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

Mean product of nuclear fusion cross section and velocity. [tex]\langle \sigma v \rangle[/tex]

Maxwell–Boltzmann probability density function:

[tex]f(v) = \sqrt{\frac{2}{\pi}\left(\frac{m}{kT}\right)^3}\, v^2 \exp \left(- \frac{mv^2}{2kT}\right)[/tex]

The mean speed is the mathematical average of the speed distribution:

[tex]\langle v \rangle = \int_0^{\infty} v \, f(v) \, dv = \sqrt{\frac{8kT}{\pi m}}[/tex]

For a mono-energy beam striking a stationary target, the cross section probability is:

[tex]P = n_2 \sigma_2 = n_2 \pi r_2^2[/tex]

And the reaction rate is:

[tex]f = n_1 n_2 \sigma_2 v_1[/tex]

Reactant number densities:

[tex]n_1, n_2[/tex]

Target total cross section:

[tex]\sigma_2 = \sigma_\text{A} + \sigma_\text{S} + \sigma_\text{L} = \pi r_2^2[/tex]

Mono-energy beam velocity:

[tex]v_1[/tex]

Aggregate area circle radius:

[tex]r_2[/tex]

Stellar nuclear fusion reaction rate (fusions per volume per time):

[tex]f = n_1 n_2 \langle \sigma v \rangle[/tex]

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

Is the mean cross section the mathematical average of the cross section distribution?:

[tex]\langle \sigma \rangle = \int_0^{\infty} \sigma \, f(\sigma) \, d\sigma = \, \text{???}[/tex]

Reference:

http://en.wikipedia.org/wiki/Cross_section_(physics)#Nuclear_physics"

http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution" [Broken]

http://en.wikipedia.org/wiki/Nuclear_fusion#Requirements"

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# Nuclear fusion cross sections

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