2e` ` 2νe ` 3γ.
In this cycle, carbon acts as a catalyst. Nitrogen and oxygen are
produced as byproducts (not all of these nuclei are consumed).
Triple-alpha reaction
The final reaction that we will discuss is the triple-alpha reaction,
4
He `4 He Ñ
8
8
Be `4 He Ñ
12
Be ` γ
C`γ
The 8 Be nucleus produced in the first reaction is unstable and
splits into two 4 He nuclei in only 2 ˆ 10´16 s. However, if the
temperature and density are high enough, a third 4 He nucleus may
fuse with the 8 Be before it decays.
Typically this requires temperatures greater than 100 MK.
Other reactions also occur, producing elements with greater atomic
mass, up to iron.
Iron is the most stable nucleus, having the lowest binding energy,
so it is not easily fused to form other elements.
Stability of stars
If fusion powers the stars, why don’t they explode like hydrogen
bombs?
Some do, but most are stable. The rate at which the nuclear
reactions proceed depends very sensitively on the temperature in
the core of the star.
If the temperature increases, more energy is generated which
makes the star expand.
This in turn lowers the temperature, reducing the energy
generation rate.
So, most stars are self-regulating.
The equilibrium is not always perfect. Some stars pulsate,
periodically expanding and contracting. Examples are RR-Lyrae
variables and Cepheid variables.
Davis/Bahcall experiment
In the late 1960s Astrophysicists Raymond
Davis and John Bahcall set out to detect
solar neutrions. Davis did the experimental
work and Bahcall did the theoretical
calculations
Davis installed a 380 m3 tank of CCl2
(perchloroethylene) 1.5 km underground at
the Homestake Gold mine in South Dakota.
Raymond Davis, Jr.
The reaction that they were looking for is
νe `
37
Cl Ñ
37
Ar ` e
Davis was able to detect neutrinos, but
only at 1{3 the rate predicted by Bahcall.
John Bahcall
Sudbury Neutrino Observatory
In 1984, The American physicist Herb Chen realized that a
neutrino detector employing heavy water would be sensitive to all
neutrinos, not just νe .
Following up on this, Canadian physicist Arthur McDonald led a
project to build such a facility, deep in a mine in Sudbury Ontario.
The Sudbury Neutrino Observatory,
has a 12 m diameter acrylic sphere
filled with heavy water, surrounded
by 9,600 photomultiplier tubes that
detect photons produced by neutrino
reactions.
SNO showed that the total neutrino
flux agreed with Bahcall’s prediction,
but that 2{3 of the νe produced in
the Sun had changed into νµ and ντ
by the time they reached Earth.
SNO