PDA

View Full Version : Moving planets around. (Gravity)



Beer w/Straw
2009-Nov-20, 01:05 PM
I've been told that if say Venus was moved to an orbit around the sun similar to Earth, that the orbits of the planets would be unstable due to effects of gravity. (Say as far away from the Earth as it could be though.)

If this is true, is there information out there that can help me understand this better?

Thanks.

swampyankee
2009-Nov-20, 01:18 PM
One place you can try is arxiv (www.arxiv.org); another is Sverre Aarseth's Institute of Astronomy (http://www.ast.cam.ac.uk/~sverre/). You could also try Piet Hut's Art of Computational Science (http://www.artcompsci.org/) site.

Stability is not necessarily an easy thing to determine in complex systems, especially as very minor changes in the initial conditions can have profound effects ("chaos"). No explanation without delving into the world of differential equations would be much more than handwaving.

Beer w/Straw
2009-Nov-20, 01:25 PM
No explanation without delving into the world of differential equations would be much more than handwaving.

Well I do have my Calculus textbook. :)

swampyankee
2009-Nov-20, 01:34 PM
Well I do have my Calculus textbook. :)

Good start :)

Jens
2009-Nov-20, 02:05 PM
I've been told that if say Venus was moved to an orbit around the sun similar to Earth, that the orbits of the planets would be unstable due to effects of gravity. (Say as far away from the Earth as it could be though.)


I'm not exactly sure if this is correct. But suppose that Venus were in an orbit very close to that of the earth. At some points in their orbits, the two would come very close to each other, and would have a strong influence on each other. And this would destabilize their orbits.

IsaacKuo
2009-Nov-20, 02:58 PM
Could Venus and Earth be placed into the sort of orbits that Saturn's shepherd moons have? I don't know the technical term for this type of orbit.

These are a pair of moons which share the same orbit, and periodically interact with each other. Most of the time, one is in a slightly lower orbit while the other is in a slightly higher orbit. The one in the lower orbit is going slightly faster, so it will eventually lap the other.

When the faster moon comes close to catching up, there's a sort of gravitational "bounce" that occurs. Gravity between the moons cause them to switch roles. The lower moon gets raised to the higher orbit and vice versa. Thus, the faster moon never quite catches up with the slower moon before the "bounce" happens.

Could the Earth and Venus be placed into such an arrangement? If so, then you'd periodically see Venus approach, 90 degrees away from the Sun (like a half moon), and then recede. Every time this happens, the length of the year would switch, getting either slightly longer or slightly shorter.

I'm not sure how often this would occur...my intuition is perhaps every few hundred or thousand years.

eburacum45
2009-Nov-20, 03:21 PM
I think that planets could exist in a 1:1 resonance, or something similar. They might even be reasonably common.

Here's a paper about this possibility
http://aps.arxiv.org/abs/astro-ph/0204091

We explore the possibility that extrasolar planets might be found in the 1:1 mean-motion resonance. There are a variety of stable co-orbtial configurations, and we specifically examine three different versions of the 1:1 resonance. These include tadpole and horseshoe type orbits, as well as a more exotic configuration which occurs when one planet has a highly eccentric orbit while the other planet moves on a nearly circular orbit.

StupendousMan
2009-Nov-20, 04:23 PM
The type of orbit to which IsaacKuo refers is called a "horseshoe orbit".

The paper which eburacum45 mentioned, Extrasolar Trojans: The Viability and Detectability of Planets in the 1:1 Resonance (http://aps.arxiv.org/abs/astro-ph/0204091), provides some numerical limits on the stability of two small bodies moving around a large body in similar orbits.

The "Trojan" arrangement, in which the two bodies mark out the vertices of an equilateral triangle with the large body, is stable as long as the two small bodies have masses which are less than 0.03812 of the total mass of the system (in other words, the two small masses are less than 3.8 percent of the sum of large body's mass plus small bodies' masses). This limit can be derived analytically.

The "horseshoe" arrangement mentioned by IsaacKuo is more difficult to analyze, so the authors provide numerical simulations to investigate its stability. They find that as long as the sum of the small bodies is less than 0.0004 of the total mass of the system, the arrangement is stable for long periods of time.

There's another case they investigate, but it involves non-circular orbits, so it's probably not of interest to the OP.

