ASTR 498N             Homework #6       due in class Tu 23 Apr  

              

R   Please review O&C §11.1, §12.1, §12.2 and §13.1, and read §12.3 and §13.2.

 

P   O&C Problems 12.1, 12.2, 13.3

 

P   The Gamow energy defined in Lecture 13 is related to the quantity b defined by O&C [ just after Eq. (10.38)] by .  In terms of E_G, O&C Eq. (10.40) for the reaction rate between nuclei A and B can be written

 

 

 

The product of the two exponential factors in the integrand produces a strongly peaked curve (“fusion window”) with a maximum at

 

                    O&C Eq. (10.41)

 

(a)   By making a Taylor expansion about E = E_G, derive the approximation

 

 

where the width of the fusion window is

                   
                     

 

(b) Using this approximation in the reaction rate integral and assuming that S(E) is approximately constant within the fusion window, show that the reaction rate is given numerically by

 

 

 

 

where μ_A is the reduced mass of the nuclei in a.m.u. and S(E₀)  is the fusion factor at the Gamow peak in units of keV barns.

 

(c)  Calculate E_G for each of the fusion reactions  and  and compare the exponential inhibition factor in the equation immediately above for the two reactions at a temperature of .