Would this produce stable orbits in other dimensions?

[anderscolingustafson]

In our Universe Centripetal Force can be expressed by the equation

Centripetal Force.png (797 Bytes) Viewed 8796 times

where F is the Centripetal Force, M is the Mass, V is the Velocity, and r is the radius. The equation for gravity is

Equation for gravity.png (1.01 KiB) Viewed 8796 times

where F is the Force of Gravity, G is a gravitational constant, M is the Mass, and d is the distance between the two objects. When there are two objects orbiting each other the centripetal force between them can be expressed by the equation

Second Equation for Centripetal Force.png (1 KiB) Viewed 8796 times

Angular Acceloration.png (595 Bytes) Viewed 8796 times

is the same as the Angular Acceleration meaning that for the Angular Acceleration the Velocity is squared. Centripetal Force is really just another form of gravity which pulls objects apart whether than pulling them together. In both the equation of Gravitational Attraction and Centripetal Force there is a value that is squared. In the equation for Centripetal Force the Velocity is squared while in the equation for Gravity the distance is squared. In order for two objects to stay in orbit around each other the Centripetal Force between them must be equal to the gravitational attraction between them.

If we assume an equivalent for Gravity in other numbers of dimensions the equation for gravity in 4d would be

4d Equation for Gravity.png (1021 Bytes) Viewed 8796 times

and in 2d the equation for gravity would be

2d Equation for gravity.png (1007 Bytes) Viewed 8796 times

So in 4d gravity would increase much more rapidly when the distance decreases while in 2d gravity would increase more slowly as distance decreases. If the equation for centripetal force was

4d equation for centripetal force.png (1005 Bytes) Viewed 8796 times

instead of

Second Equation for Centripetal Force.png (1 KiB) Viewed 8796 times

then the Centripetal Force of an object would increase more rapidly with Velocity so increasing the Velocity by a smaller amount would increase the Centripetal Force more while if the equation for Centripetal Force was

2d Equation for Centripetal Force.png (948 Bytes) Viewed 8796 times

the Centripetal Force would increase more slowly with Velocity.

If in 4d the equation for Centripetal Force was

4d equation for centripetal force.png (1005 Bytes) Viewed 8796 times

instead of

Second Equation for Centripetal Force.png (1 KiB) Viewed 8796 times

would that balance out the more rapid increase in gravitational attraction with distance and produce stable orbits in 4d? And if in 2d the equation for Centripetal Force was

2d Equation for Centripetal Force.png (948 Bytes) Viewed 8796 times

would that balance out the slower increase in gravitational attraction with distance and produce stable orbits in 2d?

[ICN5D]

Hmm, interesting. It seems like according to the rate of depreciation, 2D orbits could be wider and more stable at a greater distance than 3D. And, likewise for 4D, the stable orbits end up being closer. However, there is another effect at work here. I'm certainly no expert on the subject of 4D orbital mechanics, quickfur would be a good one to chime in on this one. But, if the drop off rates are higher in higher D spaces, then planetary accretion will happen with more violence, casting off unstable debris more often. The material that finally settles in the gravity well will be in a narrower zone of equilibrium. This means smaller planets orbiting faster closer in seems to be the general pattern. At some dimension, it would be impossible, as the stable orbit would be so close and narrow, there may not be anything left after everything got ejected. Our own solar system had somewhere around 100 planets during formation. You know those space paintings that have a ton of planets all over the place? Well, that's how it really is in a young protoplanetary nebula. All of those planets have to crash into each other, get ejected, or merge.

[Keiji]

Unfortunately, this breaks dimensional analysis - your 4D gravity equation, which results in an amount of force, is divided by unit distance, but your 4D centripetal force equation, which also results in an amount of force, is multiplied by unit distance per unit time.

I believe that's why someone ended up with tensors the last time I read about an attempt for 4D physics.