The Sun’s core reaches temperatures of about 15,000,000 ºC. This is not hot enough for fusion. From experiments it is known that at least 100,000,000 ºC are needed to overcome repulsion between protons for fusion to take place However, we observe that the Sun does sustain fusion. How is this possible then? It is due to the combination of two phenomena that fusion can happen in the Sun.

The Sun is a Main Sequence star (as classified in the Hertzprung-Russell diagram) so it mainly contains hydrogen. Thus, it generates energy through the fusion of hydrogen into helium. This process involves combining 4 protons from hydrogen to form a helium nucleus while also releasing energy in the form of neutrinos and gamma rays. The helium nucleus is more thermodynamically stable than the hydrogen protons and the energy released is given by E = mc².

This energy takes a long time to reach the surface of the Sun. Light rays are constantly colliding against molecules present in the plasma inside the Sun following a random walk behaviour. This means that light rays collide against the plasma, getting absorbed and reemitted in another random direction, losing some energy in the process.

 

Random walk of gamma rays from emitted from core

 

It takes over a 100,000 years for these rays produced at the core to reach the surface of the Sun, at just at 6,000 ºC which is very little in comparison to the core. For fusion to occur, hydrogen protons need to get really close to each other. But as protons are both positively charged, they experience electrostatic repulsion which are incredibly high at small distances between protons.

The velocities of particles in a gas follow the Maxwell-Boltzmann distribution; the number of protons decay exponentially with higher velocities. Due to the temperature not being enough, it is not possible to overcome the Coulomb barrier.

Georgiv Gamov found a way to explain the fusion of the Sun through Quantum Mechanics. In quantum mechanics protons are not point masses but are instead described by wave functions which are probabilistic. When protons approach each other, they do so without a defined position. If the wave function is collapsed and position is defined, then there would be one scenario where the protons would touch each other.

Classical physics suggests that protons do not have sufficient speed (energy) to overcome the electrostatic potential barrier. Even though particle velocities follow a Maxwell-Boltzmann distribution, and thus there are very few particles at high velocities, Gamov found that it is possible to bypass this very high velocity (energy) requirement by quantum tunnelling. This means that the proton could just go through the Coloumb barrier without having to overcome it.

Gamov discovered that although there are fewer particles at higher energies, quantum tunnelling increases at higher energies. There is a region where the probability of quantum tunnelling is maximised is known as Gamov Peak.

 

https://upload.wikimedia.org/wikipedia/commons/thumb/0/04/Gamow_peak.png/600px-Gamow_peak.png

 

However, it is important to consider that the fact that protons can get to touch each other does not necessarily imply that they will fuse. The p-p chain is the nuclear process that dominates in cool stars such as the Sun. It begins with two protons fusing to form a diproton. which is very unstable state. During this short period of the diproton, a proton can be converted into a neutron through weak nuclear interactions, forming deuterium.

 

p-p chain: www.astro.princeton.edu/~gk/A403/fusion.pdf

 

The first step of the chain is very rare, being very slow as it is dominated by weak force. The probability of the diproton producing deuterium is even lower than the probability of quantum tunnelling. However, given that there are 10^56 protons are inside the Sun’s core, fusion can occur at a steady rate for a very long time due to the slow rate imposed by the weak interaction.

 

Visual representation of possible proton interactions

 

In conclusion, without quantum tunnelling the Sun would not be able to fuse hydrogen and, as a result, it would not shine. This combined with the slow and rare weak interaction, enables the Sun to slowly fuse its contents for billions of years.

 

References

https://euro-fusion.org/fusion/fusion-conditions/

https://www.astro.princeton.edu/~gk/A403/fusion.pdf

Video by QuantumFracture on YouTube (in Spanish with possibility of English subtitles)

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