# Why would Hollywood's Planet X (at earth's L3) be unstable?

This NASA website states

"NASA is unlikely to find any use for the L3 point since it remains hidden behind the Sun at all times. The idea of a hidden "Planet-X" at the L3 point has been a popular topic in science fiction writing. The instability of Planet X's orbit (on a time scale of 150 years) didn't stop Hollywood from turning out classics like The Man from Planet X."

where "The Man from Planet X" links to IMDB: The Man from Planet X (1951). Since Planet X supposedly passes by the earth after leaving it's own sun, this should probably point to Journey to the Far Side of the Sun (1969) instead.

For the sake of discussion of the question of stability: if there hypothetically were a planet, with say a mass roughly equivalent to earth, then would it really be unstable in the same way that a low-mass object would be unstable at Earth's L3? And wouldn't that mean that both it and earth would be similarly unstable? Or does this only refer to the idea that it could't remain hidden behind the sun for hundreds of years, as a very old calculation summarized on this webpage which I found here seems to suggest. Has this calculation been repeated and reported more recently?

note: this all stared from a conversation below this question

note 2: while this question is about the statement about stability of a planet opposite earth, from this answer it seems that the quote above should point to Journey to the Far Side of the Sun (1969) and not point to The Man from Planet X (1951)

• SF wrote a good answer but then deleted it. Yes his answer describes L2 but it's also applicable to L3. However the effects are a lot more pronounced for L2. Although at SEL3 Venus and Jupiter are the perturbing influences, not the moon. Commented Feb 6, 2016 at 20:27
• OK thanks for that info. This sentence is the main question: "if there hypothetically were a planet, with say a mass roughly equivalent to earth, then would it really be unstable in the same way that a low-mass object would be unstable at Earth's L3?" We usually don't say that the orbit of the earth is "unstable" but we do say that is perturbed.
– uhoh
Commented Feb 7, 2016 at 4:14
• There are objects that follow horseshoe orbits if viewed from earth's frame. They remain at 1 A.U. from the sun, much like the earth. In that respect, their orbit is stable. But they do not keep the same position relative to the earth. Commented Feb 7, 2016 at 16:55
• Good point! I understand what you're saying but if I had to define exactly what "stable orbit" means for a horseshoe orbit that was near but not exactly in resonance I'm not sure I could. For a planet at SEL3 (my question) what does "unstable" mean? We say 2010 TK7 is at the "stable" L4 point, but it really orbits around it wildly. Here "stable" just means expected to remain associated w/ SEL4 for quite a long time.
– uhoh
Commented Feb 7, 2016 at 17:53
• Commented Jun 14, 2016 at 20:03

With the 5 Lagrange points, there is a local 0 gravitational field. Of them, L4/L5 are "hills", such that a small deviation from perfect will tend to put them back on course. L1/L2/L3 are "Saddles", which means if they can stay perfectly on point, they will remain, otherwise they will slowly start to drift. This can be seen in this illustration of the Lagrange points:

L3 will tend to remain at the right distance, but not the right relative spot in the orbit. This will eventually cause it to fall out of that alignment.

Incidentally, a current theory regarding the formation of the moon is that a Mars sized planet formed at the Earth-Sun L4 point. This object has been nicknamed "Theia". It eventually was nudged out, after millions of years. Even such a relatively stable location long term isn't stable, due to the other planets in our Solar System, which tend to gently nudge objects periodically.

• The Lagrange points are mathematical points in the restricted 3-body problem or "CR3BP", where two masses rotate in circular orbits around their center of mass, and the third body has zero mass. I'm asking about planet of similar mass to the earth. This is a whole different problem! While sometimes people still use the symbols $L_1$...$L_5$ as labels for certain areas in real orbits, that doesn't mean that the rules for the mathematical CR3BP actually apply.
– uhoh
Commented Apr 16, 2016 at 1:31
• ...thus the careful wording of my question: "...with say a mass roughly equivalent to earth, then would it really be unstable in the same way that a low-mass object would be unstable at Earth's L3?" Now it's the full-blown three-body problem. Earth and "Planet-X" (if you will) now interact with each other as "equals".
– uhoh
Commented Apr 16, 2016 at 1:44

Because Earth and the Sun are not alone in this dance. Jupiter and Venus tug on us too. They have unsynochrized periods and slightly different inclinations. If a copy of Earth were suddenly put at L3, it would soon gradually spiral into another orbit. And unavoidably collide with us. Such a copycat planet 180 degrees away would have to have a Moon like ours with identical mass and orbital parameters too, because it influences Earth's orbit around the Sun. It irregularly sometimes pulls us ahead or away when we are slightly closer to the Sun, because of the eccentricities of both Earth and the Moon. For the same reason that there is not an even number of months per year. I have not done the math, this is a simple (and rare) case where sound space intuition knows why it is impossible.

• Interesting - since both planets have roughly similar mass (and either one could be the slightly heavier), whatever behavior occurs should be equally likely to apply to both Earth as to Planet X. It's not like the one with humans is "supposed to be there" and the other know's it's not.
– uhoh
Commented Jun 15, 2016 at 1:23
• @uhoh For example, now Mars and Saturn pull us in the same direction. Six months from now they will be widely separated in our sky. It is not the same thing to be in L3 now as being here now, 300 000 000 km away. Commented Jun 15, 2016 at 8:44
• I'll try once more. When you want to describe what you think might happen, you can't just choose one and say it will spiral into the other. If one spirals, the other spirals, and maybe whichever is a little lighter might spiral a little more than which ever is a little heavier, but they dance together. Oh, and there's nothing in the question about "suddenly put" either.
– uhoh
Commented Jun 15, 2016 at 8:51
• @uhoh Well, Janus and Epimetheus, moons of Saturn, share the same orbit. They are not in opposed positions and they are set on collision course ultimately. Of course it takes two to be different, they will both be perturbed in different ways and add material to the thin ring system one day, or merge to a single moon. Commented Jun 15, 2016 at 9:04
• OK that's really helpful information! I just asked Have any co-orbital planet pairs been discovered (and not subsequently retracted)? in Astronomy SE but I didn't realize there was an example so close to home that could be studied directly! OK I'll try to do some reading about them - Thanks!
– uhoh
Commented Jun 15, 2016 at 9:37