Difference between revisions of "Kepler equation"
From Encyclopedia of Mathematics
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A transcendental equation of the form | A transcendental equation of the form | ||
| − | + | $$y-c\sin y=x.$$ | |
| − | The case < | + | 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 | + | 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 | + | 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=15304
Kepler equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kepler_equation&oldid=15304
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article