On Binary Planets, and Binary Polyhedra

Faceted Augmented Icosa

This image of binary polyhedra of unequal size was, obviously, inspired by the double dwarf planet at the center of the Pluto / Charon system. The outer satellites also orbit Pluto and Charon’s common center of mass, or barycenter, which lies above Pluto’s surface. In the similar case of the Earth / Moon system, the barycenter stays within the interior of the larger body, the Earth.

I know of one other quasi-binary system in this solar system which involves a barycenter outside the larger body, but it isn’t one many would expect: it’s the Sun / Jupiter system. Both orbit their barycenter (or that of the whole solar system, more properly, but they are pretty much in the same place), Jupiter doing so at an average orbital radius of 5.2 AU — and the Sun doing so, staying opposite Jupiter, with an orbital radius which is slightly larger than the visible Sun itself. The Sun, therefore, orbits a point outside itself which is the gravitational center of the entire solar system.

Why don’t we notice this “wobble” in the Sun’s motion? Well, orbiting binary objects orbit their barycenters with equal orbital periods, as seen in the image above, where the orbital period of both the large, tightly-orbiting rhombicosidodecahedron, and the small, large-orbit icosahedron, is precisely eight seconds. In the case of the Sun / Jupiter system, the sun completes one complete Jupiter-induced wobble, in a tight ellipse, with their barycenter at one focus, but with an orbital period of one jovian year, which is just under twelve Earth years. If the Jovian-induced solar wobble were faster, it would be much more noticeable.

[Image credit: the picture of the orbiting polyhedra above was made with software called Stella 4d, available at this website.]

Trinary Rhombicosidodecahedra

Faceted Augmented Rhombicosidodeca

This image of three rhombicosidodecahedra “orbiting” a common center was made with Stella 4d, a program you may try for free at this website.

Tidally Locked Binary Icosidodecahedra

binary icosidodecahedra

I’ve been trying to figure out for over a year how to make images like the one above, without having holes in the two polyhedra, facing each other. At last, that puzzle of polyhedral manipulation using Stella 4d (software available at this website) has been solved: use augmentation followed by faceting, rather than augmentation followed by simply hiding faces.

On the Direction of Motion of Spinning Polyhedra, the Rotating Earth, and Both the Rotation and Orbital Revolution of Other Objects in the Solar System

twistedIn which direction is the polyhedron above rotating? If you say “to the left,” you’re describing the direction faces are going when they pass right in front of you, on the side of the polyhedron which faces you. However, “to the left” won’t really do . . . for, if you consider the faces hidden on the side facing away from you, they’re going to the right. What’s more, both of these statements reverse themselves if you either turn your computer over, or stand upside-down and look at the screen. Also, if you do both these things, the situation re-reverses itself, which means it reverts to its original appearance.

Rotating objects are more often, however, described at rotating clockwise or counterclockwise. Even that, though, requires a frame of reference to be made clear. If one describes this polyhedron as rotating clockwise, what is actually meant is “rotating clockwise as viewed from above.” If you view this spinning polyhedron from below, however, it is spinning counterclockwise.

Since I live on a large, spinning ball of rock — of all solid objects in the solar system, Earth has the greatest mass and volume, both — I tend to classify rotating objects as having Earthlike or counter-Earthlike rotation, as well. Most objects in the Solar system rotate, and revolve, in the same direction as Earth, and this is consistent with current theoretical models of the formation of the Solar system from a large, rotating, gravitationally-contracting disk of dust and gas. The original proto-Solar system rotated in a certain direction, and the conservation of angular momentum has caused it to keep that same direction of spin for billions of years. Today, it shows up in the direction that planets orbit the sun, the direction that most moons orbit planets, and the direction that almost everything in the Solar system rotates on its own axis. Because one direction dominates, astronomers call it the “prograde” direction, with the small number of objects with rotation (or revolution, in the case of orbital motion) in the opposite direction designated as moving in the “retrograde” direction.

So which is which? Which non-astronomical directional terms, as used above when describing the spinning polyhedron there, should be used to describe the prograde rotation of Earth, its prograde orbital revolution around the sun, and the numerous other examples of prograde circular or elliptical motion of solar system objects? And, for the few “oddballs,” such as Neptune’s moon Triton, which non-astronomical terms should be used to describe retrogade motion? To find out, let’s take a look at Earth’s revolution around the Sun, and the Moon’s around the Earth, for those are prograde is well. This diagram is not to scale, and the view is from above the Solar, Terran, and Lunar North poles.

animation

[Image found reblogged on Tumblr, creator unknown.]

Prograde (Earthlike) motion, then, means “counterclockwise, as viewed from above the North pole.” To describe retrograde (counter-Earthlike) motion, simply substitute “clockwise” for “counterclockwise,” or “South pole” for “North pole,” but not both. Here’s the spinning Earth, as viewed from the side:

just_earth_800

[Image source: http://brianin3d.wordpress.com/2011/03/17/animated-gif-of-rotating-earth-via-povray/ ]

If you’ll go back and check the polyhedron at the top of this page, you’ll see that its spin is opposite that of this view of the Earth, and it was described as moving clockwise, viewed from above. That polyhedron, and the image of Earth above, would have the same direction of rotation, though, if either of them, but not both, were simply viewed upside-down, relative to the orientation shown.

Stella 4d, the software I use to make rotating polyhedral .gifs (such as the one that opened this post), then, has them spin, by default, in the same direction as the Earth — if the earth’s Southern hemisphere is on top! As I live in the Northern hemisphere, I wondered if that was deliberate, for the person who wrote Stella 4d, available at www.software3d.com/Stella.php, lives in Australia. Not being shy, I simply asked him if this were the case, and he answered that it was a 50/50 shot, and simply a coincidence that it came out the way it did, for he had not checked. He also told me how to make polyhedral .gifs which rotate as the Earth does, at least with the Northern hemisphere viewed at the top:  set the setting of Stella 4d to make .gifs with a negative number of rotations per .gif-loop. Sure enough, it works. Here’s an example of such a “prograde” polyhedron:

negative spin

How To Destroy a Planet

Let me make it clear from the outset that this is a purely academic exercise. I don’t REALLY want to destroy any planet, let alone the one we live on, even though the beginning of this song — http://www.youtube.com/watch?v=rR5xTgMwpiM — is my cell phone ringtone. I will admit that much.

But, seriously, how would one destroy a planet? I don’t mean kill everything on it — I mean utterly obliterate the whole ball of rock.

Well, let’s assume it isn’t a rogue planet, but one like ours. It orbits a star, in a nearly-circular orbit, and rotates on its axis. Let’s say it’s the blue one in this picture:

…and we want to know a way to destroy it — just as an interesting puzzle to solve. That’s all — I promise.

Well, planets that are closer to their stars orbit faster, which is why the planet Mercury is the fastest-moving planet in our solar system. Speed up the orbital velocity, then, and a planet’s orbit will get smaller in diameter, for faster planets orbit more quickly. In the diagram (not even close to being at scale for anything real), this is why the red planet’s velocity vector is shown as being longer (faster) than that of the blue planet.

So, speed up a planet enough, and its orbit will decay until it falls into its star, which should destroy it most effectively.

So it’s really pretty simple. On the blue planet, pile up a bunch of nuclear weapons, rocket engines — whatever you can find that invokes Newton’s Third Law of Motion — and start blasting at the position where you see the orange triangle (sundown, local time), right on the equator (unless there is axial tilt involved; correct for that if there is, and keep the blast site in the plane of the ecliptic). Keep blasting as the planet rotates for one-fourth of a rotation (counterclockwise), until the blast site is at midnight local time, and then stop. Repeat at the next sundown, and so on.

All of this blasting will speed up the blue planet’s orbital velocity. Eventually, it will end up where the red planet is, if you do this gradually enough to maintain near-circularity of the orbit (get too eccentrically elliptical, and other results may occur). Keep up this madness, and the target planet will end up slowly spiraling into its star.

That’s it. Game over.

Please do not try this at home, though. All my stuff is here.

[Later edit:  I showed this to many of my science-geek friends, and they tore it to shreds. There are a lot of mistakes here. I considered taking it down, out of embarrassment, but have decided to leave it posted to remind myself that I can, and do, screw things up. I need such reminders!]