I'm inventing a constructed world for a personal exercise in writing, and I was wondering if a theoretical cube shaped planet could have a moon. Is that physically possible? I am going on the assumption that the planet is artificial, and I intend on it being a rogue planet.
It should be possible for a moon to orbit a hypothetical cubic planet. It is true that if the moon were orbiting extremely close to the planet, weird things would be happening with gravity that would influence the orbit. However, I believe there would be a distance such that a stable orbit could be achieved. There are at least a couple reasons I believe it is possible for a moon to be in a stable orbit around a hypothetical cubic planet:
I envision a stable situation where the planet and moon could be tidally locked with each other (e.g. one side constantly facing each other) in orbit: (see http://en.wikipedia.org/wiki/Tidal_locking). This way, the varying gravity of the corners would not be impacting the orbit much. The moon is tidally locked with the earth, for example (although the earth is not tidally locked with the moon). Pluto and Charon are tidally locked with each other in this manner though.
Another example is the Rosetta orbiter which is in orbit around a comet. This comet is far from being round, and the Rosetta orbiter is currently in orbit around it (imagine the comet being the planet and the orbiter being the moon). Granted, the comet is not cubic, but it is an example of something orbiting something else that is not quite round. Here is a picture of comet 67P/Churyumov–Gerasimenko (67P), which Rosetta is orbiting:
More about the Rosetta mission can be gathered here: http://en.wikipedia.org/wiki/Rosetta_%28spacecraft%29
Note: It was pointed out that Rosetta is not necessarily in a "true orbit" around the comet, so this might not be a valid example. However, as @ilinamorato mentioned, asteroids (which usually aren't round) often do orbit one another.
A definite yes for small low-orbit and high-orbit satellites as long as the cube does not rotate. Large moons far away should be fine
The question was answered in the 2012 Physics International paper: "The Gravity Field of a Cube". (Available for free) The authors found stable orbits for a tiny moon or space probe to circle the cube. The gravity field of a cube is only distorted close to the cube and with a large moon you can presumably have a situation like the Pluto-Charon system where they each orbit a common center of mass, so cubeness shouldn't be a problem in that case. (Though I can't prove it :)
The authors also provide us with great background material for writing a sci-fi story, such as how a lake on a cube will look, and what to do if you colonize the world:
Consider now a hypothetical cube 12,000×12,000×12,000 km3, approximately the size of the Earth, with the same volume of water and atmosphere as found on the Earth, then we would approximately half fill each face with water and have an atmosphere approximately 100 km thick similar to what is assumed for the atmosphere on the Earth before reaching space. In this case then the corners and the edges of the cube, would be like vast mountain ranges several thousand km high, with their tips extending out into free space. It would therefore be very difficult to cross these mountain ranges and hence we would have six nearly independent habitable zones on each face.
There would presumably be permanent snow on the sides of these vast mountain ranges and people would live around the edges of the oceans on each face in a fairly narrow habitable zone only about 100 km wide as the cube faces rise rapidly through the atmosphere. Unfortunately climbing the approximately 3000 km high corners does not result in an improved view because the surface is still fiat in any observed direction. However the corners, being in free space, would be very suitable for launching satellites. One would also have approximately sqrt(2) x 6000 km of downhill ski run from each corner, down to the centre of each face.
In order to have a day night cycle we would also need the cube to be rotating. The sun would rise almost instantaneously over the face of a cube however, so that each face would need to be a single time zone and thus the cube as a whole would require four separate time zones, assuming the planet was rotating about the centre of an upper and lower face. The north and south faces in this case would be permanently frozen as they would receive no sunlight except that striking the oceans extending away from the surface of the cube, so there might be a permanent pool of liquid water at the two poles. Launching low orbit satellites around this cube needs special care in order to avoid certain orbital resonances that would create significant variations in the orbit.
Possible, yes. You can think of a cube-shaped planet as an ordinary spherical or ellipsoid planet, simply one with eight massive, three-sided, pyramid-shaped mountains protruding from it, equidistant from one another, connected by ridges. As long as your satellite doesn't hit one of those mountains, it can orbit just fine.
