Sun and Stars - How the Sun Works - Nuclear Fusion

People have realized how important the Sun is for thousands of years. Once people realized that the Earth and the other planets revolve around the Sun (around 1600 AD) and then Newton described an equation for how gravity works, it was possible to know how far away it is (93 million miles or 150 million kilometers) and even how "heavy" (massive) it is (about two thousand million million million million tons).

People tried to figure out how the Sun could create so much light and heat energy. Even before 1850 AD, people had figured out that if the Sun was entirely made of something like coal (and the oxygen it would need to burn) the entire Sun would completely burn itself up in only about one thousand years! Since our recorded history is at least several thousand years, and the Sun was always there, that would obviously impossible.

Other ideas were thought up, such as lots of meteorites crashing into the Sun and causing impacts that would make energy, but that too would not make enough energy.

Around 1870 AD, a guy named Helmholtz came up with a much better explanation. He Knew that water on Earth that is high on a mountain has "potential energy" which can be released (converted into some other kind of energy) by falling to a lower altitude. He knew the mass of the Sun (2 * 10³⁰ kilograms) and its size (radius about 7 * 10⁸ meters). He also knew how much energy the Sun is continuously creating and sending out into space (3.86 * 10²⁶ watts or joules/second). He made some assumptions regarding the distribution of mass inside the Sun and calculated similar to the following. Say the ENTIRE Sun's mass could fall by a height of around 37 meters (around 120 feet) in a year. The amount of potential energy that would be released would be given by G * M * M / R² * (Δheight). He knew all those numbers and then had 6.73 * 10^-11 * 2 * 10³⁰ * 2 * 10³⁰ /(7 * 10⁸)² * 37 / (3.15 * 10⁷ seconds in a year). This is 6.45 * 10²⁶ watts! He knew that the inner parts of the Sun could not fall that whole distance, and some other things and Helmholtz figured that if the Sun "collapsed" by just 37 meters radius per year, it could create all the energy by converting that potential energy into light and heat energy.

At that time, (1870) few people questioned the Bible's account where the Earth was around 6,000 years old, being Created in 4004 BC. And Helmholtz quickly calculated that, if the Sun was collapsing like that, it would be able to continue for around another 20 million years. It seemed to make reasonable sense to people in 1870! So the Helmholtz contraction theory was nearly universally accepted as the likely way that the Sun creates its energy.

However, during the following 30 years or so, geologists established that many Earth rocks are definitely millions of years old, and other researchers also confirmed that fossils of some animals were also many millions of years old. It seemed clear that Helmholtz must have been wrong, because the Sun would not have been able to last long enough!

In 1905, Einstein published his famous E = m * c² equation as part of Special Relativity. It suggested that a tiny amount of mass (m) might be converted into an enormous amount of energy (E).

By around 1925, Physicists had determined that the mass of a Hydrogen atom (one proton and one electron) is 1.008 atomic mass units (AMU). They had also determined that a Helium atom has a mass of 4.000 AMU. This seemed peculiar to them, as a Helium is really equal to 4 protons and 4 electrons, and should therefore be 4.032 AMU, four times a Hydrogen. They already realized that MAYBE four Hydrogens might be able to "fuse" together (called fusion) into a Helium, and that the 0.032 AMU that apparently disappears might become converted into energy, by Einstein's formula) an immense amount of energy for the tiny amount of mass that disappears.

Even now, it is extremely difficult for Physicists to accomplish nuclear fusion, and it has only been done in incredibly tiny amounts in amazingly expensive experimental equipment. So we don't actually know precisely what temperature is necessary for two protons to fuse together, but it is certainly many millions of degrees in any temperature scale.

The Sun is so extremely massive that there is extremely high gravitational force that acts on all of its particles that tries to pull everything toward the exact center. The result is that there is extremely high pressure that exists at the center of the Sun. Since the Sun is a ball of gas, the laws of gases are believed to apply inside it. The Ideal Gas Law is P * V = n * R * T. The pressure in an ideal gas is proportional to the temperature. The extremely high pressures due to the weight of all the overlying layers is so high that it provides the extremely high temperatures needed at the very center of the Sun for fusion to occur.

This means that fusion does NOT occur throughout the Sun but only near its exact center.

It turns out that the most obvious sequence of fusion occurring in the Sun (or any star) is not very efficient. It is usually called the Proton-Proton chain. There are several variations of what can happen. Two protons first have to be traveling at extremely high speed EXACTLY toward each other, and on the same flight path, so they don't bounce off or miss each other. Once they fuse, the resulting Helium-2 nucleus must have a decay process where one of its orbiting electrons gets pulled into the nucleus so that it now becomes a Hydrogen-2 (called Deuterium) nucleus, an isotope of Hydrogen. Now, THAT Deuterium nucleus has to hit head-on into another proton, to make a Helium-3 nucleus, which again decays into a Hydrogen-3 nucleus. And next, that new nucleus has to again have a head-on collision with yet another proton, to finally create the Helium-4 nucleus, normal Helium.

This process requires three head on collisions as well as two different decay processes, which makes it a slow process.

A variation of this works more efficiently. Like before, two protons first have to fuse, into a Helium-2. Once they fuse, there is that decay process where it becomes the deuterium nucleus, an isotope of Hydrogen. Now, also like before, THAT Deuterium nucleus has to hit head-on into another proton, to make a Helium-3 nucleus, which again decays into a Hydrogen-3 nucleus. Next, it is different! It finds another Hydrogen-3 like itself to have its head-on collision with. This results in a splattering of things, where a Helium-4 and two Hydrogen-1 nuclei are the main results, normal Helium and normal Hydrogen (which can fuse again).

