Would solar systems be "flat" in any manner in 4d?

[Vector_Graphics]

Occupying one 2d plane, two distinct 2d planes, or what?
This is assuming:
- Planets and stars can form in 4d
- The inverse square law is followed (and thus, that orbits behave identically to in 3d, as two-body motion is planar).

[quickfur]

Those are pretty big assumptions

But let's say we assume they hold, for the sake of argument. Then solar systems, or more precisely, planetary systems, assuming they coalesced from the gravitational condensation of gas that initially formed the central star and that the orbiting planets result from leftover angular momentum in the system, then probably it would be planar? I'm not 100% sure. Perhaps it could exist in two distinct planes ala the clifford double rotation. But it's hard to know without more exact parameters that one could plug into a Monte Carlo simulation to see what kinds of results are obtained.

One interesting thing about a planetary system in two 2D planes is that the planets in one plane would, as far as the other plane is concerned, occupy the stationary plane intersecting at the origin (the star), so the collective mass of those planets would effectively act as additional gravity to the planets in the other plane. Furthermore, since they would oscillate above and below the other plane within their own orthogonal plane, planets in the other plane would experience an oscillating gravitational force perpendicular to their orbital plane. The net force is addtional gravity in the direction of the star. But I'm not sure if this perpendicular oscillating force would destabilize the orbit within the plane or not. Depending on the relative orbital periods, maybe it could actually act as a stabilizer? Again, hard to say without a Monte Carlo simulation to observe the possibilities.

[Vector_Graphics]

Yeah. Usually, though, I'd imagine planets being far enough apart that it doesn't really matter that much.

[PatrickPowers]

The mundane 3D case was the great problem of the 19th century of Laplace and Lagrange so even that is well beyond me. Today the SS has been found to be (mildly) chaotic. So the 4D case is out of reach. I guess the thing to do is make a simulation and see if you can get a stable solution. My guess is that a solution with two planes could be done and could arise naturally.

[PatrickPowers]

The mundane 3D case was the great problem of the 19th century of Laplace and Lagrange so even that is well beyond me. Today the SS has been found to be (mildly) chaotic. So the 4D case is out of reach. I suppose the thing to do is make a simulation and see if you can get a stable solution. My guess is that a solution with two planes could be done and could arise naturally.

[steelpillow]

The strict 2D case (flatland) is a simple inverse law and not an inverse square law. The square arises in 3D because of the spherical surface at the orbital distance.
Nevermind. A 2D orbit in 4D is unstable. In 3D, a wobble out of the plane merely tilts the plane slightly. But in 4D it will begin to couple orbital energy into other planes in a complicated way, and the orbit soon breaks up. So you'd need some mechanism (an extra law of physics, perhaps) which prevents such wobbles from happening at all.
I guess you could then come up with something fancy, like an orbit in the wx plane and a closer-in harmonic orbit, i.e. say half or a third the period, in the yz plane. A bit like Lissajous figures in 4D.
This is possible because the distance from the star is the usual function of wxyz so being close in the y,z orbit does not mean you are close in your w,x orbit. But you do also have to take account of the wy, wz, xy and xz orbital planes as well, which is where harmonics become essential to keeping order.