I know they’re not exactly spheres (words like oblate and quadrupole come up) but they’re still pretty compact.
Protons repel each other electrostatically and nuclear forces (strong? weak?) attract protons and neutrons to each other.
But what is it about those nuclear forces that makes everything want to be a dense ball and not a chain or big loop?
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The nuclear forces – both strong and weak forces – only operate across very short distances.
In terms of your question, we are interested in the strong force. That’s the one that binds nuclear particles (protons and neutrons) together in the nucleus. (The weak force mediates certain types of nuclear decay.) As the name suggests, the strong force is *very* strong at the tiny scale where it does operate (about 10^(-15) meters), in contrast to the electromagnetic force, which is relatively weak.
The strong force is mediated by massless subatomic particles called gluons, which *mostly* only exist within the nucleus. For ELI5 purposes, you can think of the nucleons being stuck together by these gluons. One of the interesting things about gluons is that, if you begin to pull apart those nucleons, additional gluons will kind of pop into existence and add yet more binding force to that strong force interaction. They form a tiny chain-like structure called a “flux tube” which expresses a constant strong force across its entire length.
But that doesn’t quite reach the heart of your question. Why do gluons behave this way when other force-mediation particles do not? The photon mediates the electromagnetic force, so why is it that when we pull two magnetically attracted things apart, photons don’t generate more and more force to keep them together, as gluons do with the strong force?
To answer that question, we have to dip our toes into Quantum Chromodynamics (QCD). In QCD, the particles that interact with the strong force have a “color” charge, which can *sort of* be analogized to electromagnetic charge. This color charge can be red, green, blue, antired, antigreen, antiblue, and white/colorless (meaning zero net color charge), and as with electric charge, color charge is preserved across interactions. (Note: these are pretty arbitrary names; these particles aren’t actually colored in any sort of literal way.) This is where this crucial difference between gluons and photons comes into play. Photons mediate the electromagnetic force but they themselves do not have electric charge. Gluons, on the other hand, mediate the strong force *and also* carry color charge, meaning that they themselves get bound up in the same force that they mediate. This results in the phenomenon known as “color confinement,” which essentially means that these gluons are more or less spatially confined within the composite particles (hadrons) that they hold together.
So the gluons and everything else that experiences the strong force (i.e., the hadrons and their constituent quarks) are all confined within this very small range. The most energy-efficient way to do that (as another user pointed out) is to arrange everything in a single, quasi-spherical clump.
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