<p>The L1, L2, L3 points are <strong>not stable.</strong> (full stop) Small deviations grow exponentially, even in the perfectly circular restricted three-body problem. In reality, we have a non-circular many-body problem, when (1) these points are not strictly well-defined and (2) deviations from the simple case have to be accounted for to obtain the exact trajectories (and to keep any artificial satellites near these points).</p>

<p>The L1 &amp; L2 points are attractive for space missions exactly because they are unstable, since this prevents the accumulation of debris (natural and artificial in origin) and hence reduces the probability of collisions. </p>

CC BY-SA 4.0
2024-03-13T20:48:26.203Z
"L1 & L2 points are attractive for space missions..." name one mission that spends any significant time near these points, I can not think of any. There are plenty of spacecraft in very large halo orbits associated with these two points, but they are hundreds of thousands of kilometers away from the actual points. These halo orbits are used because they are far from Earth and (on the scale of decades) at least stable-enough that only very small propulsive corrections are needed to remain in them.
uhoh
CC BY-SA 3.0
2018-03-30T07:52:30.683Z
2018-03-30T07:52:30.683Z
Walter
<p>Problem solved (well, partially).  A project at <a href="https://web.archive.org/web/20170319020702/https://ccar.colorado.edu/ASEN5050/projects/projects_2016/Patel_Krishan/#conc" rel="nofollow noreferrer">colorado.edu</a> goes into horrific detail, and end with the conclusion,</p>
<blockquote>
<p>Lagrange points are a feature of the three-body problem being the
equilibrium points from the CRTBP equations of motion. The collinear
points are considered unstable and any perturbations for the exact
location will cause exponentially departure. As such, fourth-body
perturbations should be considered. We should have particular care
because the Sun-Earth-Moon system is one such case. An initial look
into the effect the Moon has on the Earth-Sun L1 and L2 points shows
little concern. The mass ratio changes as little as 1% and even in the
original case the Sun completely out-masses the other bodies. The
positions of L1 and L2 change by a few thousand kilometers which is
small in comparison to being approximately 1.5 million km away from
Earth. Looking at the individual components of acceleration did not
help. The Moon is about 2% of the Earth’s acceleration at its best
(closest) and 3/4% at its worst. Compiling all the results I conclude
that the Moon has about 1% of the effect that the Earth does on these
Lagrange points. This seems small however over time this can add up in
station keeping costs. Additionally, sensitive equipment such as the
James Webb Space Telescope require steady orbits and knowing the
perturbation will help keep it steady. In summary, the Moon does not
have a large effect but a time aspect must be considered. Further work
should include the four-body problem to completely understand changes
to these equilibrium points.</p>
</blockquote>

CC BY-SA 3.0
2018-03-30T07:44:06.140Z
This appears to be an undergraduate project (part of the degree course), i.e. not conducted by a qualified academic. Moreover, it lacks any scientific rigour.
Walter
CC BY-SA 4.0
2018-03-26T14:58:03.520Z
2024-03-13T13:10:17.507Z
2024-03-13T13:10:17.507Z
Carl Witthoft
<p>I know that the Earth-Sun Lagrange L1, L2, and L3 points are not considered stable over longer periods, especially when compared to L4 and L5... But, with the moon orbiting the Earth in the general path of the ecliptic, it would seem that the first time the moon got in between the Earth and the object at the L1 or L2 points, it would be enough to perturb the object a good bit, and the next lunar month would be even worse, and so on.</p>

<p>It seems that if the moon was taken out of the equation, the L1 and L2 points would be much more stable. And, as a result of this line of thinking, I would think that the L3 point would be much more stable than L1 and L2, since the moon would be so much farther away and unimportant, so its effect could be ignored.</p>

<p>Am I wrong in thinking this? Is the moon's influence not large enough to be a major factor? And, just to clarify, I am talking specifically about L1 and L2 with respect to a large 3rd body, like the moon, instead of some idealized 2 body only system.</p>

CC BY-SA 3.0
2018-03-26T14:50:38.493Z
Well, have you followed the equations for $L_j$ points to see just how much the moons gravitational strength at the solar-terran Lagrange points is?
Carl Witthoft
CC BY-SA 4.0
2024-03-13T16:04:53.457Z
The Lagrange points are solutions to the three body problem, no more, no less.  They also become unstable when you account for the bodies being non-rigid.  In practice, however, they are "stable enough" for some purposes.
Greg Miller
<the-moon><earth><orbital-mechanics><celestial-mechanics><lagrange-point>
How can Earth-Sun Lagrange points L1 & L2 even be semi stable considering the moon?
Tazz250
# How can Earth-Sun Lagrange points L1 & L2 even be semi stable considering the moon?