Here is that sequence in an organized way:

¹H₁ + ¹H₁ fuse to create ²H₁ (Deuterium) + e⁺ (positron) + a neutrino

That positron soon fuses with an electron in an annihilation, where the two disappear and two gamma rays are given off (as radiation).

²H₁ + ¹H₁ fuse to create ³He₂ + a third photon (more radiation).

³He₂ + ³He₂ fuse to create a ⁴He₂ + ¹H₁ + ¹H₁, so that two Hydrogen nuclei are ejected in the process.

This can happen, and when stars were very new and there was nothing other than Hydrogen around, it did. But it requires either three or five separate head-on collisions of tiny, extremely fast moving objects, which tends to take a little time to occur because of the rarity of such exact head-on collisions. It also requires a natural nuclear decay, which tends to be very slow. It is not a very "efficient" method, but it works.

In fact, the Proton-Proton chain is a valid possibility of how the Sun converts SOME of its Hydrogen into Helium in order to convert matter into energy in the Fusion process. The fact that the Sun also contains small amounts of Carbon, Oxygen and Nitrogen enables another process to be possible. It is commonly called the CNO cycle. No one actually knows which of the two processes are actually operating inside our Sun, or whether both operate, and we have no way of ever knowing, since we can never get in there where it is happening! We can collect data and then make educated assumptions, regarding one or the other.

Here is the CNO Cycle sequence. Note that it requires the existence of some previously created atoms like Carbon, Nitrogen or Oxygen, which act as catalysts for the process, meaning that they end up as they started, not being changed when it is all over.

Say we start with a ¹²C₆ standard Carbon atom nucleus. (This bigger nucleus is an easier target for a head-on collision!)

¹²C₆ + ¹H₁ fuse to create a ¹³N₇ + a photon

This ¹³N₇ spontaneously decays into a ¹³C₆ + e⁺ (a positron) + a neutrino

The positron annihilates with an electron to form two photons of radiation (three so far).

¹³C₆ + ¹H₁ fuse to create ¹⁴N₇ + a fourth photon

¹⁴N₇ + ¹H₁ fuse to create ¹⁵O₈ + a fifth photon

This ¹⁵O₈ spontaneously decays into a ¹⁵N₇ + e⁺ (a positron) + a neutrino

The positron annihilates with an electron to form two more photons of radiation (seven so far).

¹⁵N₇ + ¹H₁ fuse to create ¹²C₆ + ⁴He₂ + an eighth photon.

The end result of this cycle is to get back the original Carbon ¹²C₆ nucleus plus a new Helium nucleus. Four separate fusion steps are involved, but they each involve a more massive nucleus as a target for the Hydrogen nucleus, so the fusion process is more easily possible. However, it also turns out that the temperature needs to be even higher! The eight resulting photons carry away the radiation energy (along with some energy carried away by the neutrinos).

For you analytical types, I will add some details! Each of these steps can be analyzed if you just get the precise atomic weights of the various isotopes involved. As an example, the first step of the CNO Cycle is:

¹²C₆ + ¹H₁ fuse to create a ¹³N₇ + a photon

We know the atomic weights of three of these, so:

12.000000 AMU + 1.0078250321 AMU gives (=) 13.00573858 AMU + the photon. Since we have the Conservation of Energy and Mass, the photon must therefore have exactly as much energy as equal to 0.0020864521 AMU. One Atomic Mass Unit (AMU) is equal to 931.476 million electron-volts (MeV) of energy, so this small fraction is equal to 1.9435 MeV of energy. The photon must therefore carry away exactly that much energy, no more and no less! It also turns out that Planck discovered long ago that the amount of energy in a radiation quantum is inversely related to the wavelength of the radiation, so knowing this number tells us what "color" of light gets emitted! It turns out that this is so much energy that it is not in visible light but in waves called gamma rays, but we know the exact wavelength to look for in experiments! And it is there!

Each of the steps in either cycle can be analyzed like this. It is EXTREMELY important that every single step is "exothermic", it creates more energy that it uses up. This enables the entire sequence of steps to proceed without having to rely on any outside source of energy to drive it.

There are variations possible in the exact sequence for either the CNO or Proton-Proton cycle. In a small fraction (0.04%) of the CNO Cycle, the last step does not occur as described above but it fuses to form ¹⁶O₈ (conventional Oxygen) which then has several more steps before generating the Helium nucleus (and a ¹⁴N₇). For the Proton-Proton Cycle, the sequence described above is called pp1, and it certainly the most common sequence occurring inside our Sun. There are likely to be several variations, specifically two that involve the ³He₂ fusing with an existing ⁴He₂ to form larger nuclei. There is also a variation which creates three ⁴He₂ nuclei (also called alpha particles) in the sequence. Most of these variations are calculated to occur extremely rarely in the Sun, mostly because the Sun is not hot enough! It is all a complicated and interesting subject!

Looking at this a different way, we could multiply this number by the amount of mass that disappears in each fusion sequence and get 4.28 * 10¹² grams/second. This is 4.28 * 10⁹ kilograms or over 4 million metric tons of the Sun that disappears every second!

In case you are worried about the Sun using itself up (in disappearing at 4 million tons per second!), we can calculate that, too. We know that the Sun's mass is around 2 * 10³⁰ kilograms. Given the pp1 reaction described above, only a small portion of that could ever disappear in the process of fusing hydrogen into helium, around 1.5 * 10²⁸ kg. At a rate of using that up at 4.28 * 10⁹ kg/sec, we can easily see that it would last around 3.5 * 10¹⁸ seconds, which is about 10¹¹ years, 100 billion years! The Sun is barely a pup, only around 5 billion years old. We need not worry for a while!

For complicated reasons, Physicists think the Sun will only be as it is now for a total of around 10 billion years. Since it probably has already been operating for 5 billion, we probably have another 5 billion years before anything really drastic occurs. You might put it on your calendar!

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago