Intergalactic Planetary

Part 1: The Planetary Scale

The goal of this project was to develop a model of the solar system. To begin, we created "rough-draft" models on a google jamboard that allowed us to brainstorm our knowledge of the solar system. Next, we were given data about the solar system and tasked with creating a relational model that relates one or more aspects of the solar system to some target datapoint. Our group (of two people) chose to relate a planet's mass, gravitational pull, and distance from the sun to the number of moons a planet has. It's important to note that mass and gravitational pull are not always proportional, because the gravitational pull is measured from the planet's surface; thus planets with greater density will have a greater gravitational pull than those with equal mass but less density. (The proof for this is in the inverse square law — gravitational force decreases by the square of the distance between the masses, so a less dense and therefore physically larger planet will have less gravitational pull at its surface because this surface is farther away from the centerpoint.)

To create this model, we noted a series of relationships between the aforementioned factors:

1. Planets with greater gravitational pull and greater mass have a larger number of moons

2. Planets closer to the sun have fewer moons

3. Planets on the inside of an asteroid belt have a moderate number of moons

4. Planets on the outside of an asteroid belt have a high number of moons

The logic behind these connections is fairly straightforward. Planets with a larger gravitational pull have a greater sphere of gravitational influence, and therefore they are more likely to "catch" rogue moons. Planets close to the sun are moving quite fast, so an asteroid would have to be going very quickly to become a moon of such planets. Planets on the inside of an asteroid belt will receive a small number of moons simply because of their proximity — however, the planets on the inside will be going faster than the asteroid belt itself, meaning they are more likely to slingshot an asteroid forward rather than attract it into their sphere of influence. Conversely, planets on the outside of an asteroid belt will be going slower than the asteroid belt, so they are more likely to pull a moon into their orbit.

To further solidify the relationship, I developed a mathematical model that gives an estimate of a planet's moon count based on its descriptors. The model was essentially constructed directly from these observations — details are in the slideshow below.

Part 2: The Atomic Scale

Next, we briefly delved into the world of atoms and subatomic particles. We first learned about the parts of the atom, bonds, and each of the subatomic particles' role. The basic idea is that an atom is comprised of three particle types: protons, which carry positive electric charge and are present in the nucleus, neutrons, which carry no charge and are also found in the nucleus, and electrons, which carry negative electric charge and are present outside the nucleus. The electrons have no set position as this is the quantum phenomenon; each electron exists in all places around the nucleus at the same time (!) until we measure them.

Each type of subatomic particle also fills a role in determining the "kind" of atom. Protons determine what element the atom is — for example, helium atoms have 2 protons. Neutrons determine the isotope (or "sub-kind") — to name a few, there exist helium isotopes with 1, 2, and 4 neutrons. Finally, electrons determine an atom's ion (net electric charge). This is fairly obvious — to find an atom's net charge, simply sum the charges of its protons and electrons.

In fact, the standard conception of "particles" is entirely wrong. What we would call a particle is actually a wave, and the wave represents the probabilities of where you're most likely to observe this "particle". Let's go back to our experiment for a moment — imagine we've frozen time at the very instant that a single photon is about to hit the screen. What is the wave representation of this photon? Well, it's a wave that's very strong at the center of the screen and less strong as you go outward. But as soon as the photon is observed on the screen, something incredible happens. The wave "collapses" down to the point where we see the photon. Why? Well, because the probability of observing that photon at the point where we just saw it is 100%, so now the wave is just a single spike right at that point.

Now, photons aren't special in their wave nature. All particles, even those with mass, are actually waves! However, the wavelength of a particle with mass decreases proportional to its momentum, so the faster something moves or the heavier it is the less "wavelike" it will behave. (The name for this phenomenon is the "de Broglie wavelength", named after Louis de Broglie who conjectured this idea.) This explains why normal-sized objects, like tennis balls, don't exhibit the double-slit experiment's behavior: their wavelength is so tiny that it doesn't stretch wide enough to cover both the slits, and therefore we know with 100% certainty that the ball will fly through the slit we threw it through.

