<p>The <a href="http://en.wikipedia.org/wiki/Giant_impact_hypothesis" rel="nofollow">Giant Impact Hypothesis</a>, a theory of how the moon was created, says that once it exceeds 10% of the mass of your 'J', the orbit of an L4 or L5 (your 'T') destabilizes.</p>

<p><strong>Possible origin of Theia</strong></p>

<blockquote>
  <p>In 2004, Princeton University mathematician Edward Belbruno and astrophysicist J. Richard Gott III proposed that Theia coalesced at the L4 or L5 Lagrangian point relative to Earth (in about the same orbit and about 60° ahead or behind), similar to a trojan asteroid. Two-dimensional computer models suggest that the stability of Theia's proposed trojan orbit would have been affected when its growing mass exceeded a threshold of approximately 10% of the Earth's mass (the mass of Mars). In this scenario, gravitational perturbations by planetesimals caused Theia to depart from its stable Lagrangian location, and subsequent interactions with proto-Earth led to a collision between the two bodies.</p>
</blockquote>

CC BY-SA 3.0
2014-06-05T04:15:53.953Z
2015-02-13T22:43:40.207Z
2015-02-13T22:43:40.207Z
Mazura
<p>Note:  Answering from a comment posted on  <a href="https://space.stackexchange.com/questions/233/how-deep-is-the-force-well-of-l4-and-l5-lagrangian-points-of-earth-sun-set/4007?noredirect=1#comment21322_4007">Space Exploration</a></p>

<p>The classical stability analysis of these libration points assumes that we are examining the motion of a particle whose dynamics are perturbed by the gravitational impacts of a primary and secondary mass, so as a bottom-line-up-front type of answer, the mass of T is negligible - so any large increases in mass will negate these assumptions.  Further, the stability analysis is <em>linear stability analysis</em>, implying that the stability is only valid within a neighborhood of the equilibrium point, and very little information can be said about the non-linear behavior (However, an unstable equilibrium point will be unstable in the non-linear dynamics).</p>

<p>With that said, the critical mass value in the circular restricted three body problem (CR3BP) can be found from the following development, summarized from most major astrodynamics texts to include Vallado (1), Roy (2), Schaub (3), or the essential 1967 CR3BP text by Szebehely (4).  The linear variational equations of motion for small in-plane perturbations about the triangular libration points can be found as </p>

<p>$\ddot{\xi}=2\dot{\eta}+U_{xx}^*\xi + U_{xy}^{*}\eta\\
\ddot{\eta}=-2\dot{\xi}+U_{yx}^{*}\xi+U^{*}_{yy}\eta$</p>

<p>where $\xi,\eta$ are the perturbations in the $x$ and $y$ directions in the CR3BP synodic frame, and $U^{*}_{..}$ is are partials of an artificial pseudo-potential function.  Essentially, the characteristic equation for this linear system is found as $\Lambda^2 + \Lambda + \frac{27}{4}\mu(1-\mu)=0$, where $\Lambda = \lambda^2$, $\lambda$ being an eigenvalue of the actual characteristic equation.</p>

<p>If we let $g = 1-27\mu(1-\mu)$, the four roots of the system can be expressed as slightly complicated functions of $g$, but the eigenvalue behavior can be classified according to the value of $g$ as below:</p>

<ul>
<li>$0 &lt; g \leq 1$:  Pure imaginary eigenvalues, marginal stability</li>
<li>$g = 0$:  Repeated eigenvalues; secular terms present; unstable</li>
<li>$g &gt; 0$:  Eigenvalues with positive reals; unstable</li>
</ul>

<p>The critical $\mu$ value ($\mu_c$) comes from setting $g=0$.  Solving this, we find that $\mu_c=\frac{1}{2}\left(1\pm\frac{\sqrt{69}}{9}\right)\approx0.0385$.  Again, <em>a key assumption in this development is that the mass of the third body is negligible</em>.  A lot of systems of interest are below this critical mass value to include the Earth-Moon, Sun-Earth, Sun-Jupiter, etc.; however, some systems are definitely above this value -- consider the Pluto-Charon system with a $\mu$ value of approximately 0.1101. </p>

<p>1:  Vallado, D.A.  Fundamentals of Astrodynamics and Applications.  30 Jun, 2001.  Springer Science &amp; Business Media.</p>

<p>2:  Roy, A.E.  Orbital Motion, 4th Ed. 31 Dec 2004.  CRC Press.</p>

<p>3:  Schaub, H.P.  Analytical Mechanics of Space Systems.  2003.  AIAA.</p>

<p>4:  Szebehely, V.G.  Theory of Orbits in the Restricted Problem of Three Bodies.  June 1967.  Academic Pr.</p>

CC BY-SA 3.0
2015-02-14T16:25:13.367Z
2015-02-14T16:25:13.367Z
2017-04-13T12:58:55.210Z
jah138
<p>Narrow answer: if all masses were equal, it would be a 3-body Klemperer rosette. It is known that such trivial KR configurations are not stable in the long term.</p>

<p><a href="http://en.wikipedia.org/wiki/Klemperer_rosette" rel="nofollow">http://en.wikipedia.org/wiki/Klemperer_rosette</a></p>

CC BY-SA 3.0
2015-02-13T20:47:12.663Z
2015-02-13T20:47:12.663Z
Florin Andrei
<p>L4 and L5, the Lagrange points 60 degrees leading and trailing an orbiting body, are famous for being stable.</p>
<p>A well known example is the Trojan asteroids at the Sun-Jupiter L4 and L5. Nodding to these bodies I label the central mass S, orbiting mass J and the L4 mass T:</p>
<p><img src="https://i.sstatic.net/nG3JF.jpg" alt="enter image description here" /></p>
<p><a href="http://en.wikipedia.org/wiki/Lagrangian_point#Stability" rel="nofollow noreferrer">It's said S/J needs to be equal to or greater than 24.96 for the system to be stable</a>. For a stable system there's a ceiling on J, it can't be more than 4% of S.</p>
<p>My question: Is there a ceiling on mass T? If T were as massive as J, could the system still be stable?</p>

<orbit><lagrange-point>
Is there a ceiling for stable L4 or L5 masses?
HopDavid
# Is there a ceiling for stable L4 or L5 masses?