(Asked by: username_0 on 2018-03-23)

I know that the Earth-Sun Lagrange L1, L2, and L3 points are not considered stable over longer periods, especially when compared to L4 and L5... But, with the moon orbiting the Earth in the general path of the ecliptic, it would seem that the first time the moon got in between the Earth and the object at the L1 or L2 points, it would be enough to perturb the object a good bit, and the next lunar month would be even worse, and so on.

It seems that if the moon was taken out of the equation, the L1 and L2 points would be much more stable. And, as a result of this line of thinking, I would think that the L3 point would be much more stable than L1 and L2, since the moon would be so much farther away and unimportant, so its effect could be ignored.

Am I wrong in thinking this? Is the moon's influence not large enough to be a major factor? And, just to clarify, I am talking specifically about L1 and L2 with respect to a large 3rd body, like the moon, instead of some idealized 2 body only system.

**Tags:** the-moon, earth, orbital-mechanics, celestial-mechanics, lagrange-point

> **Comments:**
> **username_1**: Well, have you followed the equations for $L_j$ points to see just how much the moons gravitational strength at the solar-terran Lagrange points is?  
> **username_2**: The Lagrange points are solutions to the three body problem, no more, no less.  They also become unstable when you account for the bodies being non-rigid.  In practice, however, they are "stable enough" for some purposes.  

---

## 2 answers

### Answer by username_3 on 2018-03-30. Score: 5

The L1, L2, L3 points are **not stable.** (full stop) Small deviations grow exponentially, even in the perfectly circular restricted three-body problem. In reality, we have a non-circular many-body problem, when (1) these points are not strictly well-defined and (2) deviations from the simple case have to be accounted for to obtain the exact trajectories (and to keep any artificial satellites near these points).

The L1 & L2 points are attractive for space missions exactly because they are unstable, since this prevents the accumulation of debris (natural and artificial in origin) and hence reduces the probability of collisions.

> **Comments:**
> **username_4**: "L1 & L2 points are attractive for space missions..." name one mission that spends any significant time near these points, I can not think of any. There are plenty of spacecraft in very large halo orbits associated with these two points, but they are hundreds of thousands of kilometers away from the actual points. These halo orbits are used because they are far from Earth and (on the scale of decades) at least stable-enough that only very small propulsive corrections are needed to remain in them.  

---

### Answer by username_1 on 2018-03-26. Score: 2

Problem solved (well, partially). A project at [colorado.edu](https://web.archive.org/web/20170319020702/https://ccar.colorado.edu/ASEN5050/projects/projects_2016/Patel_Krishan/#conc) goes into horrific detail, and end with the conclusion,

> Lagrange points are a feature of the three-body problem being the
> equilibrium points from the CRTBP equations of motion. The collinear
> points are considered unstable and any perturbations for the exact
> location will cause exponentially departure. As such, fourth-body
> perturbations should be considered. We should have particular care
> because the Sun-Earth-Moon system is one such case. An initial look
> into the effect the Moon has on the Earth-Sun L1 and L2 points shows
> little concern. The mass ratio changes as little as 1% and even in the
> original case the Sun completely out-masses the other bodies. The
> positions of L1 and L2 change by a few thousand kilometers which is
> small in comparison to being approximately 1.5 million km away from
> Earth. Looking at the individual components of acceleration did not
> help. The Moon is about 2% of the Earth’s acceleration at its best
> (closest) and 3/4% at its worst. Compiling all the results I conclude
> that the Moon has about 1% of the effect that the Earth does on these
> Lagrange points. This seems small however over time this can add up in
> station keeping costs. Additionally, sensitive equipment such as the
> James Webb Space Telescope require steady orbits and knowing the
> perturbation will help keep it steady. In summary, the Moon does not
> have a large effect but a time aspect must be considered. Further work
> should include the four-body problem to completely understand changes
> to these equilibrium points.

> **Comments:**
> **username_3**: This appears to be an undergraduate project (part of the degree course), i.e. not conducted by a qualified academic. Moreover, it lacks any scientific rigour.  

---

