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Given that the sun’s temperature is lower than the temperature needed to fuse two protons. How does the sun provide enough energy for this fusion to happen?

Has it do something with tunneling and $E = mc^2$ ? I just can´t seem to understand the concept. If someone could simplify this for me, I would be grateful.

From this link, it says " The fusing of two protons which is the first step of the proton-proton cycle created great problems for early theorists because they recognized that the interior temperature of the sun (some 14 million Kelvins) would not provide nearly enough energy to overcome the coulomb barrier of electric repulsion between two protons. With the development of quantum mechanics, it was realized that on this scale the protons must be considered to have wave properties and that there was the possibility of tunneling through the coulomb barrier.".

I was reading through this and my belief was that the sun actually provided a high enough temperature to fuel fusion, however after reading this I am quite confused.

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    $\begingroup$ How do you mean, the Sun's temperature is too low? If it were fusion would not happen. $\endgroup$ – my2cts Oct 25 '19 at 8:27
  • $\begingroup$ From this website link , it says that " The fusing of two protons which is the first step of the proton-proton cycle created great problems for early theorists because they recognized that the interior temperature of the sun (some 14 million Kelvins) would not provide nearly enough energy to overcome the coulomb barrier of electric repulsion between two protons. ..must be considered to have wave properties and that there was the possibility of tunneling through the coulomb barrier." $\endgroup$ – dondeman Oct 25 '19 at 8:29
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    $\begingroup$ There is an potential barrier that classically would prevent fusion. When taking quantum mechanical tunneling into account the temperature and pressure are high enough. $\endgroup$ – my2cts Oct 25 '19 at 9:08
  • $\begingroup$ Which temperature do you speak of? You are aware that the surface temperature (~5777K) and the core temperature (~15 million Kelvin) are vastly different. It would help to understand your problem if you clearly specify which temperature you consider to be too low for fusion. $\endgroup$ – AtmosphericPrisonEscape Oct 25 '19 at 11:22
  • $\begingroup$ A (somewhat) related question on our sister site: astronomy.stackexchange.com/q/33276/16685 $\endgroup$ – PM 2Ring Oct 25 '19 at 15:58
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For completeness I am copying:

The proton–proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun, whereas the CNO cycle, the other known reaction, is suggested by theoretical models to dominate in stars with masses greater than about 1.3 times that of the Sun.

In general, proton–proton fusion can occur only if the kinetic energy (i.e. temperature) of the protons is high enough to overcome their mutual electrostatic or Coulomb repulsion.

In the Sun, deuterium-producing events are rare. Diprotons are the much more common result of proton–proton reactions within the star, and diprotons almost immediately decay back into two protons. Since the conversion of hydrogen to helium is slow, the complete conversion of the hydrogen in the core of the Sun is calculated to take more than ten billion years. ........

The theory that proton–proton reactions are the basic principle by which the Sun and other stars burn was advocated by Arthur Eddington in the 1920s. At the time, the temperature of the Sun was considered to be too low to overcome the Coulomb barrier. After the development of quantum mechanics, it was discovered that tunneling of the wavefunctions of the protons through the repulsive barrier allows for fusion at a lower temperature than the classical prediction.

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The first step in all the branches is the fusion of two protons into deuterium. As the protons fuse, one of them undergoes beta plus decay, converting into a neutron by emitting a positron and an electron neutrino

through the weak decay.

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This reaction is extremely slow due to it being initiated by the weak nuclear force. The average proton in the core of the Sun waits 9 billion years before it successfully fuses with another proton. It has not been possible to measure the cross-section of this reaction experimentally because of these long time scales.[7]

After it is formed, the deuterium produced in the first stage can fuse with another proton to produce the light isotope of helium, $He_3$

This process, mediated by the strong nuclear force rather than the weak force, is extremely fast by comparison to the first step. It is estimated that, under the conditions in the Sun's core, each newly created deuterium nucleus exists for only about four seconds before it is converted to $He_3$ .

In the Sun, each$He_3$ nucleus produced in these reactions exists for only about 400 years before it is converted into $He_4$ .

etc.

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    $\begingroup$ Because the diproton has such a short halflife, the decay of its proton to a neutron is very rare: around 1 in $10^{26}$ times that a diproton is formed does it manage to decay to a deuteron. $\endgroup$ – PM 2Ring Oct 25 '19 at 15:51
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You are correct, the Sun's core ISN'T at a high enough temperature for two protons to fuse together directly.

However, you've answered your question already in the details.

Quantum physics allows a proton to turn into a neutron via the weak interaction. This is an energetically unfavourable process but because of quantum tunnelling, it can happen at the required rate to allow nuclear fusion to power the Sun's output.

Quantum tunnelling is a process by which a classically impenetrable barrier (such as two protons fusing or protons turning into neutrons) is passable.

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    $\begingroup$ In the p-p chain, 2 protons have to fuse into a diproton before 1 of them decays to a neutron, and that decay has a very low probability of occurring in time before the diproton just splits up into 2 protons. There simply isn't enough energy for a free proton in the Sun's core to convert to a neutron, it has to be bound to another proton first. $\endgroup$ – PM 2Ring Oct 25 '19 at 15:57

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