Re: Stable Orbit Formulae?

In message <C-WdnQtKKr2OKRLanZ2dnUVZ_uuqnZ2d@xxxxxxxxxxxxx>, Timothy Partee
<tpartee@xxxxxxxxxxxxx> writes
>Agent Smith wrote:
>> Timothy Partee <tpartee@xxxxxxxxxxxxx> wrote in
>> news:A72dncmixMt5vBjanZ2dnUVZ_judnZ2d@xxxxxxxxxxxxx:
>>> Martin Brown wrote:
>>>> On Jan 4, 1:46 am, Timothy Partee <tpar...@xxxxxxxxxxxxx> wrote:

>>>>> 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 )
>>>>
[Snip]
>>>> 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
>>> Thank you for the references and additional background every
>>> little bit helps. =) To be honest this is all for an ultra-realistic online game that I'm working
>>>on with the help of some friends that is themed around "galaxy exploration and colonization".
>>>We're generating stellar
>>> models already that appear fairly believable, and that wasn't terribly
>>> hard, but now we're trying to populate those stars with planets, planetoids and moons and
>>>things are getting exponentially more
>>> complex. While it would be killer to have computationally correct data
>>> about the orbital interactions, we just need "believable" metrics for
>>> planetesimal orbits. Thanks to earlier comments on Hill Spheres we're
>>> playing with random model-generation for moons now that look pretty
>>> accurate. Our (known) universe is a pretty complex place, not the easiest thing to be
>>>modeling. ;) But we're trying, using the best possible
>>> data at hand. Thanks again for your input!
>> The problem is that a planet has to be within the habitable thermal band around the sun, for life
>>to exist. Because Bode's law requires consecutive orbital radii to follow a series of powers of
>>two, only one planet can fall into the habitable zone. You're gonna have to gloss it over, because
>>anything else is Pandora's Box, and you've just chaged your careers from game writer to
>>astrophysicist.

You can place a smaller planetoid at the 60 degree L4,L5 Lagrange points either side of the main body - this would allow a Jupiter mass (or maybe a bit more) body placed in the right location to shepherd at least a couple of its own Earth sized moons and a pair of Trojans 60 degrees either side that are capable of holding down an atmosphere and sit in the goldilocks zone. Masses of smaller planets kept less than 3.8% of the main one.

Anyone care to figure out how far it could be pushed before mutual interactions will cause for serious instability?

>> Unless of course, you want to write some simulation codes, in which case you can probably
>>get an NSF grant and a teaching job at your local university. Do you know how to numerically
>>solve differential equations? There's a big market for that, in science research.
>
> What rock did you crawl out from under? The Titius-Bode "law" was disproven over 150 years
>ago. Even Wikipedia knows that! http://en.wikipedia.org/wiki/Titius-Bode_Law

I think you are being too hard on AS and over reliant on Wiki here. And I don't think it is really all that disproven only that it doesn't predict all stable orbit locations (or equivalently that not all the positions need be occupied).

And even though it isn't a "Law" as such - it is a fairly good redictive heuristic for a pattern of stable orbits for planets around a large central massive object. It derives from the principle of least interaction, but for the purposes of the OPs enquiry it is about as good a practical answer as there is. Unless you really want to simulate everything from scratch.

Regards,
--
Martin Brown

--
Posted via a free Usenet account from http://www.teranews.com

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