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]

Deleted as redundant.

[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.

[quickfur]

It's well known that if gravity obeys an inverse cube law, as it would in 4D if we assume gravity is the result of the exchange of (possibly virtual) gravitons, then orbits are inherently unstable. The consequences have been worked out here on this forum before; there are 5 cases:

1) The planet's path diverges, i.e., there is no orbit, it flies off into space.
2) The planet's path spirals outwards, each iteration being a constant distance farther than the previous. Eventually it will also fly off into space, but in a more controlled way.
3) The planet's path is a perfect circle. This is the only case where there is a stable orbit, but it's a local maximum, meaning that the slightest perturbation will destroy its stability and send it into the other 4 unstable cases. I.e., a speck of dust landing on the planet from space will knock it off its perfect circular orbit into one of the other cases. Not to mention the extreme unlikelihood that a perfectly circular orbit would arise spontaneously in a hypothetical star formation scenario.
4) The planet's path spirals inwards, each iteration being a constant distance closer than the previous. Eventually, it will collide with the star it orbits.
5) The planet's path converges to the star, i.e., there is no orbit, it just crashes into the star.

The cause of the lack of stable orbits (besides the impractical perfect circular case) is that the 1/r^3 gravity well can never be perfectly balanced by the mv^2 component of the orbiting body's momentum. There is always a leftover term that will cause the orbit to be unstable. As long as gravity obeys a 1/r^3 law, stable orbits are inherently impossible.

The only way around this is to somehow force gravity to obey a 1/r^2 law instead. It would likely require an unnatural by-fiat imposition of some arbitrary law that violates the flux law or otherwise postulates a completely different mechanism for gravity. Assuming this is done (and this is a huge, huge, huge assumption), then we could have stable orbits in 4D in the analogous way to 3D orbital systems (e.g., orbital paths are conic sections, the stable ones among which would be the circular and elliptical cases).

One possibility that I've come up with in seeking a solution to this conundrum is Einstein's idea behind general relativity: in our 3D universe, it seems awfully convenient that acceleration follows a square law, and gravity also follows a square law. (And furthermore, inertial mass equals gravitational mass, even though there's no a priori reason for such an equivalence.) Einstein thus made the leap of considering what if acceleration is gravity, and gravity is acceleration. Thus, he arrived at general relativity, where gravity isn't a conventional force per se, but is caused by a curvature of space, and the path of an object under gravitational acceleration is actually its inertial path in curved spacetime. If we take this idea in its most radical form, we could postulate that if in our hypothetical 4D universe a similar thing holds, where 4D gravity is the consequence of a quadratically-varying curvature of space, then we could imagine a scenario where gravity actually obeys an inverse square law rather than the inverse cube law implied by the flux theory of force. I.e., our 4D gravity would be exactly the same as motion in an inertial frame, and it's the curvature of the space itself that causes the 1/r^2 shape of the gravity well. This would give us stable 4D orbits, and possibly also give rise to a bunch of unusual consequences that could be rather interesting to explore.

[steelpillow]

Didn't Kaluza and Klein independently study just that? They added an extra dimension to General Relativity and found that, in order to stabilise things, you had to somehow "compactify" it to tiny size. The remarkable outcome of that was that, alongside conventional General Relativity, the equations of electromagnetism just appeared. Unfortunately, so did something known as the Kaluza-Klein scalar, and nobody has ever been able to detect anything matching its possible properties.
So yeah, any extensive 4D space would have to explain gravity - or its equivalent - some other way that would allow stable "de-compactification". Things might be a bit dark in there though, unless you added some form of light back in as well.

[quickfur]

It all boils down to what your goal is.

If your goal is to derive a workable theory of physics with 4 macroscopic dimensions from first principles, then good luck, my hats off to you.

OTOH if the goal is simply to have some kind of workable, consistent physical laws that govern a universe with 4 macroscopic dimensions, then a lot more options are available. We don't have to be bound to the laws of physics that we see in 3D. (In fact, it's questionable whether an actual 4D universe would exhibit the same or analogous behaviours at all. Almost all physical theories of today are postulated in the context of observable physics, i.e., the physics of the 3D universe in which we live. Many of the underlying assumptions do not necessarily hold once you alter the number of macroscopic dimensions to 4 instead of 3, and many of the widely-accepted consequences may no longer apply.)

