Gaussian vectors

The Gaussian vectors are used in astronomy and space travel as column vectors of a matrix as an aid to coordinate transformation between coordinates of the rotating equatorial coordinate system and coordinates of the orbit plane . They provide a clear relationship between the coordinates of a celestial body on its orbit around a planet and the position in the fixed, rotating equatorial coordinate system. With the help of the 3 orbit elements inclination , the argument of the node (node ​​length) and the argument of the periapsis , the 3 unit vectors , and can be formed. ${\displaystyle i}$ ${\displaystyle \Omega }$ ${\displaystyle \omega }$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle W}$

${\displaystyle P={\begin{pmatrix}+\cos \omega \cdot \cos \Omega -\sin \omega \cdot \cos i\cdot \sin \Omega \\+\cos \omega \cdot \sin \Omega +\sin \omega \cdot \cos i\cdot \cos \Omega \\+\sin \omega \cdot \sin i\end{pmatrix}}}$

${\displaystyle Q={\begin{pmatrix}-\sin \omega \cdot \cos \Omega -\cos \omega \cdot \cos i\cdot \sin \Omega \\-\sin \omega \cdot \sin \Omega +\cos \omega \cdot \cos i\cdot \cos \Omega \\+\cos \omega \cdot \sin i\end{pmatrix}}}$

${\displaystyle W={\begin{pmatrix}+\sin i\cdot \sin \Omega \\-\sin i\cdot \cos \Omega \\+\cos i\end{pmatrix}}}$

${\displaystyle P}$points from the center of the planet to the pericenter of the orbit of the celestial body, points from the planet center in the direction of the true anomaly at 90 ° and is perpendicular to both vectors. The 3 vectors define the system of the orbital plane of the celestial body and are the column vectors of the matrix ${\displaystyle Q}$ ${\displaystyle W}$

${\displaystyle \,(P,Q,W)=R_{z}(-\Omega )R_{x}(-i)R_{z}(-\omega )}$.

About the expression

${\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}=R_{z}(-\Omega )R_{x}(-i)R_{z}(-\omega )r{\begin{pmatrix}\cos \vartheta \\\sin \vartheta \\0\end{pmatrix}}\!\,}$

the three vectors can , and the rotating equatorial coordinate system are represented, wherein the true anomaly. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle \vartheta }$

literature

• Oliver Montenbruck, Eberhard Gill: Satellite orbits: models, methods, and applications . Springer, Berlin 2000, ISBN 978-3540672807 , pp. 24-27.