# Planet 180 degrees from earth in same orbit?

Discussion in 'Astronomy, Exobiology, & Cosmology' started by Dinosaur, May 9, 2009.

1. ### DinosaurRational SkepticValued Senior Member

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This is not the first time that my intuition has been wrong.

It still seems counter-intuitive that the Earth's orbit could be stable for thousands, if not millions, of years while counter Earth would stop mirroring Earth's orbit in 150 years or less.

3. ### D HSome other guyValued Senior Member

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A ball balanced on top of another ball is metastable. (Metastable == unstable equilibrium). If the two balls are perfectly round and utterly frictionless, balancing one on top of the other is nigh impossible. If you do manage to get the ball balanced, the slightest disturbance will make it fall.

The L3 point is a metastable point, and there is no friction in space.

The easiest way to look at things regarding these libration points is to use a reference frame with origin at the Sun-Earth center of mass and rotating at the Earth's orbital rate (i.e., the Sun and Earth are stationary). The math gets hairy; I'm not going to reproduce it here. There's lots of stuff on the web on this. Key phrases (conveniently encapsulated):
• "circular restricted three body problem" Note that this link provides lots of hits at universities with course notes. Several of these are very, very good.
• "Jacobi integral" The Jacobi integral is a constant of motion in the circular restricted three body problem.
• "Lyapunov orbit" Janus and I have been cheating a bit. The L3 point is very unstable. There are various kinds of pseudo-orbits around the linear libration points. These orbits, while still unstable, are a lot less unstable than are the libration points themselves in the sense that takes a lot longer for the instability to manifest itself in one of these pseudo-orbits than it does for a test object placed close to one of the libration points and stationary with respect to the point. A Lyapunov orbit is in the orbital plane of the two bodies ...
• "halo orbit" ... while a halo orbit has out of plane components. Lyapunov and halo orbits are orbits in the sense that they form closed paths from the perspective the rotating frame.
• "Lissajous orbit" Lissajous orbits aren't really orbits in the sense that they don't form a closed path. They would remain within some bounded volume it weren't for those pesky perturbations.

If you think the libration points are counterintuitive, try to intuit the behavior of 2002 AA[sub]29[/sub]. An animated GIF (warning: 3.9 MB): http://neo.jpl.nasa.gov/2002aa29/2002aa29a.gif.

5. ### D HSome other guyValued Senior Member

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Can someone turn this broken record off?

Editorial note:
The above referred to a post that a moderator later deleted. It does not refer to this thread as a whole. Billy T's response (see below) was made after the offending post was deleted.

Last edited: Jun 2, 2009

7. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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I suggested this tread be moved to the cesspool in post 5. However before it is closed (also good idea, IMHO) I hope you will comment on my post 15 idea* that it may be possible to stabilize “180 Earth” or a small “test body” in some dynamic resonance orbit around L3. You state in post 22 that:

“…There are various kinds of pseudo-orbits around the linear libration points. These orbits, while still unstable, are a lot less unstable than are the libration points themselves in the sense that takes a lot longer for the instability to manifest itself in one of these pseudo-orbits than it does for a test object placed close to one of the libration points and stationary with respect to the point. A Lyapunov orbit is in the orbital plane of the two bodies ...”

I.e. with a resonate interaction due to third significant mass body in orbit of the sun with the period of the two body Lyapunov orbit of the “test mass,” is it possible that the instability of the Lyapunov orbit can be overcome? (Of course a 2:1 resonance can be considered, but my guess is that if the 1:1 fails, all others will too.)

I have complete faith in you and Janus 58 in these matters, but you have not clearly responded to my post 15 idea that it might be possible for a test mass to stay near Earth/sun’s L3 via some resonate interaction with a third body. (I of course want to neglect the reality that Jupiter exists etc. when asking.)

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*From post 15:
He knows much more than I do about all this, but if counter Earth were to drift away from Earth's L3 then, as he stated, there would be a force on it towards Earth (not the same direction as the towards the sun). It seems to me the component of that force which is not towards the sun is like small rocket thrusting out the back. This, I think mainly raises the orbit so it would slow its angular rate and fall back towards L3. - But again Janus knows much more than me, so I am probably wrong to even suggest there could be some sort of resonance trapping when both have roughly the same mass.

8. ### D HSome other guyValued Senior Member

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Billy: You didn't see the context that prompted me to ask about turning a broken record off. I intentionally did not quote the broken record, but now I regret not doing so as my dangling post appears to be a response to the thread. Sorry about the confusion.

9. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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OK, I understand now. You still have not commented on my post 15 suggestion that a resonance traping might be possible. Is this because you strongly think it is not but proving that it isn't, is very tough?

10. ### eburacum45Valued Senior Member

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Planets in 1:1 resonance with each other do appear to be possible;
see
http://iaus249.nju.edu.cn/Posters/Psychoyos.pdf
and
http://www.iop.org/EJ/article/1538-3881/124/1/592/202074.text.html
Mostly these references consider trojan-type orbits, tadpole and horsehoe orbits. Planets at the 180 degree L3 location are not expected to be stable, but would migrate into one or other of these stable orbits (or impact the other planet, as seems to have happened to the Earth during the formation of the Moon).

11. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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thanks. The first ref is about two planets orbiting the sun, but not because one is small and locked near the L3 of the other, I think based on quick skim of only first part. The second ref. considers the orbits you mentioned, but it too does not seem, again based on even quicker scan, to address the stabalized near L3 problem directly.

I tend to think the moon resulted from impact of another body with Earth but do not see how that "other body" could have been resonately locked with Earth prior to impact. The period is determined by the major axis so they would need to be the same for 1 to 1 resonance. If the other body had high ecentricity and apogee in Earth's path I guess it might be possible that Earth could run into it when it is slowly moving at its perigee to make the moon, but this seems imposible as on the prior orbit there would have been a very near miss with huge mutual scattering.

12. ### eburacum45Valued Senior Member

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The Mars sized impactor is thought to have formed at the L5 point and 'crept' around the Earth's orbit until it impacted at a relatively low speed. See this diagram from Wiki, where the impactor is called 'Theia' for some reason.
http://en.wikipedia.org/wiki/File:BigSplashEnglish.svg

Why did 'Theia' wander away from the L5 point? Probably because of perturbations by other bodies, esp. Jupiter. The role of other bodies in these metastable relationships is generally to destabilise them, rather than to stabilise them. That doesn't mean that it is impossible for a body to be stabilised at such a location by a planet in a completely different orbit; but that would be a very rare occurence, I should think.

You might like to play with a gravity simulator program to see if you can find such a resonance- it would be a challenge, I'm sure.
Here's Tony Dunn's Gravity Simulator:
http://www.orbitsimulator.com/gravity/articles/what.html

Last edited: Jun 6, 2009
13. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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I will not try to test, but would expect that if a "Mars Mass" were at were Earth/Sun alone's L5, then that L5 does not even exist, (I think and will assume). I.e. if a Mars mass were in exactly the Earth's orbit but leading Earth by 60 degrees, then there would be no potential hill between that their mutual gravitations could not climb; or stating this another way there is no confining potentail well.

These thoughts do make me consider an alternate way that Earth got its moon: (if there were no jupitor to throw stuff out of Earth/Sun L5.): Mass accumulates near L5 as time passes and under mutual graviation and occasional low relative speed collisions agregates to approximatelly current moon mass, MM, which happens to be the mass that abolishes the L5 well. Then MM's orbit speed is decreased by Earth's gravitational attraction and this causes it to orbit the sun with slightly less than "1AU" so is separates from Earth with slightly less orbit period.

Of course Earth is being speeded up by the force from MM, but as E is more massive, the acceleration is less so from MM's POV E is moving slightly farther away from sun and falling behind (more than 60 degrees and increasing).

After a few decades (or hunderds of years?) MM has made (300/360) of one more orbit of the sun than E has. I.e MM is between E and the sun with stong mutual gravitational interaction, so E has been slowing down and MM speeding up more during this final closest approach period and they become locked together with approximately 28 day rotation period about their Baracenter.

I.e. Earth MM are now a co-orbiting the sun system! New, no-collision, no near miss passing third body with gravity gradient ripping chunck out of Earth, theory as to how Earth got a noon.

It can easly be tested: What is the is the mass at L5 which destroys the L5 well?* Is it approximately the mass of the moon? Can Earth being overtaken by moon mass in slightly closer to sun orbit with same eccentricity as Earth and same apogee to sun line (L5 has Both these) convert to a bound system?

Please do not immediately dismiss the idea with:
"Body A can not capture body B without some third body C assisting."
That does not apply in this case as E and MM are ALREADY captured - just not nearly co located but initially 60 degrees apart. Initially E and MM are in IDENTICAL orbits (mutually captured). Jupiter is the greatest threat I see to the idea, but perhaps it can even help. I.e. push MM out of L5 before MM is large enough to destroy L5.

