wendy wrote:If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.
wendy wrote:If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.
wendy wrote: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 wrote:[...] 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.
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Higher_Order wrote: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?
PatrickPowers wrote:Higher_Order wrote: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?
Magnets tend to attract with an inverse cube law. If you attempt to get one magnet to orbit another, either they are sucked together very quickly or there is not much effect.
An inverse cube law seems like the best bet, but forces do whatever they like. Like the strong force, which gets stronger with increased distance. Question: does an inverse squared law in 4D lead to an impossible conclusion, or is it just heuristically favored?
quickfur wrote:Since momentum varies with the square of the velocity, when gravity also obeys an inverse square law this allows certain subsets of trajectories to have momentum and force balance each other out in an equillibrium.
quickfur wrote:How exactly does an object have "double" angular momentum? I.e., for it to work the way I think you're trying to describe, that means the object must be travelling (orbiting, etc.) in two different directions simultaneously. I have a hard time grasping what the actual path of the object would look like for it to satisfy such a strange constraint.
quickfur wrote:PatrickPowers wrote:Higher_Order wrote: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?
Magnets tend to attract with an inverse cube law. If you attempt to get one magnet to orbit another, either they are sucked together very quickly or there is not much effect.
Mathematically speaking, the difference lies in how the behaviour of the force as distance varies balances the momentum of the orbiting object. Any stable orbit is a delicate interplay between momentum and gravity. Since momentum varies with the square of the velocity, when gravity also obeys an inverse square law this allows certain subsets of trajectories to have momentum and force balance each other out in an equillibrium.
When gravity obeys an inverse cube law, however, there will almost always be an extra linear term in the total force experienced by the object, giving rise to a number of trajectories (in the best cases) that all have the tendency to either lose or gain average orbital distance per cycle, except for the exceptional case of the perfect circle, which in real life almost never happens because the slightest perturbation would nudge it into one of the unstable paths that eventually diverge from the star or collide into it.An inverse cube law seems like the best bet, but forces do whatever they like. Like the strong force, which gets stronger with increased distance. Question: does an inverse squared law in 4D lead to an impossible conclusion, or is it just heuristically favored?
The reason we tend to think of a 4D universe as having an inverse cube law for gravity is because of the flux conjecture: think of a force as being carried by "force carriers" emitted by the origin of the force. We assume that the strength of the force would be proportional to the density of force carriers at any point in space. All things else being equal, one would assume that there would be a constant number of force carriers emanating from the origin with constant density in all directions. Which means the farther away from the origin you are, the more spread out the force carriers will be. So the density of the force would be inversely proportional to the surface of an n-dimensional sphere of radius r, where r is the distance to the origin. In 3D space, then, this gives rise to the inverse square law. In 4D space, this gives rise to the inverse cube law.
Assuming an inverse square law for gravity in 4D by fiat won't, AFAIK, lead to any contradictions, but it would seem rather arbitrary and hard to explain.
quickfur wrote:The rotation of an orbiting body has little (or no) impact on its global path around the star being orbited. You can't stabilize an unstable orbital path by making the planet rotate differently. You can't simply combine the two angular momenta willy-nilly, i.e., the angular momentum around the star is not equivalent to the angular momentum of the planet rotating around its axis / in its rotational plane. The two do not affect each other, unless the planet breaks up into orbiting particles or orbiting dust accumulates into a spinning planet, in which case you could talk about the total angular momentum of the star system.
quickfur wrote:I don't see any way of obtaining a stable 4D orbit in this way without in some way changing the inverse cube law.
d023n wrote:the system of planet and star together would itself also be double-rotating around its center of mass with the planet behaving as a single point that is double-orbitting with 2 orbital angular momenta components. These would be combined into a single 4D angular momentum vector that still defines a single curved 1D path, and the total kinetic energy of the planet relative to the solar system's center of mass would scale with the 4^{th} power of its orbital velocity.
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PatrickPowers wrote:It's because the acceleration vector is always pointing at the center of mass, so the orbit is confined to a 2D plane defined by that acceleration vector and the velocity vector of the planet.
d023n wrote:PatrickPowers wrote:It's because the acceleration vector is always pointing at the center of mass, so the orbit is confined to a 2D plane defined by that acceleration vector and the velocity vector of the planet.
Wow. That.. is a glaringly obvious point. How did I not see that? What's worse is that I can't see any way around it. Dang.
PatrickPowers wrote:" Well, why does the acceleration vector always point at the center of mass in 3D? "
That's a good question. We know that information travels at the speed of light. So why does Newton's law of gravity works so well? It takes eight minutes for light to get here from the Sun. Shouldn't the Earth's gravitational acceleration vector point at the place the center of mass was eight minutes ago? But that doesn't work. The vector points almost exactly at the place the Sun is now.
wendy wrote:If one supposes Bohr's model, then the energy of orbit is quantised at multiples of h/L, where L is the length of the orbit.
In four and higher dimensions, the vector of rotation is different to just the orthogonal. In four dimensions, there is a 2-space orthogonal to the vector
Flattening of the sphere is at the rate of w²r/g, where w is the angular speed of rotation, and r the radius.
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