<p>I may have found a partial answer to my question in the Navigational Astronomy chapter of <a href="https://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&amp;_pageLabel=msi_portal_page_62&amp;pubCode=0002" rel="nofollow noreferrer">'American Practical Navigator' by N. Bowditch</a> . On page 228 </p>

<blockquote>
  <p>[...] nutation, an irregularity in the motion of the Earth due to the
  disturbing effect of other celestial bodies, principally the Moon</p>
</blockquote>

<p>and on page 230 </p>

<blockquote>
  <p>The precession of the Earth’s axis is the result of gravitational
  forces exerted principally by the Sun and Moon on the Earth’s
  equatorial bulge. The spinning Earth responds to these forces in the
  manner of a gyroscope. Regression of the nodes introduces certain
  irregularities known as nutation in the precessional motion.</p>
</blockquote>

<p>So the dominant effects as for predicting the orientation of the Earth's axis seem to be the Earth's ellipticity combined with the attraction of the Sun and Moon. </p>

CC BY-SA 3.0
2017-07-31T10:42:01.327Z
2017-07-31T10:42:01.327Z
James
<p>One of the simplest versions of Newton's equations of motion for the Sun, Earth and Moon can be obtained by making the approximation that the three bodies are perfect spheres. In this approximation, the pair potential between body i and j is  $$\frac{-G m_i m_j}{ r_{ij}},$$ where $r_{ij}$ is the distance between the bodies: one can solve the equations numerically and get the orbits. </p>

<p>This simple model does not include multiple, additional factors, such as </p>

<ul>
<li>the fact that the Earth is not perfectly spherical (equatorial bulge),</li>
<li>Earth's tides,</li>
<li>the influence of other celestial bodies,</li>
</ul>

<p>and others.</p>

<p>Amongst all these factors, what is the dominant one as for predicting the orbit of the Earth? In other words, what is the leading factor that should be included in the simple model above in order make its predictions for the Earth's orbit closer to the observations? </p>

CC BY-SA 3.0
2017-07-28T15:26:34.013Z
You may have to specify "dominant". The non-spherical Earth probably has a big effect on the moon, but not much on the sun. Over the long term, the cumulative effects of other planets may add up whereas the tides average out, even though tidal forces are much stronger.
James K
CC BY-SA 3.0
2017-07-28T15:53:36.713Z
James K, you are right, I revised the question. I am interested in the prediction of the Eart's orbit, e.g., the position of its center of mass, rotation speed as a function of time, and direction of axis of rotation.
James
CC BY-SA 3.0
2017-07-28T16:31:50.393Z
*one can solve the equations*  The general [three body problem](https://en.wikipedia.org/wiki/Three-body_problem) has no solution of this type.  This is a [detailed history of solution methods](http://sites.apam.columbia.edu/courses/ap1601y/Moon-Earth-Sin%20RMP.70.589.pdf) for the Earth-Moon-Sun system (it's 50 pages).
StephenG - Help Ukraine
CC BY-SA 3.0
2017-07-28T20:06:37.097Z
Obviously, I was talking about numerical solutions, for which my statement is true.
James
<celestial-mechanics><newtonian-gravity>
Equations of motion for Earth and Moon
James
# Equations of motion for Earth and Moon

(Asked by: username_0 on 2017-07-28)

One of the simplest versions of Newton's equations of motion for the Sun, Earth and Moon can be obtained by making the approximation that the three bodies are perfect spheres. In this approximation, the pair potential between body i and j is $$\frac{-G m\_i m\_j}{ r\_{ij}},$$ where $r\_{ij}$ is the distance between the bodies: one can solve the equations numerically and get the orbits.

This simple model does not include multiple, additional factors, such as

* the fact that the Earth is not perfectly spherical (equatorial bulge),
* Earth's tides,
* the influence of other celestial bodies,

and others.

Amongst all these factors, what is the dominant one as for predicting the orbit of the Earth? In other words, what is the leading factor that should be included in the simple model above in order make its predictions for the Earth's orbit closer to the observations?

**Tags:** celestial-mechanics, newtonian-gravity
> **Comments:**
> **username_1**: You may have to specify "dominant". The non-spherical Earth probably has a big effect on the moon, but not much on the sun. Over the long term, the cumulative effects of other planets may add up whereas the tides average out, even though tidal forces are much stronger.  
> **username_0**: James K, you are right, I revised the question. I am interested in the prediction of the Eart's orbit, e.g., the position of its center of mass, rotation speed as a function of time, and direction of axis of rotation.  
> **username_2**: *one can solve the equations*  The general [three body problem](https://en.wikipedia.org/wiki/Three-body_problem) has no solution of this type.  This is a [detailed history of solution methods](http://sites.apam.columbia.edu/courses/ap1601y/Moon-Earth-Sin%20RMP.70.589.pdf) for the Earth-Moon-Sun system (it's 50 pages).  
> **username_0**: Obviously, I was talking about numerical solutions, for which my statement is true.  

---

## 1 answers

### Answer by username_0 on 2017-07-31. Score: 1

I may have found a partial answer to my question in the Navigational Astronomy chapter of ['American Practical Navigator' by N. Bowditch](https://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_pageLabel=msi_portal_page_62&pubCode=0002) . On page 228

> [...] nutation, an irregularity in the motion of the Earth due to the
> disturbing effect of other celestial bodies, principally the Moon

and on page 230

> The precession of the Earth’s axis is the result of gravitational
> forces exerted principally by the Sun and Moon on the Earth’s
> equatorial bulge. The spinning Earth responds to these forces in the
> manner of a gyroscope. Regression of the nodes introduces certain
> irregularities known as nutation in the precessional motion.

So the dominant effects as for predicting the orientation of the Earth's axis seem to be the Earth's ellipticity combined with the attraction of the Sun and Moon.


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