Natural laws of a discworld (astrophysics nerd discussion)

[Systole]

WARNING: Serious astrophysics/math geek discussion ahead.

Note: For convenience, I'm going to refer to planets in our universe as spheres and ignore equatorial bulges due to rotation.




So I was pondering how to make a universe where the natural laws were mostly like ours, except that the natural shape of a planet was a disc, rather than a sphere. I didn't quite come up with a way to make a perfect disc, but I do think I may have figured out a way to make a very flat ellipsoid.

The basic idea is this: What if gravity was not spherically symmetric? What if the universe had a very specific xyz orientation, and gravity behaved differently along the different axes?

So, in our universe, the basic equation for graviational force looks like this:

http://i1257.photobucket.com/albums/ii513/pnthffr/grav1.png

which can be rearranged to

http://i1257.photobucket.com/albums/ii513/pnthffr/grav2.png

and given that r-squared is just the sum of the squares of the x, y and z distances

http://i1257.photobucket.com/albums/ii513/pnthffr/grav3.png


With me so far? Equation 3 is just an alternate way of writing how gravity works in our universe. However, let's imagine a scenario where there are specific, different gravitational constants for each of the xyz axes. Let's call them Gx, Gy, and Gz. In this case, the equation for gravity looks like:

http://i1257.photobucket.com/albums/ii513/pnthffr/grav4.png

In this situation where Gx = Gy = Gz, we have our universe, with its spherical worlds. However, the case where Gx = Gy << Gz, planets would naturally form flat ellipsoids.

http://i1257.photobucket.com/albums/ii513/pnthffr/grav5.png



However, this leads into all kinds of other interesting phenomena. Any rigid body is only going to revolve only around its z axis. By extension, a solar system is almost certainly going to exist completely in the xy plane. In other words, the discworld is going to be in a perpertual twilight, forever viewing its disc-sun edge-on. (For an idea of what this would be like, put a penny and a quarter flat on a table. Move the penny in a circle around the quarter. Now imagine you're living on the face of the penny and the quarter is your sun.)

However, in the case where a planet was somehow knocked out of its normal xy orbit, an xz or yz orbit gets ... complicated. The analogue to a perfectly circular real-universe orbit is a natural ellipse with the sun at the center. The planet maintains constant speed and constant gravitational potential at all points during its orbit, which means that it will orbit farther out when it's looking at the edge of the sun, and very close to the sun as it passes face-to-face. Life on the discworld would suck, because summer would be an inferno, fall and spring would be dim and cold, and winter would be utter darkness.




EDIT: I just realized you'd need to calculate Fx, Fy, and Fz as independent terms. Bleh.

[Yitzi]

Another idea might be to not ignore bulges due to rotation, but rather have that be the primary factor causing the disk shape. Then all you have to do is prevent gravity from collapsing the planet into a sphere. The way I'd approach it (at least to see if it works) is probably to say that rather than gravity pulling toward the center of mass of a body, it pulls toward the closest surface (in essence, if two particles are sufficiently close and sufficiently tightly bound, the gravitons can jump from one to another at minimal "cost").

You're still going to get the planet with no axial tilt viewing the sun (which would probably be a sphere, as this method means that only rocky planets are discs but gaseous bodies are spheres) edge-on, but you can introduce a bit of axial tilt and I doubt it'd be an issue.