I stumbled on The Lagrangian points during some wikipedia reading. After looking at the gravity contours, I naturally come to the conclusion that the L4 & L5 should have a wave pattern and then found the Lissajous orbit page. It states:

Orbits about Lagrangian points L4 and L5 are dynamically stable in theory so long as the ratio of the masses of the two main objects is greater than about 25, meaning the natural dynamics keep the [third object] in the vicinity of the Lagrangian point even when slightly perturbed from equilibrium.

After reading it I started to wonder if there is a maximum possible amplitude (height of the peaks and troughs relative to the orbital plane of the second object) of the pattern?

Also, if there theoretically is none, for cases where it is extremely large, say larger than the 2 times the radius of the second orbiting object, what would the respective object weight ratios have to be to keep such a perturbed orbit stable for any realistic period of time?

FYI, I'm a computer scientist who loves reading about physical cosmology but can be kinda a noob sometimes. Please forgive me if I'm asking for the wrong parameters.

Edit: Here is an animation of 2010 TK7, Earth's first trojan asteroid, showing the wave pattern I'm referring to. Recall that my question is referring to the height of the peaks and troughs relative to Earth's orbital plane. Since the video is a top-down view, the peaks and troughs are going into and out of the screen.


1 Answer 1


Is there a ceiling for stable L4 or L5 masses?

Your link says orbits about Lagrangian points L4 and L5 are dynamically stable so long as the ratio of the masses of the two main objects is greater than about 25.

Earth < 4% Sun = Stable L4,L5 points

http://en.wikipedia.org/wiki/Giant_impact_hypothesis says that the Lagrangian object would destabilize once above 10% of the Earth's mass.

L4 or L5 object < 10% Earth = Stable object

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    $\begingroup$ Thanks for your input on the second half of the question. But what about the amplitude? $\endgroup$ Commented Jul 30, 2014 at 18:09
  • $\begingroup$ No clue, I thought stuff just sat out there. Ask some math guys to plug in what is the circumference of the orbit of a L5 10% the mass of Earth into some nifty equation. Then you could plot the previous course giving you the maximum deviation before destabilization. Yes, I am making this all up. $\endgroup$
    – Mazura
    Commented Jul 30, 2014 at 23:08
  • $\begingroup$ Ha. Yes, I'm not so sure it has a relative circular orbit like the L2. This all spun off of some search on some new telescope that was to be parked in the l4 or l5. NASA or ESA had a cool animation of it in orbit; it obviously had a wave pattern while rotating around the sun. I wish I could find that animation. Bonus, a list of Lagrangian point objects found here: en.wikipedia.org/wiki/List_of_objects_at_Lagrangian_points $\endgroup$ Commented Jul 31, 2014 at 17:56
  • $\begingroup$ Bonus! Why are you and I the resident SE experts on Lagrangian points... I never even finished collage. $\endgroup$
    – Mazura
    Commented Aug 1, 2014 at 0:40
  • $\begingroup$ Probably because Astronomy SE is still in beta. I'm thinking about asking this in the Physics or Math SE. $\endgroup$ Commented Aug 1, 2014 at 15:30

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