3.3 Orbital elements

It can be show than only 6 of the 8 motion integrals obtained are independent.

The motion integrals are usually expressed in terms of the orbital elements, defined as:

• Argument of the ascending node ( ): angle between the X-axis and the direction of the ascending node, i.e. the point where the satellite cross the XY-plane (equator) with Z-velocity component positive.
• Orbit inclination (i): is the angle between the orbit plane and the XY-plane defined as the angle between the angular momentum and the Z-unitary vector .
• Perigee argument ( ): angle between the perigee and the ascending node direction, extended on the orbital plane.
• Eccentricity (e).
• Perigee pass epoch ( ).
• Major semiaxis (a).

From equations 2,4 and 5:

and

From the orbital elements is posible to compute the position and velocity of the satellite in any epoch t:

1. The mean anomaly is computed.
2. By means of the Kepler equation 10 we estimate (iteratively) the mean anomaly E:

   

3. Now, with equations 7, 8, we get the geocentric satellite distance and the true anomaly, r and :

   

4. From the polar coordinates in the orbit plane, we have to perform several rotations to get the coordinates in the equatorial reference frame:
   • Z-rotation of angle
   • X-rotation of angle -i (following the new X axis)
   • Z-rotation of angle (following the new Z axis)

   

   To get finally: