Irregular Webcomic! #2693

I could just point you to the Wikipedia article on hyperbolic trajectories, but frankly as it stands that article tells you a bunch of boring and technical stuff and none of the good stuff. And if you think that's bad, just try going to Kepler orbit, linked in the opening sentence with the intention of describing a hyperbolic trajectory in simpler terms as a type of Kepler orbit, and scroll down a bit.

While some of you are no doubt perfectly comfortable with solving second order differential equations in order to understand a joke in a webcomic, I'm going to assume that most of you would rather hear the good stuff.

Okay, let's think about how things move in space. To make it concrete, let's think about things going around the sun.

Of course for much of history people didn't realise things went around the sun at all. The Ancient Greeks, particularly Aristotle and Ptolemy, generally held to the idea that the sun, moon, planets, and stars all revolved around the Earth. This is not an entirely unreasonable assumption based on everyday observations available to civilisations without the benefit of knowledge derived from telescopic observations of the heavens. After all, it doesn't feel like the Earth is moving at all, and so if things look like they spin around the Earth roughly once a day, then hey, they probably do. It's only with advanced knowledge that we realise the Earth is spinning once a day and revolving around the sun, and we don't feel that motion because of inertia and gravity and so on.

If you think of things revolving around other things, the most natural path to assume that the objects are taking is a circle. It's simple, it's straightforward, and most of all it's neat and tidy. It's also wrong. While objects can orbit around things in circles, it's not the most common case. Planets (for example) generally move around the sun in ellipses, not circles.

An ellipse is kind of like a stretched circle, if you take the centre of the circle and pull it in half to make two "centres". Any point on a circle is the same distance from the centre of that circle. If you take any point on an ellipse and measure the distance from that point to both of the two "centres", and add those distances together, you get the same total distance. The two "centres" are called foci (which is the plural of focus - one focus, two foci). In the case of a planetary orbit, the planet moves around the ellipse and the sun is at one of the foci. What's at the other focus? Nothing!

This fact, that planets move in ellipses around the sun, was discovered in 1605 by Johannes Kepler, who we've met before in the annotation to strip #2404.

But this was 1400-odd years after Ptolemy had more than enough observations to show that the planets really did not move in simple circles, and certainly not circles around the Earth. How did Ptolemy reconcile this, without breaking his fundamental assumption that the Earth was in the centre of the universe? He invented a system of epicycles, which were smaller circles that moved around the Earth, with the sun or planet moving around the smaller circle as the small circle itself moved around the Earth. This complex jiggery-pokery was necessary to fit even the gross observed movements of the planets, let alone the fine details. In effect, although nobody realised it at the time, the addition of epicycles actually made the path of the planet more like an ellipse than a circle.

Which is kind of by the way, but interesting in itself. We're here to talk about what other sorts of shapes an orbit might have. As it turns out, a stable orbit in which an object keeps going around another object (say, the sun), is always an ellipse. It could be a circle, but a circle is actually just a special case of an ellipse, where the two foci happen to be at the same point. Planetary orbits are, for the most part, almost circular. The foci of the orbital ellipse are very close together. There are objects in the solar system with much more elliptical orbits though.

One such object is Pluto, whose orbit is so elliptical that the two foci are almost 20 astronomical units apart (i.e. 20 times the distance from the sun to the Earth), and it varies in distance from the sun between about 30 and 49 astronomical units. But even this is peanuts compared to some other objects.

Comets are chunks of frozen gases, water, dust, and rock ranging in size from a city block to a major city urban sprawl. There are bazillions of them moving slowly around the sun in the Kuiper belt, roughly were Pluto lives (in fact Pluto is a Kuiper belt object), and the Oort cloud, filling space up to some thousand times further from the sun (roughly halfway to the nearest star). They're almost impossible to observe because they're so small and so far away.

But occasionally something happens to perturb the almost-circular orbit of a comet. If it loses some of its orbital energy, it starts falling in towards the sun. Depending on how the perturbation happened, a couple of different things can occur.

Firstly, the comet can end up on a very stretched out elliptical orbit. This is the sort of thing the famous periodic comets end up doing. Comet Halley is the classic example. Its orbit is so long and stretched out that it gets inside the orbit of Venus at closest approach to the sun, before heading back out as far as Pluto again, to return to the sun again 75 and a bit years later. The orbit of Halley is roughly four times as long as it is wide.

The other possibility is that a comet can plunge in towards the sun and race past it, curving around it slightly because of the sun's gravity, but so fast that the sun can't pull it all the way back around into an elliptical orbit. If this happens, the comet will just fly out the far side of the solar system and keep going, eventually escaping from the sun altogether. This form of orbit isn't really an orbit at all, because the comet won't ever return. It's a hyperbolic trajectory.

It's called this because it follows a hyperbola, a different sort of mathematical curve related to an ellipse, but different. The simplest way to see the relation is to consider conic sections. This is just a fancy term for what you get if you cut a cone. Imagine taking an ice cream cone, resting it on a table pointy end up, and cutting it with a knife. If you cut it horizontally, then the shape made by your cut is a circle - that's straightforward enough. If you angle the knife slightly, you end up making a cut shaped like a somewhat stretched circle. This is actually an ellipse!

Now here's the thing, if you make a vertical cut, slightly to the side of the point, the shape your cut makes through the cone is a hyperbola. You can see that the hyperbola doesn't close back in on itself like the ellipse - if the cone went on forever, the hyperbola would continue forever too, with the two ends always getting further apart. So if an object is following a hyperbolic path through space, it comes in from a great distance, swings around an attracting body (like the sun), then flies back out again in a different direction that never joins back up to where it came from.

That's a hyperbolic trajectory. If you're navigating your way through space and want to get somewhere fast by slinging around some gravitational object in the way, you'd pretty much end up taking a hyperbolic trajectory.

Oh, and one more thing.