4D Orbits

[Higher_Order]

Ok, I'm a bit new to the wonderful world of the 4th dimension. I understand why, and have suspected that in this new dimension energy would decrease as the cube of the distance, rather than the square. So it was no surprise that this was true, however what was a surprise was that this makes all but a perfectly circular orbit impossible, can somebody please explain to me why this happens?

[quickfur]

Why this happens is not very obvious, but it arises from the way the equations of motion change in a fundamental, qualitative way when the denominator is r³ instead of r². Basically, you derive the Keplerian equations for orbital motion by writing out the momentum of the planet, and using Newton's equation for gravity to relate it to the mass of the star, and then try to solve for the parameters which gives you a periodic motion. You can find this derivation if you search for this topic somewhere on this forum.

It turns out that anything except r² gives rise to unstable motion. That is to say, in every dimension perfectly circular motion is possible, but 3D is the only dimension in which slight deviations from the perfect circle will still give a stable orbit (in the form of elliptical orbits). In all other dimensions, the orbit will either degrade (the planet collides with the star eventually) or diverge (the planet will fly out of orbit). In 4D, there are three kinds of planetary motions: (1) the perfect circle, which is stable in theory, but impractical because nothing is a perfect circle in real life; (2) a spiralling "orbit" in which each orbit adds a constant displacement to the distance between the star and the planet (so the planet is moving towards/away from the central star at a constant rate -- which means no long-term orbit is possible); (3) there is no orbit: the planet either flies into the star within a short time, or flies off the star's gravitational influence very quickly.

Intuitively speaking, this is caused by the way the force of gravity degrades with increasing radius. In 3D, the rate of degradation is just right, that the changing force on the planet pulls it into an elliptical orbit. In 4D, the rate of degradation is too fast: gravity weakens too quickly as you leave the star, so things will gradually fly out of the solar system, or conversely, the force of gravity increases too quickly when you approach the star, so you'll get pulled inwards and hit the star.

[quickfur]

Oh, and BTW, this topic is in the wrong forum; metaphysics is supposed to deal with non-physics based topics. This one belongs in "higher spatial dimensions".

[Higher_Order]

Ok, that makes sense. Sorry for the late reply, I've been very busy lately.

[wendy]

If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.

[Higher_Order]

If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.

Could you please explain that to me (or link to another resource), I've never heard of that before...

[quickfur]

If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.

I'm hard-pressed to find a justification for such a quantization though. Unless we're to believe that the 4D planet exhibits macroscopic wave-like properties, in which case there might be practical difficulties in living on it.

[wendy]

Stable elliptical orbits are only stable in 3d. In any other dimension, one can only have stable circular orbits. Anything else is going to sling-shot your planet into the sun or deep space, long before life forms thereon. That's why you need something else to stabalise a radiant inverse-biquadratic. In Bohr's atom, one might point to the quantum nature of planck's constant controlling action. Bohr does not specifically specify waves, but the wave model fits neatly.

You can of course, use something like a radiant repelling force, where the carriers decay. This can be used for to create an inverse-force of a higher dimension, which would make the orbits gravitate to a specific distance from the sun (varyingly for different suns and planets), based on some secondary quantity other than mass. There was some talk about a similar one based on the bayonic number in 3D. Planets of different compositions might give a different bayonic mass, which when coupled with gravitational mass, would drive the ones with more bayons further out.

[quickfur]

Stable elliptical orbits are only stable in 3d. In any other dimension, one can only have stable circular orbits. Anything else is going to sling-shot your planet into the sun or deep space, long before life forms thereon. That's why you need something else to stabalise a radiant inverse-biquadratic. In Bohr's atom, one might point to the quantum nature of planck's constant controlling action. Bohr does not specifically specify waves, but the wave model fits neatly.

You can of course, use something like a radiant repelling force, where the carriers decay. This can be used for to create an inverse-force of a higher dimension, which would make the orbits gravitate to a specific distance from the sun (varyingly for different suns and planets), based on some secondary quantity other than mass. There was some talk about a similar one based on the bayonic number in 3D. Planets of different compositions might give a different bayonic mass, which when coupled with gravitational mass, would drive the ones with more bayons further out.

Interesting idea.

Introducing a repelling force has other consequences. For example, an orbit need not be circular, but sinusoidal -- imagine if initially, the planet is moving towards the sun from a distance where gravity overwhelms the bayonic force, then when it reaches the point where the two forces balance out, it has already acquired momentum from the initial acceleration. As a result, it will move past the stable point, with the consequence that the bayonic force now kicks in and pushes it outward. So the planet will bounce back, but by the time it reaches the balance point, it again has acquired momentum from the bayonic force, so it will fly out past the balance point again. At which point the force of gravity starts pulling it back in. So the resulting motion will be harmonic, and the planet will have a sinuisoidal orbit.

A sinusoidal orbit will cause the planet to not only have seasons and climates caused by its double-rotation and its tilt wrt to the sun, but the sinusoidal variation of orbital radius will add an additional periodic warming/cooling cycle to the overall temperature on the planet's surface.

The bayonic force itself will have other consequences, such as matter of higher bayonic mass being driven to the surface of the planet, and so the interior of the planet will be stratified by bayonic mass. Unless, of course, one postulates that repulsive bayonic force only emanates from the sun (maybe as a result of thermonuclear reactions in the sun, say). Then planets will be solid but there will be no moons.

[Higher_Order]

Ok, I see what you're getting at there, and as quickfur stated before, it seems like it would work.

[quickfur]

[...] A sinusoidal orbit will cause the planet to not only have seasons and climates caused by its double-rotation and its tilt wrt to the sun, but the sinusoidal variation of orbital radius will add an additional periodic warming/cooling cycle to the overall temperature on the planet's surface.
[...]

Also, from what I can tell (correct me if I'm wrong), this sinusoidal motion would be undampened, or only slightly dampened, so most planets are likely to have orbits with significant sinuidoidal amplitude -- their orbital radius will vary by a large amount. So most planets will likely be unsuitable for life, as the change in orbital radius also means a drastic change in surface temperatures.

Another consequence of a bayonic force is that planets need not orbit at all; they could just "float" around the ideal radius where the bayonic force balances gravity, so there can be stationary planets. There can also be planets that have vertical harmonic motion but zero lateral motion, like a bobbing weight on a spring.