Or alternatively, if the goal is simply to explore the consequences of having a geometry of 4 macroscopic dimensions, we needn't concern ourselves with the nitpickings of physical laws at all. Just declare that planetary systems exist by fiat, and call it a day. All sorts of interesting geometrical insights can be obtained this way, without needing to worry about being consistent with 3D physics (why should it even be, in the first place?).

[Frisk-256]

No, Because in 4D a cloud of gas does not have to rotate around 1 plane overall. In 3D, the overall cloud of gas forming the system has to spin around 1 specific plane overall. This rotation flattens the disc and makes it so there is only material to form planets around that flat disc. In 4d a cloud of gas can rotate around both the xz and yw planes at the same time. So because there is not just 1 overall rotation, the could would not flatten, and the resulting star system would not be either. Also I know about the problems in general with 4D orbits, this is assuming 4D orbits are even possible.

[Frisk-256]

The cause of the lack of stable orbits (besides the impractical perfect circular case) is that the 1/r^3 gravity well can never be perfectly balanced by the mv^2 component of the orbiting body's momentum. There is always a leftover term that will cause the orbit to be unstable.

Are we sure the mv^2 term is the same in 4D? If it was different could it cancel out the 1/r^3?

[quickfur]

Integrating momentum always gives you an mv^2 term, regardless of dimension. This comes straight out of the definition of velocity and momentum. The point isn't so much that the mv^2 "'cancels out" the 1/r^3 term; the point is that since both are quadratic, their linear combination is also quadratic, i.e., the resulting path is always a quadratic curve, which leads to the well-known classes of orbital paths (circular, elliptical, parabolic/hyperbolic). But when you have a 1/r^3 term, the resulting path will almost always be a cubic curve, and the problem is that among cubic curves only an extremely narrow class (the perfect circle) is periodic; all others are non-periodic and end up with the planet either colliding with the star or flying out of orbit.

Now, the interesting thing here is this: suppose you're on the surface of some n-dimensional planet, and you throw a ball into the air. What path will it follow? This is a trick question, because, assuming the height of the ball's path is small enough, and the radius of the planet is large enough, the strength of the gravitational field is almost exactly the same at every point of the ball's path. Meaning that it will experience, to a small margin of error, a constant downward acceleration. So the resulting path will be parabolic, just like in 3D (!). The only time you'll notice any difference is when the height difference in the ball's path is large enough that the shape of the 1/r^(n-1) gravitational well becomes non-negligible. Then the path starts to diverge from the expected parabolic (quadratic) path.

This is where my idea about Einstein's general relativity came from. He noticed that it seemed awfully convenient that in our 3D universe, the effect of the gravitational field produces the same kind of results (parabolic trajectories) as the effect of acceleration under some externally-applied force. What if gravity is acceleration? So I extend his idea to higher dimensions as well: what if n-dimensional gravity isn't a particle-mediated force that then must obey the flux law, resulting in a 1/r^(n-1) law, but instead, was the result of the same curvature of space that gives rise to the same acceleration as in 3D general relativity? I.e., what if the quadratic (parabolic) path of the ball remains quadratic regardless of the magnitude of its height difference? Then we would get stable elliptical orbits in n-dimensional space, regardless of the value of n. Thus, we would bypass the 1/r^(n-1) flux law, by postulating a completely different mechanism of gravity that isn't related to the exchange of mediating particle force carriers.

[PatrickPowers]

Superexpert Ed Witten said that in general relativity the inverse square law arises directly from the 3D nature of space. That doesn't prove anything, but I found it a strong hint. I'd like to have someone confirm that we get inverse cube in 4D general relativity, but the physicists I deal with are very touchy about this sort of thing so I don't dare. They are angered if you ask them a question they can't answer.

[quickfur]

If the physicists you know get angered by questions they can't answer, maybe you should get to know other, less ego-centric physicists.

