DTLE

The table above shows the face-value difference between the elements determined using the Zenith Method and the published TLE orbit elements. The streak in the image is the Russian Dumsat satellite (NORAD #26086). It is indeed a LEO satellite!

Although most of the determined orbit elements appear close to the actual (TLE) counterparts, a major exception is the mean anomaly (M). This is true for two reasons. First, we had assumed that the mean anomaly is 0 at the RAAN, not the perigee, as the TLE mean anomaly is based. Second, initial orbit determination rarely determines the mean anomaly (or the argument of perigee) with any good degree of accuracy. Gauss' Method also fails to accurately determine these quantities.

We have determined four of the six Keplerian orbit elements of the satellite, but how good was this estimation? Could these orbit elements be used to predict where the satellite could be detected in the future, given the same field of view as the initial image shown in Step #1? Fortunately, this satellite is known and has published two-line element sets (TLEs) that can be used to compare how good this estimation really is. These elements can be also be propagated and compared to show the angular error between the position determined using the IOD and the position determined from the published TLE.

The figures below show the differences between the determined and the actual (TLE) orbits and ground tracks.

The Determined Orbit (Red) vs. the Actual (TLE) Orbit (Green)

The Determined Ground Track (Red) vs. the Actual (TLE) Ground Track (Green)

Finally, we must know if the determined orbit elements match the initial observations. If they do not match, we would need to adjust the determined orbit elements until the errors (residuals) between the predicted and observed are minimized. Below is a figure illustrating the result of this comparison.

Comparison between the Predicted (Red) and Observed (Green) Initial Observation

If the inclination and mean anomaly are slightly adjusted, the predicted and the observed nearly line up, as shown in the figure below.

The Final Comparison between the Predicted (Red) and the Observed (Green) Initial Observation after Determined Orbit Element Adjustment

The adjusted orbit elements (in blue) are shown alongside the actual (TLE) orbit elements for further comparison.
 

The determined inclination is shown to be closer to the actual value after the initial adjustment. If the determined elements are propagated into the future, the difference in the orbit periods will become apparent. Adjusting the semi-major axis value (a) will correct for the "along-track" residuals caused by this discrepancy. Although the orbit element adjustments can be repeated, this method is not the most accurate or practical when compared to more robust orbit determination methods.

This is only the first (initial) stage of many with respect to precise orbit determination. The final stage of orbit determination involves using statistical analysis to minimize the errors (residuals) between the observed satellite positions and the predicted (propagated) positions. The final stages require very involved matrix and calculus calculations.

The final section (to be added later) will address residuals in more detail and will include a more accurate orbit determination of another LEO satellite.