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Difference between revisions of "Kepler equation"

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A transcendental equation of the form
 
A transcendental equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552101.png" /></td> </tr></table>
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$$y-c\sin y=x.$$
  
The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552102.png" /> is important for applications; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552103.png" /> is uniquely determined from a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552105.png" />. This equation was first considered by J. Kepler (1609) in connection with the problem of planetary motion: Let the ellipse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552106.png" /> (see Fig.) with focal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552107.png" /> be a planetary orbit, with circumscribed circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552108.png" />.
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The case $0\leq c<1$ is important for applications; here $y$ is uniquely determined from a given $c$ and $x$. This equation was first considered by J. Kepler (1609) in connection with the problem of planetary motion: Let the ellipse $AQB$ (see Fig.) with focal point $D$ be a planetary orbit, with circumscribed circle $APB$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k055210a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k055210a.gif" />
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Figure: k055210a
 
Figure: k055210a
  
Then the Kepler equation gives the relation between the eccentric anomaly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k0552109.png" /> and the mean anomaly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k05521010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k05521011.png" /> being the eccentricity of the ellipse.
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Then the Kepler equation gives the relation between the eccentric anomaly $y=\angle POA$ and the mean anomaly $x$, $c$ being the eccentricity of the ellipse.
  
 
The Kepler equation plays an important role in astronomy in determining the sections of elliptic orbits of planets.
 
The Kepler equation plays an important role in astronomy in determining the sections of elliptic orbits of planets.
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====Comments====
 
====Comments====
The mean anomaly is a linear function of the time of the planet's passage at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055210/k05521012.png" />. For more details, including the corresponding equations for hyperbolic and parabolic motion, see e.g. [[#References|[a1]]].
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The mean anomaly is a linear function of the time of the planet's passage at the point $Q$. For more details, including the corresponding equations for hyperbolic and parabolic motion, see e.g. [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Fitzpatrick,  "Principles of celestial mechanics" , Acad. Press  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Fitzpatrick,  "Principles of celestial mechanics" , Acad. Press  (1970)</TD></TR></table>

Latest revision as of 22:35, 30 November 2018

A transcendental equation of the form

$$y-c\sin y=x.$$

The case $0\leq c<1$ is important for applications; here $y$ is uniquely determined from a given $c$ and $x$. This equation was first considered by J. Kepler (1609) in connection with the problem of planetary motion: Let the ellipse $AQB$ (see Fig.) with focal point $D$ be a planetary orbit, with circumscribed circle $APB$.

Figure: k055210a

Then the Kepler equation gives the relation between the eccentric anomaly $y=\angle POA$ and the mean anomaly $x$, $c$ being the eccentricity of the ellipse.

The Kepler equation plays an important role in astronomy in determining the sections of elliptic orbits of planets.

References

[1] M.F. Subbotin, "A course in celestial mechanics" , 1 , Leningrad-Moscow (1941) (In Russian)


Comments

The mean anomaly is a linear function of the time of the planet's passage at the point $Q$. For more details, including the corresponding equations for hyperbolic and parabolic motion, see e.g. [a1].

References

[a1] P.M. Fitzpatrick, "Principles of celestial mechanics" , Acad. Press (1970)
How to Cite This Entry:
Kepler equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kepler_equation&oldid=43516
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article