Now, how stable will that orbit be? If you assume that the orbit is caused by rotational motion constrained by a gravity well, the orbit will be as stable as the gravity well is. Since a gravity well is a "depression" in space-time caused by the mass of the object, you can compare the gravity well of a spherical vs. a cubical body in space by placing a ball and a block onto a rubber sheet held taut (or a very heavy such body onto a well-tucked-in bedspread). You'll notice that while the gravity well is much sharper to begin with, it soon becomes semi-circular. So depending upon the proximity of your satellite to your planet, the orbit will be more or less stable. If it is close, it will be drawn in more closely as it orbits the vertices or ridges, and flung further out as it travels over a flat plane; if it is further from the planet, those fluctuations will be diminished.
Irregular objects with satellites in stable orbits are far from uncommon. Asteroids, for instance, commonly orbit about one another. Similarly, as Jonathan has noted, the recent Rosetta mission features a man-made orbiter which is maintaining an orbit around a comet by tracing its path about the comet's nucleus.
Yes, once the orbit of the moon is far enough the shape of the planet is largely irrelevant. The orbit of the moon is not going to be as regular as it would be with a sphere but that is generally not an issue. The exception is if the plane of the moons orbit and the plane of rotation of the planet are different or the rotation of the planet is irregular. Also the sun would have effects on the moons orbit and in theory unless the planet is tidally locked the effects became chaotic and can disrupt the moons orbit. In practice, you can simply give the moon a far orbit and assume the effects are negligible for the duration of your story.
Also, while this was not asked, I see no reason why an artificial planet cannot be shaped like a cube. Stability depends on the shape of the gravity field. The shape of the gravity field depends on the shape and density profile of the object. On an artificial planet those are both decided by builders. I'd assume that under some dirt and stone there would be a cube shaped structural frame of high density metal. The rest of the planet, the core of it would be some sort of low density silicate. Surface gravity would become from the high density metal near the surface, while the silicate would provide static pressure to keep the cube from collapsing.
That said a planet like this would be much less stable than natural planets. Specifically, and relevant to the question, the planet would be vulnerable to tidal forces. A small far moon that does not cause significant tides should be fine.
You'd still need a solution to tidal stresses from the sun. But the question was just about the moon, and the answer is that a moon that is far and small enough not to cause noticeable tides would work. And note that the builders might have been satisfied with a million year stable period, and ignored the possibility of the planet falling apart after 10 million years or so. "Warranty expired. No refund. Go ho... away."
This is not a good answer. I am inclined to delete it, but for now, I suggest reading Abulafia's answer and the paper he links to, and Jonathan's answer.
So, I'm putting aside my instinctive feelings towards the concept of a constructed cubic planet here... Your question regarding can X have a satellite Y can be answered by the question: what shape is the gravity well in spacetime that is made by X? Y doesn't care what shape X is. It will just fall along the well made in spacetime by X. Now, if that well has 'corners' on it because of the geometric shape of X, and Y runs into one of those corners, then something odd will happen... Without crunching any numbers on it, I would suggest that a satellite wouldn't be possible, there wouldn't be a stable orbit. Think about what it would be like to stand on an edge of the planet, stepping over onto an adjacent face... Nope, feels like weird maths happening right there.
But this is a constructed planet, right? So... Perhaps the cubic shape doesn't correspond to density, so the edges and vertices of the planet do not present a greater mass to the centre than a point in the middle of a face. If it was made to create a spherical dent in spacetime, then a satellite could orbit it. Of course, standing on the surface of the planet would be strange, as close to vertices, gravity would pull you down at an angle to the surface. Though walking across a flat surface, it would feel steeper and steeper the closer you got to an edge.
But it still wouldn't be a planet, as planets by definition have reached hydrostatic equilibrium (ie they have a round shape). Making it a rogue planet doesn't really change anything... You've really just invented a massive spacecraft with a shape like a Borg craft.
In 1998 I drew a wooden space station for a cubical Earth.
(Full size link)
If you really want to go crazy, the cube is the least of it. Play around with other things in your setting, too.
If a planet is cube, does that mean that spacetime is warped so the shape it settles into (shaped gravity generators)? Then, the round orbital mechanics would not be applicable, or a problem. Make the orbit a hamiltonian path over the corners!
How about warped space so that close to the surface you can't perceive the cubeness and it seems flat, but connected like a spread-open box. Even light turns the corners. They would seem to be on a flat world, but the sky would look odd. Later large-scale measurements would allow the inhabitants to discover the true topology. Later still they use a fleet of hamiltonian satellites to build their GPS system.
Move over, Rincewind.