Phys 132 Homework 18 Solutions
Professor Crystal Martin
TA: Ellie Hadjiyska
Problem 18: (Phillips Problem 4.6)
The power per unit volume produced by the triple-αreaction is
"3α=(3m4−m12)c2dn12
dt ,
where
dn12
dt =33/2n3
4
τ(12∗→12) !h
2πm4kT "3
e−(m∗
12−3m4)/kT
is the number density of carbon atoms produced per unit time. The individual symbols are as
defined in the textbook. The power per unit volume at T=2×108Kand10
8kg m−3is 2.2×
1018 Wm−3. Changing the temperature only changes T3in the denominator and the exponent in
the rate equation. Decreasing the temperature to 108K decreases T3by a factor of 8 (and the rate
by a factor of 8) and the exponential factor changes from e−22 to e−44. The resulting power per
unit volume is
"3α(T=10
8K)=8
e−44
e−22 "3α(T=2×108K)≈4.9×109Wm
−3.
Decreasing the temperature by a factor of two sharply decreases the energy generation rate of the
reaction.
Altering the energy of the excited carbon is equivalent to changing the mass of the excited
state m∗
12, which only affects the exponential factor in the rate equation. At 108K, kT ≈10 KeV
and changing the excited state mass from 7.65 MeV to 7.66 MeV lowers the exponent by a factor
of ∼2.7. Therefore the overall power generated is lowered by a factor of 2.7 as well. A tiny change
in the excited state mass has a drastic effect on the production of carbon.