# Alternate orbital mechanics question

I have a question that I think is quite fascinating, but first, I don't know if it really belongs here, and second, I don't know how to phrase it in a constructive, answerable format for a Q&A site. It might belong on math stackexchange. It might be right for physics. Damn, it might be the correct fit for Worldbuilding.SE. Could you help me get it off the ground?

The idea is to find interesting alternate orbital mechanics.

The question as it stands now would be:

Let's assume an alternate universe, where $$F = G {{m_1 m_2}\over{r^2}}$$ doesn't hold true. Their universal law of gravity is different. Maybe it's $$F = G {{m_1 m_2}\over{r}}$$ or $$F = G {{m_1 m_2}\over{r^3}}$$. Maybe $$F = G {{m_1 m_2}\over{e^r}}$$ or $$F = G {m_1m_2\over \log r}$$. Maybe $$F = G {{m_1+m_2}\over{r^2}}$$. Maybe even $$F = G {m_1m_2 r^2}$$ with some alternate "strong" force holding the bodies from breaking apart, and the size of the micro-universe very limited. Maybe something that introduces a cross-product, curving the trajectories in 3rd dimension introduces other variables. These are just examples - feel free to come up with your own. Are there any that would result in some working, viable, interesting alternate orbital mechanics? Or is the product of masses over distance squared the only way to get anything like the sometimes-stable systems of bodies we have?

Please don't answer that question here, just help me get it where it belongs and into shape that won't get closed as "primarily opinion-based".

• I would try Worldbuilding, with the science (but not hard-science) tag. May 5 at 21:59

I asked a very similar question on Worldbuilding SE, and it was well received there.

I would recommend asking that type of question over there rather than on physics SE, as you don't end up in the uphill battle of "crank physics".

Not an answer to the orbit question but has math anyway.

There's a lot of mathematical history regarding power law central forces "other than inverse square force":

$${\bf F}({\bf r}) = -\frac{k}{r^a}\hat{\bf r}, \ \ a \ne 2$$

and where the total energy is then

$$E=\frac{p^2}{2m} + V(r)$$

where $$V(r) = - k \frac{1}{r^a}, \ \ a \ne 1$$

and in Yukawa potentials

$$V(r) = - g^2 \frac{e^{-\alpha m r}}{r}$$

Physics SE is one place to ask about these two types of potential, but it's a fast-paced and sometimes unforgiving & quick-closing site if something remotely similar has been asked or if somebody feels you haven't included enough research.

I've seen non-inverse square force questions in Astronomy SE as well, and it's a much easier site to ask, get feedback and adjust question if necessary before insta-closure.

So my advice would be to choose the scope of you deviation from $$1/r$$ potentials (e.g. $$a \ne 1$$ power law, Yukawa, roll-your-own...), choose from Physics SE, Astronomy SE or even History of Science and Math SE if you'd like to ask from that angle, do a little research so that you have "evidence of research" then go for it.

You will want to use caution with terms like "stable" and "closed" when describing orbits if you choose to ask about or specify those kinds of characteristics. For example "For which powers $$a$$ are the orbits stable?" may get a quick reply "Voting to close as unclear because you have not specified which kind of stability blah blah blah..." An orbit might precess (therefore not be closed) but be stable in that the average, peri and apo all stay constant and a small deviation in starting conditions does not lead to a chaotically diverging future trajectory.

Random sample in Physics SE of some fairly specific questions, there will likely be more/better examples

Bertrand's Theorem might be good to cite in your question as evidence of prior research. It says that the only potentials $$V$$ for which "all bound orbits are also closed orbits" are "normal gravity" $$a=-1$$ and harmonic oscillator $$a=+2$$.