Stellar nuclear fusion: Mean cross section and velocity theorem

[Orion1]

I am inquiring as to what the theorem function is for the mean product of cross section and velocity for stellar fusion reactions? \langle \sigma v \rangle

Mean product of nuclear fusion cross section and velocity. \langle \sigma v \rangle

Maxwell–Boltzmann probability density function:
f(v) = \sqrt{\frac{2}{\pi}\left(\frac{m}{kT}\right)^3}\, v^2 \exp \left(- \frac{mv^2}{2kT}\right)

The mean speed is the mathematical average of the speed distribution:
\langle v \rangle = \int_0^{\infty} v \, f(v) \, dv = \sqrt{\frac{8kT}{\pi m}}

For a mono-energy beam striking a stationary target, the cross section probability is:
P = n_2 \sigma_2 = n_2 \pi r_2^2

And the reaction rate is:
f = n_1 n_2 \sigma_2 v_1
Reactant number densities:
n_1, n_2
Target total cross section:
\sigma_2 = \sigma_\text{A} + \sigma_\text{S} + \sigma_\text{L} = \pi r_2^2
Mono-energy beam velocity:
v_1
Aggregate area circle radius:
r_2

Stellar nuclear fusion reaction rate (fusions per volume per time):
f = n_1 n_2 \langle \sigma v \rangle

What is the theorem and solution for the mean cross section in stellar nuclear fusion? \langle \sigma \rangle

Is the mean cross section the mathematical average of the cross section distribution?:
\langle \sigma \rangle = \int_0^{\infty} \sigma \, f(\sigma) \, d\sigma = \, \text{?}
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Reference:
http://en.wikipedia.org/wiki/Cross_section_(physics)#Nuclear_physics
http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution
http://en.wikipedia.org/wiki/Nuclear_fusion#Requirements

 

[mfb]

You don't need ⟨σ⟩ and it is not particularly well-defined anyway. The cross section depends on the velocity in a nonlinear way and the velocity has a very non-linear distribution - σ(⟨v⟩) will be completely different from ⟨σv⟩.

https://en.wikipedia.org/wiki/Gamow_factor