Orbit Determination by Optical Sightings

Following material extracted from FUNDAMENTALS OF ASTRODYNAMICS (starting from pg 117) by Bate, Mueller, and White. Apologies to authors for any errors introduced by my notes.

Historically, astronomers had to use strictly optical methods to observe celestial objects; thus, they had to determine orbits from angular data (e.g., topocentric right ascension, RA, and declination, dec).

We've previously established that we need six independent measurements to completely specify a satellites orbit. These may be the six classical orbital elements or the six components of vectors r and v at some epoch. Either way, one optical observation yields only two independent quantities such as Elevation (El) / Azimuth (Az) or RA / Dec; so, a minimum of three observations at 3 different times is required to determine an orbit.

Since astronomers had to determine the orbits of comets and asteroids using angular data only, the method present below has been in long use; it was introduced by Laplace in 1780. (See link.)

Determine Line of Sight Unit Vectors. Assume topocentric angles RA (α_i) and Dec (δ_i) for three observations. Let L_i be the corresponding Line of Sight unit vector.

DETERMINANT WORK TO BE DONE LATER.

r = ρL + R

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