<p>Answer to the NEW question:</p>

<p>the Angular Momentum Conservation Law states that, for any moving body, its angular momentum does not change unless you exercise an external force different from the central force.</p>

<p>For an orbiting body like a planet, this means that Sun's gravity, being the central force, does not modify Angular Momentum, but any other external force will do.</p>

<p>Examples of external forces are collisions or the forces made by Jupiter on another planet, or by Neptune on Pluto.</p>

<p>After the Solar System was formed, these external forces are quite small, and thus does not change greatly the Angular Momentum of any major body. But you can see how passing near a body can alter a comet's orbit.</p>

<p>Moreover, the external forces made by bodies that are in the same plane as an orbiting body does modify the <em>value</em> of its Angular Momentum, but not the <em>direction</em>. This causes that  the orbiting body changes its orbit but can not not change planes.</p>

<p>So if you add small forces from objects in the same plane, you end up with no changes to planes.</p>

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2013-12-12T15:33:37.733Z
2013-12-12T15:43:19.677Z
2013-12-12T15:43:19.677Z
Envite
<h2>Angular momentum conservation</h2>
<p>To put it in more mathematical terms, you can play with the energy and the angular momentum of a bunch of particles orbinting a central mass <span class="math-container">$M$</span>, given by</p>
<p><span class="math-container">$$E = \sum_i m_i \left(\frac{1}{2}v_i^2 - \frac{GM}{r_i}\right),$$</span></p>
<p>for the energy and</p>
<p><span class="math-container">$${\bf I} = \sum_i m_i {\bf r}_i \times {\bf v_i},$$</span></p>
<p>for the angular momentum. Now, let's try to extremize the energy for a given angular momentum, keeping in mind that the system has to conserve angular momentum, and that collisions between the particles can reduce the energy. One good way to do it is to use <a href="http://en.wikipedia.org/wiki/Lagrange_multiplier" rel="noreferrer">Lagrange multiplier</a></p>
<p><span class="math-container">$$\delta E - \lambda\cdot\delta {\bf I} = \sum_i\left[\delta {\bf v}_i \cdot \left({\bf v}_i - \lambda \cdot {\bf r}_i \right) + \delta {\bf r}_i \cdot \left( \frac{GM}{r_i^3} + \lambda \times {\bf v}_i\right)\right],$$</span></p>
<p>that requires</p>
<p><span class="math-container">$$\lambda\cdot{\bf r}_i = 0, \qquad {\bf v}_i = \lambda \times {\bf r}_i, \qquad \lambda^2 = \frac{GM}{r_i^3},$$</span></p>
<p>that means that all orbits are coplanar and circular.</p>
<h2>Is this true in general?</h2>
<p>That's the principle. Note, however, that all the planetary systems <a href="http://exoplanet.eu/diagrams/?t=s&amp;f=&amp;x=inclination&amp;xmin=&amp;xmax=&amp;y=mass&amp;ymin=min&amp;ymax=max" rel="noreferrer">do not always stay in an orbital plane</a>. Such systems can be explained by <a href="http://en.wikipedia.org/wiki/Kozai_mechanism" rel="noreferrer">Lidov-Kozai oscillations</a>, typically trigger by &quot;high-excentricity migration&quot; of hot Jupiters (<a href="http://adsabs.harvard.edu/abs/2010AAS...21536708F" rel="noreferrer">Fabrycky, 2012</a>). As far as we know now, we can say that:</p>
<ul>
<li>our Solar System is flat!</li>
<li>planetary systems observed by Kepler are mostly flat (there is kind of an observational bias, due to the transit method);</li>
<li>planetary systems observed by radial-velocity method are more or less flat (with an mean angle between 10 and 20°);</li>
<li>planetary systems with hot Jupiters are not flat in general.</li>
</ul>
<p><strong>More dirty details</strong>:</p>
<p>There is an excellent talk by <a href="http://www.eso.org/sci/meetings/garching/abs13.html" rel="noreferrer">Scott Tremaine, given at ESO last year</a> you could watch online.</p>

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2013-12-13T09:50:36.873Z
2013-12-13T09:50:36.873Z
2020-06-17T09:47:40.653Z
MBR
<p>An earlier question addressed why all planets formed in the same orbital plane, but how is this angle maintained? What prevents the planets from taking on a different orbital plane?</p>

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2013-12-12T14:52:33.943Z
Your latest edit asks a different question than has been asked previously. I have made some more edits to bring your question more in line with your recent focus and reopened your question.
called2voyage
<solar-system><planet>
Why do the planets in the Solar system stay in the same orbital plane?
Abhinav
# Why do the planets in the Solar system stay in the same orbital plane?

