Exponential Laws

[AlternateGravity]

I tried simulating planetary orbits using exponential laws such as, e^(-r), and e^(-3r), and 2^(-r) and found that they produce stable orbits in the form of flower shaped orbits. I think it's interesting that the exponential laws produce stable orbits while most of the inverse laws don't produce stable orbits as with the exponential laws Gravity decreases with distance faster with the exponential laws than it does with the inverse laws. I tried combining the exponential and inverse laws as well using r^(-r) and found that, that also produces stable orbits.

[testtubegames]

Hmm, this is going to take some math to figure out.

My knee jerk reaction is this: compared to a simple law like r^-2, the e^-r force law changes more gradually when the objects are close, and more quickly when objects are far apart. So I'd expect the objects to orbit fairly stable-ly when they are close, but less so once you pass about r=3 units distance. A quick test shows that I *could* find a circular orbit that spiraled out, but only when I got far enough away.

Screen Shot 2014-10-18 at 12.52.20 AM.png (78.03 KiB) Viewed 12064 times

(e^-r force law)

But yeah, this is all hand-wavy, I wanna do some math with it.

[AlternateGravity]

I wonder what kind of orbits (r^(-3))*(e^(-r)) would produce as that would multiply an inverse law by an exponential law.