Ph135c. Solution set #9, 6/3/10
1.
12.2
In the iron core, the number of protons is:
N
p
=
Z
Z
+
N
1
.
4
M
sun
m
p
≈
7
.
7
×
10
56
Each of these is converted into a neutron and neutrino by a process which consumes an energy of
Δ
E
≈
m
n

m
p

m
e
≈
0
.
78
MeV
, for a total energy of neutronization of:
E
n
= 6
.
1
×
10
56
MeV
Since each such process creates a single electron neutrino, the number of such neutrinos produced
is also 7
.
7
×
10
56
.
Next consider the energy released in collapse to a smaller neutron core. The change in gravitational
potential energy is:
Δ
< V >
=

3
5
GM
2
±
1
R
i

1
R
f
²
≈ 
1
.
7
×
10
58
MeV
Assuming the virial theorem holds, this corresponds to a change in total energy of:
Δ
< E >
= Δ(
< T >
+
< V >
) =
1
2
Δ
< V >
or half this amount (where we’ve used 2
< T >
+
< V >
= 0). So we see that only about 1% of
this is necessary for neutronization. Most of the rest of the energy is released as neutrinos of all types,
with a typical energy of 12
MeV
, so that a total of about 3
×
10
57
of them are produced.
12.3
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 Spring '11
 Smith
 Physics, Energy, Neutron, iron core, P2 − P1, smaller neutron core

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