(Asked by: username_0 on 2013-12-08)

An earlier question addressed why all planets formed in the same orbital plane, but how is this angle maintained? What prevents the planets from taking on a different orbital plane?

**Tags:** solar-system, planet

> **Comments:**
> **username_1**: Your latest edit asks a different question than has been asked previously. I have made some more edits to bring your question more in line with your recent focus and reopened your question.  

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## 2 answers

### Answer by username_3 on 2013-12-13. Score: 7 | Accepted answer

## Angular momentum conservation

To put it in more mathematical terms, you can play with the energy and the angular momentum of a bunch of particles orbinting a central mass $M$, given by

$$E = \sum\_i m\_i \left(\frac{1}{2}v\_i^2 - \frac{GM}{r\_i}\right),$$

for the energy and

$${\bf I} = \sum\_i m\_i {\bf r}\_i \times {\bf v\_i},$$

for the angular momentum. Now, let's try to extremize the energy for a given angular momentum, keeping in mind that the system has to conserve angular momentum, and that collisions between the particles can reduce the energy. One good way to do it is to use [Lagrange multiplier](http://en.wikipedia.org/wiki/Lagrange_multiplier)

$$\delta E - \lambda\cdot\delta {\bf I} = \sum\_i\left[\delta {\bf v}\_i \cdot \left({\bf v}\_i - \lambda \cdot {\bf r}\_i \right) + \delta {\bf r}\_i \cdot \left( \frac{GM}{r\_i^3} + \lambda \times {\bf v}\_i\right)\right],$$

that requires

$$\lambda\cdot{\bf r}\_i = 0, \qquad {\bf v}\_i = \lambda \times {\bf r}\_i, \qquad \lambda^2 = \frac{GM}{r\_i^3},$$

that means that all orbits are coplanar and circular.

## Is this true in general?

That's the principle. Note, however, that all the planetary systems [do not always stay in an orbital plane](http://exoplanet.eu/diagrams/?t=s&f=&x=inclination&xmin=&xmax=&y=mass&ymin=min&ymax=max). Such systems can be explained by [Lidov-Kozai oscillations](http://en.wikipedia.org/wiki/Kozai_mechanism), typically trigger by "high-excentricity migration" of hot Jupiters ([Fabrycky, 2012](http://adsabs.harvard.edu/abs/2010AAS...21536708F)). As far as we know now, we can say that:

* our Solar System is flat!
* planetary systems observed by Kepler are mostly flat (there is kind of an observational bias, due to the transit method);
* planetary systems observed by radial-velocity method are more or less flat (with an mean angle between 10 and 20°);
* planetary systems with hot Jupiters are not flat in general.

**More dirty details**:

There is an excellent talk by [Scott Tremaine, given at ESO last year](http://www.eso.org/sci/meetings/garching/abs13.html) you could watch online.


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### Answer by username_2 on 2013-12-12. Score: 9

Answer to the NEW question:

the Angular Momentum Conservation Law states that, for any moving body, its angular momentum does not change unless you exercise an external force different from the central force.

For an orbiting body like a planet, this means that Sun's gravity, being the central force, does not modify Angular Momentum, but any other external force will do.

Examples of external forces are collisions or the forces made by Jupiter on another planet, or by Neptune on Pluto.

After the Solar System was formed, these external forces are quite small, and thus does not change greatly the Angular Momentum of any major body. But you can see how passing near a body can alter a comet's orbit.

Moreover, the external forces made by bodies that are in the same plane as an orbiting body does modify the *value* of its Angular Momentum, but not the *direction*. This causes that the orbiting body changes its orbit but can not not change planes.

So if you add small forces from objects in the same plane, you end up with no changes to planes.


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