<p><a href="https://astronomy.stackexchange.com/questions/3595/is-there-a-ceiling-for-stable-l4-or-l5-masses/3617#3617">Is there a ceiling for stable L4 or L5 masses?</a></p>

<p>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. </p>

<p>Earth &lt; 4% Sun = Stable L4,L5 points</p>

<p><a href="http://en.wikipedia.org/wiki/Giant_impact_hypothesis" rel="nofollow noreferrer">http://en.wikipedia.org/wiki/Giant_impact_hypothesis</a> says that the Lagrangian object would destabilize once above 10% of the Earth's mass.</p>

<p>L4 or L5 object &lt; 10% Earth = Stable object</p>

CC BY-SA 3.0
2014-07-30T18:09:33.347Z
Thanks for your input on the second half of the question.  But what about the amplitude?
Matthew Cordaro
CC BY-SA 3.0
2014-07-30T23:08:20.833Z
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.
Mazura
CC BY-SA 3.0
2014-07-31T17:56:30.943Z
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: http://en.wikipedia.org/wiki/List_of_objects_at_Lagrangian_points
Matthew Cordaro
CC BY-SA 3.0
2014-08-01T00:40:40.617Z
Bonus! Why are you and I the resident SE experts on Lagrangian points... I never even finished collage.
Mazura
CC BY-SA 3.0
2014-08-01T15:30:55.223Z
Probably because Astronomy SE is still in beta.  I'm thinking about asking this in the Physics or Math SE.
Matthew Cordaro
CC BY-SA 3.0
2014-08-01T23:56:11.103Z
I'm guessing you found this, too: http://astronomy.stackexchange.com/questions/1219/how-stable-are-lissajous-orbits?rq=1 most of it is over my head so don't know if it helps.
Mazura
CC BY-SA 3.0
2014-08-29T09:49:04.423Z
I'd like to see that Lagrangian collage, Mazura :-)
Rory Alsop
CC BY-SA 3.0
2014-07-30T07:35:30.790Z
2014-07-30T07:35:30.790Z
2017-04-13T12:59:06.910Z
Mazura
<p>I stumbled on The <a href="https://en.wikipedia.org/wiki/Lagrangian_point#L4_and_L5" rel="nofollow">Lagrangian points</a> during some wikipedia reading.  After looking at the gravity contours, I naturally come to the conclusion that the L4 &amp; L5 should have a wave pattern and then found the <a href="https://en.wikipedia.org/wiki/Lissajous_orbit" rel="nofollow">Lissajous orbit</a> page.  It states:</p>

<blockquote>
  <p>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.</p>
</blockquote>

<p>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?</p>

<p>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?</p>

<p>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.</p>

<p><strong>Edit</strong>: <a href="https://www.youtube.com/watch?v=VGmdSbtkFS0" rel="nofollow">Here</a> 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.</p>

<gravity><lagrange-point><satellite>
Maximum Amplitude of a Lissajous Orbiting Object in a L4 or L5 Position
Matthew Cordaro
# Maximum Amplitude of a Lissajous Orbiting Object in a L4 or L5 Position

(Asked by: username_0 on 2014-04-22)

I stumbled on The [Lagrangian points](https://en.wikipedia.org/wiki/Lagrangian_point#L4_and_L5) 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](https://en.wikipedia.org/wiki/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](https://www.youtube.com/watch?v=VGmdSbtkFS0) 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.

**Tags:** gravity, lagrange-point, satellite

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## 1 answers

### Answer by username_1 on 2014-07-30. Score: 1

[Is there a ceiling for stable L4 or L5 masses?](https://astronomy.stackexchange.com/questions/3595/is-there-a-ceiling-for-stable-l4-or-l5-masses/3617#3617)

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

> **Comments:**
> **username_0**: Thanks for your input on the second half of the question.  But what about the amplitude?  
> **username_1**: 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.  
> **username_0**: 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: http://en.wikipedia.org/wiki/List_of_objects_at_Lagrangian_points  
> **username_1**: Bonus! Why are you and I the resident SE experts on Lagrangian points... I never even finished collage.  
> **username_0**: Probably because Astronomy SE is still in beta.  I'm thinking about asking this in the Physics or Math SE.  
> **username_1**: I'm guessing you found this, too: http://astronomy.stackexchange.com/questions/1219/how-stable-are-lissajous-orbits?rq=1 most of it is over my head so don't know if it helps.  
> **username_2**: I'd like to see that Lagrangian collage, Mazura :-)  

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