At the crux of this question is, what exactly it is about matter that determines the curvature of space. Because that's what will determine what kind of curvature results from a particular distribution of mass, whether it will be quadratic or something else. One key question in this direction is, why does the curvature produce quadratic trajectories, when the mass of a planet is proportional to the cube of its radius? Why doesn't the gravity well produce cubic trajectories instead? What is it about the matter-curvature relation that causes the gravity well to produce quadratic trajectories, when the mass of an object (a sufficiently large one, which by tidal forces must be spherical) is proportional to the cube of its radius?

Thinking about it more, there should be no reason why the curvature of space should relate to the rate at which mass increases when you increase the radius of a planet: at astronomical scales it doesn't actually matter what shape the planet is or how its mass is distributed; it's essentially a point-mass. So in a 4D space, stars and planets also ought to be treated as point masses as well. So then the question is, why is it that a point mass in 3D produces a quadratic well, whereas a point mass in 4D produces a cubic well? If curvature depends only on the magnitude of a point mass, why can't it result in a quadratic gravity well instead of a cubic one?

I guess the answer must have something to do with the nature of space and how it curves. It must be some kind of inherent property to space and how it interacts with mass, that determines the resulting curvature.. I admit ignorance in this area -- what exactly is the relationship between mass and curvature, why is it that mass causes space to curve? And why by that specific amount per unit mass? Can it be otherwise? Can a higher-dimensional space continue to generate quadratic gravity wells in response to point-masses?

//

Another way of thinking about this, is this thought experiment: suppose space was 4D yet stable orbits exist. If Einstein were to live in such a universe, what kind of theory would he have come up with to explain it? How different would General Relativity look if Einstein had invented it in a 4D universe with stable orbits? Which parts of the theory would be different? Can these differences be logically consistent, without leading to bizarre consequences that negate the other parts of a 4D physics derived in analogy from 3D?

[steelpillow]

why is it that a point mass in 3D produces a quadratic well, whereas a point mass in 4D produces a cubic well? If curvature depends only on the magnitude of a point mass, why can't it result in a quadratic gravity well instead of a cubic one?

It's a simple consequence of the way the field spreads out and weakens over distance.
You can think of a gravitational field as lines radiating out from the mass. The number of lines is proportional to the mass.
If you now imagine a spherical shell around the mass in 3-space, the gravitational force is the number of lines passing through a square meter of the shell. If you double the distance of the shell then you quadruple its area and each square meter gets only a quarter of the gravity.
In 4D the shell is a 3-sphere not a 2-sphere, so the flux density is per cubic meter and the double-sized shell gets only 1/2^3 = 1/8 the gravity per cubic meter.

[quickfur]

But the whole point of this exercise was to postulate a geometrical source of gravity rather than the field model, which we already know results in a 1/r^(n-1) law of gravity.

[steelpillow]

But the whole point of this exercise was to postulate a geometrical source of gravity rather than the field model, which we already know results in a 1/r^(n-1) law of gravity.

General Relativity is precisely the geometric expression of the field model. The field strength is just the curvature of spacetime at that point. So just do [find] "field strength" [replace] "spacetime curvature" and you have my post the way you want it.

[mr_e_man]

Another way of thinking about this, is this thought experiment: suppose space was 4D yet stable orbits exist. If Einstein were to live in such a universe, what kind of theory would he have come up with to explain it? How different would General Relativity look if Einstein had invented it in a 4D universe with stable orbits? Which parts of the theory would be different? Can these differences be logically consistent, without leading to bizarre consequences that negate the other parts of a 4D physics derived in analogy from 3D?

Recall the cosmological constant. Einstein himself (the 3D one) couldn't decide on the correct form of his equations.

You may be interested in modifications of General Relativity, such as Gauss-Bonnet gravity, or f(R) gravity. (Here R denotes the curvature scalar, not r, the distance between two gravitating particles.) I suspect that such an arbitrary function f, determining the tensor field equations, could produce an arbitrary force law in the Newtonian approximation, e.g. 1/r^2 rather than 1/r^(n-1).

[mr_e_man]

Yesterday I did some numerical simulations.

A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!

