Gamowyn: Narrow Window
Cosmology & the Sky arc, page 146 of 193. Back to The Pixelated Physics Universe
Who they are
Gamowyn holds open a narrow, glowing door between two dark walls. Fusion in the Sun squeezes through it.
Gamowyn is a character of the Cosmology & the Sky arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
In a star's core, few nuclei have enough thermal energy to climb the Coulomb barrier, and the fast ones are exponentially rare. Quantum tunnelling helps: the chance of getting through rises steeply with energy. Multiply the falling Maxwell–Boltzmann tail by the rising tunnelling factor and you get a sharp peak, the Gamow peak, where almost all the fusion happens. The tunnelling theory behind it is George Gamow's, Z. Phys. 51, 204 (1928), written for alpha decay.
$$ \begin{gathered} \text{rate} \propto e^{-E/kT}\, e^{-\sqrt{E_G/E}} \\ E_0 = \left(\frac{E_G (kT)^{2}}{4}\right)^{1/3} \\ E_G = 2\mu c^{2}(\pi\alpha Z_1 Z_2)^{2} \approx 493\ \mathrm{keV}\ (p+p) \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $E$ | collision energy in the centre-of-mass frame | J |
| $k$ | Boltzmann constant, 1.380 649 × 10⁻²³ J/K (exact) | J/K |
| $T$ | temperature | K |
| $E_G$ | Gamow energy | J |
| $E_0$ | energy of the Gamow peak | J |
| $\mu$ | reduced mass of the two nuclei | kg |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
| $\alpha$ | fine-structure constant ≈ 1/137.036 | 1 |
| $Z_1,\ Z_2$ | charge numbers of the two nuclei | 1 |
Honest caveat: Gamow's 1928 paper explained alpha decay; others applied tunnelling to fusion in stars. The 15.7 million K core temperature is a standard solar model value.
Sources
- Gamow (1928) Zur Quantentheorie des Atomkernes, Z. Phys. 51, 204
- NASA NSSDCA, Sun Fact Sheet
- Adelberger et al. (2011) Solar fusion cross sections II, Rev. Mod. Phys. 83, 195
Crossovers
Gamowyn meets Stellar Core (Nuclear Fusion). Stellar Core fuses protons in the Sun at about 15.7 million kelvin, which is too cool for most protons to climb the Coulomb barrier. Gamowyn opens the narrow window where the thermal tail and quantum tunnelling overlap, and fusion squeezes through.
Gamowyn meets Trialpha (Hoyle State). Gamowyn's window lets helium nuclei fuse in a red giant's core. Trialpha's Hoyle state makes the third helium nucleus stick, so carbon forms quickly enough.
Wear the lore
- Gamowyn – The Narrow Window Physics T-Shirt
- Related infographic poster: The pp-I Chain – Plate 069 Physics Poster
Concept keywords: Narrow Window, Gamowyn, Cosmology & the Sky, physics lore, Pixelated Physics