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