The orbit is in the opposite direction to the magnetic field (bivector). So the force of magnetism points outward, against gravity.
If the planet drifts too far inward, then it speeds up, and since the magnetic force is proportional to velocity, magnetism pushes the planet outward.
If the planet drifts too far outward, then it slows down, and since the magnetic force is proportional to velocity, magnetism relaxes and allows gravity to pull the planet inward.

If the magnetic field is not constant but is produced by the star and decreases with distance, then it should be even more helpful in stabilizing the orbit. However, given that our 3D Sun negates its magnetic field every 11 years, perhaps this shouldn't be relied on to sustain a 4D solar system.

The simulations were 2D. I haven't considered what would happen if the orbital and magnetic planes weren't aligned, or if the magnetic field wasn't planar, or if there was also an electric field.

And of course we must ask how the planet retains its charge, and how that would affect life on its surface.
Ions in space, of the same charge as the planet, would be repelled. Ions of the opposite charge would be attracted, and become part of the planet and neutralize it.

[PatrickPowers]

Yesterday I did some numerical simulations.

A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!

The orbit is in the opposite direction to the magnetic field (bivector). So the force of magnetism points outward, against gravity.
If the planet drifts too far inward, then it speeds up, and since the magnetic force is proportional to velocity, magnetism pushes the planet outward.
If the planet drifts too far outward, then it slows down, and since the magnetic force is proportional to velocity, magnetism relaxes and allows gravity to pull the planet inward.

If the magnetic field is not constant but is produced by the star and decreases with distance, then it should be even more helpful in stabilizing the orbit. However, given that our 3D Sun negates its magnetic field every 11 years, perhaps this shouldn't be relied on to sustain a 4D solar system.

The simulations were 2D. I haven't considered what would happen if the orbital and magnetic planes weren't aligned, or if the magnetic field wasn't planar, or if there was also an electric field.

Gosh this is really interesting. A wild idea I never would have considered. It might actually work. I don't have time to look at it today, but will give it a lot of thought. Working strongly in its favor is that 4D magnetic fields have no poles and are orientable. One can guarantee that the two heavenly bodies always have fields that are in a sense opposite. But is it enough?

And of course we must ask how the planet retains its charge, and how that would affect life on its surface.
Ions in space, of the same charge as the planet, would be repelled. Ions of the opposite charge would be attracted, and become part of the planet and neutralize it.

But magnetism doesn't depend on charge. Consider kitchen magnets. Planets are more or less like that.

Let me mention that I'm no expert on magnetism.

[mr_e_man]

But magnetism doesn't depend on charge. Consider kitchen magnets. Planets are more or less like that.

Let me mention that I'm no expert on magnetism.

https://en.wikipedia.org/wiki/Lorentz_force

The magnetic force on an object with charge q and velocity v is

f = q B•v,

where B is the magnetic field, which can be considered either as a bivector or as an antisymmetric matrix.

The magnetic field has many effects. I'm considering its effect on an electrically charged object. You're considering its effect on a magnetized object. (Note that some planets are not magnetic, particularly Venus.)

I think if the star and the planet are both magnetized and not charged, then the force is attractive, and thus not helpful in stabilizing the orbit. If they're oriented such that the force is repulsive, then the planet will twist around until it's attractive. And if the magnetism is too weak to rotate the planet, then it's too weak to be relevant at all.

Now I'm suspecting that my 2D simulations would be unstable when a dimension is added. The orbit itself might flip over, and align with the field.

[quickfur]

Yesterday I did some numerical simulations.

A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!

The orbit is in the opposite direction to the magnetic field (bivector). So the force of magnetism points outward, against gravity.
If the planet drifts too far inward, then it speeds up, and since the magnetic force is proportional to velocity, magnetism pushes the planet outward.
If the planet drifts too far outward, then it slows down, and since the magnetic force is proportional to velocity, magnetism relaxes and allows gravity to pull the planet inward.
[...]