PS As you can deduce from the above, I think the drawings in your link are wrong. I.e. the body at L5 does not close the 60 degrees gap but expands it by 300 degrees.
-----------------
*Or if contrary to my assumption, L5 always exists, it is no longer deep enough to keep MM from being pulled out of it by Earth.

Last edited by a moderator: Jun 6, 2009
14. ### D HSome other guyValued Senior Member

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Billy: What makes you think there is a mass that abolishes the triangular libration points (L4 and L5)? Did you read the second link cited in post #27? In particular, equations 9 and 10 describe the stability conditions. A system with Earth and Theia at the triangular points will be stable so long as the combined mass of Earth and Theia is less than about 1/25 of the mass of the Sun, or about 13,000 Earth masses.

Regarding the stability of the linear points: http://farside.ph.utexas.edu/teaching/336k/lectures/node139.html

15. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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Note: I had L5 leading but now have skimed your link and see it is L4 that is leading so tried to switch 4 and 5. - hope I got them all changed.

Here for the benefit of others from DH's link is equation 1149 results in words:
"We thus conclude that the L4 and L5 Lagrange points are stable equilibrium points, in the co-rotating frame, provided that mass m2 {the Earth} is less than about 4% of mass m1 {the sun} ..." I.e. small m3 can orbit at either L4 or L5 forever if there were no Jupiter etc.
I only suspected that they could be abolished, called it an assumption, etc. The post 30 footnote is what I more believe might be true; but it too may be wrong as only based on intuition, and the following thoughts, not analysis:

What you are telling me is that 6 masses (A,B,C,D,E,F in cyclic order with B leading A etc.) each 2000 times Earth mass in circular orbit around the sun at the corners of regular hexagon (neglecting all other planets etc.) are stable. Each is in a stable L point of two others, so I can certainly accept they are at least “meta-stable.”) I believe that they are completely stable, probably even if Jupiter is added to the system.)

As the eccentricity increases from zero, I strongly think they would go unstable, even if the major axis sun line was the same for all as the in orbit speed is changing as they speed up towards the perigee point on that line. Consider the instant when E (heavy Earth) has least speed at apogee. Then L4 of D is slightly in front of E approximately (if not exactly) at L5 of F which has been speeding up for two months. It would seem to naive me the E should be slightly falling down into this “double well” which I think is slightly closer to the sun than E’s undisturbed ellipse. That I think should increase the eccentricity of E. – a positive feedback that drives the system unstable.

Now consider the less extreme case of Sun and only Earth and mass M (= to the moon mass) which on average is at E’s L4 in same, but slightly eccentric, orbit at the instant when M is at apogee. Then the following E (always faster for the last half year than M) is less than 60 degrees behind M so E’s L4 is in front of M and slightly closer to the sun (as before with six planets). This case seems to be unstable also (I think, but may be wrong) like the case with six 2000 Earth masses with large e, but with much small growth rate for the instability. That is why my intuition suggested to me that the alternative moon formation idea of my prior post needs to be seriously considered (some analysis). However, with Earth’s current small eccentricity and moon much smaller mass, a moon mass at E’s L4 might not escape from E’s L4 well. That is why I suggested, my alternative idea for moon formation might need to call on Jupiter to help push it out of the well.

You might try running a simulation with the real earth and moon masses but e = 0.9 and see my intuition is still wrong. If it is not, lower e enough to take too long to exhibit the instability; perhaps by noting that M’s separation for E’s L4 is starting to decrease again. – Hinting that M is in some orbit about E’s L4. If that “stable e” is not too much larger than Earth’s e, maybe it is interesting enough to put Jupiter into the simulation and see if M can get helped out of the well with e near Earth’s current e?

One of the advantage of my slightly less than total ignorance about all this is that it lets me think of things you would not normally.

I hope this post makes enough sense so that you can at least tell where my intuitive ideas come from.

Last edited by a moderator: Jun 7, 2009
16. ### D HSome other guyValued Senior Member

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I never said that! I suspect such a system would be unstable.

What you are alluding to is a Klemperer rosette. This site, http://burtleburtle.net/bob/physics/kempler.html simulates various multibody configurations.

Yes and no. What does happen is that the mass ratio that denotes the boundary between stable and metastable configuration tends to zero as the eccentricity tends to one. So for a given mass, there is some eccentricity above which the triangular libration points are no longer stable. On the other hand, for a given orbit there is always some mass below which an object of negligible mass placed at the one of the triangular libration points with the appropriate velocity will remain in the vicinity of that libration point. Google "elliptical restricted three body problem" for more.

17. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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Sorry, I did jump to the six are more stable idea as each was in a stable L well, I erroneously thought, of two others.

Thanks for rest of the post too - that make it clear what happens as e increases from zero.

Do you have any ideas, facts, about how much mass could be at Earth's L4 before it would become an unstable 3 body problem. based on the "six all separated by 60degrees" are unstable, I am assuming that not even two earths can remain in the same orbit but ~60 degrees separated. I am especially interested to know if the moon's mass could be at Earth's L4 and remain there (Jupiter neglected and e=0 assumed if need be.).

18. ### D HSome other guyValued Senior Member

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You are jumping around too much, Billy. Your six earths example has a total mass of 12,000 earth masses and involves seven bodies. The Earth+Moon system has a total mass of 1.0123 earth masses, and only has three bodies. Read the second link eburacum45 cited in post #27. The ratio of the total mass of the Earth and Moon to the total mass of the Sun, Earth, and Moon Earth+Moon+Sun system is a lot less 0.03812...

19. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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Thanks. I had only very quickly skimmed his 2nd ref. where my question is clearly answered by:

“…one can ask whether it is possible to have co-orbital planets, both with masses of the order of Jupiter. In this paper, we argue that such "Trojan planets" are indeed a viable possibility. …”

Also, with a little interpretation, answered by:
“…The extent of the Jovian Trojan population is now thought to rival that of the main asteroid belt. …”

As I am reasonably sure that the mass in the asteroid belt is as large as that of the moon. However, I am surprised that with so many objects bound to Jupiter’s L4 that they have not “fused” at least into a big “dust / fragment” ball. Probably the main reason my alternative origin for the moon idea is nonsense, is that Earth’s L4 Trojans get thrown out of Earth’s L4 long before they can “fuse” to become Mm.

I guess to generalize from their two equal mass planets case criteria:
2mpl/(2mpl + M*)
That the critical ratio for stability in the Earth with moon mass, Mm, at L4 is probably something like:
(Me + Mm) / (Me +Mm + Ms)

Noting that (Me + Mm) << Mj (Jupiter’s mass) I conclude that if there were no Jupiter, the mass Mm could surely accumulate at Earth's L4, but I appear to be in the awkward position of wanting to have my cake and eat it too:
I.e. want small masses to accumulate at L4 in spite of Jupiter existing and yet have Jupiter throw the “fused” Mm out from L4 later. Only hope for this would be that one mass Mm at L4 is less stable against Jupiter’s perturbations than a multitude of small ones totaling Mm. That may be true as Mm is closer to the stability criteria limit than the small masses individually are. Perhaps the small mass fusing into Mm can be speeded by Jupiter’s disturbing their orbits about L4 to make mutual collision much more common? What do you think: Is there zero possibility that the Moon assembled at Earth’s L4 and later Jupiter threw it out?

If either you or Janus 58 think that is possible, then what do you think of my arguments that Mm could after many years advance 300 degrees more than Earth and become bound into the current Earth/moon system orbiting the sun (without the standard need for some “third body” to permit the “capture”)*?

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*I think this is the case as Mm was ALREADY “captured” at L4 – just not with close proximity to Earth. One can either take the POV that the system momentum does not change as Mm slowly moves to current Earth Moon separation OR take the POV that Jupiter is the needed “third body.”

For reasons I discussed in post 30, I think Mm advancing 300 degrees wrt Earth is more likely than falling back 60 degrees to become a bound system, but the details of how Jupiter throws Mm out of L4 may be controlling which way Earth and Mm approach each other.

Last edited by a moderator: Jun 8, 2009
20. ### frankuitaalstRegistered Member

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Planet on 180°

Hello , I'm new on this forum and saw this interesting subject .
Just wanted to say I can fully agree with Janus58, as I have simulated such behaviour of a second earth on orbitsimulator.com.
Unfortunately my status doesn't allow to give links yet

21. ### SomedumbhickRegistered Member

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Here is a side question . What if we moved Venus into Earths orbit 180 degrees ?. As far I know it's the same size and if not in Venus's orbit it might cool down to be Earth like.

22. ### SomedumbhickRegistered Member

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That is not the L3 point. The L3 point is not on the orbital plane. And a problem I see with trying to do this is that the orbits are not perfect circles they are elliptical.

23. ### Janus58Valued Senior Member

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All the Lagrange points, including the L3 point are on the orbital plane.
The vast majority of Venus's high temp comes from the greenhouse house effect of its thick atmosphere. At its present distance, it only gets ~91% more sunlight than the Earth does.