In the Gamow-peak approximation the reaction-rate integrand is dominated by

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Nuclear astrophysics and applications

Stars from the Inside, Ch Thermonuclear energy generation: reaction rates, burning stages and the elements they make

Problem

S(E)\,e^{-f(E)}

, with

S(E)

varying slowly and

f(E) = \frac{E}{k_BT} + \sqrt{\frac{E_G}{E}}.

((a)) Find the energy

E_0

at which

f(E)

has its minimum, and show that

E_0 = \left(\dfrac{E_Gk_B^2T^2}{4}\right)^{1/3}

. ((b)) Expand

f(E)

to second order about

E_0

and show that the Gaussian approximation to

e^{-f(E)}

, namely

e^{-f(E_0)}e^{-\frac12f''(E_0)(E-E_0)^2}

, has full width

\Delta = 4\sqrt{E_0k_BT/3}

between the two points where the exponent equals

-1

. ((c)) Oxygen burning proceeds partly through

^{16}\mathrm{O}+{}^{16}\mathrm{O}

. Take the atomic mass of

^{16}

O to be

15.995\,\mathrm{u}

, the fine-structure constant

\alpha \approx 1/137.0

and

m_uc^2 = 931.5\,\mathrm{MeV}

. Using

E_G = 2\mu_{red}c^2(\pi\alpha Z_1Z_2)^2

with

Z_1=Z_2=8

and

\mu_{red}

the reduced mass of the pair, compute

E_G

, then evaluate

E_0

and

\Delta

T = 2.0\times10^9\,\mathrm{K}