(Asked by: username_0 on 2014-06-01)

L4 and L5, the Lagrange points 60 degrees leading and trailing an orbiting body, are famous for being stable.

A well known example is the Trojan asteroids at the Sun-Jupiter L4 and L5. Nodding to these bodies I label the central mass S, orbiting mass J and the L4 mass T:

![enter image description here](https://i.sstatic.net/nG3JF.jpg)

[It's said S/J needs to be equal to or greater than 24.96 for the system to be stable](http://en.wikipedia.org/wiki/Lagrangian_point#Stability). For a stable system there's a ceiling on J, it can't be more than 4% of S.

My question: Is there a ceiling on mass T? If T were as massive as J, could the system still be stable?

**Tags:** orbit, lagrange-point


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## 3 answers

### Answer by username_1 on 2014-06-05. Score: 7

The [Giant Impact Hypothesis](http://en.wikipedia.org/wiki/Giant_impact_hypothesis), a theory of how the moon was created, says that once it exceeds 10% of the mass of your 'J', the orbit of an L4 or L5 (your 'T') destabilizes.

**Possible origin of Theia**

> In 2004, Princeton University mathematician Edward Belbruno and astrophysicist J. Richard Gott III proposed that Theia coalesced at the L4 or L5 Lagrangian point relative to Earth (in about the same orbit and about 60° ahead or behind), similar to a trojan asteroid. Two-dimensional computer models suggest that the stability of Theia's proposed trojan orbit would have been affected when its growing mass exceeded a threshold of approximately 10% of the Earth's mass (the mass of Mars). In this scenario, gravitational perturbations by planetesimals caused Theia to depart from its stable Lagrangian location, and subsequent interactions with proto-Earth led to a collision between the two bodies.


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### Answer by username_2 on 2015-02-14. Score: 5

Note: Answering from a comment posted on [Space Exploration](https://space.stackexchange.com/questions/233/how-deep-is-the-force-well-of-l4-and-l5-lagrangian-points-of-earth-sun-set/4007?noredirect=1#comment21322_4007)

The classical stability analysis of these libration points assumes that we are examining the motion of a particle whose dynamics are perturbed by the gravitational impacts of a primary and secondary mass, so as a bottom-line-up-front type of answer, the mass of T is negligible - so any large increases in mass will negate these assumptions. Further, the stability analysis is *linear stability analysis*, implying that the stability is only valid within a neighborhood of the equilibrium point, and very little information can be said about the non-linear behavior (However, an unstable equilibrium point will be unstable in the non-linear dynamics).

With that said, the critical mass value in the circular restricted three body problem (CR3BP) can be found from the following development, summarized from most major astrodynamics texts to include Vallado (1), Roy (2), Schaub (3), or the essential 1967 CR3BP text by Szebehely (4). The linear variational equations of motion for small in-plane perturbations about the triangular libration points can be found as

$\ddot{\xi}=2\dot{\eta}+U\_{xx}^\*\xi + U\_{xy}^{\*}\eta\\
\ddot{\eta}=-2\dot{\xi}+U\_{yx}^{\*}\xi+U^{\*}\_{yy}\eta$

where $\xi,\eta$ are the perturbations in the $x$ and $y$ directions in the CR3BP synodic frame, and $U^{\*}\_{..}$ is are partials of an artificial pseudo-potential function. Essentially, the characteristic equation for this linear system is found as $\Lambda^2 + \Lambda + \frac{27}{4}\mu(1-\mu)=0$, where $\Lambda = \lambda^2$, $\lambda$ being an eigenvalue of the actual characteristic equation.

If we let $g = 1-27\mu(1-\mu)$, the four roots of the system can be expressed as slightly complicated functions of $g$, but the eigenvalue behavior can be classified according to the value of $g$ as below:

* $0 < g \leq 1$: Pure imaginary eigenvalues, marginal stability
* $g = 0$: Repeated eigenvalues; secular terms present; unstable
* $g > 0$: Eigenvalues with positive reals; unstable

The critical $\mu$ value ($\mu\_c$) comes from setting $g=0$. Solving this, we find that $\mu\_c=\frac{1}{2}\left(1\pm\frac{\sqrt{69}}{9}\right)\approx0.0385$. Again, *a key assumption in this development is that the mass of the third body is negligible*. A lot of systems of interest are below this critical mass value to include the Earth-Moon, Sun-Earth, Sun-Jupiter, etc.; however, some systems are definitely above this value -- consider the Pluto-Charon system with a $\mu$ value of approximately 0.1101.

1: Vallado, D.A. Fundamentals of Astrodynamics and Applications. 30 Jun, 2001. Springer Science & Business Media.

2: Roy, A.E. Orbital Motion, 4th Ed. 31 Dec 2004. CRC Press.

3: Schaub, H.P. Analytical Mechanics of Space Systems. 2003. AIAA.

4: Szebehely, V.G. Theory of Orbits in the Restricted Problem of Three Bodies. June 1967. Academic Pr.


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### Answer by username_3 on 2015-02-13. Score: 3

Narrow answer: if all masses were equal, it would be a 3-body Klemperer rosette. It is known that such trivial KR configurations are not stable in the long term.

<http://en.wikipedia.org/wiki/Klemperer_rosette>


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