<blockquote>
<p>Are all solutions of the 3-body problem restricted to 2 dimensions?</p>
</blockquote>
<p>The five stationary solutions to the three body problem are restricted to a plane. That does not mean that all solutions are restricted to a plane.</p>
<blockquote>
<p>Has any work been done on 3-dimensional orbits?</p>
</blockquote>
<p>NASA is proposing to build its <a href="https://en.wikipedia.org/wiki/Lunar_Gateway" rel="noreferrer">Lunar Gateway</a> as a vehicle in a &quot;<a href="https://en.wikipedia.org/wiki/Near-rectilinear_halo_orbit" rel="noreferrer">Near-Rectilinear Halo Orbit</a>&quot; (NRHO) about the Moon. These halo orbits only exist in the presence of a third body, in this case, the Earth. The orbit is nearly orthogonal to the Moon's orbit about the Earth.</p>

CC BY-SA 4.0
2020-08-08T16:34:11.170Z
2020-08-08T16:34:11.170Z
David Hammen
<p>As <a href="https://astronomy.stackexchange.com/a/38342/7982">@DavidHammen already pointed out</a> there is no restriction to two dimensions, the Lagrange points and the 2D planar orbits possible around some of them were &quot;low hanging fruit&quot; because they could be solved using pencil and paper or &quot;<a href="https://hsm.stackexchange.com/q/6347/3869">writing with feathers using light from burning animal fat</a>&quot; so to speak.</p>
<p>Still, the stability of 3D three body orbits was at least suspected a long time ago but I don't yet know when the first halo orbit was <em>proven</em> to be stable.</p>
<p>For more on this, including some stunning visuals and videos, see</p>
<ul>
<li><a href="http://www.gg.caltech.edu/%7Emwl/publications/papers/dynamicalThreeBody.pdf" rel="nofollow noreferrer">Dynamical Systems, the Three-Body Problem and Space Mission Design</a> by &quot;KoLoMaRo&quot;</li>
<li><a href="https://en.wikipedia.org/wiki/Aleksandr_Lyapunov" rel="nofollow noreferrer">Aleksandr Lyapunov</a> and <a href="https://www.semanticscholar.org/paper/Lyapunov-%27-s-stability-theory-%E2%80%94-100-years-on-*-Parks/28d683191e11b09287c0882a89cbe4bf5a43285b" rel="nofollow noreferrer">Lyapunov's stability theory — 100 years on</a></li>
<li><a href="https://space.stackexchange.com/q/15268/12102">Are some Halo Orbits actually Stable?</a></li>
<li><a href="https://space.stackexchange.com/q/41870/12102">How does a butterfly orbit move in 3D? Way to generate or visualize?</a></li>
<li><a href="https://space.stackexchange.com/q/24019/12102">What is a near rectilinear halo orbit?</a></li>
<li><a href="https://space.stackexchange.com/q/20590/12102">How to best think of the State Transition Matrix, and how to use it to find periodic Halo orbits?</a></li>
<li><a href="https://space.stackexchange.com/q/27534/12102">What sort of orbital elements are used to describe halo orbits?</a></li>
</ul>

CC BY-SA 4.0
2020-08-09T03:04:37.253Z
2020-08-09T03:04:37.253Z
uhoh
<p>I am not a physicist.</p>
<p>Suppose there are two massive bodies orbiting each other. Their orbit will presumably define a plane.  As far as I know, the existing solutions to the 3-body problem are limited to a plane.</p>
<p>Has any work been done on 3-dimensional orbits? For example two bodies are orbiting one another and a third body enters the system at right angle to their plane?</p>
<p>Specifically (1) Are there any solutions even for special cases of this? (2) Would such 3-D systems inevitability decay into 2 dimensions?</p>

CC BY-SA 4.0
2020-08-08T18:51:31.103Z
Cris Moore found some pretty amazing n-body orbits, both planar (eg a [figure 8](http://tuvalu.santafe.edu/~moore/figure8-3.loop.gif)) & 3D. See http://tuvalu.santafe.edu/~moore/gallery.html
PM 2Ring
<orbit>
Are all solutions of the 3-body problem restricted to 2 dimensions?
chasly - supports Monica
# Are all solutions of the 3-body problem restricted to 2 dimensions?

(Asked by: username_0 on 2020-08-08)

I am not a physicist.

Suppose there are two massive bodies orbiting each other. Their orbit will presumably define a plane. As far as I know, the existing solutions to the 3-body problem are limited to a plane.

Has any work been done on 3-dimensional orbits? For example two bodies are orbiting one another and a third body enters the system at right angle to their plane?

Specifically (1) Are there any solutions even for special cases of this? (2) Would such 3-D systems inevitability decay into 2 dimensions?

**Tags:** orbit
> **Comments:**
> **username_1**: Cris Moore found some pretty amazing n-body orbits, both planar (eg a [figure 8](http://tuvalu.santafe.edu/~moore/figure8-3.loop.gif)) & 3D. See http://tuvalu.santafe.edu/~moore/gallery.html  

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## 2 answers

### Answer by username_2 on 2020-08-08. Score: 7

> Are all solutions of the 3-body problem restricted to 2 dimensions?

The five stationary solutions to the three body problem are restricted to a plane. That does not mean that all solutions are restricted to a plane.

> Has any work been done on 3-dimensional orbits?

NASA is proposing to build its [Lunar Gateway](https://en.wikipedia.org/wiki/Lunar_Gateway) as a vehicle in a "[Near-Rectilinear Halo Orbit](https://en.wikipedia.org/wiki/Near-rectilinear_halo_orbit)" (NRHO) about the Moon. These halo orbits only exist in the presence of a third body, in this case, the Earth. The orbit is nearly orthogonal to the Moon's orbit about the Earth.


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### Answer by username_3 on 2020-08-09. Score: 2

As [@DavidHammen already pointed out](https://astronomy.stackexchange.com/a/38342/7982) there is no restriction to two dimensions, the Lagrange points and the 2D planar orbits possible around some of them were "low hanging fruit" because they could be solved using pencil and paper or "[writing with feathers using light from burning animal fat](https://hsm.stackexchange.com/q/6347/3869)" so to speak.

Still, the stability of 3D three body orbits was at least suspected a long time ago but I don't yet know when the first halo orbit was *proven* to be stable.

For more on this, including some stunning visuals and videos, see

* [Dynamical Systems, the Three-Body Problem and Space Mission Design](http://www.gg.caltech.edu/%7Emwl/publications/papers/dynamicalThreeBody.pdf) by "KoLoMaRo"
* [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov) and [Lyapunov's stability theory — 100 years on](https://www.semanticscholar.org/paper/Lyapunov-%27-s-stability-theory-%E2%80%94-100-years-on-*-Parks/28d683191e11b09287c0882a89cbe4bf5a43285b)
* [Are some Halo Orbits actually Stable?](https://space.stackexchange.com/q/15268/12102)
* [How does a butterfly orbit move in 3D? Way to generate or visualize?](https://space.stackexchange.com/q/41870/12102)
* [What is a near rectilinear halo orbit?](https://space.stackexchange.com/q/24019/12102)
* [How to best think of the State Transition Matrix, and how to use it to find periodic Halo orbits?](https://space.stackexchange.com/q/20590/12102)
* [What sort of orbital elements are used to describe halo orbits?](https://space.stackexchange.com/q/27534/12102)


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