Gamow Quantum Tunneling Factor, Gamow Energy & Stellar Gamow Peak

Explain how the Sun fuses protons at 15 MK despite a 550 keV Coulomb barrier!

At the Sun's core temperature T = 1.5e7 K, thermal energy k_B * T is only 1.29 keV—over 400 times weaker than the classical proton-proton Coulomb repulsion barrier! George Gamow proved in 1928 that nuclei tunnel quantum-mechanically through the electrostatic barrier with WKB transmission probability P(E) = EXP(-SQRT(E_G / E)), governed by Gamow Energy E_G = 2 * (mu * c^2) * (PI() * (1 / 137.036) * Z_1 * Z_2)^2!

Convolving the exponentially falling Maxwell-Boltzmann tail EXP(-E / k_B T) with the exponentially rising Gamow tunneling curve EXP(-SQRT(E_G / E)) creates a narrow thermonuclear reaction window centered at the Gamow Peak Energy: E_0 = ((SQRT(E_G) * k_B T) / 2)^(2 / 3)! For solar proton-proton fusion (Z_1 = Z_2 = 1, reduced mass mu c^2 = 469,140 keV), E_G = 493.13 keV and fusion peaks near E_0 = 5.898 keV (~4.6 * k_B T)!

Gamow Energy E_G (keV) from Nuclear Charges Z1, Z2 & Reduced Mass

Evaluates E_G = 2 * mu_keV * (PI() * (1 / 137.036) * Z1 * Z2)^2 using fine-structure constant alpha = 1 / 137.036.

=IF(OR(B2 < 1, B3 < 1, B4 <= 0), "Invalid Nuclear Input", ROUND(2 * B4 * ((PI() * (1 / 137.036) * B2 * B3) ^ 2), 2))

Stellar Gamow Peak Energy E_0 (keV) at Thermal Energy kT

Computes E_0 = ((SQRT(E_G) * kT) / 2)^(2 / 3) from Gamow energy and core thermal energy B5 (keV).

=IF(B5 <= 0, "Thermal kT Must Be > 0", ROUND(((SQRT(2 * B4 * ((PI() * (1 / 137.036) * B2 * B3) ^ 2)) * B5) / 2) ^ (2 / 3), 3))

Further reading

• Microsoft Support — EXP function