Doesn't this just amount to introducing some arbitrary repulsive force between the star and the planet, and balancing it against gravity? Giving two opposing forces in spherical distribution over a common center, this would create a local minimum in the shape of a spherical shell, so any path on along this shell would be (meta)stable. Whether the source of this opposing force is the magnetic field or some other arbitrary source, strictly speaking isn't relevant anymore. It might as well be some ad hoc fiat force introduced just for the purpose of preventing orbital collapse, and it would function more-or-less the same way. This situation is quite different from 3D where momentum interacts with gravity in such a way that elliptical orbits are stable, with no other forces at play.

[mr_e_man]

No, it's not arbitrary. It obeys the known laws of physics (or, the most natural generalization of the known 3D laws). It's true that the existence and magnitude of the field are arbitrary (maybe it's produced by some distant supermassive object), but its effect is strictly pre-defined, by the equation in my previous post. Also, the field doesn't have a weird "shape"; it's constant (it doesn't vary over time) and uniform (it doesn't vary with location, at least within the solar system).

The magnetic field is not spherically symmetric! If the planet orbits the wrong way, then the magnetism makes it more unstable.

[quickfur]

Hmm that's interesting. What if in 4D magnetism is weak but gravity is strong, so that magnetism works at the astronomical scale whereas gravity works only at the atomic scale? Would have interesting consequences

[PatrickPowers]

But magnetism doesn't depend on charge. Consider kitchen magnets. Planets are more or less like that.

Let me mention that I'm no expert on magnetism.

https://en.wikipedia.org/wiki/Lorentz_force

The magnetic force on an object with charge q and velocity v is

f = q B•v,

where B is the magnetic field, which can be considered either as a bivector or as an antisymmetric matrix.

The magnetic field has many effects. I'm considering its effect on an electrically charged object. You're considering its effect on a magnetized object. (Note that some planets are not magnetic, particularly Venus.)

I've never heard tell of a heavenly body with a static charge. Even here on the surface of the Earth how often does one come across any object with a static charge. It is theoretically possible but static charges tend not to persist. A positively charged heavenly body would attract negative ions until it was neutral again. If you could have such a static charge then all you would have to do would be to have two heavenly bodies with the same polarity and they would always repel one another. That would definitely stabilize the orbit.

I've always seen those laws about charged particles applied to elementary particles and ions. Molecules might be as large as that goes.

I think if the star and the planet are both magnetized and not charged, then the force is attractive, and thus not helpful in stabilizing the orbit. If they're oriented such that the force is repulsive, then the planet will twist around until it's attractive. And if the magnetism is too weak to rotate the planet, then it's too weak to be relevant at all.

Now I'm suspecting that my 2D simulations would be unstable when a dimension is added. The orbit itself might flip over, and align with the field.

You are definitely correct that nothing like this will work in odd dimensional spaces. Magnetic fields in odd dimensional spaces are not orientable so they will move about until they attract one another maximally and reduce their potential energy. Even dimensional spaces are a different matter. Magnetic fields are orientable (they surely are in 4D. I think this is true in all even dimensional spaces but I'm not sure.) That means that two magnets can be fundamentally incompatible. The attraction between then can't be particularly strong. I'm not optimistic but think this is worth looking into. Unlike charge the magnetic fields of heavenly bodies can be extremely strong and stable.

[steelpillow]

I've never heard tell of a heavenly body with a static charge. Even here on the surface of the Earth how often does one come across any object with a static charge. It is theoretically possible but static charges tend not to persist.

A planet bathed in the solar wind will pick up a small charge due to the relative mobilities of electrons and protons; electrons will tend to escape first, with a small positive charge persisting as long as the star is active.
The Earth's atmosphere is a good deal more positively charged than that, but most of it is cancelled out by a buildup of electrons in the ground. The effect of that is of a giant capacitor with a strong vertical electric field (kilovolts per metre). The field can break down spectacularly, which we call a lightning strike.
Any exo-atmospheric forces arising from the residual positive charge will be minuscule compared to the action of the exo-atmospheric magnetic field on the electrical currents flowing in the planet's fluid atmosphere or solid surface. As an orbital stabiliser, natural electric charge is just not a contender. For artificial systems, current flows in coils are orders of magnitude more practical.
In nature, something like an intermediate-scale (i.e. between quantum and cosmic) Chameleon force would be necessary. This is the proposed fifth force of nature which in our 3D universe acts on the cosmic scale to cause the expansion of the universe (which is accelerating, so the "cosmological constant" is not actually a constant. I don't see why it shouldn't act more locally in 4D. After all, electrostatics are long-rage but magnetostatics are relatively short. Gravity is also long-range, so why not a partner which is relatively short?

