Re: Stable Orbit Formulae?

On Jan 4, 1:46 am, Timothy Partee <tpar...@xxxxxxxxxxxxx> wrote:
Greg Neill wrote:
"Timothy Partee" <tpar...@xxxxxxxxxxxxx> wrote in message
news:c7WdnY4sp7RM8uDanZ2dnUVZ_ramnZ2d@xxxxxxxxxxxxxxxx

    Given two (planetesimal) orbital bodies whose Mass << Stellar Body,
what is the minimum permissible distance between the two bodies such
that they will not disrupt one another's orbital parameters?

There will always be perturbations due to mutual
interactions causing changes to the orbital
parameters.  Whether or not these perturbations
can lead to disruptive changes (collisions or
ejections) can be difficult to determine because
it can depend upon resonances in the periods of
the bodies which can amplify, damp, or induce
cyclical changes.

You might want to try a search on orbital resonances
and peruse the topic.

    Hmmm, interesting. However, it would seem that our own solar
system's planets have a negligible effect upon each other in their
(nearly) concentric orbits around Sol. There must be a threshold of

We still can't prove long term stability though many have tried.

distance/mass (Gravitational Force) at which effects of one orbital body
become negligible to the other. In stellar formation modeling this
equation, threshold, formula or what-have-you must exist and be defined

There is nothing so simple that will work. The closest formula to
meeting your requirements in general is Ovendens principle (which is
anyway more of a conjecture) and states that given enough time planets
will evolve into orbits where they avoid mutual interaction (or get
flung out of the solar system). His "principle of least interaction"
so you have to minimise the time average of his mutual interaction
expression:

<R> = sum i,j ( m[i]m[j]/(r[i]-r[j]) ) (i != j )

Whether or not you beieve this formula to be correct (AFAIK noone has
proved it) it seems to roughly predict the right sort of behaviour in
numerical simulations. A brief discussion of it exists in A E Roys
book orbital motion.

There are basically no easy answers. Some systems which intuitively
you might expect to be stable are unstable and vice versa. A
discussion of one part of the restricted 3 body problem with some
exploration of the regions of stability is online at:

http://www.iop.org/EJ/article/1538-3881/124/4/2332/202196.text.html

Most of the others require subscription access. This has some charts
for the regions of stability obtained by numerical simulation for one
restricted version of the 1:1 resonance problem.

    I'm just looking for a general and believable "magic number" that
says "orbital bodies A and B's semi-major axii should be at least X km
distant from one another given A, B and stellar masses and assuming a
perfectly circular, flat, non-eccentric orbits".

Nothing quite like that exists either, although again you could
conjecture that a variant of Bodes law might give you a sporting
chance of arranging a toy solar system that will not fall apart too
easily under perturbations. Several orbital simulators exist on the
web and you could try out a few ideas in one of those.

Even resonances can be good or bad. Jupiter and Saturn appear to be
roughly phase locked 5:2 on orbital period. Pluto and Neptune by 2:3
(and appear stable despite the fact that Pluto is in an eccentric
orbit that goes inside Neptunes more circular path). And 3 of the
Trojan moons of Jupiter are locked together in a 3:2:1 stable
resonance.

hugely ignorant on my part of some peculiarity of orbital physics...
Which is the whole reason I'm asking the question here. =)

You have asked one of those simple sounding questions for which no
simple answer exists.

There are research papers around on orbital stability proofs from an
analytical perspective but they are incomprehensible to all but
specialists in the field. Numerical simulations are more accessible if
you can find them. If you are serious about this I'd recommend A E
Roys Orbital Motion for an introduction (warning not an easy read)

Regards,
Martin Brown
.

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