Lecture 2
Energy Generation in the Sun
In this lecture, we take a big step back from the endpoint of lecture 1 and fill in more of the details about nuclear fusion in the sun. We begin by defining the concept of nuclear fusion more clearly:
Nuclear Fusion: Two atomic nuclei fuse together to form a different type of nucleus. Energy is released.
An example of this is the (here-simplified) reaction
Hydrogen + hydrogen à deuterium + energy
This is one of the reactions that take place in the core of the Sun. Two hydrogen nuclei combined and formed something new, a deuterium nucleus.
What’s in an atom?
Atoms are composed of three different “ingredients,” protons, neutrons, and electrons. Protons have a charge of +1, neutrons are electrically neutral, and electrons have a charge of -1. Together, protons and neutrons are known as “nucleons.” These are the particles that make up the nucleus of an atom.
There are actually three different forms of hydrogen. They all contain one proton and one electron, but the number of neutrons varies. The simplest atom of all is hydrogen with no neutrons. Hydrogen with one neutron is called “deuterium” or “hydrogen-2” and hydrogen with two neutrons is called “tritium” or “hydrogen-3.” No matter how many neutrons you jam into the nucleus, it is still hydrogen. The single proton in the nucleus is what makes it hydrogen. These different forms of hydrogen are called “isotopes” of hydrogen.
Below we will encounter two different forms of helium: helium-3, containing two protons and one neutron, and helium-4, containing two each of protons and neutrons. Helium is defined by the presence of two protons in its nucleus. It also has two electrons.
Chemical vs. nuclear reactions
The reactions we see on Earth are generally chemical reactions. These are fundamentally different from nuclear reactions. In a chemical reaction, the nuclei of the atoms involved remain intact and separated. The positively charged nuclei repel one another, but the combined negative charge of their associated electrons serve as a sort of glue to keep the whole thing together. If two hydrogen atoms are combined in a chemical reaction, the molecule H2 results. Also known as molecular hydrogen, the two protons (hydrogen nuclei) remain separate, although the atoms are bound together. Although they are joined together for the time being, the two atoms still exist and maintain their integrity.
In a nuclear reaction, the nuclei of the two atoms join together and become a single nucleus of something new. When two hydrogen atoms are combined in a nuclear reaction, an atom of deuterium (hydrogen-2) results. The two original hydrogen atoms no longer exist.
Why Aren’t All Reactions Nuclear?
Nuclear fusion reactions are difficult to achieve because the positively-charged nuclei repel one another due to electrostatic repulsion. The closer you try to push them together, the more powerfully they repel one another. In fact, if it weren’t for a second type of force, no atoms beyond hydrogen could exist. Fortunately for us, there is another force involved, the strong nuclear force. When two nucleons (either two protons, two neutrons, or one of each) are brought closely enough together, they feel an attraction due to this force. This attraction is so strong that it completely overwhelms any repulsion due to like charges. Just as gravity is a force that all matter feels just because it is matter, the strong nuclear force is a force that all nucleons can feel just because they are nucleons. The catch is that the nucleons have to be extremely close for the strong nuclear force to take over, within 10-15 meters. This is the force that binds together the multiple protons in the nucleus of an atom of a heavier-than-hydrogen element.
So, there is a force that can make two protons stick together if they can be brought close enough together. How do we bring them within this distance? It’s not so easy – try to move them together, and the repulsion they feel for one another pushes them apart.
The key is temperature. The higher the temperature, the faster the atoms are flying around. At the most extreme high temperatures, atoms are separated into electrons and nuclei, individually swimming around at high speeds. If two nuclei are moving too slowly, even if they are on a collision course, they will be stopped by their mutual repulsion before they can interact. However, if they move fast enough, they can get within 10-15 meters before the repulsion can stop them. Then the strong nuclear force takes over and the nuclei can fuse. At temperatures above about 10 million K, protons are moving fast enough to overcome their repulsion and fuse together. These temperatures are found at the core of the Sun.
Reactions in the Sun
Step 1: hydrogen-1 + hydrogen-1 à hydrogen-2 + position + neutrino
We have already discussed hydrogen-1 and hydrogen-2 above. A positron is just like an electron, but with a positive charge. It is an anti-matter particle. We will talk about neutrinos next week, in lecture 3.
Two things happened in this reaction. First, two protons came close enough together to fuse. Second, one of those protons turned into a neutron (!), forming hydrogen-2 (which contains one proton and one neutron). The “extra” particles on the right are formed in that conversion process:
Proton à neutron + positron + neutrino
We will talk about this sort of reaction more in the future, but for now you should know that a neutrino is formed whenever a neutron is formed. The positron is necessary to carry away the positive charge that was in the proton. The charge can’t just be destroyed.
Where does the energy come from in step 1?
Associated with each hydrogen-1 nucleus was an electron. After the nuclei fuse into hydrogen-2, those two electrons are still out there. One of these electrons will annihilate with the positron. They are a matter and anti-matter pair and their annihilation will result in energy. This will leave the hydrogen-2 nucleus with the one electron that it requires.
Step 2: hydrogen-1 + hydrogen-2 à helium-3 + energy
Much more straightforward. Two protons and one neutron on the left and two protons and one neutron on the right. The hydrogen-2 in this reaction is that created in step 1.
Step 3: helium-3 + helium-3 à helium-4 + hydrogen-1 + hydrogen-1 + energy
Again, fairly straightforward.
Why doesn’t the Sun explode, what with all of this energetic nuclear active in its core?
Unlike a bomb, the Sun has a tremendous amount of mass (2 x 1030 kg). The combined gravity of the whole thing keeps it together. The pressure pushing outward due to the high temperatures at the core is precisely balanced by the gravity pushing inwards. It is an extremely stable equilibrium. Should it get slight perturbed, it will get pushed back to this equilibrium point:
If the temperature rises slightly at the core due to increased reactions & energy generation, the pressure at the core will rise. The increased pressure will cause the star to expand somewhat in the interior, which will then cause the temperatures to cool and the reactions to slow down. Thus equilibrium is restored.
Finally, one calculation:
How many fusion reactions occur in the Sun each second?
Luminosity = 4 x 1026 J/s
One Reaction Produces 4 x 10-12 J (from E=mc2 on the “lost” mass)
How many reactions?
(4 x 1026 J/s) / (4 x 10-12 J/reaction)
à 1038 reactions each second!