Archive of Astronomy Questions and Answers

Where can I get information about computing the orbits of natural satellites?

Natural satellites are ones which orbit planets and were not put there by humans, unlike 'artificial satellites'. Computing satellite orbits is a very old area of physics. In 3-dimensional space, you need to specify six orbital elements, which can be found by direct observation. Three of these specify the orientation of the orbit in space relative to the gravitating body holding the satellite in place. The other three include a measure of the linear size of the ellipse's major axis, its eccentricity and the location of the satellite in time along its orbit.

Inclination (i) 
     angle between the plane of the Ecliptic and the plane of the orbit.
 
Longitude of the Ascending Node (o) 
     states the position in the orbit where the elliptical path of the planet passes through the plane of the ecliptic, from below
     the plane to above the plane. 

Longitude of Perihelion (p) 
     states the position in the orbit where the planet is closest to the Sun.
 
Mean distance (a) 
     the value of the semi-major axis of the orbit - measured in Astronomical        Units for the major planets. 


Eccentricity (e) 
     eccentricity of the ellipse which describes the orbit 

Mean Longitude (L) 
     Position of the planet in the orbit on the date of the elements.

With these elements, you can create an ephemeris for where the object will be at any time in its orbit, and if you want, projected on the sky in a particular coordinate system ( RA and Dec or Altitude-Azimuth). The formula you need to convert the observations to orbital elements are somewhat complex and you often have to iterate your solutions for the six elements as better data comes in. For asteroids and comets, you need at least three measurements spanning several weeks to obtain a reliable orbit good enouh for you to re-acquire the new object in later years. Sometimes astronomers actually lose asteroids because only one or two observatiosn were available. Here is a eb site that lets you calculate planetary positions in the sky with fair accuracy, and which shows you how orbital elements are used Planet Orbits

For artificial satellites, my best suggestion is to look at the Satellite Passes site where they predict which satellites will be visible from various cities, and get in touch with the author of this site for further help. There's a great little JAVA Applet that lets you experiment with satellite orbits.