Expert: Philip Stahl - 11/5/2011

Question
Hi Philip, since fusion is a fast chain reaction, I don't understand why hydrogen will remain unconverted to helium in the sun for billions of years. I would have expected the chain reaction to spread so rapidly that the sun would become, in effect, a massive nuclear bomb. What is slowing the process of fusion down to such a controlled state?

Answer
Hello,

The problem is one first broached and explored by Prf. Martin Schwarschild in his excellent monograph 'Stellar Evolution' (Dover, 1958). The reason for this has to do with the Coulomb (electrostatic) repulsion between the potentially fusing H-nuclei. Thus, each proton, having (+) charge tends to repel any other proton within a discrete sphere or distance around it. (Recall from your basic physics, like charges repel, unlike attract - and that's what essentially obtains here)

In order for thermonuclear fusion to be realized, the Coulomb barrier must be overcome. 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 ('matter') wave arising from a single proton (p+) of form:

U(x) ~ sin(kx)


where x is the particle's linear displacement (e.g. in 1-D) and k, the wave number vector(k= 2π/L). Here L denotes the wavelength.

Now, though the associated kinetic 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
!
!
! -V-->o
! ! !
! ! !
! ! !
!-------------> x

with the "barrier" at height V, we visualize the particle (o) moving from the left side of the E-axis "tunneling" over to the right side where it may have wave function, U(x) ~sin (kx + φ), where φ denotes a phase angle.

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

It's important here to point out 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 respect of proton fusion in the Sun, Martin Schwarzschild once calculated that the probability of the Coulomb potential barrier being overcome (by tunnelling) is about 1 fusion every 14 billion (14 x 10^9) years.

In temporal terms, we'd thus expect about one proton-proton fusion every 14 billion years (or more than the age of the universe) on this basis!

Clearly, an "offset" is required to reduce the probability, since clearly the Sun and other stars are shining by fusion.

This 'offset' arrives via enormously high density of protons, e.g. in the core, which: i) increases the probability enormously, since so many more protons are in extremely close proximity, and enhances temperatures to the point they can be sustained, and continue - thereby building up other fusion reactions to finish the initial one.

Hopefully, this sheds some light on the seeming paradox!