Astrophysics II
Tutorial 3
1)
Find the classical distance of closest approach for two protons with an energy of
2 keV. Estimate the probability that the protons penetrate the Coulomb barrier
that tends to keep them apart. Compare this probability with the corresponding
probability for two 4He nuclei with the same energy.
2)
Show that:
1/ 2
d
E  EG 
exp[−
−
 ]=0
dE
kT  E 
1/ 3
implies:
3)
 E (kT ) 2 
E0 =  G

4


Show that:
E  EG 
 EG 
 E − E0 
−
 ]  exp[−3 
 ]  exp[− 
 ]
kT  E 
 4kT 
 /2 
4
with  = 1/ 2 1/ 3 EG1/ 6 (kT )5 / 6
3 2
1/ 2
1/ 3
2
exp[−
4)
For a Clayton model of the sun, show that the density at r = a is 0.53 c , and
that a sphere of radius a contains 0.26M.
5)
In the lecture notes, we found that the mass function for a Clayton model star
with large mass concentration towards the center can be found from:
1
(4 ) 2
r2
G m( r ) 2 =
G c2  r 5 exp[− 2 ] dr  .
2
3
a
0
r
Show that this results in the following expression for the mass function m(r ) :
m( r ) =
6)
4
c a 3 ( x)
3
with x = r / a and  2 ( x) = 6 − 3exp[ − x 2 ]( x 4 + 2 x 2 + 2)
Now show that the mass function m(r ) in problem 1 has the right limiting
behaviour:
m( r ) →
r →0
7)
4
c r 3
3
with
a 3 ( x) → r 3
r →0
Calculate the power per kilogram produced by helium burning when the density
is 108 kg/m3and the temperature is 2x108 K. By how much would this power
change if the excitation energy of the 0+ state of 12C were at 7.66 MeV instead
of 7.65 MeV?