Hyperion (not the book)

Hyperion Credit: NASA / JPL / SSI / Gordan Ugarkovic

Hyperion is one of Saturn’s many moons. It is also known as Saturn VI for that reason. There are two exciting things about Hyperion; one being its shape that resembles an egg or more mathematically, non-ellipsoid. Secondly, it looks like a sponge! It has many craters all over its surface, giving it this resemblance.

It was the first non-round moon to be discovered. This was in 1848 by three different astronomers; William Lassel and George and William Bond. They named it after the Greek god of watchfulness, it is now watched over by the Cassini spacecraft. This is how we have photographs and most of the data about Saturn, and it’s moons. The orbiter made several flybys of the moon. It took nearly all of the pictures of the solar systems second-largest irregular moon. However, it was Voyager 2 that first found the irregular texture and ridges over the surface in its flyby many years before.

Since then, we have learnt of the low density of the moon, the fact that the moon is half porous and its chaotic rotation of approximately 21 days. Out of all these facts we know, I am incredibly interested in the fact that it has a chaotic rotation.

A chaotic rotation or chaos, in general, is a term from the study of dynamical systems. If you ask for a formal definition, you get something like this: the property of a complex system whose behaviour is so unpredictable as to appear random, owing to high sensitivity to small changes in conditions. A chaotic system could be something as easy as a pendulum pointing vertically upwards as it would be incredibly sensitive to initial conditions.

To talk about a chaotic rotation or orbit, would mean talking about a system of differential equations. Don’t worry that sounds like a *very* complex thing to understand. Still, they are literally some instructions relating to the rate of change of a variable. So let us have a look:

$$ A\,\frac{d\omega_a}{dt} = \left( \omega_b\,\omega_c \,+\, Rbc\right)(B \,-\, C) $$

This is the first equations. It tells you about an $\omega_a$, which is Maths language for velocity in a certain direction, much like on an axis. You can think of $a$ as the $x$ direction. It says that it is proportional to the speeds in the other two directions and some constants. One of which, $R$, relates to the radius of the object. The constants $A$, $B$, $C$ relate to quantities that the object possess’, like eccentricity or mass. The $a, b$, and $c$ relate to the cosine of the angle of rotation of the object concerning a preset axis. These are known as the directional cosines, as you can see which direction the object is pointing.

$$ B\,\frac{d\omega_b}{dt} = \left( \omega_c\,\omega_a \,+\, Rca\right)(C \,-\, A) $$

This is the second equation, which tells you about a velocity in the direction of $b$, or even $y$ if you prefer. The constants hold the same meaning as above, so nothing drastic changes and the equation looks very similar to equation 1.

To get information out of these instructions, we can either; painstakingly sit down with a piece of paper and solve them, or we can go and plug them into a computer and shout ‘SOLVE!’ at it loudly (while clicking on the run script button). Here is what you get if you shout at the computer:

(NB! $xa$ is $\omega_a$ and $xb$ is $\omega_b$)

An ellipse! We have a nice curved ellipse, but it looks like somebody has sketched in an art class? Exactly, that’s why its a chaotic system. While it was being monitored by Cassini, it saw that the orbit started to deviate very slightly from what was considered normal. The model I chose to plot was one developed for Euler, so it is only valid for smallish values of t. However, we only needed a few rotations t show that Hyperion has a chaotic orbit.

The orbit changes and morphs, although changes in the initial conditions don’t seem to change the diagram much, it produced an entirely different orbit. This change also is tough to simulate as you would need to know everything about the conditions on Hyperion for every moment you want the model to run.

Things that went wrong, the forgotten plots.

This project didn’t go totally to plan the whole time. I thought here I would share some plots that were created along the line. I was using MatLab for my plots and the MatCont package to get some dynamical phase portraits that didn’t cause me to have any vector fields etc. over my plots. Plus it was a lot easier than writing my own scripts. Here are the forgotten plots which I think are very interesting in their individual rights.

The first was when I tried to plot $t$ against $\omega_a$ to see what would happen, and I seemed to get two sine waves out. I quite liked this plot but didn’t quite think it was right. The second was when I tried to plot $\omega_a$ against $\omega_b$ and came out with an orbit where Hyperion zoomed through the middle, just for fun.

If this your first saunter into Dynamical systems, look at them more. They are incredibly impressive. If you are looking for something a bit meatier; check out the MathOff pitch I did a few months ago about Arnolds Cat map. I’ll link it below and add the direct PDF of the article.

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