Environmental Engineering Reference
In-Depth Information
proton, then, we can write the rate of collisions as
f
coll
¼ðv
=LÞ
ð
2
:
27
Þ
and the rate of fusion reactions per proton as
f
fus
¼
T
Tf
coll
¼
T
Tðv
=LÞ:
ð
2
:
28
Þ
assuming that the geometric collision corresponds to coming
within the classical turning point
r
2
of the second proton, this spacing was found to be
1113 f, and, for simplicity, we ignore the He ions. We earlier found that the density of
protons in the core of the sun is
N
p
¼
3.106
10
31
m
3
and that the most probable
velocity was 0.498
10
6
m/s at 15million K.
So
We will calculate
L
L¼
1/(
n s
)
¼
1/
n pr
2
2
¼
8.28 nm. (In terms of interparticle spacings, this is
large, about 260 spacings). So,
f
coll
¼ v
/
L¼
0.6
10
14
/s and
f
fus
¼
T
Tf
coll
¼
1.0
10
8
T
0.6
10
14
/s
¼
0.6
10
6
T
/s. We can use this to estimate the power radiated
by the sun, by multiplying by the number of participating protons
N
cp
in the core of
the sun and by the energy release per reaction, and in this way get a value for the
reaction probability
T
.
The number of protons is
N
cp
¼
6.87
10
56
, taking the core as radius
r R
S
/4, if
we take the proton density as constant at 3.106
10
31
m
3
. This number is too large,
and we earlier estimated, in text following Equation 2.3, the number of protons in the
core as
N
cp
¼
2.51
10
56
, but mentioned that the number was still too large. Here, we
take the effective number of participating protons as 1.58
10
56
. Then, with energy
release per fusion as 26.2MeV, we get
P¼
3.96
10
50
T
watts. (The p
-
p cycle is
(Equation 1.10a,b) initiated by the fusion reaction we are considering. The
rst
reaction is quickly followed by several others with larger
T
(reaction probability)
values, with a total release of energy in the sun of 26.2MeV per initial fusion). By
forcing our resulting power
P¼
3.96
10
50
T
watts equal to the known power
3.82
10
26
watts, we estimate that the fusion reaction probability
T
as 9.6
10
25
(but we will adjust it to 8
10
24
later). So this reaction, we predict, is 23
-
24 orders of
magnitude slower than a more straightforward reaction such as D
þ
D and D
þ
T.
It turns out that this estimate is a reasonably accurate prediction of
T
for the p
-
p
reaction! We have made a simpli
ed analysis; on the other hand, it has been stated
that the reaction probability
T
, which is also referred to as the astrophysical
S
factor, is
much smaller for the p
p reaction than for straight reactions not involving generation
of neutrinos. To quote Atzeni (Section 1.3.3), The p
-
p reaction involves a low
probability beta-decay, resulting in a value of
S
about 25 orders of magnitude smaller
than that of the DT (Deuterium-Tritium) reaction.
In the literature, this is described as a small astrophysical cross-section, re
ect-
ing the nature of the nuclear reaction that is involved, an inverse beta decay, known to
be a slow process. The
first product, after the Gamow tunneling step, is an excited
state,
2
He, of two protons, which must stabilize by emitting a positron and neutrino
(inverse beta decay), else return to two separate protons. Note that the energy
difference in the stabilization is positive energy 1.293 keV going to
2.22MeV, the
latter is the binding energy of the deuteron, see Figure 2.3.
-
