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::<math> (x, y, z, v_x, v_y, v_z) \Longrightarrow  (a, e, i, \Omega, \omega, M) </math>
::<math> (x, y, z, v_x, v_y, v_z) \Longrightarrow  (a, e, i, \Omega, \omega, M) \qquad \mbox{(1)}</math>


::<math> \vec c= \vec r \times \vec v \Longrightarrow p=\frac{c^2}{\mu} \Longrightarrow p \qquad \mbox{(2)}</math>


::<math> \vec c= \vec r \times \vec v \Longrightarrow p=\frac{c^2}{\mu} \Longrightarrow p </math>
::<math> v^2=\mu (2/r -1/a)  \Longrightarrow  a \qquad \mbox{(3)}</math>
::<math> v^2=\mu (2/r -1/a)  \Longrightarrow  a </math>
 
::<math> p=a(1-e^2)  \Longrightarrow  e </math>
::<math> p=a(1-e^2)  \Longrightarrow  e \qquad \mbox{(4)}</math>
::<math> \vec c = c \vec S  \Longrightarrow \Omega=\arctan(-c_x/c_y); i=arcs(c_z/c) \Longrightarrow  \Omega, i </math>
 
 
::<math> \vec c = c \vec S  \Longrightarrow \Omega=\arctan(-c_x/c_y); i=arcs(c_z/c) \Longrightarrow  \Omega, i \qquad \mbox{(5)}</math>




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\end{array}
\end{array}
\right)  
\right)  
\Rightarrow  \omega+V
\Rightarrow  \omega+V
\qquad \mbox{(6)}
</math>  
</math>  




::<math> r=\frac{p}{1+e\cos(V)} \Longrightarrow \omega, V </math>
::<math> r=\frac{p}{1+e\cos(V)} \Longrightarrow \omega, V \qquad \mbox{(7)}</math>




::<math> \tan(E/2)=(\frac{1-e}{1+e})^{1/2}\tan(V/2)  \Longrightarrow  E </math>
::<math> \tan(E/2)=(\frac{1-e}{1+e})^{1/2}\tan(V/2)  \Longrightarrow  E \qquad \mbox{(8)}</math>




::<math> M= E -e \sin E \Longrightarrow M </math>
::<math> M= E -e \sin E \Longrightarrow M \qquad \mbox{(9)}</math>




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== Calculation of the position and velocity of the satellite from its orbital elements ==
== Calculation of the position and velocity of the satellite from its orbital elements ==


:::[[File:Osculating_Elem_Fig_1.png|none|700px]]
:::[[File:Osculating_Elem_Fig_1.png|none|700px]] (10)




: where
: where


:::[[File:Osculating_Elem_Fig_2.png|none|700px]]
:::[[File:Osculating_Elem_Fig_2.png|none|700px]] (11)





Revision as of 14:41, 5 August 2011


FundamentalsFundamentals
Title Osculating Elements
Author(s) J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
Level Advanced
Year of Publication 2011
Logo gAGE.png


A scheme with the necessary calculations to obtain the osculating orbital elements starting from the position and velocity of the satellite, and vice-versa, is provided as follows (see figure 1):


Calculation of the orbital elements of the satellite from its position and velocity

[math]\displaystyle{ (x, y, z, v_x, v_y, v_z) \Longrightarrow (a, e, i, \Omega, \omega, M) \qquad \mbox{(1)} }[/math]
[math]\displaystyle{ \vec c= \vec r \times \vec v \Longrightarrow p=\frac{c^2}{\mu} \Longrightarrow p \qquad \mbox{(2)} }[/math]
[math]\displaystyle{ v^2=\mu (2/r -1/a) \Longrightarrow a \qquad \mbox{(3)} }[/math]
[math]\displaystyle{ p=a(1-e^2) \Longrightarrow e \qquad \mbox{(4)} }[/math]


[math]\displaystyle{ \vec c = c \vec S \Longrightarrow \Omega=\arctan(-c_x/c_y); i=arcs(c_z/c) \Longrightarrow \Omega, i \qquad \mbox{(5)} }[/math]


[math]\displaystyle{ \left( \begin{array}{l} x\\ y\\ z \end{array} \right) = R \left( \begin{array}{l} r \cos(V)\\ r \sin(V)\\ 0 \end{array} \right) = r \left( \begin{array}{l} \cos \Omega \cos(\omega+V)- \sin \Omega \sin(\omega+V) \cos i\\ \sin \Omega \cos(\omega+V)+ \cos \Omega \sin(\omega+V) \cos i\\ \sin(\omega+V) \sin i \end{array} \right) \Rightarrow \omega+V \qquad \mbox{(6)} }[/math]


[math]\displaystyle{ r=\frac{p}{1+e\cos(V)} \Longrightarrow \omega, V \qquad \mbox{(7)} }[/math]


[math]\displaystyle{ \tan(E/2)=(\frac{1-e}{1+e})^{1/2}\tan(V/2) \Longrightarrow E \qquad \mbox{(8)} }[/math]


[math]\displaystyle{ M= E -e \sin E \Longrightarrow M \qquad \mbox{(9)} }[/math]



Calculation of the position and velocity of the satellite from its orbital elements

Osculating Elem Fig 1.png
(10)


where
Osculating Elem Fig 2.png
(11)


Figure 1: Orbit in space.