With this in mind, we can approach the oft-cited "Schrödinger's Cat" experiment. The fact of the matter is, this thought experiment is incredibly misused! Unlike the common interpretation, the cat is not both alive and dead at the same time until a human opens the box! In fact, this was Schrödinger's original message when he conceived the experiment: it's fallacious to assume that the cat is somehow both alive and dead at the same time simply because no one has looked at it yet. To see exactly why this is, let's walk through the experiment step-by-step.

We start with a cat in a sealed metal box, containing a jar of poison and a geiger-counter-activated hammer. If the geiger counter detects a single particle radioactively decaying, it releases the hammer to break the jar and the cat owners now have to call PETA. If the particle doesn't decay, the cat stays happily alive. The important part is that the radioactive decay of a particle follows the same "wave nature" we've seen before — it can either decay or not decay, and in fact there's an exactly 50% chance (completely and entirely random) of the particle decaying.

Now, this is where the confusion begins. The common explanation goes something like, "Because no one has observed this particle, it is still in a wave-state (it hasn't collapsed yet) and so therefore the cat is in a half-alive, half-dead state until someone looks to see whether the particle decayed or not. Once the particle is observed, its state collapses and therefore the cat becomes either 100% dead or 100% alive." However, this neglects an incredibly important fact: the particle has already been observed. Not by a person, but by the geiger counter! Because the particle is instantly observed by the geiger counter, its waveform collapses instantly and therefore the particle is either fully stable or fully decaying (not a superposition of both!). This means that the hammer either falls or doesn't fall, and thus the cat is either alive or dead long before you open the box.

Additionally, there is the matter of the cat's wavelength. The cat is such a massive object (compared to the atomic scale) that its "wavelength" is very tiny — therefore, even if it was put into a state of superposition it wouldn't be a difference between "alive" and "dead" but rather something like "the instant after death" and "the instant after staying alive".

Despite its misinterpretations, the experiment can provide a good analogy for quantum superposition. In the quantum world, a particle takes on all possible states until it is "observed" (affects an outside system), which then collapses its state onto a single possibility. The same is "true" of the cat: to the animal-abusing scientists outside the box, the cat takes on both the state of being dead and alive (since there is no way for them to know which one it is), and only when the box is opened does the cat's state become definitive to us.

Luckily, there's much more to quantum mechanics than cat abuse, but these experiments cover the core ideas. The key concept is that particles are waves which exhibit every possible state of that particle, until it is "observed" by some outside system. Once the particle is observed, its state collapses down into a single point.

graphics created with help from https://www.falstad.com/ripple/

Momentum

Earlier, I mentioned that a particle's wavelength decreases proportional to its momentum. But massless particles, like photons, have no mass and thus no momentum — so wouldn't that be a division by zero? The answer is that the classical definition of momentum is wrong when objects approach the speed of light. In fact, the correct definition of momentum is:

E² = (pc)² + (m₀c²)²

Where E is the energy of the particle, p is the momentum, m₀ is the resting mass and c is the speed of light. The classical definition of momentum can actually be derived from this, if we assume the object's velocity to be small (relative to the speed of light), but the more important observation is that even when m₀ is 0 there are still terms left over to give the particle momentum.

Reflection

This was a very enjoyable project for me since I really love this type of physics (both atomic and cosmic). In particular, I think I did well with critical thinking. Coming up with the mathematical model for moon count was extremely difficult and required a lot of problem solving to make it accurate. Additionally, even more critical thinking went into writing this description of quantum mechanics, through which I learned a great deal and had to think to explain things I didn't understand at first. Additionally, I think I did well with conscientious learning. I put a lot of extra time into these units, and I actually found myself doing a lot of research regarding quantum mechanics during my free time! I also worked hard and didn't give up, even when some of the calculation was difficult.

As always, there were ways to improve. I think our collaboration wasn't as strong as it could've been this time, since my groupmate and I didn't do as much direct teamwork as usual. We divided up our tasks, did independent research, and then convened for the presentation, but while that approach works it's not exactly "collaboration" since we weren't really working together as much (aside from calculations on Model 5). This also ties into our communication. Most likely, the reason our collaboration wasn't as central as possible was that we weren't really communicating a lot with each other. Because of that, we naturally did more of an "independent work together" strategy than true collaboration.

While there were some minor collaborative improvements to be made, I think this project was (overall) a big success. I learned a lot of things, worked hard, and we ended up with a professional final product.