The Gamow-peak approximation gives

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

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

Problem

\langle\sigma v\rangle \propto S(E_0)\,\tau^2e^{-\tau}

, with

\tau \equiv 3E_0/k_BT

and

E_0 = (E_Gk_B^2T^2/4)^{1/3}

the Gamow peak energy;

S(E_0)

varies only slowly with temperature and may be treated as constant. ((a)) Show that

\tau \propto T^{-1/3}

, and use this to derive the temperature exponent

\nu \equiv \mathrm{d}\ln\langle\sigma v\rangle/\mathrm{d}\ln T

, obtaining

\nu = (\tau-2)/3

. ((b)) Helium-burnt cores can also make magnesium through

\alpha + {}^{20}\mathrm{Ne} \to {}^{24}\mathrm{Mg}

. Take the atomic masses

m({}^4\mathrm{He}) = 4.0026\,\mathrm{u}

m({}^{20}\mathrm{Ne}) = 19.9924\,\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=2

Z_2=10

, compute

E_G

E_0

\tau

and

\nu

for this reaction at

T = 1.5\times10^9\,\mathrm{K}