Expert: Philip Stahl - 12/17/2004

Question
I am doing research on stars for a book I am writing, but I am also an engineer so feel free to provide a technical answer.

In my research I came across a statement that the core of the sun is not quite hot enough for nuclear fusion to take place.  The fusion takes place because the "extra" energy is provided due to the quantum properties of atomic nuclei and the uncertainty principle.  The protons "borrow" a bit of energy from the vaccuum and are able to overcome their repulsion and escape into the strong force range, and hence "stick" together, even before the proper core temperature is reached.

Is this commonplace in most stars?  And does this happen at every stage of a star's life?  Such as, during a Red Giant phase when helium, oxygen, carbon, silicon, etc. are being formed at all layers of the star, is the fusion process at the proper temperature or is the uncertainty principle facilitating the fusion process a bit early?

Thank you for your time and for any explaination you can provide.

Answer
Hello.

I think that two separate considerations are being conflated in the reference you cited. That is, high temperature and the force of Coulomb repulsion (between two like charges, e.g. protons).

All the evidence certainly discloses the Sun's core temperature is more than hot enough to permit nuclear fusion to occur. However, the core density is not high enough to uniformly override Coulomb forces - say to ensure a probability of 1.0 for every fusion, of every particle.

So the Coulomb forces are still going to be there, irrespective of the temperature. And this is going to apply to all the various stellar types - excepting collapsars, and stars with totally degnerate cores (wherein other processes must also enter).

What I am saying is that an integral part of all nuclear fusion processes in the stars is precisely to overcome the energy barrier called "the Coulomb barrier". Fortunately, quantum mechanics allows for a certain non-vanishing probability that a particle (say proton) of kinetic energy K, can overcome a barrier of energy V ("barrier potential"), via the process of "tunnelling".

Note that tunnelling is a general feature of low mass systems, such as single proton (H) states.

Consider a deBroglie wave arising from (p+) of form: U(x) ~ sin(kx) where x is the linear dimension along discplacement and k, the wave number vector (k= 2 pi/lambda).

Now, though the associated energy K < V (the barrier "height") the wavefunction is *non-zero* within the barrier, e.g.

U(x_b)~ exp(-cx)

So, visualizing axes for this:


E
!
!
!       ___
!   O   !  !
!       !  !
!       !  !
!------------------------------>  x

with the "barrier" at heigh V, we visualize the particle on the left side (o) "tunneling" over to the right side - where it may have wave function, U(x) ~sin (kx + phi), where phi denotes a phase angle.

Note that if the barrier is not too much higher than the incident energy, and if the mass is small, then tunnelling is significant.

Note that the penetration of the barrier is a direct result of the *wave nature of matter*! In effect, this wave nature - which is uniquely quantum mechanical in origin- allows a higher energy barrier to be penetrated by a lower energy particle, something totally without parallel in classical, Newtonian physics!

In the case of the Red Giant stars, the primary fusion process is known as the "triple alpha":

He4 + He4 + He4 ->   C12          (Summary 0f 3-alpha)

Bear in mind that for helium, there are now *three* Coulomb terms for its Schrodinger eqn. solution. Thus, Coulomb repulsion will enter here as well, and in order for the neecessary fusion to occur, tunnelling will be required.(Again, it must be born in mind that the probability is extremely small, but because so many fusion particles exist in the core, they compensate for the small numerical probability!)

Extrapolating the above to "every stage of a star's life" is both complex and perhaps - assuming too much - given the present stage of our knowledge. For one thing, what's been described above applies mainly to the non-degenerate case, e.g. non-degenerate matter.

Once a core becomes "degenerate", then numerous other factors enter that complicate the whole picture, including (but not limited to): core contraction releasing large amounts of gravitational energy to heat it adiabatically, neutrino losses cooling the central region of degenerate cores.

Up to now, no one is certain of how these antagonistic processes govern stellar cores in the realm beyond Red Giants, and fusion....say, past that of helium nuclei fusion.

You will notice in going over my response that I've avoided any mention of the "Heisenberg Principle". This is because the energy for pair-production (e.g. of pions), usually discussed in conjunction with the *energy-time* form of the principle (delta E * delta t > = h/2 pi) , should not be confused with the wave properties of matter responsible for tunnelling.

I hope the above information proves useful.