[mr_e_man]

A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!
[...]
The simulations were 2D. I haven't considered what would happen if the orbital and magnetic planes weren't aligned, or if the magnetic field wasn't planar, or if there was also an electric field.

Now I'm suspecting that my 2D simulations would be unstable when a dimension is added. The orbit itself might flip over, and align with the field.

I did more simulations, and it still works. Whether it's 2D or 3D or 4D, whether there's a (weak, constant, uniform) electric field, whether there's a second orthogonal component of the magnetic field... it is stable, in some generality.

[mr_e_man]

I've never heard tell of a heavenly body with a static charge. Even here on the surface of the Earth how often does one come across any object with a static charge. It is theoretically possible but static charges tend not to persist. A positively charged heavenly body would attract negative ions until it was neutral again. [...]

I've always seen those laws about charged particles applied to elementary particles and ions. Molecules might be as large as that goes.

Agreed. Even in a different dimension, it does seem very unlikely, that a planet could hold such a huge amount of charge that the Lorentz force is comparable to the gravitational force. I'm thinking about (the 4D equivalent of) the square-cube law, but I'm not sure about the details.

If you could have such a static charge then all you would have to do would be to have two heavenly bodies with the same polarity and they would always repel one another. That would definitely stabilize the orbit.

Definitely not!

The gravitational force is (a/r^3)(-r/r), and the electrostatic force is (b/r^3)(+r/r), where a and b are constants, and r is the vector from the star to the planet, and r is its magnitude, i.e. the distance. The sum of these forces is just ((b-a)/r^3))(r/r); it's equivalent to a single 1/r^3 force, which we already know is unstable. That is, unless a=b, in which case the planet drifts away in a straight line.

[PatrickPowers]

Yes, I figured that out later.

[mr_e_man]

I think if the star and the planet are both magnetized and not charged, then the force is attractive, and thus not helpful in stabilizing the orbit. If they're oriented such that the force is repulsive, then the planet will twist around until it's attractive. And if the magnetism is too weak to rotate the planet, then it's too weak to be relevant at all.

Now I'm suspecting that my 2D simulations would be unstable when a dimension is added. The orbit itself might flip over, and align with the field.

You are definitely correct that nothing like this will work in odd dimensional spaces. Magnetic fields in odd dimensional spaces are not orientable so they will move about until they attract one another maximally and reduce their potential energy. Even dimensional spaces are a different matter. Magnetic fields are orientable (they surely are in 4D. I think this is true in all even dimensional spaces but I'm not sure.) That means that two magnets can be fundamentally incompatible. The attraction between then can't be particularly strong. I'm not optimistic but think this is worth looking into. Unlike charge the magnetic fields of heavenly bodies can be extremely strong and stable.

You got me thinking.
And calculating. I derived some complicated formulas for the interaction between two magnets. Should I start a new topic?
But the details don't seem to matter here.

Any bivector in 4D, such as a magnet's moment (I hesitate to call it a "dipole moment"), can be written in the form

M = A e₁e₂ + B e₃e₄

where A and B are scalars and the e's are orthonormal vectors. Its wedge-square is

M∧M = 2AB e₁e₂e₃e₄,

and this quadvector doesn't change when M is rotated (though it does change when M is reflected). The sign of AB tells whether M is right-handed or left-handed. So, yes, two magnets can be "fundamentally incompatible" in some sense. However, attraction between them is just as possible as repulsion (of the same strength). That's because M can be rotated and end up as -M. Rotate by an angle θ in the e₁e₃ plane:

M(θ) = A (e₁cosθ + e₃sinθ)e₂ + B (-e₁sinθ + e₃cosθ)e₄
M(180°) = - M(0°)