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Orion1
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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
http://en.wikipedia.org/wiki/Nuclear_fusion#Requirements
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