Why 15 million K is sufficient for fusion in solar core?

[jarekduda]

Wikipedia says that solar core has 15 millions of Kelvins ( https://en.wikipedia.org/wiki/Solar_core ), what translates into ~1.4 keV energy per degree of freedom.
For nuclear fusion we need to take the two nuclei to distance of range of nuclear forces: ~1fm ( https://en.wikipedia.org/wiki/Nuclear_force ), Coulomb potential for proton-proton (kE*e*e/r) is ~1.4 MeV in 1fm.
So thermally these protons would reach 1pm distance, but fusion distance would require 1000x larger energy.
Even worse - it is for 1D, in 3D they not only would need 1000x larger energy, but also have velocity precisely pointing the 10^-15m size second nucleus. Otherwise, it would just bounce from the repulsion and fly away.

How is it explained that fusion is actually happening in such relatively low temperature?
"Because tunneling" might be a good explanation for electrons, but for much heavier protons we should understand trajectory, forces.

Maybe electron assistance might be crucial there (?) - e.g. Coulomb says that p - e - p symmetric initial situation should collapse - fuse into deuteron. Electron remaining between the two closing nuclei could screen their repulsion.
More realistically, if electron would have low angular momentum trajectory ( https://en.wikipedia.org/wiki/Michał_Gryziński#Free-fall_atomic_model ): ellipse degenerated into a segment in some direction, if another nucleus would approach from this direction, this electron could stay between them by performing successive backscatterings - screening their repulsion.

 

[e.bar.goum]

This paper will be of interest: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.2954&rep=rep1&type=pdf
The seminal work is of Salpeter 1954: http://www.publish.csiro.au/PH/pdf/PH540373
Indeed, electron screening is found to increase reaction rates above the estimate from "because tunneling". However, "because tunneling" is sufficient to explain the existence of any fusion without the need for electron screening. You do need electron screening for precision cross sections, however.

 

[mfb]

The 1.4 keV are an average. The particle energies have a wide distribution with a tails to more than 10 keV. Keep in mind that for any given collision, a fusion process is incredibly unlikely - the proton "lifetime" is 10 billion years, or something like 10³⁴ nucleus collisions. A 1 in 10³⁴ chance of fusion for every collision.

The power density in the core is just 280 W/m³. A human has a higher power density than the core of sun.

 

[Gerhard Mueller]

Due to fusion processes, the plasma in the suns core is not in thermal equilibrium. The fusion products itself have a high energy, which is transferred via collisions to protons and other nuclei. At least, the fusion rate is increased by this "chain reaction" type of processes. Calculating the stationary velocity distribution under sun core conditions should be possible. (I don't know if somebody has calculated this yet.)

 

[mfb]

At least, the fusion rate is increased by this "chain reaction" type of processes.

Do you have a source showing that it is relevant? The pp fusion is a weak process - it is rare even at higher energies, while the fusion products thermalize in just a few collisions.

 

[Gerhard Mueller]

Braz. J. Phys. vol.29 n.1 São Paulo Mar. 1999
http://dx.doi.org/10.1590/S0103-97331999000100014
Thermal distributions in stellar plasmas, nuclear reactions and solar neutrinos
IV Reaction rates and modified thermal distribution
The dramatic effect of small deviations from the Maxwellian distribution on the rate can be appreciated by ...

 

[mfb]

I don't see any mention of fusion products there.
The studied deviation from a Maxwell distribution is purely an effect of plasma physics.

 

[Gerhard Mueller]

The rate, mentioned in this article is the fusion rate. And the article says, that small deviations from a Maxwell distribution have a dramatic effect on the rate. The article shows, that the "chain reaction effect", which means that fusion reactions indirectly trigger further fusion reactions, is relevant.

 

[mfb]

Fusion products are not a small deviation from the Maxwell distribution. They are an utterly irrelevant deviation at the 10^-30 level. The small deviations the article is discussing are a few percent.

The article shows, that the "chain reaction effect", which means that fusion reactions indirectly trigger further fusion reactions, is relevant.

Where?

 

[Gerhard Mueller]

Fusion products disturb the Maxwell distribution of the particles which have not been fused. The article says, that this is relevant for the fusion rate.

 

[mfb]

No. The article says the deviations discussed in the article are relevant for fusion rates. The deviations discussed in the article are not from fusion products.
See the ~30 orders of magnitude difference between those effects.

 

[Gerhard Mueller]

The article considers the influence of thermonuclear reactions on the velocity distribution and shows that they cause small deviations from the Maxwell distribution. It is a fundamental statement of the article, that these small deviations have a strong effect on the fusion rate.
Text extracted from the article:
(Table 1 reports the values of E0 in units of kT (t) for several reactions.
Table 1. Most effective energies for thermonuclear reactions and exponents g that characterize the change of the thermal average ávsñ to the leading order in d, when the energy distribution changes by a factor exp{ -d(E/kT)2 }: áv sñd = áv sñ0 exp{ -dg}. )
...
In the next section, we shall demonstrate how even such small deviations from the Maxwell-Boltzmann distribution can be very important for solar physics.
IV Reaction rates and modified thermal distribution
--------------------------------------------
(This chapter explains the effect of the velocity distribution on the thermonuclear reaction rates.)

 

[mfb]

What you quote is exactly what I am saying, and in contradiction to your claims.

Plasma effects (unrelated to fusion) lead to a deviation from MB, this deviation has an impact on the fusion rate. This is what the article describes.

 

[Gerhard Mueller]

My claim is, that due to fusion processes, the suns interior is not in thermal equilibrium. And the deviation of thermal equilibrium has an influence on the fusion rate.

Plasma effects are all processes (fusion included) with influence on the velocity distribution. Fusion causes an enormous energy flux from the suns interior to the surface. This energy flux also has an important influence on the velocity distribution.
Apart from that, the article says: Fusion processes cause "Gamov peaks" in the velocity distribution. But other processes additionally modify the velocity distribution.