For reference, two Earths would have a mass ratio of 0.000006 in a combined (two Earths plus Sun) system, so -- IF one could ignore the perturbations from other planets -- one might set up a stable arrangement with the two Earths sharing an orbit in some way.

I have my doubts about the validity of ignoring perturbations, though.

mugaliens
2009-Nov-21, 03:34 PM
Well I do have my Calculus textbook. :)

Excellent! Turn to page 4,096, and...

Seriously - the interrelated nature of the planet's orbits encompasses several different disciplines, including orbital dynamics (http://en.wikipedia.org/wiki/Orbit_(dynamics))and several subsets of general relativity (http://en.wikipedia.org/wiki/General_Relativity).

I'd start with orbital mechanics (http://en.wikipedia.org/wiki/Orbital_mechanics), as that doesn't require much math. Once you've mastered that, study system dynamics (http://en.wikipedia.org/wiki/Dynamical_system_(definition)), where you'll need your calculus text. Then ask a librarian to help you locate some advanced orbital theory texts.

It's pretty hairy stuff.

Beer w/Straw
2009-Nov-21, 04:03 PM
Thank you for the reply.

But I was under the impression that Newtonian mechanics would be good enough and not have to delve into relativity.

As I believe only Mercury, because it is so close to the Sun, needs a realativistic explanation.

frankuitaalst
2009-Nov-21, 05:21 PM
This OP has been answered hereabove very profoundly .
I would like to add that a similar discussion was here :
http://www.bautforum.com/space-astronomy-questions-answers/83935-co-orbitals-moons-extrapolation-planets.html a while ago .

Based on the formulas above one indeed can argue such an orbit may be stable , ie Earth would allow a second Earth like planet sharing its orbit , in a dynamically stable way . The second body will move to an horseshoe coorbital orbit which is dynamically stable .

chornedsnorkack
2009-Nov-21, 07:51 PM
For reference, two Earths would have a mass ratio of 0.000006 in a combined (two Earths plus Sun) system, so -- IF one could ignore the perturbations from other planets -- one might set up a stable arrangement with the two Earths sharing an orbit in some way.

I have my doubts about the validity of ignoring perturbations, though.

But do perturbations matter here?

The inner satellites of Saturn are pretty massive and perturb each other strongly, as shown by two pairs of 1-2 resonances and Cassini gap. Janus and Epimetheus nevertheless share an orbit.

frankuitaalst
2009-Nov-22, 09:33 AM
Could Venus and Earth be placed into the sort of orbits that Saturn's shepherd moons have? I don't know the technical term for this type of orbit.

These are a pair of moons which share the same orbit, and periodically interact with each other. Most of the time, one is in a slightly lower orbit while the other is in a slightly higher orbit. The one in the lower orbit is going slightly faster, so it will eventually lap the other.

When the faster moon comes close to catching up, there's a sort of gravitational "bounce" that occurs. Gravity between the moons cause them to switch roles. The lower moon gets raised to the higher orbit and vice versa. Thus, the faster moon never quite catches up with the slower moon before the "bounce" happens.

Could the Earth and Venus be placed into such an arrangement? If so, then you'd periodically see Venus approach, 90 degrees away from the Sun (like a half moon), and then recede. Every time this happens, the length of the year would switch, getting either slightly longer or slightly shorter.

I'm not sure how often this would occur...my intuition is perhaps every few hundred or thousand years.
Eburacum45 and StupendousMan are right when they say such a system evolves to a horseshoe orbit in which both planets interact dynamically so that we get in a situation both planets seem to approach each other and then recede again in a rather regular way . The period of this depends upon the initial conditions and perturbations of other planets ( Jupiter having the biggest influence ) . As you say even the length of the year varies .
I've put up a simulation and animation of this, but the filesize is simply to big to post here , so I'll try to post a link . The 1/2 period I get in this setup is about 400 years . I also gave the second Earth a real moon like ours .
One gets a good visualisation of such a system in a rotating frame in which one of the bodies is representated in a fix position . In this case the relative motion of the second body becomes very clear .

frankuitaalst
2009-Nov-22, 07:05 PM
I've been told that if say Venus was moved to an orbit around the sun similar to Earth, that the orbits of the planets would be unstable due to effects of gravity. (Say as far away from the Earth as it could be though.)

If this is true, is there information out there that can help me understand this better?

Thanks.
This question is already answered above with lots of references .
Here's a reference which gives an animation of what happens to such a system .
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1258915637/1#1